statistical-physics.bigb

= Statistical physics
{wiki=Statistical_mechanics}

= One-dimensional freely jointed chain
{parent=Statistical physics}
{wiki=Ideal_chain}

A chain of $n$ links of length $a$, each pointing independently right or left, has extension
$$
l=(n_+-n_-)a=(2n_+-n)a
$$
and multiplicity $n!/(n_+!n_-!)$.

= Fixed-tension partition function of a one-dimensional chain
{title2=$Z=(2\cosh(\tau a))^n$}
{parent=One-dimensional freely jointed chain}

Weighting each configuration by $e^{\tau l}$ gives
$$
Z=(e^{\tau a}+e^{-\tau a})^n
=(2\cosh(\tau a))^n,
\qquad
L=\partial_\tau\log Z=na\tanh(\tau a).
$$

= Entropic Hooke law for a one-dimensional chain
{parent=Fixed-tension partition function of a one-dimensional chain}

The physical tension is $f=k_BT\tau$. At small extension,
$$
f\sim\frac{k_BT}{na^2}L.
$$
At fixed tension the extension is $L=na\tanh(fa/(k_BT))$ and decreases as temperature rises.

= State degeneracy
{parent=Statistical physics}
{wiki=Degenerate_energy_levels}

The degeneracy $g(l)$ of a <macrostate> labelled by $l$ is the number of distinct <microstates> having that value of $l$.

= Degeneracy factor
{parent=State degeneracy}

A degeneracy factor counts the distinct microscopic states that share the same macroscopic quantum numbers or energy.

= Microstate
{parent=State degeneracy}
{wiki}

A microstate is a complete microscopic specification of a system.

= Macrostate
{parent=State degeneracy}
{wiki}

A macrostate groups together all <microstates> that share prescribed macroscopic data.

= Quantum ideal-gas statistics
{parent=Statistical physics}

An ideal quantum gas has mean occupation
$$[\exp((E-\mu)/(k_BT))\mp1]^{-1},$$
with one sign for bosons and the other for fermions.

= Quantum statistics
{synonym}

= Maxwell-Boltzmann distribution
{parent=Quantum ideal-gas statistics}
{c}
{wiki=Maxwell%E2%80%93Boltzmann_distribution}

When $E-\mu\gg k_BT$, either quantum occupation law reduces to the classical factor
$$
e^{-(E-\mu)/(k_BT)}.
$$

= Bose-Einstein distribution
{parent=Quantum ideal-gas statistics}
{c}
{wiki=Bose%E2%80%93Einstein_statistics}

The mean occupation of a bosonic one-particle state of energy $E$ is
$$
n(E)=\frac1{e^{\beta(E-\mu)}-1},
\qquad \beta=(k_BT)^{-1}.
$$

= Bose-Einstein condensation
{c}
{parent=Bose-Einstein distribution}
{wiki}

Bose-Einstein condensation is the macroscopic occupation of the lowest one-particle state by bosons below a critical temperature.

= Phonon
{parent=Bose-Einstein distribution}
{wiki}

A phonon is a bosonic quantum of a crystal's normal modes. Because phonon number is not conserved in thermal equilibrium, its <chemical potential> is zero.

= Phonon dispersion relation
{parent=Phonon}
{wiki=Phonon#Phonon_dispersion}

A phonon dispersion relation gives its angular frequency $\omega$ as a function of its <wavevector> $k$. For an isotropic power law, $\omega=C|k|^\alpha$.

= Density of states for a two-dimensional power-law phonon dispersion
{parent=Phonon dispersion relation}

For two polarizations in area $A$ and $\omega=C|k|^\alpha$, counting <wavevector> states in a disk gives
$$
g(\omega)=\frac{A}{\pi\alpha C^{2/\alpha}}
\omega^{2/\alpha-1}.
$$

= Debye model
{c}
{parent=Phonon}
{wiki}

The Debye model replaces the crystal's vibrational spectrum by a continuum of phonon modes up to a cutoff chosen to preserve the total number of vibrational degrees of freedom.

= Debye frequency
{title2=$\omega_D$}
{c}
{parent=Debye model}
{wiki=Debye_model#Debye_frequency}

The Debye frequency is the upper frequency cutoff fixed by equating the continuum mode count to the crystal's number of vibrational degrees of freedom.

