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Physics · Article

Phase transitions and order parameters

Ask how a magnetic sample can be locally ordered yet have zero signed average. Use explicit free-energy landscapes and exact finite spin distributions to distinguish continuous onset, first-order coexistence, metastability and finite-size rounding.

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After reading, you can

  • Choose an order parameter and distinguish its distribution, magnitude and signed average.
  • Identify equilibrium by global free-energy comparison, not by finding just one stationary point.
  • Explain continuous onset and a first-order jump using two explicitly assigned stable polynomial models.
  • Separate coexistence, metastability, spinodals, nucleation and observed hysteresis.
  • Reconcile finite-system analyticity with symmetry breaking in a stated thermodynamic-limit procedure.
  • Recognize the spatial, fluctuation, symmetry and topological limits of a scalar mean-field description.
01

A magnet with no preferred sign

Imagine a small model magnet whose twenty labelled moments can point either up or down. Interactions favour agreement, but no external field favours up over down. If an equilibrium calculation gives zero average magnetization, must the sample be disordered?

Not necessarily. A distribution can put most of its weight near a large positive magnetization and an equally large negative one. Their signed average cancels. A narrow distribution centred at zero and a two-peaked distribution can therefore have the same signed average but very different physical organization.

We will first use a landscape to understand competing states, then count every configuration sector of an assigned finite model. The landscape is an illustrative phenomenology; the finite model is an all-to-all interaction model, not a simulation of a particular piece of iron. Neither assigns a switching time or a measured transition temperature.

02

Name the control and the order

A control is something specified from outside, such as temperature, pressure or an applied field. An order parameter is an observable or collective variable chosen to distinguish the states of interest. Its meaning must be defined before interpreting whether it is zero.

For our spins, write sᵢ = +1 or −1, M = Σᵢsᵢ and m = M/N. Thus m lies between −1 and +1. The sign distinguishes the two aligned orientations; its magnitude measures net alignment. The average ⟨m⟩, the average magnitude ⟨|m|⟩ and the full probability distribution P(m) answer different questions.

For a first reading, follow the landscapes, compare their minima, and return to the complete finite distributions. The marked return paths explain thermodynamic derivatives, convexity, limit order and spatial physics. Preparation links cover equilibrium, probability, derivatives and free energy; this lesson defines the needed physical terms locally.

Three descriptions that must not be confused in the assigned spin model.
QuantityMeaningWhat zero does not establish
m for one configurationSigned fraction imbalance, M/NIt does not describe the whole equilibrium distribution.
⟨m⟩Probability-weighted signed meanIt does not rule out two oppositely ordered peaks.
⟨|m|⟩Probability-weighted magnitudeA positive value alone does not prove a thermodynamic phase transition.
P(m)Weights of all allowed magnetizationsA finite-size change of shape alone is not a nonanalytic free energy.
03

What makes a transition sharp?

An equilibrium phase is a macroscopic thermodynamic state with characteristic properties. A phase transition is a nonanalytic change in the appropriate equilibrium thermodynamic potential in the relevant thermodynamic limit. Nonanalytic means that a smooth power-series description fails there; it need not mean that the order parameter itself jumps.

At a first-order boundary, at least one first derivative of the potential with respect to a natural control is discontinuous. Entropy is conjugate to temperature, volume to pressure, and magnetization to its field with the appropriate sign and normalization. Two competing equilibrium branches have equal potential at coexistence, but their slopes differ.

At a continuous transition the chosen order parameter can approach zero continuously while a response or higher derivative becomes singular. A rapid but smooth finite-system change is a crossover, not by itself proof of a true singularity. Real finite samples can show very sharp, useful transition signatures without literally possessing the infinite-system nonanalyticity.

04

Read a landscape as a model

Let η be a real dimensionless scalar collective coordinate. Unlike the spin imbalance m, η in this polynomial model is not restricted to the interval −1 to +1. Assign the dimensionless potential per unit Φ(η;t,h) = tη²/2 + η⁴/4 − hη. The controls t and h are dimensionless coefficients; positive h favours positive η.

At fixed controls, solve ∂Φ/∂η = tη + η³ − h = 0 for every stationary point. Positive curvature ∂²Φ/∂η² = t + 3η² identifies a local minimum; negative curvature identifies a local maximum. Zero curvature requires examining the next nonzero term, not automatically declaring a maximum or a minimum.

Compare the potential values at all local minima. The lowest value identifies equilibrium within this homogeneous model; a higher local minimum is metastable. A state can resist sufficiently small local disturbances and still fail to be the global equilibrium. The positive quartic term bounds this assigned potential below as |η| grows.

05

Two wells emerge continuously

Set h = 0. The stationary equation factors as η(t + η²) = 0. For t > 0, η = 0 is the unique minimum. For t < 0, the minima are η = ±√(−t), each with Φ = −t²/4, while η = 0 is a local maximum.

The two selected branches approach zero continuously as t rises to zero from below. At t = 0 the potential is η⁴/4: η = 0 is still a stable minimum even though its quadratic curvature vanishes. The equilibrium minimum Φ_eq is −t²/4 below zero and zero above zero. Its first derivative with respect to t is continuous; its second derivative changes at zero in this model.

If one chooses t = (T − T_c)/T_scale with a positive temperature scale, this describes a simple mean-field continuous onset at T_c. Here mean-field means treating the collective state as spatially uniform rather than resolving its fluctuating local environment. No dimensional calibration is assigned to our plotted numbers. The square-root onset follows from our polynomial assumptions, not a universal measured exponent for every magnet or fluid.

Two minima approach the same zero state

Compare five assigned quartic landscapes and both equilibrium branches. The coordinate η and coefficients are dimensionless, not a calibrated magnetization or temperature.

Five complete assigned potential curves

  • t = -1
  • t = -0.25
  • t = 0
  • t = 0.25
  • t = 1
Five complete assigned potential curvesShared linear axes retain every assigned point from η = −1.6 to +1.6; the finite drawing window is not a bound on the model coordinate.Φ (dimensionless potential)0123-1.5-1-0.500.511.5η (dimensionless, unrestricted model coordinate)

A closer view of the central wells

  • t = -1
  • t = -0.25
  • t = 0
  • t = 0.25
  • t = 1
A closer view of the central wellsSame five curves, using only their supplied samples with |η| ≤ 1.1. This is a stated smaller horizontal window, not a new model or vertical clipping.Φ (dimensionless potential)-0.2500.250.50.751-1-0.500.51η (dimensionless; central window only)

Both signs meet continuously at t = 0

  • Positive minimum
  • Negative minimum
  • Unique zero minimum
Both signs meet continuously at t = 0Both signs are global minima below zero. At and above zero the coincident branches are shown once as the zero minimum.Equilibrium η (dimensionless)-1-0.500.51-1-0.500.5t (dimensionless control)
  • At t = 0, Φ = η⁴/4 has a stable minimum even though the quadratic curvature vanishes.
  • A finite graph window does not restrict η to −1..+1. The bounded spin imbalance m is a different variable.
  • The square-root onset belongs to this homogeneous mean-field assignment; it is not a universal exponent.

Expandable complete sampled-data tables follow. The linked original body table remains in its original section.

Three descriptions that must not be confused in the assigned spin model.

Exact zero-field minima of the assigned quartic model; all entries are dimensionless.

Every assigned zero-field quartic curve sample. Both controls and potential are dimensionless; no Kelvin or joule scale is assigned.
Every assigned zero-field quartic curve sample. Both controls and potential are dimensionless; no Kelvin or joule scale is assigned.
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000
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1.00000000.100000000.0050250000
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1.00000000.720000000.32638464
1.00000000.740000000.34876644
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1.00000000.800000000.42240000
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1.00000000.840000000.47726784
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1.00000000.960000000.67313664
1.00000000.980000000.71079204
1.00000001.00000000.75000000
1.00000001.02000000.79080804
1.00000001.04000000.83326464
1.00000001.06000000.87741924
1.00000001.08000000.92332224
1.00000001.10000000.97102500
1.00000001.12000001.0205798
1.00000001.14000001.0720400
1.00000001.16000001.1254598
1.00000001.18000001.1808944
1.00000001.20000001.2384000
1.00000001.22000001.2980336
1.00000001.24000001.3598534
1.00000001.26000001.4239184
1.00000001.28000001.4902886
1.00000001.30000001.5590250
1.00000001.32000001.6301894
1.00000001.34000001.7038448
1.00000001.36000001.7800550
1.00000001.38000001.8588848
1.00000001.40000001.9404000
1.00000001.42000002.0246672
1.00000001.44000002.1117542
1.00000001.46000002.2017296
1.00000001.48000002.2946630
1.00000001.50000002.3906250
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1.00000001.54000002.5919216
1.00000001.56000002.6974022
1.00000001.58000002.8062032
1.00000001.60000002.9184000
Every assigned zero-field selected branch. Both signs are retained below zero; coincident zero branches are listed once in each signed column.
Every assigned zero-field selected branch. Both signs are retained below zero; coincident zero branches are listed once in each signed column.
tNegative selected ηPositive selected ηMinimum ΦSelected curvatureZero-state classification
-1.0000000-1.00000001.0000000-0.250000002.0000000local maximum
-0.99000000-0.994987440.99498744-0.245025001.9800000local maximum
-0.98000000-0.989949490.98994949-0.240100001.9600000local maximum
-0.97000000-0.984885780.98488578-0.235225001.9400000local maximum
-0.96000000-0.979795900.97979590-0.230400001.9200000local maximum
-0.95000000-0.974679430.97467943-0.225625001.9000000local maximum
-0.94000000-0.969535970.96953597-0.220900001.8800000local maximum
-0.93000000-0.964365080.96436508-0.216225001.8600000local maximum
-0.92000000-0.959166300.95916630-0.211600001.8400000local maximum
-0.91000000-0.953939200.95393920-0.207025001.8200000local maximum
-0.90000000-0.948683300.94868330-0.202500001.8000000local maximum
-0.89000000-0.943398110.94339811-0.198025001.7800000local maximum
-0.88000000-0.938083150.93808315-0.193600001.7600000local maximum
-0.87000000-0.932737910.93273791-0.189225001.7400000local maximum
-0.86000000-0.927361850.92736185-0.184900001.7200000local maximum
-0.85000000-0.921954450.92195445-0.180625001.7000000local maximum
-0.84000000-0.916515140.91651514-0.176400001.6800000local maximum
-0.83000000-0.911043360.91104336-0.172225001.6600000local maximum
-0.82000000-0.905538510.90553851-0.168100001.6400000local maximum
-0.81000000-0.900000000.90000000-0.164025001.6200000local maximum
-0.80000000-0.894427190.89442719-0.160000001.6000000local maximum
-0.79000000-0.888819440.88881944-0.156025001.5800000local maximum
-0.78000000-0.883176090.88317609-0.152100001.5600000local maximum
-0.77000000-0.877496440.87749644-0.148225001.5400000local maximum
-0.76000000-0.871779790.87177979-0.144400001.5200000local maximum
-0.75000000-0.866025400.86602540-0.140625001.5000000local maximum
-0.74000000-0.860232530.86023253-0.136900001.4800000local maximum
-0.73000000-0.854400370.85440037-0.133225001.4600000local maximum
-0.72000000-0.848528140.84852814-0.129600001.4400000local maximum
-0.71000000-0.842614980.84261498-0.126025001.4200000local maximum
-0.70000000-0.836660030.83666003-0.122500001.4000000local maximum
-0.69000000-0.830662390.83066239-0.119025001.3800000local maximum
-0.68000000-0.824621130.82462113-0.115600001.3600000local maximum
-0.67000000-0.818535280.81853528-0.112225001.3400000local maximum
-0.66000000-0.812403840.81240384-0.108900001.3200000local maximum
-0.65000000-0.806225770.80622577-0.105625001.3000000local maximum
-0.64000000-0.800000000.80000000-0.102400001.2800000local maximum
-0.63000000-0.793725390.79372539-0.0992250001.2600000local maximum
-0.62000000-0.787400790.78740079-0.0961000001.2400000local maximum
-0.61000000-0.781024970.78102497-0.0930250001.2200000local maximum
-0.60000000-0.774596670.77459667-0.0900000001.2000000local maximum
-0.59000000-0.768114570.76811457-0.0870250001.1800000local maximum
-0.58000000-0.761577310.76157731-0.0841000001.1600000local maximum
-0.57000000-0.754983440.75498344-0.0812250001.1400000local maximum
-0.56000000-0.748331480.74833148-0.0784000001.1200000local maximum
-0.55000000-0.741619850.74161985-0.0756250001.1000000local maximum
-0.54000000-0.734846920.73484692-0.0729000001.0800000local maximum
-0.53000000-0.728010990.72801099-0.0702250001.0600000local maximum
-0.52000000-0.721110260.72111026-0.0676000001.0400000local maximum
-0.51000000-0.714142840.71414284-0.0650250001.0200000local maximum
-0.50000000-0.707106780.70710678-0.0625000001.0000000local maximum
-0.49000000-0.700000000.70000000-0.0600250000.98000000local maximum
-0.48000000-0.692820320.69282032-0.0576000000.96000000local maximum
-0.47000000-0.685565460.68556546-0.0552250000.94000000local maximum
-0.46000000-0.678233000.67823300-0.0529000000.92000000local maximum
-0.45000000-0.670820390.67082039-0.0506250000.90000000local maximum
-0.44000000-0.663324960.66332496-0.0484000000.88000000local maximum
-0.43000000-0.655743850.65574385-0.0462250000.86000000local maximum
-0.42000000-0.648074070.64807407-0.0441000000.84000000local maximum
-0.41000000-0.640312420.64031242-0.0420250000.82000000local maximum
-0.40000000-0.632455530.63245553-0.0400000000.80000000local maximum
-0.39000000-0.624499800.62449980-0.0380250000.78000000local maximum
-0.38000000-0.616441400.61644140-0.0361000000.76000000local maximum
-0.37000000-0.608276250.60827625-0.0342250000.74000000local maximum
-0.36000000-0.600000000.60000000-0.0324000000.72000000local maximum
-0.35000000-0.591607980.59160798-0.0306250000.70000000local maximum
-0.34000000-0.583095190.58309519-0.0289000000.68000000local maximum
-0.33000000-0.574456260.57445626-0.0272250000.66000000local maximum
-0.32000000-0.565685420.56568542-0.0256000000.64000000local maximum
-0.31000000-0.556776440.55677644-0.0240250000.62000000local maximum
-0.30000000-0.547722560.54772256-0.0225000000.60000000local maximum
-0.29000000-0.538516480.53851648-0.0210250000.58000000local maximum
-0.28000000-0.529150260.52915026-0.0196000000.56000000local maximum
-0.27000000-0.519615240.51961524-0.0182250000.54000000local maximum
-0.26000000-0.509901950.50990195-0.0169000000.52000000local maximum
-0.25000000-0.500000000.50000000-0.0156250000.50000000local maximum
-0.24000000-0.489897950.48989795-0.0144000000.48000000local maximum
-0.23000000-0.479583150.47958315-0.0132250000.46000000local maximum
-0.22000000-0.469041580.46904158-0.0121000000.44000000local maximum
-0.21000000-0.458257570.45825757-0.0110250000.42000000local maximum
-0.20000000-0.447213600.44721360-0.0100000000.40000000local maximum
-0.19000000-0.435889890.43588989-0.00902500000.38000000local maximum
-0.18000000-0.424264070.42426407-0.00810000000.36000000local maximum
-0.17000000-0.412310560.41231056-0.00722500000.34000000local maximum
-0.16000000-0.400000000.40000000-0.00640000000.32000000local maximum
-0.15000000-0.387298330.38729833-0.00562500000.30000000local maximum
-0.14000000-0.374165740.37416574-0.00490000000.28000000local maximum
-0.13000000-0.360555130.36055513-0.00422500000.26000000local maximum
-0.12000000-0.346410160.34641016-0.00360000000.24000000local maximum
-0.11000000-0.331662480.33166248-0.00302500000.22000000local maximum
-0.10000000-0.316227770.31622777-0.00250000000.20000000local maximum
-0.090000000-0.300000000.30000000-0.00202500000.18000000local maximum
-0.080000000-0.282842710.28284271-0.00160000000.16000000local maximum
-0.070000000-0.264575130.26457513-0.00122500000.14000000local maximum
-0.060000000-0.244948970.24494897-0.000900000000.12000000local maximum
-0.050000000-0.223606800.22360680-0.000625000000.10000000local maximum
-0.040000000-0.200000000.20000000-0.000400000000.080000000local maximum
-0.030000000-0.173205080.17320508-0.000225000000.060000000local maximum
-0.020000000-0.141421360.14142136-0.000100000000.040000000local maximum
-0.010000000-0.100000000.10000000-0.0000250000000.020000000local maximum
00000marginal quartic minimum
0.0100000000000.010000000sole minimum
0.0200000000000.020000000sole minimum
0.0300000000000.030000000sole minimum
0.0400000000000.040000000sole minimum
0.0500000000000.050000000sole minimum
0.0600000000000.060000000sole minimum
0.0700000000000.070000000sole minimum
0.0800000000000.080000000sole minimum
0.0900000000000.090000000sole minimum
0.100000000000.10000000sole minimum
0.110000000000.11000000sole minimum
0.120000000000.12000000sole minimum
0.130000000000.13000000sole minimum
0.140000000000.14000000sole minimum
0.150000000000.15000000sole minimum
0.160000000000.16000000sole minimum
0.170000000000.17000000sole minimum
0.180000000000.18000000sole minimum
0.190000000000.19000000sole minimum
0.200000000000.20000000sole minimum
0.210000000000.21000000sole minimum
0.220000000000.22000000sole minimum
0.230000000000.23000000sole minimum
0.240000000000.24000000sole minimum
0.250000000000.25000000sole minimum
0.260000000000.26000000sole minimum
0.270000000000.27000000sole minimum
0.280000000000.28000000sole minimum
0.290000000000.29000000sole minimum
0.300000000000.30000000sole minimum
0.310000000000.31000000sole minimum
0.320000000000.32000000sole minimum
0.330000000000.33000000sole minimum
0.340000000000.34000000sole minimum
0.350000000000.35000000sole minimum
0.360000000000.36000000sole minimum
0.370000000000.37000000sole minimum
0.380000000000.38000000sole minimum
0.390000000000.39000000sole minimum
0.400000000000.40000000sole minimum
0.410000000000.41000000sole minimum
0.420000000000.42000000sole minimum
0.430000000000.43000000sole minimum
0.440000000000.44000000sole minimum
0.450000000000.45000000sole minimum
0.460000000000.46000000sole minimum
0.470000000000.47000000sole minimum
0.480000000000.48000000sole minimum
0.490000000000.49000000sole minimum
0.500000000000.50000000sole minimum

Scroll each graph or table locally to keep its labels readable. Complete text tables remain available without interpreting color.

