Microcanonical, canonical, and grand canonical ensembles

One small sample, three kinds of contact

Imagine two tiny adsorption sites inside a rigid sample. A site can be empty, or hold one particle in a ground or excited internal level. A detector records the sample’s energy and particle number. We can insulate the sample, let it exchange energy with a temperature reservoir, or also let particles enter and leave. Which quantities stay fixed, and which detector readings should fluctuate?

The three choices lead to the microcanonical, canonical and grand canonical ensembles. An ensemble is a probability measure over the states allowed by a physical model and a preparation. It is not a requirement to build many actual copies of the sample. Nor does writing an equilibrium distribution prove that a single trajectory reaches it or samples it within a useful time.

On a first reading, follow the contacts, the two-site model, its complete probability table, the chemical-potential comparison and the finite-reservoir example. Return to the sections marked “Further” for fluctuation identities, thermodynamic limits, quantum and classical measures, and alternative controls. We introduce the needed terms locally. Useful published preparation is “Physical systems, states, boundaries, and environments”, “Work, kinetic energy, potential energy, and power”, and “Conservation of mass, momentum, and energy”.

Controls are not the same as instantaneous readings

E is the sample energy, V its volume and N its particle number. Temperature T describes thermal equilibrium with a reservoir. The chemical potential μ is the thermodynamic quantity controlling particle exchange; it has units of energy per particle. For one species in equilibrium, exchanging particles between two bodies is balanced when their chemical potentials agree. Different species require their own particle constraints and chemical potentials.

Write β = 1/(kB T), where kB = 1.380649×10⁻²³ J/K. Thus β has units J⁻¹ and βE is dimensionless. We use positive absolute temperatures in the numerical examples. A weight is an unnormalized positive number; probabilities result only after division by the total weight over the allowed state space.

Fixed T does not mean fixed E. A thermostat can absorb and supply energy while maintaining essentially the same temperature. Similarly, fixed μ does not mean fixed N. In the grand ensemble, N is an integer-valued fluctuating quantity even when its mean happens to be an integer. Energy and particles are conserved for the combined sample and environment; the open sample alone need not retain either.

A finite model with an explicit allowed state space

Label the two physical sites A and B. Each has three local alternatives: 0 means empty, g means one particle in its ground internal level, and e means one particle in its excited level. At most one particle may occupy each site. This capacity is an imposed adsorption constraint; it is not a deduction from whether the particles are bosons or fermions. Each listed local alternative has degeneracy one.

The complete record ge means A is occupied in its ground level and B in its excited level. Its particle number is N = 2 and its energy is ε. In general E = kε, where k counts occupied excited levels. The model has no intersite interaction energy. The sites are physically distinguished, but particles are not given additional artificial names. We do not divide the list of site occupations by N!.

Set T₀ = 300 K and choose the fixed level spacing ε = kB T₀ ln 2 ≈ 2.87097888508×10⁻²¹ J. This is an assigned illustrative energy, not a measured material constant. Once set, ε stays fixed when we vary T or take a temperature derivative. At T₀, the excited-to-ground Boltzmann weight ratio is exp(−βε) = 1/2.

The energy model neglects intersite coupling. Using all the listed records in an equilibrium measure also assumes an appropriate preparation or weak redistribution mechanism. If a particle is permanently trapped at one site, that extra constraint must be retained instead of assuming that thermal contact alone makes the other site accessible. The model’s Hamiltonian—its energy function, or energy operator in a quantum description—does not specify the actual transition dynamics.

Microcanonical: restrict first, then weight

For the discrete microcanonical example, fix N = 1 and E = ε. Exactly one site is occupied and that particle is excited. The complete compatible records are 0e and e0. The uniform microcanonical measure gives each probability 1/2 and assigns zero to every other record. The sample energy and particle number are exact in this chosen finite sector.

The general quantum or discrete rule is a uniform measure on the specified allowed energy and number sector. Degeneracy counts different orthogonal alternatives within that sector. In a classical continuum, exactly H = E usually has zero ordinary phase-space volume; one uses a stated shell, such as E ≤ H < E+δE, and an appropriate normalized measure. A shell’s width must not be silently replaced by a cumulative volume below E.