= Debye temperature
{title2=$T_D$}
{c}
{parent=Debye frequency}
{wiki=Debye_model#Debye_temperature}

The Debye temperature is $T_D=\hbar\omega_D/k_B$.

= Two-dimensional power-law Debye model
{parent=Debye model}

For two polarizations, $n$ atoms in area $A$, and $\omega=C|k|^\alpha$, mode counting gives
$$
\omega_D=C\left(\frac{4\pi n}{A}\right)^{\alpha/2}.
$$
At low temperature its energy scales as $T^{1+2/\alpha}$ and its constant-volume heat capacity as $T^{2/\alpha}$.

= Density of states
{title2=$g(E)$}
{parent=Quantum ideal-gas statistics}
{wiki=Density_of_states}

The density of states $g(E)$ is defined so that $g(E)\,dE$ counts one-particle states with energies in an interval of width $dE$. A sum over closely spaced states then becomes an integral weighted by $g(E)$.

= Fermi-Dirac distribution
{parent=Quantum ideal-gas statistics}
{c}
{wiki=Fermi%E2%80%93Dirac_statistics}

The mean occupation of a fermionic one-particle state of energy $E$ is
$$
f(E)=\frac{1}{e^{\beta(E-\mu)}+1},
\qquad \beta=(k_BT)^{-1}.
$$
At zero temperature this becomes the step function $\mathbf 1_{E<E_F}$.

= Fermi gas
{parent=Fermi-Dirac distribution}
{c}
{wiki=Fermi_gas}

A Fermi gas consists of noninteracting fermions whose one-particle states are occupied according to the Fermi-Dirac distribution.

= Fermi energy
{title2=$E_F$}
{c}
{parent=Fermi gas}
{wiki}

The Fermi energy is the energy of the highest occupied one-particle state of a noninteracting Fermi gas at zero temperature. Equivalently, it is the zero-temperature <chemical potential>.

= Fermi sea
{c}
{parent=Fermi energy}
{wiki}

The Fermi sea is the zero-temperature many-fermion state in which every one-particle state below the <Fermi energy> is occupied and every state above it is empty.

= Ultrarelativistic ideal Fermi gas
{parent=Fermi gas}

For spin-one-half particles in three dimensions with dispersion $\varepsilon=pc$, the <density of states> is
$$
g(\varepsilon)=\frac{V\varepsilon^2}{\pi^2\hbar^3c^3}.
$$
At zero temperature,
$$
E_F=\hbar c\left(3\pi^2\frac NV\right)^{1/3},
\qquad
E=\frac34NE_F,
\qquad
p=\frac E{3V}.
$$

= Degeneracy pressure
{parent=Fermi gas}
{wiki}

Degeneracy pressure is the pressure of a fermionic system that remains at zero temperature because the <Pauli exclusion principle> forces particles to occupy states with nonzero momentum.

= Pauli paramagnetism
{c}
{parent=Fermi gas}
{wiki}

Pauli paramagnetism is the weak positive magnetic response produced when a magnetic field shifts the two spin Fermi surfaces of a degenerate electron gas. At zero temperature, if $g(E_F)$ includes both spin states, its total magnetic moment and susceptibility are
$$
M=\mu_B^2B\,g(E_F),
\qquad
\chi=\mu_B^2g(E_F).
$$

= Magnetic susceptibility
{title2=$\chi$}
{parent=Pauli paramagnetism}
{wiki}

Magnetic susceptibility measures the linear change of magnetization or magnetic moment with an applied magnetic field. With the relevant convention stated explicitly, $\chi=\partial M/\partial B$.

= Two-dimensional free-electron density of states
{parent=Fermi gas}

For nonrelativistic electrons in area $A$, with spin degeneracy $g_s=2$ and energy $E=\hbar^2k^2/(2m)$, the density of one-particle states is independent of energy:
$$
g(E)=\frac{Am}{\pi\hbar^2}.
$$

= Low-temperature particle-number cancellation for constant density of states
{parent=Two-dimensional free-electron density of states}

When the density of states is constant and the chemical potential is fixed at $E_F$, the thermal gain above $E_F$ cancels the thermal depletion below it after the energy range is extended to the whole real line. The finite lower edge leaves only an exponentially small correction of order $e^{-\beta E_F}$.