Exact zero-field minima of the assigned quartic model; all entries are dimensionless.
tEquilibrium η valuesMinimum ΦCharacter of η = 0
−1−1 and +1−0.25Local maximum
−0.25−0.5 and +0.5−0.015625Local maximum
000Stable quartic minimum; zero curvature
+0.2500Positive-curvature minimum
+100Positive-curvature minimum
06

A field selects a sign

Now hold t = −1 and vary h. The term −hη tilts the landscape. At h = 0, the minima η = −1 and +1 have equal potential. A sufficiently small positive field selects the positive minimum as the global equilibrium, while the negative well remains locally stable but metastable. Reversing h reverses the preferred sign.

In the homogeneous minimum-selection description, sweeping through h = 0 at this negative t produces a jump between the two equilibrium signs. This is a first-order field boundary, even though varying t through zero at h = 0 gave continuous onset. Transition type must specify the controls and the path, not just the name of the material.

On a nonsingular stationary branch, differentiating tη + η³ = h gives dη/dh = 1/(t + 3η²). At t = −1 and η = ±1, the selected local slope is 1/2. This is the slope within one well, not the derivative of a curve that smoothly joins the two equilibrium signs across coexistence.

07

Coexistence is not a spinodal

A spinodal of this homogeneous polynomial model marks the disappearance of a metastable local minimum when it merges with a neighbouring maximum. Both the first derivative and the curvature vanish there. At t = −1 this gives η = ±1/√3 and h = η³ − η, with |h| = 2/(3√3) ≈ 0.384900.

These thresholds are not the coexistence field h = 0. Starting in the negative well and increasing h can leave that well metastable for positive fields before it disappears. A plot must distinguish global minima, metastable minima and unstable stationary points; a line through every root is not an equilibrium response curve.

A static landscape supplies neither a waiting time nor an experimental hysteresis width. Escape depends on fluctuations, interfaces, defects, boundary conditions and the control-sweep protocol. In a spatial material, nucleation may occur well before a homogeneous mean-field spinodal is reached; that spinodal need not be a sharply observable thermodynamic boundary.

Equal-depth coexistence is not the loss of a well

Tilt the quartic landscape at t = −1, then inspect every stationary root rather than joining only selected states.

Three tilts of the same quartic model

  • h = -0.3
  • h = 0
  • h = 0.3
Three tilts of the same quartic modelNegative, zero and positive h use the same axes. Positive h lowers the positive-η well.Φ (dimensionless potential)-0.500.5-1.5-1-0.500.511.5η (dimensionless, unrestricted model coordinate)

Stationary roots, with no false jump connector

  • Global minima
  • Metastable local minima
  • Local maxima
  • Exact spinodal endpoints
  • Equal-depth coexistence h = 0
Stationary roots, with no false jump connectorEach glyph is a stationary root. Blue circles are global minima, orange squares metastable minima, red crosses local maxima, violet diamonds exact spinodal endpoints and green rings coexisting minima.Stationary η (dimensionless)-1-0.500.51-0.5-0.2500.250.5h (dimensionless control; t = −1)
  • The green rings at h = 0 mark equal-depth coexistence. The stable selected sign changes there; no connecting equilibrium line is drawn through the jump.
  • Violet diamonds use the exact endpoints h = ±2/(3√3), η = ∓1/√3, not the nearest h-grid sample. They are stationary inflections with zero curvature.
  • A metastable well can persist after coexistence. Static curves supply neither a switching time nor a measured hysteresis width.

Expandable complete sampled-data tables follow. The linked original body table remains in its original section.

Exact zero-field minima of the assigned quartic model; all entries are dimensionless.