Insulation alone does not force a uniform preparation over all nominally compatible states. Extra conserved quantities, disconnected regions or a deliberately prepared state can matter. Microcanonical equilibrium is a statistical description with its own physical applicability conditions, rather than a theorem that follows from writing E, V and N on a box.

Canonical: reservoir multiplicity becomes a Boltzmann weight

Now fix N and V, but put the sample in weak thermal contact with a large equilibrium reservoir at T. If the sample is in a particular state i of energy Eᵢ, the reservoir has energy Etotal−Eᵢ. Under a uniform measure for the combined isolated system, that sample state receives a weight proportional to the reservoir multiplicity ΩB(Etotal−Eᵢ). This is a count of compatible reservoir states, not an equal-weight assumption for all sample energies.

Write the reservoir entropy as SB = kB ln ΩB. Expanding it about a reference energy gives a leading change −Eᵢ/T, up to an irrelevant reference constant. When the sample’s energy exchanges hardly change the reservoir temperature, the resulting weight is exp(−βEᵢ). Therefore pᵢ = exp(−βEᵢ)/Z_N, where Z_N is the sum of these weights over the fixed-N sample states. Weak coupling lets us treat the total energy as sample plus reservoir energy; the finite-reservoir and strong-coupling limits need care.

For our two-site sample at N = 1, the four records are 0g, g0, 0e and e0. At T₀ their weights are 1, 1, 1/2 and 1/2, so Z₁ = 3. Each ground record has probability 1/3 and each excited record 1/6. The total probability of sample energy ε is 1/3, not 1/6, because that energy has two compatible records.

The normalization Z_N is called the canonical partition function. We need it here to specify a probability measure. The next subject develops partition functions as computational tools; one does not need those later techniques to normalize this short list.

Grand canonical: particle exchange brings in μ

Allow energy and particles to cross the fixed-volume sample boundary. A particular state with Eᵢ and Nᵢ leaves the reservoir with Etotal−Eᵢ and Ntotal−Nᵢ. At fixed reservoir volume, its entropy differential is dSB = (dEB−μ dNB)/T. The sample therefore produces the leading reservoir entropy change (−Eᵢ+μNᵢ)/T. The plus sign on μNᵢ in the exponent follows from the reservoir losing those particles.

The grand canonical probability is pᵢ = exp[−β(Eᵢ−μNᵢ)]/Ξ. The dimensionless normalizer Ξ sums over all allowed microscopic records, including every allowed N. Equivalently, Ξ = Σ_N exp(βμN) Z_N, provided the state measures and particle identities in each fixed-N sector have been defined consistently. This is a weighted sum of sectors, not an instruction to choose one mean particle number and discard the others.

For convenience define the activity z = exp(βμ). At T₀ choose μ₀ = kB T₀ ln(2/3) ≈ −1.67941498813×10⁻²¹ J, so z = 2/3. An individual site then has weights 1 for 0, 2/3 for g and 1/3 for e. The single-site normalizer is 2; independent sites give Ξ = 4. The complete records and their probabilities are shown below. A negative μ is consistent with a nonzero particle population.

Compare every complete record

There are 3² = 9 complete site records before either ensemble restriction. The same physical alphabet is used in all three columns, but their constraints and weights differ. A zero in the microcanonical or canonical column marks a record excluded by that ensemble’s exact sector; it does not say that the record is impossible in every preparation.

The grand ensemble has P(N = 0) = 1/4, P(N = 1) = 1/2 and P(N = 2) = 1/4. Its mean is ⟨N⟩ = 1, the same as the canonical fixed number, but half of its observations have a different N. Agreement of the mean is much weaker than equality of distributions.

Equal means can hide different spreads

An expectation ⟨X⟩ is the probability-weighted average of a reading X. Its variance Var(X) = ⟨(X−⟨X⟩)²⟩ describes the spread across the stated measure; the standard deviation is its square root. These are statistical statements, not a time law for a detector trace.