= Quadratic low-temperature energy correction for constant density of states
{parent=Low-temperature particle-number cancellation for constant density of states}

For constant density of states, weighting the difference between the Fermi-Dirac distribution and the zero-temperature step by $E-E_F$ produces a convergent integral that scales as $\beta^{-2}$. Hence the leading low-temperature energy correction is proportional to $T^2$.

= Degeneracy pressure of a two-dimensional Fermi gas
{parent=Fermi gas}

At zero temperature, a two-dimensional free Fermi gas satisfies
$$
N=gE_F,
\qquad
E_{\rm tot}=\frac12NE_F,
\qquad
p=-\left(\frac{\partial E_{\rm tot}}{\partial A}\right)_N
=\frac{NE_F}{2A}.
$$

= Classical limit of Bose--Einstein and Fermi--Dirac statistics
{parent=Quantum ideal-gas statistics}

When $E-\mu\gg k_BT$, the additive $\mp1$ is negligible and both quantum distributions reduce to the Maxwell--Boltzmann factor $e^{-(E-\mu)/(k_BT)}$.

= Nonrelativistic Maxwell--Boltzmann number density
{parent=Classical limit of Bose--Einstein and Fermi--Dirac statistics}
{wiki=Maxwell–Boltzmann_statistics}

For particles of mass $m$, spin degeneracy $g_s$, and energy $mc^2+p^2/(2m)$,
$$
n=g_s\left(\frac{2\pi mk_BT}{h^2}\right)^{3/2}
e^{(\mu-mc^2)/(k_BT)}.
$$

= Planck photon distribution
{c}
{parent=Statistical physics}
{wiki=Planck%27s_law}

For photons in thermal equilibrium at temperature $T$, the number density per frequency interval is
$$
\frac{dn}{d\nu}=\frac{8\pi\nu^2}{c^3}\frac1{e^{h\nu/(k_BT)}-1}.
$$

= Temperature scaling of thermal photon number and energy densities
{parent=Planck photon distribution}

Changing variables to $x=h\nu/(k_BT)$ in the Planck distribution gives
$$
n=\alpha T^3,
\qquad
\rho=\xi T^4,
$$
where $\alpha$ and $\xi$ are temperature-independent constants.

= Thermodynamics
{parent=Statistical physics}
{wiki}

Thermodynamics relates energy, entropy, temperature, pressure, volume, work, and heat through state functions and process laws.

= Boltzmann constant
{title2=$k_B$}
{c}
{parent=Thermodynamics}
{wiki}

The Boltzmann constant converts temperature into an energy scale.

= Temperature
{parent=Thermodynamics}
{wiki}

Temperature is the intensive variable thermodynamically conjugate to entropy; at equilibrium it determines the direction of heat transfer.

= Thermal equilibrium
{parent=Thermodynamics}
{wiki}

Systems are in thermal equilibrium when no net heat flows between them; their temperatures are equal.

= Entropy
{parent=Thermodynamics}
{wiki}

Entropy is an extensive state function satisfying $dS=\delta Q_{\rm rev}/T$ for a reversible transfer of heat.

= Second law of thermodynamics
{c}
{parent=Thermodynamics}
{wiki}

The second law forbids cyclic devices whose sole effect is complete conversion of heat from one reservoir into work or unassisted heat transfer from cold to hot. For a closed system it implies nondecrease of total entropy.

= Clausius statement of the second law
{c}
{parent=Second law of thermodynamics}
{wiki=Second_law_of_thermodynamics#Clausius_statement}

No cyclic device can have as its sole effect the transfer of heat from a colder reservoir to a hotter reservoir.

= Kelvin-Planck statement of the second law
{c}
{parent=Second law of thermodynamics}
{wiki=Second_law_of_thermodynamics#Kelvin_statement}

No cyclic device can have as its sole effect the extraction of heat from one reservoir and its complete conversion into work.

= Carnot theorem
{c}
{parent=Second law of thermodynamics}
{wiki=Carnot%27s_theorem_(thermodynamics)}

No heat engine operating between two fixed reservoirs is more efficient than a reversible engine, and all reversible engines between those reservoirs have the same efficiency.