Every assigned tilted quartic curve sample at t = −1. The real coordinate η is not a bounded spin imbalance.
Every assigned tilted quartic curve sample at t = −1. The real coordinate η is not a bounded spin imbalance.
hηΦ
-0.30000000-1.6000000-0.12160000
-0.30000000-1.5800000-0.16419676
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-0.30000000-1.5400000-0.24167836
-0.30000000-1.5200000-0.27671296
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-0.30000000-1.2600000-0.54168156
-0.30000000-1.2400000-0.54974656
-0.30000000-1.2200000-0.55636636
-0.30000000-1.2000000-0.56160000
-0.30000000-1.1800000-0.56550556
-0.30000000-1.1600000-0.56814016
-0.30000000-1.1400000-0.56955996
-0.30000000-1.1200000-0.56982016
-0.30000000-1.1000000-0.56897500
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-0.30000000-1.0200000-0.55559196
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0-1.60000000.35840000
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0.300000000.060000000-0.019796760
0.300000000.080000000-0.027189760
0.300000000.10000000-0.034975000
0.300000000.12000000-0.043148160
0.300000000.14000000-0.051703960
0.300000000.16000000-0.060636160
0.300000000.18000000-0.069937560
0.300000000.20000000-0.079600000
0.300000000.22000000-0.089614360
0.300000000.24000000-0.099970560
0.300000000.26000000-0.11065756
0.300000000.28000000-0.12166336
0.300000000.30000000-0.13297500
0.300000000.32000000-0.14457856
0.300000000.34000000-0.15645916
0.300000000.36000000-0.16860096
0.300000000.38000000-0.18098716
0.300000000.40000000-0.19360000
0.300000000.42000000-0.20642076
0.300000000.44000000-0.21942976
0.300000000.46000000-0.23260636
0.300000000.48000000-0.24592896
0.300000000.50000000-0.25937500
0.300000000.52000000-0.27292096
0.300000000.54000000-0.28654236
0.300000000.56000000-0.30021376
0.300000000.58000000-0.31390876
0.300000000.60000000-0.32760000
0.300000000.62000000-0.34125916
0.300000000.64000000-0.35485696
0.300000000.66000000-0.36836316
0.300000000.68000000-0.38174656
0.300000000.70000000-0.39497500
0.300000000.72000000-0.40801536
0.300000000.74000000-0.42083356
0.300000000.76000000-0.43339456
0.300000000.78000000-0.44566236
0.300000000.80000000-0.45760000
0.300000000.82000000-0.46916956
0.300000000.84000000-0.48033216
0.300000000.86000000-0.49104796
0.300000000.88000000-0.50127616
0.300000000.90000000-0.51097500
0.300000000.92000000-0.52010176
0.300000000.94000000-0.52861276
0.300000000.96000000-0.53646336
0.300000000.98000000-0.54360796
0.300000001.0000000-0.55000000
0.300000001.0200000-0.55559196
0.300000001.0400000-0.56033536
0.300000001.0600000-0.56418076
0.300000001.0800000-0.56707776
0.300000001.1000000-0.56897500
0.300000001.1200000-0.56982016
0.300000001.1400000-0.56955996
0.300000001.1600000-0.56814016
0.300000001.1800000-0.56550556
0.300000001.2000000-0.56160000
0.300000001.2200000-0.55636636
0.300000001.2400000-0.54974656
0.300000001.2600000-0.54168156
0.300000001.2800000-0.53211136
0.300000001.3000000-0.52097500
0.300000001.3200000-0.50821056
0.300000001.3400000-0.49375516
0.300000001.3600000-0.47754496
0.300000001.3800000-0.45951516
0.300000001.4000000-0.43960000
0.300000001.4200000-0.41773276
0.300000001.4400000-0.39384576
0.300000001.4600000-0.36787036
0.300000001.4800000-0.33973696
0.300000001.5000000-0.30937500
0.300000001.5200000-0.27671296
0.300000001.5400000-0.24167836
0.300000001.5600000-0.20419776
0.300000001.5800000-0.16419676
0.300000001.6000000-0.12160000
Every stationary point on the assigned h grid at t = −1. Classifications distinguish global, metastable and unstable points; grid samples are not exact spinodal locations.
Every stationary point on the assigned h grid at t = −1. Classifications distinguish global, metastable and unstable points; grid samples are not exact spinodal locations.
hηΦCurvatureClassificationLocal branch slope dη/dh
-0.50000000-1.1914879-0.801718803.2589301global minimum0.30684917
-0.49000000-1.1884090-0.789819303.2369479global minimum0.30893299
-0.48000000-1.1853091-0.777950693.2148730global minimum0.31105428
-0.47000000-1.1821878-0.766113193.1927039global minimum0.31321414
-0.46000000-1.1790447-0.754307013.1704391global minimum0.31541372
-0.45000000-1.1758794-0.742532373.1480769global minimum0.31765424
-0.44000000-1.1726915-0.730789503.1256158global minimum0.31993696
-0.43000000-1.1694805-0.719078623.1030539global minimum0.32226318
-0.42000000-1.1662460-0.707399963.0803895global minimum0.32463427
-0.41000000-1.1629877-0.695753783.0576209global minimum0.32705167
-0.40000000-1.1597049-0.684140293.0347460global minimum0.32951686
-0.39000000-1.1563972-0.672559763.0117631global minimum0.33203142
-0.38000000-1.1530641-0.661012432.9886701global minimum0.33459698
-0.380000000.523311120.080680037-0.17843642local maximum-5.6042372
-0.380000000.629752930.0803323970.18976628metastable minimum5.2696402
-0.37000000-1.1497050-0.649498572.9654650global minimum0.33721524
-0.370000000.481934570.075671611-0.30321720local maximum-3.2979660
-0.370000000.667770460.0738269560.33775218metastable minimum2.9607507
-0.36000000-1.1463196-0.638018422.9421457global minimum0.33988800
-0.360000000.452895130.071003186-0.38465800local maximum-2.5997120
-0.360000000.693424430.0670152360.44251234metastable minimum2.2598240
-0.35000000-1.1429071-0.626572272.9187098global minimum0.34261713
-0.350000000.428896410.066597275-0.44814362local maximum-2.2314275
-0.350000000.714010680.0599749900.52943377metastable minimum1.8888104
-0.34000000-1.1394670-0.615160372.8951553global minimum0.34540461
-0.340000000.407834890.062415573-0.50101210local maximum-1.9959598
-0.340000000.731632140.0527447990.60585676metastable minimum1.6505552
-0.33000000-1.1359988-0.603783022.8714798global minimum0.34825249
-0.330000000.388750830.058434028-0.54661839local maximum-1.8294299
-0.330000000.747247970.0453489900.67513859metastable minimum1.4811774
-0.32000000-1.1325018-0.592440492.8476808global minimum0.35116295
-0.320000000.371110420.054635765-0.58683117local maximum-1.7040676
-0.320000000.761391350.0378047270.73915038metastable minimum1.3529047
-0.31000000-1.1289753-0.581133082.8237558global minimum0.35413827
-0.310000000.354580440.051008130-0.62281814local maximum-1.6056051
-0.310000000.774394890.0301249510.79906232metastable minimum1.2514668
-0.30000000-1.1254188-0.569861092.7997023global minimum0.35718083
-0.300000000.338936240.047541210-0.65536667local maximum-1.5258634
-0.300000000.786482540.0223198750.85566436metastable minimum1.1686825
-0.29000000-1.1218315-0.558624812.7755176global minimum0.36029316
-0.290000000.324017840.044226990-0.68503731local maximum-1.4597745
-0.290000000.797813630.0143978180.90951976metastable minimum1.0994813
-0.28000000-1.1182127-0.547424562.7511988global minimum0.36347792
-0.280000000.309706450.041058833-0.71224575local maximum-1.4040098
-0.280000000.808506230.00636572760.96104697metastable minimum1.0405319
-0.27000000-1.1145617-0.536260662.7267431global minimum0.36673789
-0.270000000.295910930.038031144-0.73731016local maximum-1.3562813
-0.270000000.81865073-0.00177048191.0105671metastable minimum0.98954344
-0.26000000-1.1108777-0.525133442.7021475global minimum0.37007602
-0.260000000.282559520.035139136-0.76048036local maximum-1.3149584
-0.260000000.82831814-0.0100056981.0583328metastable minimum0.94488234
-0.25000000-1.1071599-0.514043222.6774089global minimum0.37349543
-0.250000000.269594440.032378667-0.78195652local maximum-1.2788435
-0.250000000.83756544-0.0183354451.1045476metastable minimum0.90534806
-0.24000000-1.1034075-0.502990362.6525241global minimum0.37699940
-0.240000000.256968320.029746118-0.80190185local maximum-1.2470354
-0.240000000.84643915-0.0267557631.1493777metastable minimum0.87003600
-0.23000000-1.0996196-0.491975192.6274897global minimum0.38059140
-0.230000000.244641700.027238303-0.82045132local maximum-1.2188414
-0.230000000.85497789-0.0352631131.1929616metastable minimum0.83824996
-0.22000000-1.0957953-0.480998082.6023023global minimum0.38427512
-0.220000000.232581260.024852397-0.83771787local maximum-1.1937193
-0.220000000.86321408-0.0438543121.2354156metastable minimum0.80944419
-0.21000000-1.0919338-0.470059412.5769581global minimum0.38805443
-0.210000000.220758520.022585886-0.85379703local maximum-1.1712386
-0.210000000.87117525-0.0525264781.2768390metastable minimum0.78318412
-0.20000000-1.0880339-0.459159542.5514534global minimum0.39193348
-0.200000000.209148850.020436517-0.86877028local maximum-1.1510523
-0.200000000.87888507-0.0612769801.3173169metastable minimum0.75911879
-0.19000000-1.0840948-0.448298862.5257843global minimum0.39591663
-0.190000000.197730770.018402270-0.88270763local maximum-1.1328779
-0.190000000.88636398-0.0701034101.3569233metastable minimum0.73696131
-0.18000000-1.0801152-0.437477782.4999467global minimum0.40000853
-0.180000000.186485360.016481326-0.89566963local maximum-1.1164831
-0.180000000.89362986-0.0790035501.3957230metastable minimum0.71647456
-0.17000000-1.0760942-0.426696692.4739362global minimum0.40421414
-0.170000000.175395820.014672044-0.90770892local maximum-1.1016748
-0.170000000.90069838-0.0879753491.4337727metastable minimum0.69746062
-0.16000000-1.0720305-0.415956032.4477484global minimum0.40853871
-0.160000000.164447120.012972941-0.91887143local maximum-1.0882915
-0.160000000.90758342-0.0970169061.4711230metastable minimum0.67975282
-0.15000000-1.0679230-0.405256232.4213787global minimum0.41298786
-0.150000000.153625700.011382677-0.92919743local maximum-1.0761975
-0.150000000.91429732-0.106126451.5078187metastable minimum0.66320969
-0.14000000-1.0637703-0.394597722.3948221global minimum0.41756756
-0.140000000.142919260.0099000435-0.93872226local maximum-1.0652778
-0.140000000.92085109-0.115302321.5439002metastable minimum0.64771026
-0.13000000-1.0595712-0.383980982.3680734global minimum0.42228420
-0.130000000.132316550.0085239464-0.94747699local maximum-1.0554346
-0.130000000.92725465-0.124542971.5794036metastable minimum0.63315040
-0.12000000-1.0553242-0.373406462.3411274global minimum0.42714463
-0.120000000.121807260.0072534011-0.95548898local maximum-1.0465845
-0.120000000.93351693-0.133846941.6143616metastable minimum0.61943992
-0.11000000-1.0510278-0.362874662.3139784global minimum0.43215616
-0.110000000.111381790.0060875219-0.96278229local maximum-1.0386564
-0.110000000.93964602-0.143212861.6488039metastable minimum0.60650026
-0.10000000-1.0466805-0.352386072.2866204global minimum0.43732663
-0.100000000.101031260.0050255156-0.96937805local maximum-1.0315893
-0.100000000.94564927-0.152639441.6827576metastable minimum0.59426264
-0.090000000-1.0422807-0.341941222.2590473global minimum0.44266448
-0.0900000000.0907473110.0040666749-0.97529478local maximum-1.0253310
-0.0900000000.95153341-0.162125451.7162475metastable minimum0.58266655
-0.080000000-1.0378267-0.331540642.2312525global minimum0.44817877
-0.0800000000.0805220900.0032103736-0.98054858local maximum-1.0198373
-0.0800000000.95730456-0.171669731.7492961metastable minimum0.57165851
-0.070000000-1.0333165-0.321184882.2032291global minimum0.45387926
-0.0700000000.0703481430.0024560622-0.98515342local maximum-1.0150703
-0.0700000000.96296838-0.181271191.7819243metastable minimum0.56119107
-0.060000000-1.0287484-0.310874502.1749699global minimum0.45977648
-0.0600000000.0602183670.0018032636-0.98912124local maximum-1.0109984
-0.0600000000.96853005-0.190928761.8141514metastable minimum0.55122192
-0.050000000-1.0241203-0.300610112.1464672global minimum0.46588181
-0.0500000000.0501259470.0012515704-0.99246217local maximum-1.0075951
-0.0500000000.97399435-0.200641461.8459950metastable minimum0.54171328
-0.040000000-1.0194300-0.290392302.1177128global minimum0.47220756
-0.0400000000.0400643090.00080064206-0.99518455local maximum-1.0048387
-0.0400000000.97936573-0.210408341.8774717metastable minimum0.53263119
-0.030000000-1.0146754-0.280221722.0886983global minimum0.47876708
-0.0300000000.0300270730.00045020287-0.99729512local maximum-1.0027122
-0.0300000000.98464830-0.220228481.9085968metastable minimum0.52394513
-0.020000000-1.0098539-0.270099022.0594145global minimum0.48557490
-0.0200000000.0200080100.00020004003-0.99879904local maximum-1.0012024
-0.0200000000.98984586-0.230101021.9393845metastable minimum0.51562751
-0.010000000-1.0049630-0.260024882.0298518global minimum0.49264679
-0.0100000000.0100010000.000050002501-0.99969994local maximum-1.0003002
-0.0100000000.99496199-0.240025131.9698481metastable minimum0.50765336
0-1.0000000-0.250000002.0000000global minimum0.50000000
000-1.0000000local maximum-1.0000000
01.0000000-0.250000002.0000000global minimum0.50000000
0.010000000-0.99496199-0.240025131.9698481metastable minimum0.50765336
0.010000000-0.0100010000.000050002501-0.99969994local maximum-1.0003002
0.0100000001.0049630-0.260024882.0298518global minimum0.49264679
0.020000000-0.98984586-0.230101021.9393845metastable minimum0.51562751
0.020000000-0.0200080100.00020004003-0.99879904local maximum-1.0012024
0.0200000001.0098539-0.270099022.0594145global minimum0.48557490
0.030000000-0.98464830-0.220228481.9085968metastable minimum0.52394513
0.030000000-0.0300270730.00045020287-0.99729512local maximum-1.0027122
0.0300000001.0146754-0.280221722.0886983global minimum0.47876708
0.040000000-0.97936573-0.210408341.8774717metastable minimum0.53263119
0.040000000-0.0400643090.00080064206-0.99518455local maximum-1.0048387
0.0400000001.0194300-0.290392302.1177128global minimum0.47220756
0.050000000-0.97399435-0.200641461.8459950metastable minimum0.54171328
0.050000000-0.0501259470.0012515704-0.99246217local maximum-1.0075951
0.0500000001.0241203-0.300610112.1464672global minimum0.46588181
0.060000000-0.96853005-0.190928761.8141514metastable minimum0.55122192
0.060000000-0.0602183670.0018032636-0.98912124local maximum-1.0109984
0.0600000001.0287484-0.310874502.1749699global minimum0.45977648
0.070000000-0.96296838-0.181271191.7819243metastable minimum0.56119107
0.070000000-0.0703481430.0024560622-0.98515342local maximum-1.0150703
0.0700000001.0333165-0.321184882.2032291global minimum0.45387926
0.080000000-0.95730456-0.171669731.7492961metastable minimum0.57165851
0.080000000-0.0805220900.0032103736-0.98054858local maximum-1.0198373
0.0800000001.0378267-0.331540642.2312525global minimum0.44817877
0.090000000-0.95153341-0.162125451.7162475metastable minimum0.58266655
0.090000000-0.0907473110.0040666749-0.97529478local maximum-1.0253310
0.0900000001.0422807-0.341941222.2590473global minimum0.44266448
0.10000000-0.94564927-0.152639441.6827576metastable minimum0.59426264
0.10000000-0.101031260.0050255156-0.96937805local maximum-1.0315893
0.100000001.0466805-0.352386072.2866204global minimum0.43732663
0.11000000-0.93964602-0.143212861.6488039metastable minimum0.60650026
0.11000000-0.111381790.0060875219-0.96278229local maximum-1.0386564
0.110000001.0510278-0.362874662.3139784global minimum0.43215616
0.12000000-0.93351693-0.133846941.6143616metastable minimum0.61943992
0.12000000-0.121807260.0072534011-0.95548898local maximum-1.0465845
0.120000001.0553242-0.373406462.3411274global minimum0.42714463
0.13000000-0.92725465-0.124542971.5794036metastable minimum0.63315040
0.13000000-0.132316550.0085239464-0.94747699local maximum-1.0554346
0.130000001.0595712-0.383980982.3680734global minimum0.42228420
0.14000000-0.92085109-0.115302321.5439002metastable minimum0.64771026
0.14000000-0.142919260.0099000435-0.93872226local maximum-1.0652778
0.140000001.0637703-0.394597722.3948221global minimum0.41756756
0.15000000-0.91429732-0.106126451.5078187metastable minimum0.66320969
0.15000000-0.153625700.011382677-0.92919743local maximum-1.0761975
0.150000001.0679230-0.405256232.4213787global minimum0.41298786
0.16000000-0.90758342-0.0970169061.4711230metastable minimum0.67975282
0.16000000-0.164447120.012972941-0.91887143local maximum-1.0882915
0.160000001.0720305-0.415956032.4477484global minimum0.40853871
0.17000000-0.90069838-0.0879753491.4337727metastable minimum0.69746062
0.17000000-0.175395820.014672044-0.90770892local maximum-1.1016748
0.170000001.0760942-0.426696692.4739362global minimum0.40421414
0.18000000-0.89362986-0.0790035501.3957230metastable minimum0.71647456
0.18000000-0.186485360.016481326-0.89566963local maximum-1.1164831
0.180000001.0801152-0.437477782.4999467global minimum0.40000853
0.19000000-0.88636398-0.0701034101.3569233metastable minimum0.73696131
0.19000000-0.197730770.018402270-0.88270763local maximum-1.1328779
0.190000001.0840948-0.448298862.5257843global minimum0.39591663
0.20000000-0.87888507-0.0612769801.3173169metastable minimum0.75911879
0.20000000-0.209148850.020436517-0.86877028local maximum-1.1510523
0.200000001.0880339-0.459159542.5514534global minimum0.39193348
0.21000000-0.87117525-0.0525264781.2768390metastable minimum0.78318412
0.21000000-0.220758520.022585886-0.85379703local maximum-1.1712386
0.210000001.0919338-0.470059412.5769581global minimum0.38805443
0.22000000-0.86321408-0.0438543121.2354156metastable minimum0.80944419
0.22000000-0.232581260.024852397-0.83771787local maximum-1.1937193
0.220000001.0957953-0.480998082.6023023global minimum0.38427512
0.23000000-0.85497789-0.0352631131.1929616metastable minimum0.83824996
0.23000000-0.244641700.027238303-0.82045132local maximum-1.2188414
0.230000001.0996196-0.491975192.6274897global minimum0.38059140
0.24000000-0.84643915-0.0267557631.1493777metastable minimum0.87003600
0.24000000-0.256968320.029746118-0.80190185local maximum-1.2470354
0.240000001.1034075-0.502990362.6525241global minimum0.37699940
0.25000000-0.83756544-0.0183354451.1045476metastable minimum0.90534806
0.25000000-0.269594440.032378667-0.78195652local maximum-1.2788435
0.250000001.1071599-0.514043222.6774089global minimum0.37349543
0.26000000-0.82831814-0.0100056981.0583328metastable minimum0.94488234
0.26000000-0.282559520.035139136-0.76048036local maximum-1.3149584
0.260000001.1108777-0.525133442.7021475global minimum0.37007602
0.27000000-0.81865073-0.00177048191.0105671metastable minimum0.98954344
0.27000000-0.295910930.038031144-0.73731016local maximum-1.3562813
0.270000001.1145617-0.536260662.7267431global minimum0.36673789
0.28000000-0.808506230.00636572760.96104697metastable minimum1.0405319
0.28000000-0.309706450.041058833-0.71224575local maximum-1.4040098
0.280000001.1182127-0.547424562.7511988global minimum0.36347792
0.29000000-0.797813630.0143978180.90951976metastable minimum1.0994813
0.29000000-0.324017840.044226990-0.68503731local maximum-1.4597745
0.290000001.1218315-0.558624812.7755176global minimum0.36029316
0.30000000-0.786482540.0223198750.85566436metastable minimum1.1686825
0.30000000-0.338936240.047541210-0.65536667local maximum-1.5258634
0.300000001.1254188-0.569861092.7997023global minimum0.35718083
0.31000000-0.774394890.0301249510.79906232metastable minimum1.2514668
0.31000000-0.354580440.051008130-0.62281814local maximum-1.6056051
0.310000001.1289753-0.581133082.8237558global minimum0.35413827
0.32000000-0.761391350.0378047270.73915038metastable minimum1.3529047
0.32000000-0.371110420.054635765-0.58683117local maximum-1.7040676
0.320000001.1325018-0.592440492.8476808global minimum0.35116295
0.33000000-0.747247970.0453489900.67513859metastable minimum1.4811774
0.33000000-0.388750830.058434028-0.54661839local maximum-1.8294299
0.330000001.1359988-0.603783022.8714798global minimum0.34825249
0.34000000-0.731632140.0527447990.60585676metastable minimum1.6505552
0.34000000-0.407834890.062415573-0.50101210local maximum-1.9959598
0.340000001.1394670-0.615160372.8951553global minimum0.34540461
0.35000000-0.714010680.0599749900.52943377metastable minimum1.8888104
0.35000000-0.428896410.066597275-0.44814362local maximum-2.2314275
0.350000001.1429071-0.626572272.9187098global minimum0.34261713
0.36000000-0.693424430.0670152360.44251234metastable minimum2.2598240
0.36000000-0.452895130.071003186-0.38465800local maximum-2.5997120
0.360000001.1463196-0.638018422.9421457global minimum0.33988800
0.37000000-0.667770460.0738269560.33775218metastable minimum2.9607507
0.37000000-0.481934570.075671611-0.30321720local maximum-3.2979660
0.370000001.1497050-0.649498572.9654650global minimum0.33721524
0.38000000-0.629752930.0803323970.18976628metastable minimum5.2696402
0.38000000-0.523311120.080680037-0.17843642local maximum-5.6042372
0.380000001.1530641-0.661012432.9886701global minimum0.33459698
0.390000001.1563972-0.672559763.0117631global minimum0.33203142
0.400000001.1597049-0.684140293.0347460global minimum0.32951686
0.410000001.1629877-0.695753783.0576209global minimum0.32705167
0.420000001.1662460-0.707399963.0803895global minimum0.32463427
0.430000001.1694805-0.719078623.1030539global minimum0.32226318
0.440000001.1726915-0.730789503.1256158global minimum0.31993696
0.450000001.1758794-0.742532373.1480769global minimum0.31765424
0.460000001.1790447-0.754307013.1704391global minimum0.31541372
0.470000001.1821878-0.766113193.1927039global minimum0.31321414
0.480000001.1853091-0.777950693.2148730global minimum0.31105428
0.490000001.1884090-0.789819303.2369479global minimum0.30893299
0.500000001.1914879-0.801718803.2589301global minimum0.30684917

Scroll each graph or table locally to keep its labels readable. Complete text tables remain available without interpreting color.

08

A first-order jump with even symmetry

An odd power is not required for a first-order transition. Assign a second, independently stable dimensionless potential Ψ(η;a) = aη²/2 − η⁴/4 + η⁶/6 at zero field. The positive sixth-order term stabilizes the negative quartic term for large |η|. This remains symmetric under η → −η.

Stationary points satisfy η(a − η² + η⁴) = 0. For a nonzero root, put q = η²; then q² − q + a = 0. When these roots exist, the larger q branch has positive curvature except at its endpoint. But existence and local stability alone do not tell us which state has the lowest potential.

At a = 3/16, η = 0 and η = ±√3/2 have equal potential zero. For a just above this value the zero state is global; just below it the two nonzero states are global. The equilibrium magnitude therefore jumps from zero to √3/2 ≈ 0.866025. The competing phases coexist at equal potential, rather than at the point where a well first appears.

09

Locate all three thresholds

For the sextic model, the ordered local minimum and its intervening maximum merge at a = 1/4. At that exact point the nonzero stationary points η = ±1/√2 are stationary inflections, not surviving local minima. For slightly smaller a, the ordered wells exist but remain metastable until a reaches 3/16.

The zero state remains a local minimum for positive a. Below coexistence it is metastable until it loses stability at a = 0. At a = 0, the leading nonzero term near zero is −η⁴/4, so zero is a degenerate local maximum despite its zero curvature.

At coexistence the intermediate maxima lie at η = ±1/2 with Ψ = 1/96 ≈ 0.0104167. That dimensionless barrier is measured above the equal minima. Turning it into an energy or an escape rate requires additional scale and dynamical assumptions; no Kelvin, joule or second value follows from these curves.

An even potential can jump before either endpoint

The negative quartic is stabilized by a positive sixth-order term. Compare equal-depth coexistence with the separate losses of local stability.

Seven assigned even sextic landscapes

  • a = -0.05
  • a = 0
  • a = 0.125
  • a = 0.1875
  • a = 0.225
  • a = 0.25
  • a = 0.3
Seven assigned even sextic landscapesAll seven supplied curves and all their assigned samples share the same linear axes.Ψ (dimensionless potential)-0.100.10.2-1-0.500.51η (dimensionless scalar coordinate)

The coexistence barrier on a closer scale

  • a = 3/16
  • Three equal global minima
  • Dimensionless barrier 1/96
The coexistence barrier on a closer scaleAt a = 3/16, zero and ±√3/2 are equal global minima. The orange squares at ±1/2 lie 1/96 above them. This panel uses a stated smaller horizontal window.Ψ (dimensionless potential)00.0050.01-1-0.500.51η (dimensionless; |η| ≤ 1)

All stationary roots through three distinct thresholds

  • Global minima
  • Metastable local minima
  • Local maxima (including degenerate zero)
  • Stationary-inflection endpoints
  • Three coexisting minima at a = 3/16
All stationary roots through three distinct thresholdsBlue circles: global minima; orange squares: metastable minima; red crosses: maxima including degenerate zero; violet diamonds: stationary-inflection endpoints. Green rings mark the three coexisting minima at a = 3/16.Stationary η (dimensionless)-1-0.500.51-0.0500.1250.18750.250.3a (dimensionless control)
  • At a = 1/4, the nonzero endpoints ±1/√2 are stationary inflections, not remaining local minima. At a = 0, zero is a degenerate local maximum because the leading nonzero term is negative quartic.
  • The green coexistence line a = 3/16 lies between the red zero-state endpoint and violet ordered-state endpoint. The path may be followed in either control direction; no kinetic direction is assigned.
  • The barrier 1/96 is dimensionless. Without an energy calibration and dynamics it predicts neither joules nor seconds.

Expandable complete sampled-data tables follow. The linked original body table remains in its original section.

Different meanings of the assigned sextic thresholds; a decreases from the zero-state side.