Our canonical and grand preparations both have mean energy ε/3. Their energy variances differ: 2ε²/9 and 5ε²/18. Particle-number fluctuations supply an additional route to energy variation in the grand ensemble. The microcanonical example instead fixes energy ε; it is not chosen to have the same mean energy as those two examples.

Changing μ controls weights, not one exact N

At fixed T₀, increasing μ increases z and favors occupied-site records relative to empty ones. For this finite-capacity model, a site is occupied with probability f = z(1+1/2)/[1+z(1+1/2)]. The two sites are independent in the grand ensemble, so their total particle number has probabilities P(N) = C(2,N) fᴺ(1−f)^(2−N). This is a consequence of the stated noninteracting site model, not a universal number distribution for every open sample.

The three activities in the table produce different mean particle numbers. At z = 2/3 the probability of exactly one particle is only 1/2 despite the integer mean 1. At either lower or higher activity, that exact-one probability drops. In the large positive-μ limit the model saturates at two particles because of its explicit site capacities. A gas with no such occupancy cap need not have that saturation law.

A small reservoir need not produce exact canonical weights

Use a separate two-level model to test the reservoir approximation. Two sample units and six reservoir units each have levels 0 and ε; the combined isolated system has exactly three excitations. All C(8,3) = 56 compatible complete strings are assigned equal probability. The sample excitation count k can be 0, 1 or 2.

For one specified sample arrangement with k excitations, the compatible reservoir count is C(6,3−k). To obtain the probability of the sample’s energy record, multiply by C(2,k), then divide by 56. The joint macrostate weights are 20, 30 and 6. The reservoir is altered appreciably when this small sample takes energy from it.

For comparison choose a canonical product preparation with the same mean sample excitation density, p = 3/8. Its excited-to-ground weight ratio is p/(1−p) = 3/5, so βmatch ε = ln(5/3). This comparison uses its own matched temperature, not T₀. It gives probabilities 25/64, 15/32 and 9/64; the exact finite-reservoir distribution differs despite the matched mean.

The per-sample-state reservoir weights 20, 15 and 6 have successive ratios 3/4 and 2/5. A single Boltzmann factor for equally spaced sample energies would give the same ratio at both steps. This simple failure shows what the large-reservoir approximation has removed; it predicts no exchange rate or equilibration time.

Conditioning is exact; ensemble equivalence is a different claim

Conditioning means restricting a probability measure to a specified event and renormalizing. In the grand two-site example, condition on N = 1. Since exp(βμN) is then the same factor for every retained record, it cancels, leaving exactly the canonical fixed-N probabilities. For example, P(0e|N=1) = (1/12)/(1/2) = 1/6.

Similarly, the canonical measure conditioned on its exact energy E = ε is uniform on 0e and e0: (1/6)/(1/3) = 1/2 for each. In a finite discrete model these are exact conditional identities when the specified events have nonzero probability. A finite-width energy shell can retain unequal Boltzmann weights; equality is exact at one degenerate energy, or approximate when their variation across the shell is negligible.

These identities do not say that the unconditional measures are identical. Nor do they describe physically switching a wall or filtering a sample without disturbing it. Thermodynamic-limit ensemble equivalence concerns agreement of appropriately matched predictions as system size grows; it needs additional hypotheses.

Further: response and equilibrium fluctuations

For a canonical equilibrium measure with a temperature-independent Hamiltonian and fixed V and N, differentiation of the normalized weights gives ∂⟨E⟩/∂β = −Var(E). Since dβ/dT = −1/(kB T²), Var(E) = kB T² C_(V,N), where C_(V,N) = (∂⟨E⟩/∂T)_(V,N). The variance is nonnegative. The identity is not a claim that every isolated system has the same energy variance or heat capacity.

For one fixed particle in our two-site model, C_(V,N) = kB(βε)² exp(−βε)/[1+exp(−βε)]². The site-choice degeneracy is temperature independent and cancels. At T₀ this becomes (2/9)kB(ln 2)², consistent with Var(E) = 2ε²/9. Keeping ε fixed during differentiation is essential.