= Thermodynamic temperature
{parent=Carnot theorem}
{wiki=Thermodynamic_temperature}

Absolute thermodynamic temperature can be defined so that a reversible engine exchanging heats $Q_h$ and $Q_c$ with reservoirs at $T_h$ and $T_c$ satisfies
$$
\frac{Q_c}{Q_h}=\frac{T_c}{T_h},
\qquad
\eta=1-\frac{T_c}{T_h}.
$$
A choice of one reference temperature fixes the scale.

= Heat
{parent=Thermodynamics}
{wiki}

Heat is energy transferred because of a temperature difference.

= Heat capacity
{parent=Heat}
{wiki}

Heat capacity is the heat required per unit temperature change under a specified constraint.

= Heat capacity at constant volume
{title2=$C_V$}
{parent=Heat capacity}
{wiki=Heat_capacity#Constant-volume_heat_capacity}

The heat capacity at constant volume is $C_V=(\partial E/\partial T)_V$ when particle number and other conserved quantities are also fixed.

= Dulong-Petit law
{c}
{parent=Heat capacity at constant volume}
{wiki=Dulong%E2%80%93Petit_law}

In the classical high-temperature limit, each independent harmonic mode contributes $k_B$ to the constant-volume heat capacity.

= Third law of thermodynamics
{c}
{parent=Thermodynamics}
{wiki}

The third law implies that the entropy of a system with a nondegenerate ground state approaches zero as its temperature approaches absolute zero. In particular, ordinary equilibrium heat capacities approach zero.

= Pressure
{parent=Thermodynamics}
{wiki}

Pressure is normal force per unit area and is thermodynamically conjugate to volume.

= Equation of state
{parent=Pressure}
{wiki=Equation_of_state}

An equation of state is a relation among thermodynamic state variables such as pressure, volume, temperature, energy, and particle number.

= Equation of state of an ideal ultrarelativistic gas
{parent=Equation of state}

An isotropic ideal gas whose particles obey the ultrarelativistic dispersion $\varepsilon=pc$ satisfies
$$
pV=\frac E3.
$$

= Adiabatic equation of state
{parent=Equation of state}
{wiki=Adiabatic_process#Ideal_gas_(reversible_process)}

An adiabatic equation of state relates pressure and volume along a process with no heat transfer. It commonly has the form $pV^\gamma=\text{constant}$.

= Volume
{parent=Thermodynamics}
{wiki=Volume_(thermodynamics)}

Volume is the extensive measure of the space occupied by a thermodynamic system.

= Internal energy
{parent=Thermodynamics}
{wiki}

Internal energy is the energy stored in the microscopic degrees of freedom of a system.

= Free energy
{parent=Thermodynamics}
{wiki=Thermodynamic_free_energy}

A free energy is a thermodynamic potential whose decrease governs equilibrium under specified environmental constraints.

= Chemical potential
{parent=Thermodynamics}
{wiki}

The chemical potential $\mu=(\partial E/\partial N)_{S,V}$ is the energy cost of adding a particle under fixed entropy and volume.

= Chemical equilibrium
{parent=Chemical potential}
{wiki}

At chemical equilibrium, the sum of chemical potentials weighted by each reaction's stoichiometric coefficients vanishes.

= Photon chemical potential
{parent=Chemical equilibrium}

The chemical potential of photons in thermal equilibrium is zero because their number is not conserved.

= Ideal gas
{parent=Thermodynamics}
{wiki}

An ideal gas obeys $pV=Nk_BT$ and neglects intermolecular interactions except during elastic collisions.

= Specific-heat ratio
{title2=$\gamma$}
{parent=Ideal gas}
{wiki=Heat_capacity_ratio}

The specific-heat ratio of an ideal gas is $\gamma=C_p/C_V>1$.

= Nondegenerate gas
{parent=Ideal gas}
{wiki=Classical_ideal_gas}

A gas is nondegenerate when its occupation numbers are low enough that <quantum statistics> reduce to the <Maxwell-Boltzmann distribution>. A common criterion is $n\lambda_{\rm th}^3\ll1$, where $\lambda_{\rm th}$ is the thermal de Broglie wavelength.

= Internal energy of an ideal gas
{parent=Ideal gas}

For a fixed amount of ideal gas, internal energy depends only on temperature. If $C_V$ is constant, $dE=C_V\,dT$ and hence $E=C_VT$ after a choice of energy zero.