Every assigned even sextic curve sample. The control a and potential Ψ are dimensionless, not measured temperatures or energies.
Every assigned even sextic curve sample. The control a and potential Ψ are dimensionless, not measured temperatures or energies.
aηΨ
-0.050000000-1.2500000-0.013631185
-0.050000000-1.2250000-0.037281055
-0.050000000-1.2000000-0.056736000
-0.050000000-1.1750000-0.072439578
-0.050000000-1.1500000-0.084803935
-0.050000000-1.1250000-0.094211197
-0.050000000-1.1000000-0.10101483
-0.050000000-1.0750000-0.10554099
-0.050000000-1.0500000-0.10808979
-0.050000000-1.0250000-0.10893661
-0.050000000-1.0000000-0.10833333
-0.050000000-0.97500000-0.10650955
-0.050000000-0.95000000-0.10367375
-0.050000000-0.92500000-0.10001449
-0.050000000-0.90000000-0.095701500
-0.050000000-0.87500000-0.090886815
-0.050000000-0.85000000-0.085705810
-0.050000000-0.82500000-0.080278256
-0.050000000-0.80000000-0.074709333
-0.050000000-0.77500000-0.069090605
-0.050000000-0.75000000-0.063500977
-0.050000000-0.72500000-0.058007612
-0.050000000-0.70000000-0.052666833
-0.050000000-0.67500000-0.047524980
-0.050000000-0.65000000-0.042619247
-0.050000000-0.62500000-0.037978490
-0.050000000-0.60000000-0.033624000
-0.050000000-0.57500000-0.029570252
-0.050000000-0.55000000-0.025825622
-0.050000000-0.52500000-0.022393078
-0.050000000-0.50000000-0.019270833
-0.050000000-0.47500000-0.016452983
-0.050000000-0.45000000-0.013930102
-0.050000000-0.42500000-0.011689812
-0.050000000-0.40000000-0.0097173333
-0.050000000-0.37500000-0.0079959869
-0.050000000-0.35000000-0.0065076849
-0.050000000-0.32500000-0.0052333818
-0.050000000-0.30000000-0.0041535000
-0.050000000-0.27500000-0.0032483252
-0.050000000-0.25000000-0.0024983724
-0.050000000-0.22500000-0.0018847233
-0.050000000-0.20000000-0.0013893333
-0.050000000-0.17500000-0.00099531051
-0.050000000-0.15000000-0.00068716406
-0.050000000-0.12500000-0.00045102437
-0.050000000-0.10000000-0.00027483333
-0.050000000-0.075000000-0.00014850549
-0.050000000-0.050000000-0.000064059896
-0.050000000-0.025000000-0.000015722616
-0.05000000000
-0.0500000000.025000000-0.000015722616
-0.0500000000.050000000-0.000064059896
-0.0500000000.075000000-0.00014850549
-0.0500000000.10000000-0.00027483333
-0.0500000000.12500000-0.00045102437
-0.0500000000.15000000-0.00068716406
-0.0500000000.17500000-0.00099531051
-0.0500000000.20000000-0.0013893333
-0.0500000000.22500000-0.0018847233
-0.0500000000.25000000-0.0024983724
-0.0500000000.27500000-0.0032483252
-0.0500000000.30000000-0.0041535000
-0.0500000000.32500000-0.0052333818
-0.0500000000.35000000-0.0065076849
-0.0500000000.37500000-0.0079959869
-0.0500000000.40000000-0.0097173333
-0.0500000000.42500000-0.011689812
-0.0500000000.45000000-0.013930102
-0.0500000000.47500000-0.016452983
-0.0500000000.50000000-0.019270833
-0.0500000000.52500000-0.022393078
-0.0500000000.55000000-0.025825622
-0.0500000000.57500000-0.029570252
-0.0500000000.60000000-0.033624000
-0.0500000000.62500000-0.037978490
-0.0500000000.65000000-0.042619247
-0.0500000000.67500000-0.047524980
-0.0500000000.70000000-0.052666833
-0.0500000000.72500000-0.058007612
-0.0500000000.75000000-0.063500977
-0.0500000000.77500000-0.069090605
-0.0500000000.80000000-0.074709333
-0.0500000000.82500000-0.080278256
-0.0500000000.85000000-0.085705810
-0.0500000000.87500000-0.090886815
-0.0500000000.90000000-0.095701500
-0.0500000000.92500000-0.10001449
-0.0500000000.95000000-0.10367375
-0.0500000000.97500000-0.10650955
-0.0500000001.0000000-0.10833333
-0.0500000001.0250000-0.10893661
-0.0500000001.0500000-0.10808979
-0.0500000001.0750000-0.10554099
-0.0500000001.1000000-0.10101483
-0.0500000001.1250000-0.094211197
-0.0500000001.1500000-0.084803935
-0.0500000001.1750000-0.072439578
-0.0500000001.2000000-0.056736000
-0.0500000001.2250000-0.037281055
-0.0500000001.2500000-0.013631185
0-1.25000000.025431315
0-1.22500000.00023457035
0-1.2000000-0.020736000
0-1.1750000-0.037923953
0-1.1500000-0.051741435
0-1.1250000-0.062570572
0-1.1000000-0.070764833
0-1.0750000-0.076650364
0-1.0500000-0.080527289
0-1.0250000-0.082670986
0-1.0000000-0.083333333
0-0.97500000-0.082743922
0-0.95000000-0.081111247
0-0.92500000-0.078623860
0-0.90000000-0.075451500
0-0.87500000-0.071746190
0-0.85000000-0.067643310
0-0.82500000-0.063262631
0-0.80000000-0.058709333
0-0.77500000-0.054074980
0-0.75000000-0.049438477
0-0.72500000-0.044866987
0-0.70000000-0.040416833
0-0.67500000-0.036134355
0-0.65000000-0.032056747
0-0.62500000-0.028212865
0-0.60000000-0.024624000
0-0.57500000-0.021304627
0-0.55000000-0.018263122
0-0.52500000-0.015502453
0-0.50000000-0.013020833
0-0.47500000-0.010812358
0-0.45000000-0.0088676016
0-0.42500000-0.0071741875
0-0.40000000-0.0057173333
0-0.37500000-0.0044803619
0-0.35000000-0.0034451849
0-0.32500000-0.0025927568
0-0.30000000-0.0019035000
0-0.27500000-0.0013577002
0-0.25000000-0.00093587240
0-0.22500000-0.00061909827
0-0.20000000-0.00038933333
0-0.17500000-0.00022968551
0-0.15000000-0.00012466406
0-0.12500000-0.000060399373
0-0.10000000-0.000024833333
0-0.075000000-0.0000078804932
0-0.050000000-0.0000015598958
0-0.025000000-9.7615560e-8
000
00.025000000-9.7615560e-8
00.050000000-0.0000015598958
00.075000000-0.0000078804932
00.10000000-0.000024833333
00.12500000-0.000060399373
00.15000000-0.00012466406
00.17500000-0.00022968551
00.20000000-0.00038933333
00.22500000-0.00061909827
00.25000000-0.00093587240
00.27500000-0.0013577002
00.30000000-0.0019035000
00.32500000-0.0025927568
00.35000000-0.0034451849
00.37500000-0.0044803619
00.40000000-0.0057173333
00.42500000-0.0071741875
00.45000000-0.0088676016
00.47500000-0.010812358
00.50000000-0.013020833
00.52500000-0.015502453
00.55000000-0.018263122
00.57500000-0.021304627
00.60000000-0.024624000
00.62500000-0.028212865
00.65000000-0.032056747
00.67500000-0.036134355
00.70000000-0.040416833
00.72500000-0.044866987
00.75000000-0.049438477
00.77500000-0.054074980
00.80000000-0.058709333
00.82500000-0.063262631
00.85000000-0.067643310
00.87500000-0.071746190
00.90000000-0.075451500
00.92500000-0.078623860
00.95000000-0.081111247
00.97500000-0.082743922
01.0000000-0.083333333
01.0250000-0.082670986
01.0500000-0.080527289
01.0750000-0.076650364
01.1000000-0.070764833
01.1250000-0.062570572
01.1500000-0.051741435
01.1750000-0.037923953
01.2000000-0.020736000
01.22500000.00023457035
01.25000000.025431315
0.12500000-1.25000000.12308757
0.12500000-1.22500000.094023633
0.12500000-1.20000000.069264000
0.12500000-1.17500000.048365109
0.12500000-1.15000000.030914815
0.12500000-1.12500000.016530991
0.12500000-1.10000000.0048601667
0.12500000-1.0750000-0.0044238017
0.12500000-1.0500000-0.011621039
0.12500000-1.0250000-0.017006924
0.12500000-1.0000000-0.020833333
0.12500000-0.97500000-0.023329860
0.12500000-0.95000000-0.024704997
0.12500000-0.92500000-0.025147298
0.12500000-0.90000000-0.024826500
0.12500000-0.87500000-0.023894628
0.12500000-0.85000000-0.022487060
0.12500000-0.82500000-0.020723569
0.12500000-0.80000000-0.018709333
0.12500000-0.77500000-0.016535918
0.12500000-0.75000000-0.014282227
0.12500000-0.72500000-0.012015425
0.12500000-0.70000000-0.0097918333
0.12500000-0.67500000-0.0076577926
0.12500000-0.65000000-0.0056504974
0.12500000-0.62500000-0.0037988027
0.12500000-0.60000000-0.0021240000
0.12500000-0.57500000-0.00064056441
0.12500000-0.550000000.00064312760
0.12500000-0.525000000.0017241097
0.12500000-0.500000000.0026041667
0.12500000-0.475000000.0032892041
0.12500000-0.450000000.0037886484
0.12500000-0.425000000.0041148750
0.12500000-0.400000000.0042826667
0.12500000-0.375000000.0043087006
0.12500000-0.350000000.0042110651
0.12500000-0.325000000.0040088057
0.12500000-0.300000000.0037215000
0.12500000-0.275000000.0033688623
0.12500000-0.250000000.0029703776
0.12500000-0.225000000.0025449642
0.12500000-0.200000000.0021106667
0.12500000-0.175000000.0016843770
0.12500000-0.150000000.0012815859
0.12500000-0.125000000.00091616313
0.12500000-0.100000000.00060016667
0.12500000-0.0750000000.00034368201
0.12500000-0.0500000000.00015469010
0.12500000-0.0250000000.000038964884
0.1250000000
0.125000000.0250000000.000038964884
0.125000000.0500000000.00015469010
0.125000000.0750000000.00034368201
0.125000000.100000000.00060016667
0.125000000.125000000.00091616313
0.125000000.150000000.0012815859
0.125000000.175000000.0016843770
0.125000000.200000000.0021106667
0.125000000.225000000.0025449642
0.125000000.250000000.0029703776
0.125000000.275000000.0033688623
0.125000000.300000000.0037215000
0.125000000.325000000.0040088057
0.125000000.350000000.0042110651
0.125000000.375000000.0043087006
0.125000000.400000000.0042826667
0.125000000.425000000.0041148750
0.125000000.450000000.0037886484
0.125000000.475000000.0032892041
0.125000000.500000000.0026041667
0.125000000.525000000.0017241097
0.125000000.550000000.00064312760
0.125000000.57500000-0.00064056441
0.125000000.60000000-0.0021240000
0.125000000.62500000-0.0037988027
0.125000000.65000000-0.0056504974
0.125000000.67500000-0.0076577926
0.125000000.70000000-0.0097918333
0.125000000.72500000-0.012015425
0.125000000.75000000-0.014282227
0.125000000.77500000-0.016535918
0.125000000.80000000-0.018709333
0.125000000.82500000-0.020723569
0.125000000.85000000-0.022487060
0.125000000.87500000-0.023894628
0.125000000.90000000-0.024826500
0.125000000.92500000-0.025147298
0.125000000.95000000-0.024704997
0.125000000.97500000-0.023329860
0.125000001.0000000-0.020833333
0.125000001.0250000-0.017006924
0.125000001.0500000-0.011621039
0.125000001.0750000-0.0044238017
0.125000001.10000000.0048601667
0.125000001.12500000.016530991
0.125000001.15000000.030914815
0.125000001.17500000.048365109
0.125000001.20000000.069264000
0.125000001.22500000.094023633
0.125000001.25000000.12308757
0.18750000-1.25000000.17191569
0.18750000-1.22500000.14091816
0.18750000-1.20000000.11426400
0.18750000-1.17500000.091509641
0.18750000-1.15000000.072242940
0.18750000-1.12500000.056081772
0.18750000-1.10000000.042672667
0.18750000-1.07500000.031689480
0.18750000-1.05000000.022832086
0.18750000-1.02500000.015825107
0.18750000-1.00000000.010416667
0.18750000-0.975000000.0063771713
0.18750000-0.950000000.0034981276
0.18750000-0.925000000.0015909834
0.18750000-0.900000000.00048600000
0.18750000-0.875000000.000031153361
0.18750000-0.850000000.000091065104
0.18750000-0.825000000.00054596228
0.18750000-0.800000000.0012906667
0.18750000-0.775000000.0022336133
0.18750000-0.750000000.0032958984
0.18750000-0.725000000.0044103565
0.18750000-0.700000000.0055206667
0.18750000-0.675000000.0065804886
0.18750000-0.650000000.0075526276
0.18750000-0.625000000.0084082286
0.18750000-0.600000000.0091260000
0.18750000-0.575000000.0096914668
0.18750000-0.550000000.010096253
0.18750000-0.525000000.010337391
0.18750000-0.500000000.010416667
0.18750000-0.475000000.010339985
0.18750000-0.450000000.010116773
0.18750000-0.425000000.0097594063
0.18750000-0.400000000.0092826667
0.18750000-0.375000000.0087032318
0.18750000-0.350000000.0080391901
0.18750000-0.325000000.0073095870
0.18750000-0.300000000.0065340000
0.18750000-0.275000000.0057321436
0.18750000-0.250000000.0049235026
0.18750000-0.225000000.0041269955
0.18750000-0.200000000.0033606667
0.18750000-0.175000000.0026414082
0.18750000-0.150000000.0019847109
0.18750000-0.125000000.0014044444
0.18750000-0.100000000.00091266667
0.18750000-0.0750000000.00051946326
0.18750000-0.0500000000.00023281510
0.18750000-0.0250000000.000058496134
0.1875000000
0.187500000.0250000000.000058496134
0.187500000.0500000000.00023281510
0.187500000.0750000000.00051946326
0.187500000.100000000.00091266667
0.187500000.125000000.0014044444
0.187500000.150000000.0019847109
0.187500000.175000000.0026414082
0.187500000.200000000.0033606667
0.187500000.225000000.0041269955
0.187500000.250000000.0049235026
0.187500000.275000000.0057321436
0.187500000.300000000.0065340000
0.187500000.325000000.0073095870
0.187500000.350000000.0080391901
0.187500000.375000000.0087032318
0.187500000.400000000.0092826667
0.187500000.425000000.0097594063
0.187500000.450000000.010116773
0.187500000.475000000.010339985
0.187500000.500000000.010416667
0.187500000.525000000.010337391
0.187500000.550000000.010096253
0.187500000.575000000.0096914668
0.187500000.600000000.0091260000
0.187500000.625000000.0084082286
0.187500000.650000000.0075526276
0.187500000.675000000.0065804886
0.187500000.700000000.0055206667
0.187500000.725000000.0044103565
0.187500000.750000000.0032958984
0.187500000.775000000.0022336133
0.187500000.800000000.0012906667
0.187500000.825000000.00054596228
0.187500000.850000000.000091065104
0.187500000.875000000.000031153361
0.187500000.900000000.00048600000
0.187500000.925000000.0015909834
0.187500000.950000000.0034981276
0.187500000.975000000.0063771713
0.187500001.00000000.010416667
0.187500001.02500000.015825107
0.187500001.05000000.022832086
0.187500001.07500000.031689480
0.187500001.10000000.042672667
0.187500001.12500000.056081772
0.187500001.15000000.072242940
0.187500001.17500000.091509641
0.187500001.20000000.11426400
0.187500001.22500000.14091816
0.187500001.25000000.17191569
0.22500000-1.25000000.20121257
0.22500000-1.22500000.16905488
0.22500000-1.20000000.14126400
0.22500000-1.17500000.11739636
0.22500000-1.15000000.097039815
0.22500000-1.12500000.079812241
0.22500000-1.10000000.065360167
0.22500000-1.07500000.053357448
0.22500000-1.05000000.043503961
0.22500000-1.02500000.035524326
0.22500000-1.00000000.029166667
0.22500000-0.975000000.024201390
0.22500000-0.950000000.020420003
0.22500000-0.925000000.017633952
0.22500000-0.900000000.015673500
0.22500000-0.875000000.014386622
0.22500000-0.850000000.013637940
0.22500000-0.825000000.013307681
0.22500000-0.800000000.013290667
0.22500000-0.775000000.013495332
0.22500000-0.750000000.013842773
0.22500000-0.725000000.014265825
0.22500000-0.700000000.014708167
0.22500000-0.675000000.015123457
0.22500000-0.650000000.015474503
0.22500000-0.625000000.015732447
0.22500000-0.600000000.015876000
0.22500000-0.575000000.015890686
0.22500000-0.550000000.015768128
0.22500000-0.525000000.015505360
0.22500000-0.500000000.015104167
0.22500000-0.475000000.014570454
0.22500000-0.450000000.013913648
0.22500000-0.425000000.013146125
0.22500000-0.400000000.012282667
0.22500000-0.375000000.011339951
0.22500000-0.350000000.010336065
0.22500000-0.325000000.0092900557
0.22500000-0.300000000.0082215000
0.22500000-0.275000000.0071501123
0.22500000-0.250000000.0060953776
0.22500000-0.225000000.0050762142
0.22500000-0.200000000.0041106667
0.22500000-0.175000000.0032156270
0.22500000-0.150000000.0024065859
0.22500000-0.125000000.0016974131
0.22500000-0.100000000.0011001667
0.22500000-0.0750000000.00062493201
0.22500000-0.0500000000.00027969010
0.22500000-0.0250000000.000070214884
0.2250000000
0.225000000.0250000000.000070214884
0.225000000.0500000000.00027969010
0.225000000.0750000000.00062493201
0.225000000.100000000.0011001667
0.225000000.125000000.0016974131
0.225000000.150000000.0024065859
0.225000000.175000000.0032156270
0.225000000.200000000.0041106667
0.225000000.225000000.0050762142
0.225000000.250000000.0060953776
0.225000000.275000000.0071501123
0.225000000.300000000.0082215000
0.225000000.325000000.0092900557
0.225000000.350000000.010336065
0.225000000.375000000.011339951
0.225000000.400000000.012282667
0.225000000.425000000.013146125
0.225000000.450000000.013913648
0.225000000.475000000.014570454
0.225000000.500000000.015104167
0.225000000.525000000.015505360
0.225000000.550000000.015768128
0.225000000.575000000.015890686
0.225000000.600000000.015876000
0.225000000.625000000.015732447
0.225000000.650000000.015474503
0.225000000.675000000.015123457
0.225000000.700000000.014708167
0.225000000.725000000.014265825
0.225000000.750000000.013842773
0.225000000.775000000.013495332
0.225000000.800000000.013290667
0.225000000.825000000.013307681
0.225000000.850000000.013637940
0.225000000.875000000.014386622
0.225000000.900000000.015673500
0.225000000.925000000.017633952
0.225000000.950000000.020420003
0.225000000.975000000.024201390
0.225000001.00000000.029166667
0.225000001.02500000.035524326
0.225000001.05000000.043503961
0.225000001.07500000.053357448
0.225000001.10000000.065360167
0.225000001.12500000.079812241
0.225000001.15000000.097039815
0.225000001.17500000.11739636
0.225000001.20000000.14126400
0.225000001.22500000.16905488
0.225000001.25000000.20121257
0.25000000-1.25000000.22074382
0.25000000-1.22500000.18781270
0.25000000-1.20000000.15926400
0.25000000-1.17500000.13465417
0.25000000-1.15000000.11357107
0.25000000-1.12500000.095632553
0.25000000-1.10000000.080485167
0.25000000-1.07500000.067802761
0.25000000-1.05000000.057285211
0.25000000-1.02500000.048657139
0.25000000-1.00000000.041666667
0.25000000-0.975000000.036084203
0.25000000-0.950000000.031701253
0.25000000-0.925000000.028329265
0.25000000-0.900000000.025798500
0.25000000-0.875000000.023956935
0.25000000-0.850000000.022669190
0.25000000-0.825000000.021815494
0.25000000-0.800000000.021290667
0.25000000-0.775000000.021003145
0.25000000-0.750000000.020874023
0.25000000-0.725000000.020836138
0.25000000-0.700000000.020833167
0.25000000-0.675000000.020818770
0.25000000-0.650000000.020755753
0.25000000-0.625000000.020615260
0.25000000-0.600000000.020376000
0.25000000-0.575000000.020023498
0.25000000-0.550000000.019549378
0.25000000-0.525000000.018950672
0.25000000-0.500000000.018229167
0.25000000-0.475000000.017390767
0.25000000-0.450000000.016444898
0.25000000-0.425000000.015403938
0.25000000-0.400000000.014282667
0.25000000-0.375000000.013097763
0.25000000-0.350000000.011867315
0.25000000-0.325000000.010610368
0.25000000-0.300000000.0093465000
0.25000000-0.275000000.0080954248
0.25000000-0.250000000.0068766276
0.25000000-0.225000000.0057090267
0.25000000-0.200000000.0046106667
0.25000000-0.175000000.0035984395