In a grand equilibrium ensemble, Var(N) = kB T (∂⟨N⟩/∂μ)_(T,V). For the two independent sites, ⟨N⟩ = 2f and the derivative gives Var(N) = 2f(1−f), equal to 1/2 at the central activity. This is a response of an equilibrium family of distributions, not a rate at which particles cross a boundary.

The canonical energy identity cannot simply be relabelled with a derivative at fixed μ. For a Hamiltonian and number operator independent of T and μ, (∂⟨E⟩/∂T)_(V,μ) = [Var(E)−μ Cov(E,N)]/(kB T²), where Cov(E,N)=⟨EN⟩−⟨E⟩⟨N⟩. Energy and number can fluctuate together. The later fluctuation-and-response subject develops these cross responses.

Further: finite baths, coupling and equilibration

If an equilibrium reservoir has a smooth entropy and positive heat capacity CB, expand around a reference reservoir energy. For sample energy change δE, the logarithm of its reservoir weight changes by −δE/(kB T) − δE²/(2kB T² CB) plus higher terms. The canonical weight retains only the linear term. The quadratic term follows from ∂²SB/∂EB² = −1/(T² CB). It matters when the exchanged energies sample appreciable curvature.

A reservoir must remain approximately at the assigned T and μ over the relevant exchange range. A large particle count alone is not a sufficient test: susceptibilities, phases, conservation laws and accessible exchanges matter. The finite six-unit reservoir illustrates changing weights directly without assigning it a universal bulk temperature.

For strong sample–environment coupling, the interaction energy cannot be neglected in Htotal. An equilibrium reduced sample state is then obtained from the combined equilibrium state; it need not be exp(−βHsample) with the uncoupled sample Hamiltonian. Effective potentials or a Hamiltonian of mean force may be useful, but one must specify their temperature dependence and boundary.

Weak coupling also does not prove equilibration. Dynamics, conserved sectors, preparation and observation time determine whether the equilibrium measure is useful. A disconnected state space or a long-lived nonequilibrium preparation may prevent the sampling assumed in a practical application. Counting equilibrium weights supplies no transport coefficient or clock.

Further: what can agree as a system grows?

Now consider N physically localized two-level units, all present; this is a fixed-N model without empty adsorption sites. Choose N = 3, 30 or 300 and a uniform microcanonical sector with q = N/3 excitations. Compare it with independent canonical units at T₀, where each unit is excited with probability p = 1/3. The mean energy is matched.

A specified single unit has excitation probability q/N = 1/3 in both descriptions, even at small N. For two specified distinct units, however, the microcanonical joint probability is q(q−1)/[N(N−1)], whereas the canonical probability is 1/9. Fixing q creates a weak anticorrelation. The difference is −2/[9(N−1)], which tends to zero for this fixed-size local question.

The total energy behaves differently. In the exact microcanonical sector its variance is zero; canonically Var(E) = (2N/9)ε². The standard deviation of q/N is nevertheless √[2/(9N)] canonically and tends to zero. Thus small local correlations and fluctuations of energy density can vanish while the two total-energy distributions retain different constraints.

Further: the conditions behind ensemble equivalence

For ordinary stable systems with sufficiently short-range, additive interactions, take a thermodynamic limit with particle number and volume growing at fixed density. Away from singular regimes, a strictly concave, smooth entropy density can select one matching equilibrium energy for a given T; the corresponding conditions for particle density relate canonical and grand canonical descriptions. One then expects agreement for suitable intensive or fixed-size local observables. Existence of the thermodynamic limit and the relevant state-space measure is part of the hypothesis.

The underlying geometry is visible in the large-system weight Ω(E)exp(−βE). If s(u)=lim (ln Ω(Nu))/N with u=E/N, its leading logarithm per particle is s(u)−βu. A unique stable interior maximum obeys s′(u)=β. Strict concavity makes that matching unique in the regular case. A zero curvature or several competing maxima changes the fluctuation analysis; one cannot assert a universal 1/√N law without its model conditions.

At a first-order transition, one temperature can support more than one equilibrium energy or phase. Concave but non-strict entropy regions can give partial equivalence of equilibrium macrostate sets; phase fractions and finite-system distributions need their actual constraints. This does not automatically mean that all bulk thermodynamic relations fail. A genuinely nonconcave microcanonical entropy can instead produce energies not realized as canonical equilibrium energies for any T. Long-range, nonadditive systems are important settings where this can occur.