= Mayer relation
{c}
{parent=Ideal gas}
{wiki=Mayer%27s_relation}

For an ideal gas of $N$ particles,
$$
C_p-C_V=Nk_B.
$$
Thus, with $\gamma=C_p/C_V$, one has $Nk_B/C_V=\gamma-1$.

= Dieterici equation
{parent=Thermodynamics}
{c}
{wiki=Dieterici_equation_of_state}

The Dieterici equation of state
$$
p=\frac{k_BT}{v-b}\exp\left(-\frac{a}{k_BTv}\right)
$$
models excluded volume through $b$ and attraction through $a$. Its critical point has $v_c=2b$ and $T_c=a/(4bk_B)$.

= Intensive and extensive thermodynamic quantities
{parent=Thermodynamics}

An extensive quantity scales in proportion to the amount of material, while an intensive quantity is unchanged when the system is replicated. Energy, entropy, volume, and particle number are extensive; temperature, pressure, and chemical potential are intensive.

= Euler theorem for homogeneous functions
{parent=Thermodynamics}
{c}
{wiki=Euler%27s_homogeneous_function_theorem}

If $f(\lambda x)=\lambda^kf(x)$, differentiation with respect to $\lambda$ at one gives
$$
\sum_ix_i\frac{\partial f}{\partial x_i}=kf.
$$

= Gibbs-Duhem equation
{c}
{parent=Euler theorem for homogeneous functions}
{wiki=Gibbs–Duhem_equation}

Extensivity and $dE=T\,dS-p\,dV+\mu\,dN$ give
$$
E=TS-pV+\mu N,
\qquad
S\,dT-V\,dp+N\,d\mu=0.
$$

= Gibbs free energy
{parent=Thermodynamics}
{c}
{wiki}

The Gibbs free energy is $G=E-TS+pV$. For a simple system of fixed composition,
$$
dG=-S\,dT+V\,dp.
$$

= Chemical-potential balance for a reaction
{parent=Gibbs free energy}

For a reaction with stoichiometric changes $\nu_i$, varying the reaction extent at fixed temperature and pressure gives
$$
dG=\left(\sum_i\nu_i\mu_i\right)d\xi.
$$
Equilibrium requires $\sum_i\nu_i\mu_i=0$.

= Phase coexistence curve
{parent=Gibbs free energy}

Two phases coexist where their molar Gibbs free energies agree. Differentiating that equality along the coexistence curve relates its slope to the entropy and volume jumps.

= First-order phase transition
{parent=Phase coexistence curve}
{wiki=Phase_transition#First-order_phase_transitions}

At a first-order phase transition, the Gibbs free energy is continuous while at least one first derivative, such as entropy or volume, jumps. A nonzero entropy jump produces latent heat $L=T\Delta S$.

= Latent heat
{parent=Phase coexistence curve}
{wiki}

The latent heat from phase $\alpha$ to phase $\beta$ is
$$
L=T(S_\beta-S_\alpha)=H_\beta-H_\alpha
$$
per mole or per particle, according to the normalization used.

= Clausius-Clapeyron relation
{parent=Latent heat}
{c}
{wiki=Clausius–Clapeyron_relation}

Along a first-order coexistence curve,
$$
\frac{dp}{dT}
=\frac{S_\beta-S_\alpha}{V_\beta-V_\alpha}
=\frac{L}{T(V_\beta-V_\alpha)}.
$$
At a critical point the two phases merge, their entropy jump vanishes, and the latent heat tends to zero.

= Thermodynamic conjugate variable
{parent=Thermodynamics}
{wiki=Conjugate_variables_(thermodynamics)}

Two thermodynamic variables are conjugate when their product has units of energy and they occur as a paired term in a thermodynamic differential, such as pressure and volume or tension and length.

= First law of thermodynamics
{parent=Thermodynamics}
{wiki}

For a simple compressible system with fixed particle number, $dE=T\,dS-p\,dV$.

= Adiabatic process
{parent=First law of thermodynamics}
{wiki}

An adiabatic process exchanges no heat with its surroundings, so $\delta Q=0$. A reversible adiabatic process is isentropic.

