Canonical partition function ($Z(\beta )$)

The partition function $Z(\beta )={\sum }_s{e}^{-\beta E_s}$ is the sum of the Boltzmann weights over individual microstates. Dividing by it makes their probabilities sum to one. When summing distinct energies instead, include the state degeneracy: $Z={\sum }_{E}g(E)e^{-\beta E}$. For two independent distinguishable two-level systems,The middle coefficient counts two microstates, exactly as in the preceding figure. For $N$ such units, $Z_N=Z_1^N$; this factorization requires additive energy and independence, and generally fails with interactions.

The usefulness goes beyond normalization. For fixed energies, derivatives of the natural logarithm of the partition function yield measurable quantities:Here $U$ is internal energy, the second expression is the energy variance, and $S$ is Gibbs entropy. For one two-level system at fixed $\Delta$, its heat capacity $C=\partial U/\partial T$ and entropy areAt $\beta \Delta =\ln 3$, these give $U/\Delta =0.25$, $C/k_B\simeq 0.2263$ and $S/k_B\simeq 0.5623$. For independent units, all three quantities add.

The figure shows why heat capacity peaks. At low temperature, the excited energy is rarely occupied; at high temperature, the probabilities are almost equal and further warming barely changes internal energy. The largest response lies between these limits, at $k_{B}T/\Delta \simeq 0.4168$, where $C/k_B\simeq 0.4392$. Meanwhile entropy rises from zero toward $k_B\ln 2$, reflecting two equally likely alternatives. This high-temperature limit belongs to the stipulated two-level system; extra accessible energy eigenstates would change it.

Counting permutations of identical classical particles as different states produces a spurious entropy of mixing when otherwise identical gases are combined. The dilute ideal gas partition function $Z_N=Z_1^N/N!$ removes that label multiplicity and gives an extensive entropy in the thermodynamic limit. For different species, genuinely distinct composition information still produces a entropy of mixing. quantum statistics supplies the underlying identical-particle counting.

The single-molecule state sum $q={\sum }_s{e}^{-\beta E_s}$ for specified energy references and external constraints. In a dilute classical ideal gas, indistinguishability gives $Q_N=q^N/N!$; translation supplies the factor proportional to volume.

If energies are measured upward from a species reference ${\epsilon }_0$, its absolute single-particle sum is $q_{\mathrm{abs}}=e^{-\beta {\epsilon }_0}q$. The ideal-gas chemical potential is then ${\epsilon }_0-k_{B}T\ln (q/N)$. A zero-point factor can be included either in $q$ or in the reference energy, but not both.

For a free molecule in a uniform ideal gas, $q_i$ is proportional to volume. The ratio $f_i=q_i/V$ is independent of the container volume at fixed temperature and has units of inverse volume.

The electronic partition function includes electronic excitation energies and degeneracies $g_i$. If only an exactly degenerate ground manifold of size $g$ is populated, $q_{\mathrm{el}}=g$ and its molar entropy is $R\log g$. Resolved splittings require a state sum rather than replacing all components by one degeneracy.

A vibrational partition function sums over molecular vibrational energy levels. A single quantum harmonic oscillator, with energies measured relative to its ground level and vibrational temperature ${\theta }_v=h\nu /k$, has $q_{\mathrm{vib}}=(1-e^{-{\theta }_v/T}{)}^{-1}$. Retaining its absolute zero-point energy multiplies this by $e^{-{\theta }_v/(2T)}$. That factor shifts the internal energy but contributes no entropy.

The harmonic-oscillator entropy contribution for one vibrational coordinate, with $x={\theta }_{\mathrm{vib}}/T$. Contributions from independent normal modes add, with one term for each component of a degenerate mode. A temperature-independent zero-point energy does not change this entropy.

The rotational partition function sums Boltzmann factors over rotational energy levels, including their degeneracies and the appropriate indistinguishability convention. For a linear rigid rotor at high temperature, $q_{\mathrm{rot}}=8{\pi }^{2}IkT/(\sigma h^2)$, with moment of inertia $I$ and rotational symmetry number $\sigma$.

The entropy contribution from populated molecular rotational states. A nonlinear rigid rotor has $q_{\mathrm{rot}}\propto T^{3/2}$ at high temperature, giving $S_{\mathrm{rot},\mathrm{m}}=R(\ln q_{\mathrm{rot}}+3/2)$. Its rotational symmetry number removes redundant orientations.

For a free molecule of mass $m$ in a large three-dimensional volume $V$, replacing the translational state sum by an integral gives the displayed partition function. For identical classical molecules the many-molecule canonical partition function is $q_{\mathrm{trans}}^N/N!$, not simply $q_{\mathrm{trans}}^N$. Differentiating its Helmholtz free energy at fixed volume gives the ideal-gas translational entropy.

The trace of imaginary-time evolution is the canonical partition function of a quantum system. In a coordinate basis it is the integral of the diagonal Euclidean path integral kernel. Inserting a symmetry operator instead gives a weighted thermal trace.

For two equal-frequency quantum harmonic oscillators, the ordinary thermal trace is $r/(1-r{)}^2$ and the trace with the parity operator is $r/(1+r{)}^2$, where $r=e^{-\beta \omega }$. Total excitation $N$ has degeneracy $N+1$ and parity $(-1{)}^N$. Therefore the traces expand as ${\sum }_{n\ge 1}(\pm 1{)}^{n-1}n{r}^n$. The alternating weights come from parity insertion in a bosonic state space, not fermionic statistics.