0.25000000-0.150000000.0026878359
0.25000000-0.125000000.0018927256
0.25000000-0.100000000.0012251667
0.25000000-0.0750000000.00069524451
0.25000000-0.0500000000.00031094010
0.25000000-0.0250000000.000078027384
0.2500000000
0.250000000.0250000000.000078027384
0.250000000.0500000000.00031094010
0.250000000.0750000000.00069524451
0.250000000.100000000.0012251667
0.250000000.125000000.0018927256
0.250000000.150000000.0026878359
0.250000000.175000000.0035984395
0.250000000.200000000.0046106667
0.250000000.225000000.0057090267
0.250000000.250000000.0068766276
0.250000000.275000000.0080954248
0.250000000.300000000.0093465000
0.250000000.325000000.010610368
0.250000000.350000000.011867315
0.250000000.375000000.013097763
0.250000000.400000000.014282667
0.250000000.425000000.015403938
0.250000000.450000000.016444898
0.250000000.475000000.017390767
0.250000000.500000000.018229167
0.250000000.525000000.018950672
0.250000000.550000000.019549378
0.250000000.575000000.020023498
0.250000000.600000000.020376000
0.250000000.625000000.020615260
0.250000000.650000000.020755753
0.250000000.675000000.020818770
0.250000000.700000000.020833167
0.250000000.725000000.020836138
0.250000000.750000000.020874023
0.250000000.775000000.021003145
0.250000000.800000000.021290667
0.250000000.825000000.021815494
0.250000000.850000000.022669190
0.250000000.875000000.023956935
0.250000000.900000000.025798500
0.250000000.925000000.028329265
0.250000000.950000000.031701253
0.250000000.975000000.036084203
0.250000001.00000000.041666667
0.250000001.02500000.048657139
0.250000001.05000000.057285211
0.250000001.07500000.067802761
0.250000001.10000000.080485167
0.250000001.12500000.095632553
0.250000001.15000000.11357107
0.250000001.17500000.13465417
0.250000001.20000000.15926400
0.250000001.22500000.18781270
0.250000001.25000000.22074382
0.30000000-1.25000000.25980632
0.30000000-1.22500000.22532832
0.30000000-1.20000000.19526400
0.30000000-1.17500000.16916980
0.30000000-1.15000000.14663357
0.30000000-1.12500000.12727318
0.30000000-1.10000000.11073517
0.30000000-1.07500000.096693386
0.30000000-1.05000000.084847711
0.30000000-1.02500000.074922764
0.30000000-1.00000000.066666667
0.30000000-0.975000000.059849828
0.30000000-0.950000000.054263753
0.30000000-0.925000000.049719890
0.30000000-0.900000000.046048500
0.30000000-0.875000000.043097560
0.30000000-0.850000000.040731690
0.30000000-0.825000000.038831119
0.30000000-0.800000000.037290667
0.30000000-0.775000000.036018770
0.30000000-0.750000000.034936523
0.30000000-0.725000000.033976763
0.30000000-0.700000000.033083167
0.30000000-0.675000000.032209395
0.30000000-0.650000000.031318253
0.30000000-0.625000000.030380885
0.30000000-0.600000000.029376000
0.30000000-0.575000000.028289123
0.30000000-0.550000000.027111878
0.30000000-0.525000000.025841297
0.30000000-0.500000000.024479167
0.30000000-0.475000000.023031392
0.30000000-0.450000000.021507398
0.30000000-0.425000000.019919563
0.30000000-0.400000000.018282667
0.30000000-0.375000000.016613388
0.30000000-0.350000000.014929815
0.30000000-0.325000000.013250993
0.30000000-0.300000000.011596500
0.30000000-0.275000000.0099860498
0.30000000-0.250000000.0084391276
0.30000000-0.225000000.0069746517
0.30000000-0.200000000.0056106667
0.30000000-0.175000000.0043640645
0.30000000-0.150000000.0032503359
0.30000000-0.125000000.0022833506
0.30000000-0.100000000.0014751667
0.30000000-0.0750000000.00083586951
0.30000000-0.0500000000.00037344010
0.30000000-0.0250000000.000093652384
0.3000000000
0.300000000.0250000000.000093652384
0.300000000.0500000000.00037344010
0.300000000.0750000000.00083586951
0.300000000.100000000.0014751667
0.300000000.125000000.0022833506
0.300000000.150000000.0032503359
0.300000000.175000000.0043640645
0.300000000.200000000.0056106667
0.300000000.225000000.0069746517
0.300000000.250000000.0084391276
0.300000000.275000000.0099860498
0.300000000.300000000.011596500
0.300000000.325000000.013250993
0.300000000.350000000.014929815
0.300000000.375000000.016613388
0.300000000.400000000.018282667
0.300000000.425000000.019919563
0.300000000.450000000.021507398
0.300000000.475000000.023031392
0.300000000.500000000.024479167
0.300000000.525000000.025841297
0.300000000.550000000.027111878
0.300000000.575000000.028289123
0.300000000.600000000.029376000
0.300000000.625000000.030380885
0.300000000.650000000.031318253
0.300000000.675000000.032209395
0.300000000.700000000.033083167
0.300000000.725000000.033976763
0.300000000.750000000.034936523
0.300000000.775000000.036018770
0.300000000.800000000.037290667
0.300000000.825000000.038831119
0.300000000.850000000.040731690
0.300000000.875000000.043097560
0.300000000.900000000.046048500
0.300000000.925000000.049719890
0.300000000.950000000.054263753
0.300000000.975000000.059849828
0.300000001.00000000.066666667
0.300000001.02500000.074922764
0.300000001.05000000.084847711
0.300000001.07500000.096693386
0.300000001.10000000.11073517
0.300000001.12500000.12727318
0.300000001.15000000.14663357
0.300000001.17500000.16916980
0.300000001.20000000.19526400
0.300000001.22500000.22532832
0.300000001.25000000.25980632
All stationary points at every assigned sextic a sample. Zero at a = 0 is a degenerate local maximum; nonzero roots at a = 1/4 are stationary inflections.
All stationary points at every assigned sextic a sample. Zero at a = 0 is a degenerate local maximum; nonzero roots at a = 1/4 are stationary inflections.
aηΨCurvatureClassification
-0.050000000-1.0235832-0.108938922.2954451global minimum
-0.05000000000-0.050000000local maximum
-0.0500000001.0235832-0.108938922.2954451global minimum
-0.047500000-1.0224655-0.107630702.2808712global minimum
-0.04750000000-0.047500000local maximum
-0.0475000001.0224655-0.107630702.2808712global minimum
-0.045000000-1.0213418-0.106325342.2662780global minimum
-0.04500000000-0.045000000local maximum
-0.0450000001.0213418-0.106325342.2662780global minimum
-0.042500000-1.0202121-0.105022852.2516654global minimum
-0.04250000000-0.042500000local maximum
-0.0425000001.0202121-0.105022852.2516654global minimum
-0.040000000-1.0190763-0.103723262.2370330global minimum
-0.04000000000-0.040000000local maximum
-0.0400000001.0190763-0.103723262.2370330global minimum
-0.037500000-1.0179343-0.102426572.2223805global minimum
-0.03750000000-0.037500000local maximum
-0.0375000001.0179343-0.102426572.2223805global minimum
-0.035000000-1.0167861-0.101132792.2077078global minimum
-0.03500000000-0.035000000local maximum
-0.0350000001.0167861-0.101132792.2077078global minimum
-0.032500000-1.0156315-0.0998419372.1930146global minimum
-0.03250000000-0.032500000local maximum
-0.0325000001.0156315-0.0998419372.1930146global minimum
-0.030000000-1.0144704-0.0985540242.1783005global minimum
-0.03000000000-0.030000000local maximum
-0.0300000001.0144704-0.0985540242.1783005global minimum
-0.027500000-1.0133029-0.0972690652.1635654global minimum
-0.02750000000-0.027500000local maximum
-0.0275000001.0133029-0.0972690652.1635654global minimum
-0.025000000-1.0121287-0.0959870722.1488088global minimum
-0.02500000000-0.025000000local maximum
-0.0250000001.0121287-0.0959870722.1488088global minimum
-0.022500000-1.0109477-0.0947080592.1340307global minimum
-0.02250000000-0.022500000local maximum
-0.0225000001.0109477-0.0947080592.1340307global minimum
-0.020000000-1.0097600-0.0934320382.1192305global minimum
-0.02000000000-0.020000000local maximum
-0.0200000001.0097600-0.0934320382.1192305global minimum
-0.017500000-1.0085653-0.0921590252.1044080global minimum
-0.01750000000-0.017500000local maximum
-0.0175000001.0085653-0.0921590252.1044080global minimum
-0.015000000-1.0073636-0.0908890332.0895630global minimum
-0.01500000000-0.015000000local maximum
-0.0150000001.0073636-0.0908890332.0895630global minimum
-0.012500000-1.0061548-0.0896220762.0746951global minimum
-0.01250000000-0.012500000local maximum
-0.0125000001.0061548-0.0896220762.0746951global minimum
-0.010000000-1.0049388-0.0883581692.0598039global minimum
-0.01000000000-0.010000000local maximum
-0.0100000001.0049388-0.0883581692.0598039global minimum
-0.0075000000-1.0037154-0.0870973262.0448892global minimum
-0.007500000000-0.0075000000local maximum
-0.00750000001.0037154-0.0870973262.0448892global minimum
-0.0050000000-1.0024845-0.0858395632.0299505global minimum
-0.005000000000-0.0050000000local maximum
-0.00500000001.0024845-0.0858395632.0299505global minimum
-0.0025000000-1.0012461-0.0845848932.0149876global minimum
-0.002500000000-0.0025000000local maximum
-0.00250000001.0012461-0.0845848932.0149876global minimum
0-1.0000000-0.0833333332.0000000global minimum
0000marginal local maximum
01.0000000-0.0833333332.0000000global minimum
0.0025000000-0.99874607-0.0820848981.9849874global minimum
0.0025000000-0.0500627750.0000015651140-0.0049874371local maximum
0.0025000000000.0025000000metastable minimum
0.00250000000.0500627750.0000015651140-0.0049874371local maximum
0.00250000000.99874607-0.0820848981.9849874global minimum
0.0050000000-0.99748421-0.0808396041.9699495global minimum
0.0050000000-0.0708890200.0000062709912-0.0099494937local maximum
0.0050000000000.0050000000metastable minimum
0.00500000000.0708890200.0000062709912-0.0099494937local maximum
0.00500000000.99748421-0.0808396041.9699495global minimum
0.0075000000-0.99621428-0.0795974671.9548858global minimum
0.0075000000-0.0869316390.000014133616-0.014885780local maximum
0.0075000000000.0075000000metastable minimum
0.00750000000.0869316390.000014133616-0.014885780local maximum
0.00750000000.99621428-0.0795974671.9548858global minimum
0.010000000-0.99493615-0.0783585031.9397959global minimum
0.010000000-0.100508960.000025169218-0.019795897local maximum
0.010000000000.010000000metastable minimum
0.0100000000.100508960.000025169218-0.019795897local maximum
0.0100000000.99493615-0.0783585031.9397959global minimum
0.012500000-0.99364970-0.0771227281.9246794global minimum
0.012500000-0.112517920.000039394282-0.024679434local maximum
0.012500000000.012500000metastable minimum
0.0125000000.112517920.000039394282-0.024679434local maximum
0.0125000000.99364970-0.0771227281.9246794global minimum
0.015000000-0.99235477-0.0758901591.9095360global minimum
0.015000000-0.123418050.000056825550-0.029535971local maximum
0.015000000000.015000000metastable minimum
0.0150000000.123418050.000056825550-0.029535971local maximum
0.0150000000.99235477-0.0758901591.9095360global minimum
0.017500000-0.99105123-0.0746608131.8943651global minimum
0.017500000-0.133482070.000077480032-0.034365076local maximum
0.017500000000.017500000metastable minimum
0.0175000000.133482070.000077480032-0.034365076local maximum
0.0175000000.99105123-0.0746608131.8943651global minimum
0.020000000-0.98973893-0.0734347081.8791663global minimum
0.020000000-0.142887540.00010137501-0.039166305local maximum
0.020000000000.020000000metastable minimum
0.0200000000.142887540.00010137501-0.039166305local maximum
0.0200000000.98973893-0.0734347081.8791663global minimum
0.022500000-0.98841773-0.0722118611.8639392global minimum
0.022500000-0.151757700.00012852805-0.043939201local maximum
0.022500000000.022500000metastable minimum
0.0225000000.151757700.00012852805-0.043939201local maximum
0.0225000000.98841773-0.0722118611.8639392global minimum
0.025000000-0.98708746-0.0709922901.8486833global minimum
0.025000000-0.160182240.00015895701-0.048683298local maximum
0.025000000000.025000000metastable minimum
0.0250000000.160182240.00015895701-0.048683298local maximum
0.0250000000.98708746-0.0709922901.8486833global minimum
0.027500000-0.98574797-0.0697760131.8333981global minimum
0.027500000-0.168228840.00019268003-0.053398113local maximum
0.027500000000.027500000metastable minimum
0.0275000000.168228840.00019268003-0.053398113local maximum
0.0275000000.98574797-0.0697760131.8333981global minimum
0.030000000-0.98439909-0.0685630491.8180832global minimum
0.030000000-0.175950060.00022971557-0.058083152local maximum
0.030000000000.030000000metastable minimum
0.0300000000.175950060.00022971557-0.058083152local maximum
0.0300000000.98439909-0.0685630491.8180832global minimum
0.032500000-0.98304067-0.0673534161.8027379global minimum
0.032500000-0.183387700.00027008240-0.062737905local maximum
0.032500000000.032500000metastable minimum
0.0325000000.183387700.00027008240-0.062737905local maximum
0.0325000000.98304067-0.0673534161.8027379global minimum
0.035000000-0.98167251-0.0661471331.7873618global minimum
0.035000000-0.190575640.00031379961-0.067361850local maximum
0.035000000000.035000000metastable minimum
0.0350000000.190575640.00031379961-0.067361850local maximum
0.0350000000.98167251-0.0661471331.7873618global minimum
0.037500000-0.98029446-0.0649442201.7719544global minimum
0.037500000-0.197541840.00036088662-0.071954446local maximum
0.037500000000.037500000metastable minimum
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0.21250000-0.832856030.00903771870.53729833metastable minimum
0.21250000-0.553489690.013878948-0.23729833local maximum
0.21250000000.21250000global minimum
0.212500000.553489690.013878948-0.23729833local maximum
0.212500000.832856030.00903771870.53729833metastable minimum
0.21500000-0.828904620.00990069990.51416574metastable minimum
0.21500000-0.559389960.014265967-0.23416574local maximum
0.21500000000.21500000global minimum
0.215000000.559389960.014265967-0.23416574local maximum
0.215000000.828904620.00990069990.51416574metastable minimum
0.21750000-0.824789410.0107553260.49055513metastable minimum
0.21750000-0.565440040.014661340-0.23055513local maximum
0.21750000000.21750000global minimum
0.217500000.565440040.014661340-0.23055513local maximum
0.217500000.824789410.0107553260.49055513metastable minimum
0.22000000-0.820490760.0116012830.46641016metastable minimum
0.22000000-0.571659790.015065384-0.22641016local maximum
0.22000000000.22000000global minimum
0.220000000.571659790.015065384-0.22641016local maximum
0.220000000.820490760.0116012830.46641016metastable minimum
0.22250000-0.815984830.0124382140.44166248metastable minimum
0.22250000-0.578073320.015478453-0.22166248local maximum
0.22250000000.22250000global minimum
0.222500000.578073320.015478453-0.22166248local maximum
0.222500000.815984830.0124382140.44166248metastable minimum
0.22500000-0.811242190.0132657180.41622777metastable minimum
0.22500000-0.584710280.015900949-0.21622777local maximum
0.22500000000.22500000global minimum
0.225000000.584710280.015900949-0.21622777local maximum
0.225000000.811242190.0132657180.41622777metastable minimum
0.22750000-0.806225770.0140833330.39000000metastable minimum
0.22750000-0.591607980.016333333-0.21000000local maximum
0.22750000000.22750000global minimum
0.227500000.591607980.016333333-0.21000000local maximum
0.227500000.806225770.0140833330.39000000metastable minimum
0.23000000-0.800887850.0148905240.36284271metastable minimum
0.23000000-0.598814370.016776142-0.20284271local maximum
0.23000000000.23000000global minimum
0.230000000.598814370.016776142-0.20284271local maximum
0.230000000.800887850.0148905240.36284271metastable minimum
0.23250000-0.795165120.0156866560.33457513metastable minimum
0.23250000-0.606392970.017230011-0.19457513local maximum
0.23250000000.23250000global minimum
0.232500000.606392970.017230011-0.19457513local maximum
0.232500000.795165120.0156866560.33457513metastable minimum
0.23500000-0.788970520.0164709610.30494897metastable minimum
0.23500000-0.614431050.017695706-0.18494897local maximum
0.23500000000.23500000global minimum
0.235000000.614431050.017695706-0.18494897local maximum
0.235000000.788970520.0164709610.30494897metastable minimum
0.23750000-0.782178620.0172424860.27360680metastable minimum
0.23750000-0.623054250.018174181-0.17360680local maximum
0.23750000000.23750000global minimum
0.237500000.623054250.018174181-0.17360680local maximum
0.237500000.782178620.0172424860.27360680metastable minimum
0.24000000-0.774596670.0180000000.24000000metastable minimum
0.24000000-0.632455530.018666667-0.16000000local maximum
0.24000000000.24000000global minimum
0.240000000.632455530.018666667-0.16000000local maximum
0.240000000.774596670.0180000000.24000000metastable minimum
0.24250000-0.765899820.0187418270.20320508metastable minimum
0.24250000-0.642959920.019174840-0.14320508local maximum
0.24250000000.24250000global minimum
0.242500000.642959920.019174840-0.14320508local maximum
0.242500000.765899820.0187418270.20320508metastable minimum
0.24500000-0.755453950.0194654820.16142136metastable minimum
0.24500000-0.655201740.019701184-0.12142136local maximum
0.24500000000.24500000global minimum
0.245000000.655201740.019701184-0.12142136local maximum
0.245000000.755453950.0194654820.16142136metastable minimum
0.24750000-0.741619850.0201666670.11000000metastable minimum
0.24750000-0.670820390.020250000-0.090000000local maximum
0.24750000000.24750000global minimum
0.247500000.670820390.020250000-0.090000000local maximum
0.247500000.741619850.0201666670.11000000metastable minimum
0.25000000-0.707106780.0208333330stationary inflection spinodal
0.25000000000.25000000global minimum
0.250000000.707106780.0208333330stationary inflection spinodal
0.25250000000.25250000global minimum
0.25500000000.25500000global minimum
0.25750000000.25750000global minimum
0.26000000000.26000000global minimum
0.26250000000.26250000global minimum
0.26500000000.26500000global minimum
0.26750000000.26750000global minimum
0.27000000000.27000000global minimum
0.27250000000.27250000global minimum
0.27500000000.27500000global minimum
0.27750000000.27750000global minimum
0.28000000000.28000000global minimum
0.28250000000.28250000global minimum
0.28500000000.28500000global minimum
0.28750000000.28750000global minimum
0.29000000000.29000000global minimum
0.29250000000.29250000global minimum
0.29500000000.29500000global minimum
0.29750000000.29750000global minimum
0.30000000000.30000000global minimum