= Reversible ideal-gas adiabat
{parent=Adiabatic process}

For a fixed amount of ideal gas with constant heat capacities, the first law and $\delta Q=0$ give
$$
TV^{\gamma-1}=\text{constant},
\qquad
pV^\gamma=\text{constant},
$$
where $\gamma=C_p/C_V$.

= Enthalpy
{parent=Thermodynamics}
{wiki}

Enthalpy is $H=E+pV$ and satisfies $dH=T\,dS+V\,dp$ for a simple compressible system.

= Enthalpy Maxwell relation
{parent=Enthalpy}

Since $T=(\partial H/\partial S)_p$ and $V=(\partial H/\partial p)_S$, equality of mixed derivatives gives $(\partial T/\partial p)_S=(\partial V/\partial S)_p$.

= Heat capacity at constant pressure
{parent=Enthalpy}
{wiki=Heat_capacity}

At fixed pressure, supplied heat equals enthalpy change; for constant $C_p$, $Q=C_p\Delta T$.

= Diatomic ideal-gas enthalpy
{parent=Thermodynamics}

With translational and rotational modes active but vibration frozen, a diatomic ideal gas has $E=5Nk_BT/2$ and $H=7Nk_BT/2$.

= Irreversible adiabatic piston compression
{parent=Thermodynamics}

After a sudden increase to constant external pressure $p_1$, an insulated gas obeys $\Delta E=p_1(V_0-V_1)$ rather than a reversible adiabatic power law.

= Constant-pressure heating of an ideal gas
{parent=Thermodynamics}

At fixed pressure, $Q=\Delta H$ and the expansion work is $p\Delta V=Nk_B\Delta T$.

= Thermodynamic cycle
{parent=Thermodynamics}
{wiki}

A thermodynamic cycle returns a working substance to its initial state. Its net internal-energy change is zero, so the net work output equals net heat input.

= Thermal efficiency
{title2=$\eta=W/Q_{\rm in}$}
{parent=Thermodynamic cycle}
{wiki=Thermal_efficiency}

The thermal efficiency of a heat engine is the net work output divided by heat input. For a cycle absorbing $Q_{\rm in}$ and rejecting the positive amount $Q_{\rm out}$,
$$
\eta=1-\frac{Q_{\rm out}}{Q_{\rm in}}.
$$

= Otto cycle
{c}
{parent=Thermodynamic cycle}
{wiki}

The ideal Otto cycle has adiabatic compression, constant-volume heat addition, adiabatic expansion, and constant-volume heat rejection. For an ideal gas with compression ratio $r=V_1/V_2$ and specific-heat ratio $\gamma$,
$$
\eta=1-\frac1{r^{\gamma-1}}.
$$

= Diesel cycle
{c}
{parent=Thermodynamic cycle}
{wiki}

The ideal Diesel cycle consists of reversible adiabatic compression, constant-pressure heat addition, reversible adiabatic expansion, and constant-volume heat rejection.

= Gibbs entropy
{title2=$S=-k_B\sum_np_n\log p_n$}
{parent=Statistical physics}
{c}
{wiki=Entropy_(statistical_thermodynamics)}

For a discrete probability distribution over microstates, the Gibbs entropy is
$$
S=-k_B\sum_n p_n\log p_n.
$$

= Maximum-entropy derivation of equilibrium ensembles
{parent=Gibbs entropy}

Maximizing the <Gibbs entropy> under normalization alone gives a uniform distribution. Adding a fixed mean-energy constraint gives the Boltzmann distribution $p_n=Z^{-1}e^{-\beta E_n}$.

= Microcanonical ensemble
{parent=Statistical physics}
{wiki}

The microcanonical ensemble assigns equal probability to accessible states in a narrow energy shell at fixed energy, volume, and particle number.

= Grand canonical ensemble
{parent=Statistical physics}
{wiki}

At fixed temperature, volume, and chemical potential, the grand canonical ensemble permits both energy and particle number to fluctuate and weights a state by $e^{-\beta(E-\mu N)}$.

= Grand canonical partition function
{title2=$\mathcal Z$}
{parent=Grand canonical ensemble}
{wiki=Grand_canonical_ensemble}

The grand canonical partition function is
$$
\mathcal Z=\sum_s e^{-\beta(E_s-\mu N_s)}.
$$

= Grand potential
{title2=$\Phi$}
{parent=Grand canonical ensemble}
{wiki=Grand_potential}

The grand potential is
$$
\Phi=E-TS-\mu\langle N\rangle=-k_BT\log\mathcal Z.
$$
For a homogeneous extensive system, $\Phi=-pV$.