With a smooth microcanonical entropy, a negative microcanonical heat capacity is possible in appropriate finite or nonadditive settings. It cannot be reproduced as a negative canonical heat capacity for a fixed, temperature-independent Hamiltonian, because the canonical variance identity enforces nonnegativity. Ensemble choice can therefore change a physical conclusion, not merely the convenience of calculation.

Equivalence is not equality of every probability measure or every fluctuation distribution. A global quantity fixed exactly in one ensemble can fluctuate in another at every finite size. Even precise asymptotic measure-level notions may compare logarithmic or per-particle distances rather than pointwise probabilities. Always name the observable, matching controls, limit and phase assumptions.

Further: quantum states and classical measures

In a finite quantum description, a density operator is a normalized positive operator encoding measurement probabilities. The microcanonical choice is ρmc = Π_(E,N)/Tr Π_(E,N), with Π the projector onto the allowed energy and number subspace. For a number-conserving Hamiltonian, [H,N]=0, the fixed-N canonical choice is ρN = Π_N exp(−βH)/Z_N and the grand choice is ρG = exp[−β(H−μN)]/Ξ. Tr means the trace over the relevant orthogonal states.

These are equilibrium statistical state choices, not all possible pure quantum vectors. A specially prepared coherent or pure state need not equal one of them. Degenerate basis states contribute separately to a trace, and probabilities for a degenerate energy add over that subspace. Our diagonal occupation model is deliberately simple; it does not require treating every normalized superposition as a new mutually exclusive counted record.

For classical structureless identical particles, the fixed-N phase measure in d spatial dimensions is commonly written [1/(N!h^(dN))]∏ d^d rᵢ d^d pᵢ in the dilute semiclassical regime. It makes normalized phase counts dimensionless and removes artificial particle-label permutations. Include actual internal degeneracies. This gas measure does not impose an extra N! division on our physically identified lattice sites or replace exact Bose–Fermi occupation rules.

A finite shell count, the cumulative phase volume below E and an inverse-energy density of states are distinct. A narrow-shell approximation uses g(E)δE only when g varies little over the interval. Both canonical integration and the grand sum over number sectors need a consistent measure. The next partition-function subject treats these integrations and normalizers in detail.

Further: allowed parameters and other controls

Every proposed ensemble must normalize. Our finite-capacity two-site grand model has a finite Ξ for every finite μ at positive T. For an ideal bosonic mode with unbounded occupation and energy εr per particle, its geometric grand sum requires μ<εr; a finite-site saturation result cannot be exported to that model. Equilibrium photons whose number is not conserved have μ=0 under the appropriate matter–radiation conditions. Quantum population laws are developed in the later statistics subject.

Positive T is appropriate for ordinary systems with energy unbounded above. Some bounded-spectrum systems permit negative inverse temperature when the equilibrium distribution is normalizable; those preparations need their own spectral and reservoir conditions. They are not included in our numerical examples, and negative T is not a continuation below absolute zero.

A piston introduces a different control: an isothermal–isobaric ensemble fixes T, external pressure p and N while V can fluctuate. Its microstate-and-volume weights involve exp[−β(E+pV)] with a specified measure. Grand canonical does not mean that every thermodynamic variable is allowed to fluctuate; the ensemble used throughout our open-site example still has fixed V.

More general descriptions can constrain charge, magnetization or several species numbers and exchange their conjugate quantities with reservoirs. Charged-particle exchange may require electrochemical potentials and charge constraints. Identify what the actual boundary and preparation allow before selecting a familiar formula.

Choose an ensemble from the physical question

A nearly isolated gas over a suitable observation interval is often treated microcanonically. A closed sample in a stable thermostat often uses the canonical ensemble. An adsorption layer or a small observation region exchanging particles with a much larger equilibrated environment may use a grand description. These choices require the relevant equilibrium, reservoir and coupling assumptions, not just a label such as “open system”.