Scroll each graph or table locally to keep its labels readable. Complete text tables remain available without interpreting color.

Different meanings of the assigned sextic thresholds; a decreases from the zero-state side.
aWhat occursWhat it does not establish
1/4Ordered extrema meet at stationary inflectionsNot coexistence or an assigned experimental switching point
3/16Zero and nonzero minima have equal global potentialNot disappearance of either well
0Zero state is a degenerate local maximumNot zero curvature implying a stable state
10

Count a finite collective magnet

Return to bounded spins. Assign E_N = −J M²/(2N) − H M with J > 0. Here J and H have energy units; H includes the moment-to-field conversion and is not directly a magnetic field in tesla. The factor 1/N makes the all-to-all interaction energy extensive. This is the Curie–Weiss model, not a nearest-neighbour lattice.

Let β = 1/(k_B T), K = βJ and b = βH. If k of the N labelled spins are positive, then M = 2k − N and m = (2k − N)/N. Exactly C(N,k) configurations have that k, where C is the binomial coefficient. Their sector weight is w_k = C(N,k) exp[K(2k − N)²/(2N) + b(2k − N)].

Sum all sectors k = 0,…,N to get Z_N = Σ_kw_k, then P_k = w_k/Z_N. No Gaussian approximation or selection of just two most likely sectors is needed for N = 20 or 80. The self terms included in M² contribute a state-independent energy offset at fixed J; they do not change these probabilities.

11

Zero mean can hide two peaks

At b = 0, the weights obey w_k = w_(N−k). Every configuration sector at m has a partner at −m with exactly the same probability. Consequently ⟨m⟩ = 0 for either assigned finite size, including distributions with two well-separated peaks.

Compare the complete distributions at K = 0.5, 1 and 1.5. Increasing K strengthens alignment relative to thermal agitation. At K = 1.5 the distribution is concentrated away from zero, but the equal positive and negative weights still cancel in the signed mean. A histogram resolves what that single mean conceals.

The larger sample has a greater mean magnitude at K = 1.5, yet both signed means remain exactly zero. Positive ⟨|m|⟩ alone is not a phase-transition test: even weakly coupled finite spins fluctuate away from zero. Sample size, full distribution, field and limit procedure matter.

Zero signed mean can conceal two ordered peaks

Every finite sector is shown for N = 20 and N = 80 at zero field. The three K values change the distribution, not the exact symmetry of its signed mean.

N = 20: complete discrete probabilities

  • K = 0.5
  • K = 1
  • K = 1.5
N = 20: complete discrete probabilitiesAll 21 sectors for each K are stems from a true zero-probability baseline. The horizontal positions are discrete allowed m values.Probability P(m), not density00.050.10.15-1-0.500.51m = (2k − N)/N (dimensionless)

N = 80: complete probabilities on the same linear scale

  • K = 0.5
  • K = 1
  • K = 1.5
N = 80: complete probabilities on the same linear scaleAll 81 sectors for each K are retained. This panel uses the same linear probability scale as N = 20; these masses are not a continuous density.Probability P(m), not density00.050.10.15-1-0.500.51m = (2k − N)/N (dimensionless)

N = 20: the same nonzero tails, logarithmic view

  • K = 0.5
  • K = 1
  • K = 1.5
N = 20: the same nonzero tails, logarithmic viewThe same 21 sectors and same exact probabilities. The stem floor is 10⁻¹⁶, not zero; equal vertical increments represent equal probability ratios.Probability (logarithmic scale)10⁻¹⁶10⁻¹²10⁻⁸10⁻⁴1-1-0.500.51Discrete m (dimensionless)

N = 80: no tail is truncated or renormalized

  • K = 0.5
  • K = 1
  • K = 1.5
N = 80: no tail is truncated or renormalizedThe same 81 sectors and logarithmic range. The smallest assigned probability is about 2.84636 × 10⁻¹⁶, strictly above the displayed floor.Probability (logarithmic scale)10⁻¹⁶10⁻¹²10⁻⁸10⁻⁴1-1-0.500.51Discrete m (dimensionless)
  • Blue circles: K = 0.5; orange squares: K = 1; green triangles: K = 1.5. All panels have b = 0. Symmetry gives ⟨m⟩ = 0, even when the peaks are away from zero.
  • Stems mark probability masses at allowed m values. There is no smooth density interpolation, Gaussian replacement, two-peak truncation or tail renormalization.
  • The complete tables retain every k and its full exact integer binomial multiplicity. No rounded count is used to replace a configuration count.

Expandable complete sampled-data tables follow. The linked original body table remains in its original section.

Three descriptions that must not be confused in the assigned spin model.

Exact finite sums at K = 1.5 and b = 0, rounded here; m and b are dimensionless.