= Equivalence of statistical ensembles
{parent=Statistical physics}
{wiki=Equivalence_of_ensembles}

For an additive system with short-range interactions and a regular extensive entropy, microcanonical, canonical, and grand canonical predictions agree for bulk observables in the thermodynamic limit because relative fluctuations vanish. Equivalence can fail for finite systems, long-range interactions, or at singular phase-coexistence points.

= Canonical ensemble
{parent=Statistical physics}
{wiki}

At fixed temperature, the canonical ensemble assigns a state of energy $E_s$ probability proportional to $e^{-\beta E_s}$.

= Canonical partition function
{parent=Canonical ensemble}
{wiki=Partition_function_(statistical_mechanics)}

The canonical partition function
$$Z(\beta)=\sum_s e^{-\beta E_s}$$
normalizes Boltzmann probabilities and generates equilibrium thermodynamic quantities.

= Partition function
{synonym}

= Thermodynamic derivatives of the canonical partition function
{parent=Canonical partition function}

Canonical differentiation gives
$$
F=-\beta^{-1}\log Z,\qquad
\langle E\rangle=-\partial_\beta\log Z,
\qquad
(\Delta E)^2=\partial_\beta^2\log Z.
$$

= Canonical energy fluctuation
{parent=Thermodynamic derivatives of the canonical partition function}
{wiki=Canonical_ensemble\#Energy_fluctuations}

Canonical energy variance equals $\partial_\beta^2\log Z=k_BT^2C_V$.

= Particle in a finite two-dimensional harmonic trap
{parent=Canonical partition function}

For
$$
H=\frac{p_x^2+p_y^2}{2m}+\frac\kappa2(x^2+y^2),
\qquad x^2+y^2<R^2,
$$
put $a=\kappa R^2/2$. The normalized classical one-particle partition function is
$$
Z_1=\frac{4\pi^2m}{h^2\beta^2\kappa}
\left(1-e^{-\beta a}\right).
$$
Its mean total and potential energies are
$$
\langle E\rangle=\frac2\beta-\frac a{e^{\beta a}-1},
\qquad
\langle V\rangle=\frac1\beta-\frac a{e^{\beta a}-1}.
$$

= Hard-wall correction to harmonic equipartition
{parent=Particle in a finite two-dimensional harmonic trap}

If $a\gg k_BT$, the wall is thermally inaccessible and $\langle E\rangle\sim2k_BT$, as four quadratic terms predict. If $a\ll k_BT$, the spatial distribution is nearly uniform in the disk and
$$
\langle E\rangle
=k_BT+\frac{\kappa R^2}{4}+O\left(\frac{\kappa^2R^4}{k_BT}\right),
$$
so only the two kinetic quadratic terms contribute at leading order.

= Classical ideal-gas partition function
{parent=Canonical partition function}

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

= Thermal wavelength
{parent=Classical ideal-gas partition function}
{wiki=Thermal_de_Broglie_wavelength}

The three-dimensional translational momentum integral is $1/\lambda^3$ per unit volume, where
$$\lambda=\sqrt{\frac{2\pi\hbar^2}{mk_BT}}.$$

= Canonical partition function for an ultrarelativistic gas in a power-law trap
{parent=Classical ideal-gas partition function}

For energy $pc+U(x)$,
$$
Z_1=\frac{(k_BT)^3}{\pi^2(\hbar c)^3}
\int e^{-U(x)/(k_BT)}d^3x.
$$
If $U(r)=r^{2n}/V^{2n/3}$, then
$$
Z_1=\frac4{\pi(\hbar c)^3}
V(k_BT)^{3+3/(2n)}I_n,
\qquad
I_n=\int_0^\infty u^2e^{-u^{2n}}du.
$$
Consequently $pV=Nk_BT$, while the mean energy and variance are $NAk_BT$ and $NA(k_BT)^2$ with $A=3+3/(2n)$.