For any application, specify the sample boundary, microscopic model, fixed external parameters, conserved quantities and reservoir controls. Define the allowed states and their measure. Normalize the appropriate weights, sum over degeneracies for the requested observable, and check finite-size or reservoir corrections against the precision of the question.

What to carry into partition functions

The microcanonical ensemble fixes an energy description and number sector; canonical and grand canonical measures replace selected exact constraints by reservoir-controlled weights. Their relationship follows from compatible reservoir states, conditional probabilities and, in suitable limits, concentration around matched equilibrium states.

“Partition functions” develops efficient ways to compute the normalizers and thermodynamic quantities. “Fluctuations and response functions” develops susceptibilities and cross responses. “Bose–Einstein and Fermi–Dirac statistics” changes the many-particle state space, while phase-transition and nonequilibrium subjects address regimes where a simple regular equilibrium picture needs more care. None of those later routes is required to normalize the finite cases here.

Worked examples

Try it yourself

1. A rigid sample exchanges energy but cannot exchange particles. Which of the three ensembles is the usual equilibrium choice?

   Show the answer and reasoning

   Canonical, at specified T,V,N.

   Thermal contact permits sample-energy fluctuations. Particle impermeability fixes N; a large stable reservoir and suitable equilibrium/coupling assumptions are still needed.

2. Which sample quantities are exact in the chosen discrete microcanonical sector?

   Show the answer and reasoning

   N=1 and E=ε; V is fixed by the rigid boundary.

   The compatible site arrangement can change without changing these assigned totals. This exact discrete sector should not be confused with a finite-width classical energy shell.

3. What does the complete record eg mean, and what are N and E?

   Show the answer and reasoning

   A is excited occupied, B is ground occupied; N=2 and E=ε.

   The sites are ordered physical locations. Swapping their internal occupations produces a different complete record, although energy and number remain the same.

4. Why is the local capacity one particle per site not a universal Bose or Fermi rule?

   Show the answer and reasoning

   It is an explicit adsorption constraint of this model.

   The model lists its allowed local states before choosing an ensemble. A different microscopic state space, spin content or interaction could allow different occupations.

5. At T₀ and N=1, is exp(−βε)=1/2 already the probability of one excited record?

   Show the answer and reasoning

   No; its normalized probability is (1/2)/3=1/6.

   The exponential is a relative weight. Both ground and both excited records contribute to the fixed-N normalizer.

6. Find the canonical probability of energy 0 at T₀ and N=1.

   Show the answer and reasoning

   2/3.

   The two ground records each have probability 1/3. Probabilities of a degenerate energy add; one must not treat the energy level as just one microscopic record.

7. Find the grand probability of the empty sample 00 at μ₀,T₀.

   Show the answer and reasoning

   1/4.

   Its unnormalized weight is 1, since E=N=0. The complete two-site grand normalizer is 4.

8. Find the grand probability of ee at μ₀,T₀.

   Show the answer and reasoning

   1/36.

   Its weight is z²exp(−2βε)=(4/9)(1/4)=1/9. Divide by Ξ=4; it has two particles and two excitations.

9. Find P(N=2) in the central grand preparation.

   Show the answer and reasoning

   1/4.

   Sum gg, ge, eg, ee probabilities 1/9+1/18+1/18+1/36. That number-sector event contains several different energies.

10. Condition the canonical N=1 distribution on E=ε. What are the probabilities of 0e and e0?

    Show the answer and reasoning

    1/2 each.

    The retained energy event has probability 1/3; each complete excited record had probability 1/6. Dividing yields the uniform degenerate energy sector.

11. Would conditioning a canonical measure on a wide energy interval always make every retained state equally likely?

    Show the answer and reasoning

    No.

    Different retained energies keep their exp(−βE) factors. Exact uniformity holds within one degenerate energy sector; approximate uniformity needs a sufficiently narrow variation in βE.

12. For activity z=1/6 at T₀, find the occupied-site probability f and mean particle number.

    Show the answer and reasoning

    f=1/5 and ⟨N⟩=2/5.

    The occupied local weight is (1/6)(3/2)=1/4, versus empty weight 1. Normalize by 5/4, then multiply the occupied probability by two sites.