Complete finite sector probabilities at N = 20, K = 0.50000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Complete finite sector probabilities at N = 20, K = 0.50000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Positive-spin count kMmExact configuration multiplicityProbability
0-20-1.000000010.00010120364
1-18-0.90000000200.00078279201
2-16-0.800000001900.0031784814
3-14-0.7000000011400.0090084499
4-12-0.6000000048450.019986999
5-10-0.50000000155040.036900784
6-8-0.40000000387600.058822447
7-6-0.30000000775200.082902956
8-4-0.200000001259700.10491794
9-2-0.100000001679600.12040495
10001847560.12598600
1120.100000001679600.12040495
1240.200000001259700.10491794
1360.30000000775200.082902956
1480.40000000387600.058822447
15100.50000000155040.036900784
16120.6000000048450.019986999
17140.7000000011400.0090084499
18160.800000001900.0031784814
19180.90000000200.00078279201
20201.000000010.00010120364
Complete finite sector probabilities at N = 20, K = 1.0000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Complete finite sector probabilities at N = 20, K = 1.0000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Positive-spin count kMmExact configuration multiplicityProbability
0-20-1.000000010.0070142253
1-18-0.90000000200.020982160
2-16-0.800000001900.036414401
3-14-0.7000000011400.048750907
4-12-0.6000000048450.056466232
5-10-0.50000000155040.060147122
6-8-0.40000000387600.061134988
7-6-0.30000000775200.060717473
8-4-0.200000001259700.059843890
9-2-0.100000001679600.059111258
10001847560.058834686
1120.100000001679600.059111258
1240.200000001259700.059843890
1360.30000000775200.060717473
1480.40000000387600.061134988
15100.50000000155040.060147122
16120.6000000048450.056466232
17140.7000000011400.048750907
18160.800000001900.036414401
19180.90000000200.020982160
20201.000000010.0070142253
Complete finite sector probabilities at N = 20, K = 1.5000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Complete finite sector probabilities at N = 20, K = 1.5000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Positive-spin count kMmExact configuration multiplicityProbability
0-20-1.000000010.11190995
1-18-0.90000000200.12946710
2-16-0.800000001900.096035567
3-14-0.7000000011400.060732446
4-12-0.6000000048450.036722772
5-10-0.50000000155040.022568336
6-8-0.40000000387600.014626553
7-6-0.30000000775200.010236766
8-4-0.200000001259700.0078576973
9-2-0.100000001679600.0066803853
10001847560.0063248471
1120.100000001679600.0066803853
1240.200000001259700.0078576973
1360.30000000775200.010236766
1480.40000000387600.014626553
15100.50000000155040.022568336
16120.6000000048450.036722772
17140.7000000011400.060732446
18160.800000001900.096035567
19180.90000000200.12946710
20201.000000010.11190995
Complete finite sector probabilities at N = 80, K = 0.50000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Complete finite sector probabilities at N = 80, K = 0.50000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Positive-spin count kMmExact configuration multiplicityProbability
0-80-1.000000012.8463577e-16
1-78-0.97500000808.4823008e-15
2-76-0.9500000031601.2796828e-13
3-74-0.92500000821601.3029405e-12
4-72-0.9000000015815801.0070745e-11
5-70-0.87500000240400166.3018615e-11
6-68-0.850000003005002003.3250630e-10
7-66-0.8250000031767164001.5212897e-9
8-64-0.80000000289875371506.1599974e-9
9-62-0.775000002319002972002.2421446e-8
10-60-0.7500000016464921101207.4262988e-8
11-58-0.72500000104776770644002.2604015e-7
12-56-0.70000000602466431203006.3740963e-7
13-54-0.675000003151362870908000.0000016765122
14-52-0.6500000015081522310774000.0000041365035
15-50-0.6250000066358698167405600.0000096210693
16-48-0.60000000269582211305085250.000021184165
17-46-0.575000001014897736677968000.000044319453
18-44-0.550000003552142078372888000.000088383621
19-42-0.5250000011591200466269424000.00016849096
20-40-0.5000000035353161422121743200.00030782261
21-38-0.47500000101009032634633552000.00054014941
22-36-0.45000000270887860247426344000.00091218314
23-34-0.42500000683108517145683824000.0014851775
24-32-0.400000001622382728220999082000.0023350463
25-30-0.375000003634137311215037943680.0035502130
26-28-0.350000007687598158339503342400.0052265126
27-26-0.3250000015375196316679006684800.0074587767
28-24-0.3000000029103050170856691224800.010329241
29-22-0.2750000052184779616708549782400.013893578
30-20-0.2500000088714125348404534630080.018166059
31-18-0.22500000143087298949039571984000.023105964
32-16-0.20000000219102426515716844600500.028607679
33-14-0.17500000318694438568315410328000.034496871
34-12-0.15000000440548194491494831924000.040534590
35-10-0.12500000579006198474536064814400.046430150
36-8-0.10000000723757748093170081018000.051862323
37-6-0.075000000860684889624310366616000.056506987
38-4-0.050000000973932901416982783276000.060068060
39-2-0.0250000001048850816910596843528000.062307776
40001075072087333361764616200.063072120
4120.0250000001048850816910596843528000.062307776
4240.050000000973932901416982783276000.060068060
4360.075000000860684889624310366616000.056506987
4480.10000000723757748093170081018000.051862323
45100.12500000579006198474536064814400.046430150
46120.15000000440548194491494831924000.040534590
47140.17500000318694438568315410328000.034496871
48160.20000000219102426515716844600500.028607679
49180.22500000143087298949039571984000.023105964
50200.2500000088714125348404534630080.018166059
51220.2750000052184779616708549782400.013893578
52240.3000000029103050170856691224800.010329241
53260.3250000015375196316679006684800.0074587767
54280.350000007687598158339503342400.0052265126
55300.375000003634137311215037943680.0035502130
56320.400000001622382728220999082000.0023350463
57340.42500000683108517145683824000.0014851775
58360.45000000270887860247426344000.00091218314
59380.47500000101009032634633552000.00054014941
60400.5000000035353161422121743200.00030782261
61420.5250000011591200466269424000.00016849096
62440.550000003552142078372888000.000088383621
63460.575000001014897736677968000.000044319453
64480.60000000269582211305085250.000021184165
65500.6250000066358698167405600.0000096210693
66520.6500000015081522310774000.0000041365035
67540.675000003151362870908000.0000016765122
68560.70000000602466431203006.3740963e-7
69580.72500000104776770644002.2604015e-7
70600.7500000016464921101207.4262988e-8
71620.775000002319002972002.2421446e-8
72640.80000000289875371506.1599974e-9
73660.8250000031767164001.5212897e-9
74680.850000003005002003.3250630e-10
75700.87500000240400166.3018615e-11
76720.9000000015815801.0070745e-11
77740.92500000821601.3029405e-12
78760.9500000031601.2796828e-13
79780.97500000808.4823008e-15
80801.000000012.8463577e-16
Complete finite sector probabilities at N = 80, K = 1.0000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Complete finite sector probabilities at N = 80, K = 1.0000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Positive-spin count kMmExact configuration multiplicityProbability
0-80-1.000000014.7135837e-8
1-78-0.97500000805.2325045e-7
2-76-0.9500000031600.0000030150174
3-74-0.92500000821600.000012021565
4-72-0.9000000015815800.000037308203
5-70-0.87500000240400160.000096111467
6-68-0.850000003005002000.00021405592
7-66-0.8250000031767164000.00042385534
8-64-0.80000000289875371500.00076159135
9-62-0.775000002319002972000.0012612413
10-60-0.7500000016464921101200.0019487559
11-58-0.72500000104776770644000.0028371248
12-56-0.70000000602466431203000.0039235270
13-54-0.675000003151362870908000.0051890433
14-52-0.6500000015081522310774000.0066007589
15-50-0.6250000066358698167405600.0081156084
16-48-0.60000000269582211305085250.0096850913
17-46-0.575000001014897736677968000.011260009
18-44-0.550000003552142078372888000.012794564
19-42-0.5250000011591200466269424000.014249415
20-40-0.5000000035353161422121743200.015593551
21-38-0.47500000101009032634633552000.016805052
22-36-0.45000000270887860247426344000.017870915
23-34-0.42500000683108517145683824000.018786215
24-32-0.400000001622382728220999082000.019552844
25-30-0.375000003634137311215037943680.020178075
26-28-0.350000007687598158339503342400.020673099
27-26-0.3250000015375196316679006684800.021051682
28-24-0.3000000029103050170856691224800.021329005
29-22-0.2750000052184779616708549782400.021520711
30-20-0.2500000088714125348404534630080.021642176
31-18-0.22500000143087298949039571984000.021707977
32-16-0.20000000219102426515716844600500.021731530
33-14-0.17500000318694438568315410328000.021724869
34-12-0.15000000440548194491494831924000.021698535
35-10-0.12500000579006198474536064814400.021661534
36-8-0.10000000723757748093170081018000.021621358
37-6-0.075000000860684889624310366616000.021584022
38-4-0.050000000973932901416982783276000.021554127
39-2-0.0250000001048850816910596843528000.021534908
40001075072087333361764616200.021528290
4120.0250000001048850816910596843528000.021534908
4240.050000000973932901416982783276000.021554127
4360.075000000860684889624310366616000.021584022
4480.10000000723757748093170081018000.021621358
45100.12500000579006198474536064814400.021661534
46120.15000000440548194491494831924000.021698535
47140.17500000318694438568315410328000.021724869
48160.20000000219102426515716844600500.021731530
49180.22500000143087298949039571984000.021707977
50200.2500000088714125348404534630080.021642176
51220.2750000052184779616708549782400.021520711
52240.3000000029103050170856691224800.021329005
53260.3250000015375196316679006684800.021051682
54280.350000007687598158339503342400.020673099
55300.375000003634137311215037943680.020178075
56320.400000001622382728220999082000.019552844
57340.42500000683108517145683824000.018786215
58360.45000000270887860247426344000.017870915
59380.47500000101009032634633552000.016805052
60400.5000000035353161422121743200.015593551
61420.5250000011591200466269424000.014249415
62440.550000003552142078372888000.012794564
63460.575000001014897736677968000.011260009
64480.60000000269582211305085250.0096850913
65500.6250000066358698167405600.0081156084
66520.6500000015081522310774000.0066007589
67540.675000003151362870908000.0051890433
68560.70000000602466431203000.0039235270
69580.72500000104776770644000.0028371248
70600.7500000016464921101200.0019487559
71620.775000002319002972000.0012612413
72640.80000000289875371500.00076159135
73660.8250000031767164000.00042385534
74680.850000003005002000.00021405592
75700.87500000240400160.000096111467
76720.9000000015815800.000037308203
77740.92500000821600.000012021565
78760.9500000031600.0000030150174
79780.97500000805.2325045e-7
80801.000000014.7135837e-8
Complete finite sector probabilities at N = 80, K = 1.5000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Complete finite sector probabilities at N = 80, K = 1.5000000, b = 0. Every integer k is retained; values are probabilities, not a continuous probability density.
Positive-spin count kMmExact configuration multiplicityProbability
0-80-1.000000010.0035969481
1-78-0.97500000800.014873965
2-76-0.9500000031600.032733960
3-74-0.92500000821600.051111504
4-72-0.9000000015815800.063689560
5-70-0.87500000240400160.067546489
6-68-0.850000003005002000.063500306
7-66-0.8250000031767164000.054418255
8-64-0.80000000289875371500.043389504
9-62-0.775000002319002972000.032692939
10-60-0.7500000016464921101200.023564796
11-58-0.72500000104776770644000.016409392
12-56-0.70000000602466431203000.011128993
13-54-0.675000003151362870908000.0074009774
14-52-0.6500000015081522310774000.0048537294
15-50-0.6250000066358698167405600.0031545691
16-48-0.60000000269582211305085250.0020404093
17-46-0.575000001014897736677968000.0013182699
18-44-0.550000003552142078372888000.00085349393
19-42-0.5250000011591200466269424000.00055531407
20-40-0.5000000035353161422121743200.00036400785
21-38-0.47500000101009032634633552000.00024092786
22-36-0.45000000270887860247426344000.00016133655
23-34-0.42500000683108517145683824000.00010950184
24-32-0.400000001622382728220999082000.000075447630
25-30-0.375000003634137311215037943680.000052847724
26-28-0.350000007687598158339503342400.000037680823
27-26-0.3250000015375196316679006684800.000027379607
28-24-0.3000000029103050170856691224800.000020295230
29-22-0.2750000052184779616708549782400.000015361030
30-20-0.2500000088714125348404534630080.000011881256
31-18-0.22500000143087298949039571984000.0000093980086
32-16-0.20000000219102426515716844600500.0000076071014
33-14-0.17500000318694438568315410328000.0000063045756
34-12-0.15000000440548194491494831924000.0000053524946
35-10-0.12500000579006198474536064814400.0000046569283
36-8-0.10000000723757748093170081018000.0000041537005
37-6-0.075000000860684889624310366616000.0000037991272
38-4-0.050000000973932901416982783276000.0000035640064
39-2-0.0250000001048850816910596843528000.0000034297702
40001075072087333361764616200.0000033861239
4120.0250000001048850816910596843528000.0000034297702
4240.050000000973932901416982783276000.0000035640064
4360.075000000860684889624310366616000.0000037991272
4480.10000000723757748093170081018000.0000041537005
45100.12500000579006198474536064814400.0000046569283
46120.15000000440548194491494831924000.0000053524946
47140.17500000318694438568315410328000.0000063045756
48160.20000000219102426515716844600500.0000076071014
49180.22500000143087298949039571984000.0000093980086
50200.2500000088714125348404534630080.000011881256
51220.2750000052184779616708549782400.000015361030
52240.3000000029103050170856691224800.000020295230
53260.3250000015375196316679006684800.000027379607
54280.350000007687598158339503342400.000037680823
55300.375000003634137311215037943680.000052847724
56320.400000001622382728220999082000.000075447630
57340.42500000683108517145683824000.00010950184
58360.45000000270887860247426344000.00016133655
59380.47500000101009032634633552000.00024092786
60400.5000000035353161422121743200.00036400785
61420.5250000011591200466269424000.00055531407
62440.550000003552142078372888000.00085349393
63460.575000001014897736677968000.0013182699
64480.60000000269582211305085250.0020404093
65500.6250000066358698167405600.0031545691
66520.6500000015081522310774000.0048537294
67540.675000003151362870908000.0074009774
68560.70000000602466431203000.011128993
69580.72500000104776770644000.016409392
70600.7500000016464921101200.023564796
71620.775000002319002972000.032692939
72640.80000000289875371500.043389504
73660.8250000031767164000.054418255
74680.850000003005002000.063500306
75700.87500000240400160.067546489
76720.9000000015815800.063689560
77740.92500000821600.051111504
78760.9500000031600.032733960
79780.97500000800.014873965
80801.000000010.0035969481

Scroll each graph or table locally to keep its labels readable. Complete text tables remain available without interpreting color.

Exact finite sums at K = 1.5 and b = 0, rounded here; m and b are dimensionless.
N⟨m⟩⟨|m|⟩⟨m²⟩∂⟨m⟩/∂b
2000.7845010.66101013.2202
8000.8451030.72087657.6701
12

A finite field response is smooth

Positive b favours positive M. At K = 1.5 and b = +0.025, the exact mean m is approximately 0.314899 for N = 20 and 0.801851 for N = 80. Negative b produces the opposite means. The larger sample selects a sign much more sharply over this field range, but both finite curves remain smooth.

Differentiate the normalized finite sum at fixed K: ∂⟨m⟩/∂b = [⟨M²⟩ − ⟨M⟩²]/N = N Var(m). This is a nonnegative, dimensionless response. Differentiation with respect to the energy H instead adds a factor β; a susceptibility with respect to a field in tesla would need the moment conversion as well.

At zero field the derivative is about 13.2202 for N = 20 and 57.6701 for N = 80. A local tangent need not predict the finite change at b = 0.025 accurately, especially for the sharper large sample. Use the full finite sum for that change rather than extrapolating a zero-field slope.

A larger finite magnet selects a sign more sharply, but smoothly

Compare the exact finite sums at K = 1.5. The equilibrium mean and its local slope are different quantities, and both stay finite here.

Smooth mean response at two finite sizes

  • N = 20
  • N = 80
Smooth mean response at two finite sizesBlue solid line and circles: N = 20. Orange dashed line and squares: N = 80. No infinite-system jump is inserted at b = 0.⟨m⟩ (dimensionless)-1-0.500.51-0.1-0.0500.050.1b = βH (dimensionless energy bias)

Variance determines the local derivative

  • N = 20
  • N = 80
Variance determines the local derivativeBoth sizes have a nonnegative derivative from the complete finite distribution. This is a local derivative, not a finite-change straight-line law.d⟨m⟩/db = N Var(m) (dimensionless)0204060-0.1-0.0500.050.1b = βH (dimensionless; K = 1.5)
  • The larger system is more sensitive near zero field, but the finite partition sum is positive and analytic at all assigned real controls.
  • H here is an energy per moment, not a field automatically measured in tesla. Differentiation against H adds β; a physical magnetic-field conversion also needs a moment calibration.
  • Taking N → ∞ before b → 0⁺ is different from setting b = 0 at every finite N. The plotted curves do not silently exchange these limits.

Expandable complete sampled-data tables follow. The linked original body table remains in its original section.

Exact finite sums at K = 1.5 and b = 0, rounded here; m and b are dimensionless.

All assigned finite field rows at K = 1.5. The derivative is against dimensionless b = βH; no infinite-size jump is interpolated into either finite curve.
All assigned finite field rows at K = 1.5. The derivative is against dimensionless b = βH; no infinite-size jump is interpolated into either finite curve.
Nb⟨m⟩⟨|m|⟩⟨m²⟩Var(m)∂⟨m⟩/∂b
20-0.10000000-0.788348400.845335850.743985830.122492622.4498524
20-0.095000000-0.775333580.841965150.739274180.138132022.7626405
20-0.090000000-0.760653240.838482980.734427240.155833893.1166777
20-0.085000000-0.744092010.834894000.729451680.175778763.5155752
20-0.080000000-0.725417310.831205980.724358300.198128033.9625606
20-0.075000000-0.704381960.827430510.719162860.223008924.4601783
20-0.070000000-0.680728580.823583730.713887080.250495695.0099137
20-0.065000000-0.654195890.819687030.708559530.280587275.6117454
20-0.060000000-0.624527550.815767720.703216530.313181876.2636375
20-0.055000000-0.591483550.811859610.697902880.348050096.9610017
20-0.050000000-0.554854310.808003310.692672260.384808967.6961791
20-0.045000000-0.514477130.804246230.687587230.422900528.4580104
20-0.040000000-0.470254420.800642120.682718620.461579409.2315881
20-0.035000000-0.422172520.797250050.678144210.499914579.9982914
20-0.030000000-0.370319360.794132780.673946520.5368101010.736202
20-0.025000000-0.314899010.791354450.670209880.5710485011.420970
20-0.020000000-0.256240680.788977670.667016590.6013573112.027146
20-0.015000000-0.194800040.787060320.664442630.6264955712.529911
20-0.010000000-0.131151200.785652000.662553180.6453525412.907051
20-0.0050000000-0.0659686520.784790910.661398370.6570465113.140930
20000.784501140.661009830.6610098313.220197
200.00500000000.0659686520.784790910.661398370.6570465113.140930
200.0100000000.131151200.785652000.662553180.6453525412.907051
200.0150000000.194800040.787060320.664442630.6264955712.529911
200.0200000000.256240680.788977670.667016590.6013573112.027146
200.0250000000.314899010.791354450.670209880.5710485011.420970
200.0300000000.370319360.794132780.673946520.5368101010.736202
200.0350000000.422172520.797250050.678144210.499914579.9982914
200.0400000000.470254420.800642120.682718620.461579409.2315881
200.0450000000.514477130.804246230.687587230.422900528.4580104
200.0500000000.554854310.808003310.692672260.384808967.6961791
200.0550000000.591483550.811859610.697902880.348050096.9610017
200.0600000000.624527550.815767720.703216530.313181876.2636375
200.0650000000.654195890.819687030.708559530.280587275.6117454
200.0700000000.680728580.823583730.713887080.250495695.0099137
200.0750000000.704381960.827430510.719162860.223008924.4601783
200.0800000000.725417310.831205980.724358300.198128033.9625606
200.0850000000.744092010.834894000.729451680.175778763.5155752
200.0900000000.760653240.838482980.734427240.155833893.1166777
200.0950000000.775333580.841965150.739274180.138132022.7626405
200.100000000.788348400.845335850.743985830.122492622.4498524
80-0.10000000-0.886356860.886359130.789681520.00405303850.32424308
80-0.095000000-0.884718090.884722510.786867830.00414173100.33133848
80-0.090000000-0.883042590.883051230.784001320.00423710040.33896803
80-0.085000000-0.881327040.881343950.781080180.00434282160.34742573
80-0.080000000-0.879566040.879599160.778102360.00446593690.35727495
80-0.075000000-0.877750210.877815150.775065520.00462008180.36960655
80-0.070000000-0.875862410.875989850.771966770.00483181520.38654521
80-0.065000000-0.873870370.874120690.768802470.00515304420.41224354
80-0.060000000-0.871712230.872204210.765567610.00568539220.45483138
80-0.055000000-0.869268170.870235590.762255050.00662789410.53023153
80-0.050000000-0.866304870.868207790.758854090.00836996130.66959691
80-0.045000000-0.862367350.866110220.755348470.0116710200.93368160
80-0.040000000-0.856569820.863927010.751713430.0180015691.4401255
80-0.035000000-0.847198390.861635050.747912640.0301675252.4134020
80-0.030000000-0.830982040.859203260.743897000.0533658484.2692678
80-0.025000000-0.801850850.856597630.739612690.0966479077.7318326
80-0.020000000-0.749159660.853802860.735036150.1737959513.903676
80-0.015000000-0.656269840.850880850.730268270.2995781723.966253
80-0.010000000-0.502959500.848079790.725710320.4727420637.819365
80-0.0050000000-0.277926620.845929640.722217900.6449746951.597976
80000.845102690.720875880.7208758857.670070
800.00500000000.277926620.845929640.722217900.6449746951.597976
800.0100000000.502959500.848079790.725710320.4727420637.819365
800.0150000000.656269840.850880850.730268270.2995781723.966253
800.0200000000.749159660.853802860.735036150.1737959513.903676
800.0250000000.801850850.856597630.739612690.0966479077.7318326
800.0300000000.830982040.859203260.743897000.0533658484.2692678
800.0350000000.847198390.861635050.747912640.0301675252.4134020
800.0400000000.856569820.863927010.751713430.0180015691.4401255
800.0450000000.862367350.866110220.755348470.0116710200.93368160
800.0500000000.866304870.868207790.758854090.00836996130.66959691
800.0550000000.869268170.870235590.762255050.00662789410.53023153
800.0600000000.871712230.872204210.765567610.00568539220.45483138
800.0650000000.873870370.874120690.768802470.00515304420.41224354
800.0700000000.875862410.875989850.771966770.00483181520.38654521
800.0750000000.877750210.877815150.775065520.00462008180.36960655
800.0800000000.879566040.879599160.778102360.00446593690.35727495
800.0850000000.881327040.881343950.781080180.00434282160.34742573
800.0900000000.883042590.883051230.784001320.00423710040.33896803
800.0950000000.884718090.884722510.786867830.00414173100.33133848
800.100000000.886356860.886359130.789681520.00405303850.32424308

Scroll each graph or table locally to keep its labels readable. Complete text tables remain available without interpreting color.