= Most likely radius in an isotropic power-law trap
{parent=Canonical partition function for an ultrarelativistic gas in a power-law trap}

The radial density is proportional to $r^2e^{-r^{2n}/(V^{2n/3}k_BT)}$, whose mode is
$$
r=V^{1/3}\left(\frac{k_BT}{n}\right)^{1/(2n)}.
$$

= Sackur-Tetrode equation
{parent=Classical ideal-gas partition function}
{c}
{wiki=Sackur–Tetrode_equation}

In the classical dilute regime, a monatomic ideal gas has entropy
$$S=Nk_B\left[\log\left(\frac{V}{N\lambda^3}\right)+\frac52\right].$$

= Spin-1 paramagnet
{parent=Canonical ensemble}

For $N$ independent spins with $s_z\in\{-1,0,1\}$ and one-spin Hamiltonian $-\mu Bs_z$,
$$
Z=\left(1+2\cosh(\beta\mu B)\right)^N.
$$
Writing $x=\beta\mu B$, its heat capacity is
$$
C=Nk_Bx^2\frac{2(\cosh x+2)}
{(1+2\cosh x)^2}.
$$

= Independent symmetric three-level system
{parent=Spin-1 paramagnet}

For $N$ noninteracting units with energies $-\epsilon,0,\epsilon$, put $x=\beta\epsilon$. Then
$$
Z=(1+2\cosh x)^N,
\qquad
E=-\frac{2N\epsilon\sinh x}{1+2\cosh x},
$$
and
$$
S=Nk_B\left[
\log(1+2\cosh x)
-\frac{2x\sinh x}{1+2\cosh x}
\right].
$$

= Negative temperature
{parent=Canonical ensemble}
{wiki}

Negative absolute temperature can occur only for a system with an energy spectrum bounded above. Since $\beta=\partial S/\partial E<0$, higher-energy states are more populated than lower-energy states.

= Magnetic-field reversal and negative temperature
{parent=Negative temperature}

Instantaneously reversing a Hamiltonian $H(B)$ satisfying $H(-B)=-H(B)$ leaves populations unchanged but rewrites them as a canonical distribution with $\beta'=-\beta$. When the partition function is unchanged, the free energy changes sign and the heat capacity is unchanged.

= Heat flow from negative to positive temperature
{parent=Negative temperature}

A negative-temperature system is hotter than every positive-temperature system. If heat $\delta Q$ passes from it to a positive-temperature body, the total entropy change is
$$
\delta S_{\rm total}
=\delta Q(\beta_{\rm positive}-\beta_{\rm negative})>0.
$$

= Equipartition theorem
{parent=Canonical ensemble}
{wiki}

Each independent quadratic term in a classical Hamiltonian contributes $k_BT/2$ to the mean energy.

= Nonrelativistic equilibrium number density
{parent=Canonical ensemble}

In the dilute Maxwell--Boltzmann regime, a species has number density proportional to $g(mk_BT)^{3/2}e^{(\mu-mc^2)/(k_BT)}$.

= Boltzmann factor
{title2=$e^{-\beta E}$}
{parent=Canonical ensemble}
{c}
{wiki}

At inverse temperature $\beta=1/(k_BT)$, the Boltzmann factor $e^{-\beta E}$ is the relative <canonical ensemble> weight of a <microstate> with energy $E$.

= Ideal-gas atmosphere in uniform gravity
{parent=Statistical physics}

An isothermal noninteracting gas in potential $mgz$ has an exponential vertical density and scale height $k_BT/(mg)$.

= Barometric formula
{parent=Ideal-gas atmosphere in uniform gravity}
{wiki}

For constant gravitational acceleration and temperature,
$$p(z)=p(0)e^{-mgz/(k_BT)}.$$

= Scale height of an isothermal atmosphere
{parent=Barometric formula}
{wiki=Scale_height}

The pressure and density decay length in an isothermal atmosphere is $H=k_BT/(mg)$.

= Hydrostatic equilibrium
{parent=Ideal-gas atmosphere in uniform gravity}
{wiki}

A static fluid in a uniform downward gravitational field obeys $dp/dz=-\rho g$.

= Ideal gas free energy
{parent=Statistical physics}
{wiki=Ideal_gas}

Up to volume-independent terms, $F=-Nk_BT\log V$, so $-\partial F/\partial V=Nk_BT/V$.