13. At z=8/3, which is more probable: N=0 or N=2?

    Show the answer and reasoning

    N=2, with 16/25 versus 1/25.

    The occupied probability is 4/5, so the two independent-site probabilities are f² and (1−f)². Higher μ favors higher N, within the specified capacity.

14. Does μ<0 require the sample to contain no particles?

    Show the answer and reasoning

    No; the central μ₀<0 example has mean particle number 1.

    Occupation weights depend on β,μ,all state energies and degeneracies together. A sign of μ alone is not an occupancy rule.

15. In the finite-bath example, find P(k=0).

    Show the answer and reasoning

    20/56=5/14.

    The sample has one all-ground record; the bath must carry all three excitations in C(6,3)=20 ways. The normalization remains the full combined fixed-total count 56.

16. For sample k=1, why is the bath weight 15 multiplied by 2?

    Show the answer and reasoning

    Two distinct sample records have that energy.

    15 is the reservoir count for one specified sample arrangement. An energy macrorecord includes both choices for which sample unit is excited.

17. Is the matched finite-bath canonical comparison at T₀?

    Show the answer and reasoning

    No; it satisfies βmatch ε=ln(5/3).

    Its mean excitation probability is 3/8, so the canonical adjacent weight ratio is (3/8)/(5/8)=3/5. The adsorption and growing-unit examples instead use β₀ε=ln 2.

18. For N=3,q=1 fixed two-level units, find the probability that two specified units are both excited.

    Show the answer and reasoning

    0 microcanonically;1/9 in the matched independent canonical preparation.

    The fixed sector has only one excitation to share. Matching the single-unit probability 1/3 cannot produce simultaneous excitation of two units within that sector.

19. For N=300,q=100, write the microcanonical pair probability exactly.

    Show the answer and reasoning

    33/299.

    Use 100×99/(300×299). This approaches the canonical 1/9 as N grows through the specified matching sequence, but is not equal at this finite N.

20. At N=30, compare the standard deviations of the excitation fraction q/N.

    Show the answer and reasoning

    Microcanonical 0; canonical about 0.0860663.

    Exact q removes that variance. In the independent canonical model Var(q)=2N/9, so its fractional standard deviation is √[2/(9N)].

21. Does canonical energy-density concentration make its total-energy variance exactly zero at finite N?

    Show the answer and reasoning

    No.

    In the matching two-level model Var(E)=(2N/9)ε² even while the standard deviation of E/N tends to zero. These are different normalizations of the random variable.

22. Why must an ideal unbounded bosonic mode be checked for grand-trace convergence?

    Show the answer and reasoning

    Its occupation sum is infinite and requires μ below that mode energy at positive T.

    Its ratio is exp[−β(εr−μ)]. A finite-capacity site has no such infinite geometric tail, so its parameter range cannot simply be transferred.

23. What must be specified before claiming microcanonical and canonical equivalence?

    Show the answer and reasoning

    The matching state, observable, thermodynamic limit, measure and regularity/phase conditions.

    Strict concavity and an appropriate unique equilibrium branch support the simple matching argument. Global constrained fluctuations or nonconcave entropy can invalidate a stronger claim.

24. Does a weak thermal contact determine the relaxation time?

    Show the answer and reasoning

    No.

    An equilibrium weight formula omits transition rates, transport dynamics, accessible communicating sectors and observation time. Those are additional physical inputs.

What to carry forward

• An ensemble specifies a probability measure together with a physical model, constraints and preparation; it is not a dynamical clock.
• Microcanonical fixes an energy description and particle sector; canonical fixes T,V,N; grand canonical fixes T,V,μ while E and N fluctuate.
• Reservoir multiplicity produces Boltzmann weights only under its equilibrium, coupling and size conditions. Count degeneracy for the requested macrorecord.
• A mean particle number is not an exact integer constraint. Conditioning can recover a constrained measure without making the unconditional ensembles equal.
• Matching local observables and concentrating energy density do not erase all finite-size or global fluctuation differences.
• Ensemble equivalence needs a specified limit and appropriate concavity, phase and observable conditions; finite baths and nonadditive systems require explicit care.