Canonical differentiation gives

Canonical energy variance equals ${\partial }_{\beta }^2\log Z=k_B{T}^2{C}_V$.

At fixed particle number and volume, a classical monatomic ideal gas has $U=3N{k}_{B}T/2$ and $C_V=3N{k}_B/2$. The canonical energy fluctuation identity $\mathrm{Var}(E)=k_B{T}^2{C}_V$ therefore gives ${\sigma }_U/U=\sqrt{2/(3N)}$. Absolute fluctuations grow as $\sqrt{N}$, while relative fluctuations decrease as $N^{-1/2}$. A particle reservoir changes the fluctuation problem because energy and particle-number fluctuations are then correlated.

Forput $a=\kappa R^2/2$. The normalized classical one-particle partition function isIts mean total and potential energies are

If $a\gg k_{B}T$, the wall is thermally inaccessible and $⟨E⟩\sim 2k_{B}T$, as four quadratic terms predict. If $a\ll k_{B}T$, the spatial distribution is nearly uniform in the disk andso only the two kinetic quadratic terms contribute at leading order.

For $N$ indistinguishable noninteracting classical particles, $Z_N=z_1^N/N!$, where $z_1$ is the normalized one-particle phase-space integral.

For a structureless particle with Hamiltonian $p^2/(2m)+ar$, $a>0$, the classical phase space integral gives the displayed one-particle partition function. Its spatial integral is $4\pi {\int }_0^{\infty }{r}^2{e}^{-ar/(k_{B}T)}dr=8\pi (k_{B}T/a{)}^3$, while the momentum integral gives ${\lambda }_T^{-3}$. For $N$ indistinguishable dilute particles, $Z_N=Z_1^N/N!$, $F\simeq -N{k}_{B}T[\ln (Z_1/N)+1]$, $U=(9/2)N{k}_{B}T$ and $S\simeq N{k}_B[\ln (Z_1/N)+11/2]$. These thermodynamic formulas use leading Stirling accuracy; the classical phase-space integral additionally requires temperatures above the discrete trap level scale.

For a dilute classical trapped gas, $\mu \simeq k_{B}T\ln (N/Z_1)$ with energy zero at the potential minimum. Thus $z=e^{\mu /(k_{B}T)}=N/Z_1$. Quantum occupation effects begin when $z$ is no longer small, giving the displayed order-of-magnitude particle threshold. Equivalently, the central density $n(0)=N/[8\pi (k_{B}T/a{)}^3]$ satisfies $n(0){\lambda }_T^3\sim 1$. This is a degeneracy crossover criterion rather than an exact Bose condensation threshold. In the continuum approximation, spinless bosons have excited-state saturation $N_c=\zeta (9/2)Z_1$, from the trap density of states proportional to $E^{7/2}$. A quantum ground energy should be subtracted consistently when discreteness matters.

At fixed atom number, the entropy of the classical ideal gas in a spherical linear trap depends on temperature and strength through $Z_1\propto T^{9/2}{a}^{-3}$. Reversible adiabatic change therefore holds this product constant, giving the displayed relation. Reducing trap strength lowers the temperature while increasing cloud size. Its peak classical phase-space occupation $N/Z_1$ remains constant, so this cooling alone does not make the gas more quantum degenerate. Heat exchange, particle loss or irreversible entropy production would change this relation.

For the potential $U=r^{2n}/V^{2n/3}$, rescaling $r=V^{1/3}(kT{)}^{1/(2n)}u$ in the classical one-particle partition function gives the formula above. The parameter-conjugate force is $P=NkT/V$, the internal energy is $(3/2+3/(2n))NkT$, and $C_P-C_V=Nk$. The kinetic term contributes $3NkT/2$, while the potential term contributes $3NkT/(2n)$. For indistinguishable particles the $N!$ factor changes free energy and entropy but not these force and heat-capacity formulas.

The three-dimensional translational momentum integral is $1/{\lambda }^3$ per unit volume, where

Equating thermal de Broglie wavelength $h/\sqrt{2\pi m{k}_{B}T}$ to mean particle spacing $n^{-1/3}$ gives the displayed quantum-degeneracy scale. It is an onset estimate rather than a universal phase-transition temperature. Different typical-momentum conventions change its order-one factor; density, mass, spin degeneracy and interactions determine the detailed crossover or condensation criterion.

The translational concentration scale for a nonrelativistic particle in three dimensions, per internal state. It is the inverse cube of the thermal de Broglie wavelength. A classical component with number density $n$ and external one-particle energy $U$ has chemical potential $\mu =k_{B}T\ln (n/n_Q)+U$. An additional internal degeneracy must be included explicitly; it is not hidden in this definition. The dilute classical condition is $n/n_Q\ll 1$ for each occupied internal component.

For energy $pc+U(x)$,If $U(r)=r^{2n}/V^{2n/3}$, thenConsequently $pV=N{k}_{B}T$, while the mean energy and variance are $NA{k}_{B}T$ and $NA(k_{B}T{)}^2$ with $A=3+3/(2n)$.

The radial density is proportional to $r^2{e}^{-r^{2n}/(V^{2n/3}{k}_{B}T)}$, whose mode is

In the classical dilute regime, a monatomic ideal gas has entropy