13

Return path: why finite sums have no singularity

For any fixed finite N and finite real K,b, Z_N is a finite sum of strictly positive exponentials with positive integer coefficients. It is positive and real analytic. Hence ln Z_N and the finite moments are analytic on this real-control domain. A polynomial minimum-selection calculation must not be passed off as the exact finite partition sum.

A sequence of analytic finite-system functions can approach a nonanalytic limit nonuniformly. Increasing size can narrow a smooth crossover until a sharp thermodynamic boundary emerges only after taking the limit. Complex zeros of finite partition functions provide one way to study how that limiting singularity develops; finite positive-real Z_N has no zero on the domain just specified.

This argument is for the assigned finite discrete model. It is not a blanket theorem covering every finite system with an unbounded or continuous state space, every singular interaction or every zero-temperature limit. State the partition function, its convergence domain and the limit being taken.

14

Return path: select a phase in the right order

Use the binomial multiplicity to form a large-N dimensionless free energy per spin, up to a constant: f(m) = −Km²/2 − bm + [(1+m)/2]ln[(1+m)/2] + [(1−m)/2]ln[(1−m)/2]. The endpoint terms use the continuous convention 0 ln 0 = 0.

For an interior stationary point, f′(m) = −Km − b + atanh(m) = 0, or m = tanh(Km + b). Here tanh x = [exp(x) − exp(−x)]/[exp(x) + exp(−x)], and atanh is its inverse for |m| < 1. At b = 0, expansion near zero gives a quadratic coefficient (1−K)/2 and a positive quartic coefficient 1/12. The limiting onset is at K = 1.

For K > 1, taking b → 0 at each finite N keeps the signed mean zero, so taking N → ∞ afterwards still gives zero. Taking N → ∞ with a positive field first and only then b → 0⁺ selects the positive ordered branch. A negative selecting field gives the negative branch. The symmetric mixture and an individually selected equilibrium phase are different limiting states, not contradictory answers.

15

Return path: derivatives and latent heat

The conjugate variable for h in the assigned landscape is η: on a smooth selected minimum, ∂Φ_eq/∂h = −η_eq. Thus the sign change at t < 0 gives a jump in this first derivative. That directly demonstrates the first-order field boundary within the model.

Latent heat is a temperature-path statement: for coexisting phases at positive transition temperature, the reversible latent heat per unit is T times the entropy difference with a specified transfer direction. Obtaining entropy requires the temperature derivative of a dimensionful free energy with the correct fixed controls.

Our coefficient a has no assigned Kelvin calibration or energy prefactor, so the sextic jump does not supply a numerical latent heat. Even a first-order field transition need not carry a nonzero entropy jump: symmetry-related signs can have equal entropy. A jump in the relevant conjugate derivative is the more general criterion, not a universal requirement of heat release on every crossing.

16

Return path: mixtures and convexity

A nonconvex Landau curve is a homogeneous or coarse-grained model before allowing all possible spatial phase separation. It is not automatically the exact equilibrium free energy as a function of a conserved extensive quantity. If mixtures are allowed and interface costs are subextensive in the macroscopic limit, equilibrium replaces a nonconvex region by the appropriate convex envelope.

For a fluid at coexistence, the two bulk phases have equal temperature, pressure and chemical potential. If the total average density is constrained between their coexistence densities, changing the phase fractions can satisfy that constraint. The phase densities need not sweep continuously through unstable homogeneous values; the mixture changes instead.

This is why a local minimum, a homogeneous spinodal, a common-tangent coexistence construction and the exact equilibrium constrained potential describe related but distinct objects. Finite interfaces, long-range forces, geometry and constraints can modify the simple macroscopic mixture argument. Keep the uniform polynomial calculation labelled as such.

17

Return path: space, interfaces and clocks

To discuss a domain wall or a droplet, let the order parameter vary in space and include a gradient cost, schematically ∫[f(η) + κ|∇η|²/2] dV with κ > 0. The local free-energy density f and the coefficient κ must carry compatible units. The uniform zero-dimensional curves above do not contain this interface information.

Nucleation compares a bulk free-energy advantage with the cost of making an interface. A critical droplet is a free-energy barrier along a spatial transformation path, not simply the central maximum of a one-coordinate graph. Predicting its rate additionally requires a kinetic law, noise model and prefactor.

Hysteresis records dependence on history and observation protocol. It may reflect delayed nucleation, pinning or other dynamics. It does not by itself locate the equilibrium coexistence line, and an equilibrium landscape does not promise that a finite-duration experiment visits both signs with equilibrium weights.

18

Other orders and the model boundary

One scalar is a teaching model, not the universal form of order. A vector can describe an orientation; a tensor can distinguish orientational order without a preferred head-to-tail direction; density contrasts or spatial Fourier components can distinguish other structures. Some descriptions require complex amplitudes, correlations or other observables rather than a single real sign.

A nonzero field conjugate to an order parameter explicitly favours one state and can smooth a symmetry-breaking temperature crossing. Whether a transition remains depends on the model and the chosen path. Likewise, not every transition fits an ordinary local-order-parameter picture: topological and Berezinskii–Kosterlitz–Thouless transitions require other diagnostics, and quantum transitions involve a zero-temperature limit rather than an ordinary thermal scan.

Close to a continuous transition, long-distance fluctuations can invalidate a simple homogeneous mean-field prediction. Critical exponents, universality, correlations and renormalization have their own next subject. Here the essential skill is narrower: define the variable, state the controls and approximation, compare the correct equilibrium states, and keep finite observations separate from thermodynamic limits.

Worked examples

Find all quartic equilibrium states

At t = −0.25 and h = 0, which stationary states are equilibrium minima?

  1. Factor η(t + η²) = 0 to obtain η = 0 and η = ±0.5.
  2. The curvature at zero is −0.25, so zero is a local maximum. At either nonzero root it is −0.25 + 3(0.25) = 0.5 > 0.
  3. Substitute into Φ: each minimum has Φ = −t²/4 = −0.015625. Both are equally global at zero field.

η = −0.5 and +0.5 are degenerate equilibrium minima; η = 0 is not.

Zero curvature is not a classification

Is η = 0 stable when the quartic model has t = h = 0?

  1. The first and second derivatives vanish at zero.
  2. Inspect Φ = η⁴/4 directly. Every nonzero η has a greater value than Φ(0) = 0.
  3. The positive fourth-order term makes zero a minimum despite the missing quadratic restoring term.

Yes: it is a stable, quartically flat minimum, not a maximum or an ordinary positive-curvature well.

A local slope does not join two phases

Find dη/dh at the positive quartic minimum for t = −1 and h = 0. Does it describe crossing to the negative phase?

  1. The positive stationary minimum is η = +1.
  2. Differentiate tη + η³ = h to obtain dη/dh = 1/(t + 3η²) = 1/2.
  3. This follows one local branch. Global equilibrium changes sign discontinuously when the selecting field crosses zero.

The local branch slope is 0.5. It is not a finite tangent through the first-order equilibrium jump.

Separate a spinodal from coexistence

Where does the negative metastable quartic well disappear as h increases at t = −1?

  1. At the merger, solve −η + η³ − h = 0 together with −1 + 3η² = 0.
  2. Choose the negative-coordinate root η = −1/√3. Substitution gives h = +2/(3√3) ≈ +0.384900.
  3. The equal-depth coexistence field was h = 0, so it is a different threshold.

The negative well disappears at the positive spinodal field about 0.384900, not at coexistence; no switching time is assigned.

Derive the sextic coexistence point

Find a nonzero stationary minimum with Ψ equal to Ψ(0).

  1. Put q = η². Stationarity gives a = q − q².
  2. Insert that into Ψ = aq/2 − q²/4 + q³/6 to obtain Ψ = q²/4 − q³/3.
  3. Set this equal to zero for q > 0: q = 3/4. Hence a = 3/4 − 9/16 = 3/16 and η = ±√3/2.

At a = 3/16, zero and both η = ±√3/2 are equal global minima. The order magnitude jumps by about 0.866025.

Measure the dimensionless barrier

At sextic coexistence, what is the intervening barrier height?

  1. For a = 3/16, the smaller nonzero stationary q is 1/4, giving η = ±1/2.
  2. Evaluate Ψ = (3/16)(1/4)/2 − (1/4)²/4 + (1/4)³/6 = 1/96.
  3. The three coexisting minima have Ψ = 0, so the barrier above either neighbouring minimum is 1/96.

The barrier is about 0.0104167 in the model’s dimensionless potential, not a specified number of joules or seconds.

Use symmetry before doing a sum

At b = 0, what is ⟨m⟩ for N = 80 and K = 1.5, even if the distribution is bimodal?

  1. Pair sector k with N−k. Their binomial multiplicities and M² interaction terms are identical.
  2. Their magnetizations are opposite. The bM term is absent, so paired contributions to the signed numerator cancel.
  3. The partition sum is positive, and the central zero-m sector contributes no signed magnetization.

⟨m⟩ = 0 exactly. This does not mean the full distribution is concentrated at zero.

Convert a finite variance into a response

For N = 20, K = 1.5 and b = 0, use ⟨m²⟩ ≈ 0.661009832 and symmetry to find the slope against b.

  1. Symmetry gives ⟨m⟩ = 0, so Var(m) = ⟨m²⟩.
  2. Differentiate the complete normalized weights: ∂⟨m⟩/∂b = N Var(m).
  3. Multiply by 20: the slope is about 13.2201966. The derivative is with respect to dimensionless b, not tesla.

The zero-field slope is approximately 13.2202 per unit b; a finite field change still needs the full smooth response.

Know what a landscape cannot measure

Does the sextic jump at a = 3/16 determine a latent heat?

  1. The model specifies only dimensionless Ψ and coefficient a.
  2. Entropy needs a temperature derivative of a dimensionful free energy. Neither its energy scale nor the temperature dependence of a is supplied.
  3. Equal-depth minima establish coexistence in the model, but do not provide those missing physical inputs.

No numerical latent heat follows. Supply the energy and temperature mapping, fixed controls and phase entropy difference before calculating it.

Try it yourself

  1. Two finite distributions both have ⟨m⟩ = 0. Must they describe the same degree of alignment?

    Show the answer and reasoning

    No. One can be narrow at zero and the other bimodal away from zero.

    The signed mean discards the distribution’s shape. Compare P(m), magnitude and correlations with the model and sample size stated.

  2. Does ⟨|m|⟩ > 0 in a finite sample prove a sharp thermodynamic transition?

    Show the answer and reasoning

    No.

    Finite disordered spins also fluctuate. A singular limiting free energy needs a limit analysis, not just a nonzero average magnitude.

  3. A stationary state has positive curvature but higher potential than another minimum. Is it the global equilibrium?

    Show the answer and reasoning

    No; it is metastable within this homogeneous model.

    Curvature tests sufficiently small local changes. Global equilibrium requires comparing all permitted states and their potentials.

  4. Why is zero stable for Φ = η⁴/4 despite zero curvature?

    Show the answer and reasoning

    Every nonzero η increases the potential.

    A second-derivative test is inconclusive when that derivative vanishes. The positive leading fourth-order term decides this case.

  5. May the polynomial coordinate η be treated as the bounded spin magnetization m without further justification?

    Show the answer and reasoning

    No.

    Our η is an unrestricted phenomenological variable. The separately defined m = M/N lies in [−1,+1]; a mapping and validity range would need to be supplied.

  6. Can the quartic model have continuous onset along h = 0 and a first-order field crossing at t < 0?

    Show the answer and reasoning

    Yes.

    The first path makes the selected magnitude approach zero continuously. The second crosses between opposite selected signs with a discontinuous field-conjugate derivative.

  7. At t = −1, is h = 2/(3√3) the equal-depth coexistence field?

    Show the answer and reasoning

    No. Coexistence is at h = 0.

    The positive spinodal is where the negative metastable well merges with its maximum. Equal global depths and loss of local stability are different conditions.

  8. Must an odd term appear in every Landau model with a first-order transition?

    Show the answer and reasoning

    No.

    The assigned even sextic potential has equal-depth zero and nonzero minima at a = 3/16, with a discontinuous selected magnitude.

  9. Classify η = 0 at a = 0 in the sextic model.

    Show the answer and reasoning

    It is a degenerate local maximum.

    The leading nonzero small-η term is −η⁴/4; the stabilizing sixth-order term is smaller sufficiently close to zero. Zero curvature alone did not classify it.

  10. Are the nonzero stationary points at a = 1/4 still local minima?

    Show the answer and reasoning

    No; they are stationary inflections.

    The minimum and maximum have merged. Their vanishing curvature needs the higher-order local expansion, not a continued stable-well label.

  11. Can the assigned finite spin Z_N vanish at finite real K,b?

    Show the answer and reasoning

    No.

    It is a finite sum of strictly positive terms. Its real logarithm and finite moments are analytic there, despite possible sharp-looking changes.

  12. For K > 1, does setting b = 0 at every finite N select the positive phase as N grows?

    Show the answer and reasoning

    No; symmetry keeps the signed mean zero.

    Select the positive phase by taking the large-size limit with a positive field first and then sending that field to zero. State this different order explicitly.

  13. Does N Var(m) directly give the derivative against an applied field measured in tesla?

    Show the answer and reasoning

    No; here it is the derivative against b = βH.

    For the energy H the derivative includes β. A physical magnetic-field derivative also requires the assigned moment-to-field conversion.

  14. Does a barrier of 1/96 predict a switching time of 96 seconds?

    Show the answer and reasoning

    No.

    The potential is dimensionless and no dynamics is assigned. Interface physics, energy scale, temperature, kinetic law and prefactor would be needed for a rate.

  15. Must a fluid constrained between two coexistence densities occupy an unstable uniform intermediate density?

    Show the answer and reasoning

    Not if spatial phase separation is allowed under the stated macroscopic assumptions.

    A mixture can meet the average-density constraint by changing its phase fractions while the bulk phase densities remain at their coexistence values.

  16. Does η ∝ (−t)^(1/2) prove every continuous transition has exponent 1/2?

    Show the answer and reasoning

    No.

    That exponent follows from the specified homogeneous quartic model. Long-distance fluctuations, symmetry, dimension and interactions can change critical behaviour.

What to carry forward

  • An order parameter needs a defined observable, distribution, controls and limiting procedure; a zero signed mean need not mean a disordered finite distribution.
  • Stationarity and local stability do not establish global equilibrium. Compare all allowed minima and possible mixtures.
  • The assigned quartic model has continuous zero-field onset and a first-order field boundary below onset; the path matters.
  • The even sextic model separates coexistence from both metastability endpoints, without requiring an odd term or assigning a switching clock.
  • Every assigned finite spin partition sum is positive and analytic at finite real controls; thermodynamic symmetry breaking involves a specified size/field limit order.
  • Static scalar landscapes do not by themselves supply interfaces, nucleation rates, latent heat, universal critical exponents or every possible kind of order.

References

Sources for the concepts and teaching approaches in this article.

  1. University of Cambridge2 sources
    1. Statistical Physics: phase transitions and Landau theoryDavid TongCourse notesdamtp.cam.ac.ukRead source
    2. Statistical Field Theory: continuous symmetries and the Kosterlitz–Thouless transitionDavid TongCourse notesStatistical Field Theory: continuous symmetries and the Kosterlitz–Thouless transitionVector and complex order parametersKosterlitz–Thouless transition and topological defectsBeyond simple symmetry classificationdamtp.cam.ac.uk
  2. Lecture Notes on Statistical Physics: thermodynamics and phase transitionsLydéric BocquetCourse notes · PDFphys.ens.psl.euRead source
  3. Statistical Mechanics I, Spring 2020: supplementary notes on Landau theoryS. TeitelCourse notes · PDFpas.rochester.eduRead source
  4. Large Random Systems, 2019 notes: the Curie–Weiss modelAntti Kemppainen and Kalle KytöläCourse notes · PDFmath.aalto.fiRead source
  5. M.Phys Option in Theoretical Physics, 2010–2011: order parameters and liquid crystalsJohn Chalker and Andre LukasCourse notes · PDFwww-thphys.physics.ox.ac.ukRead source