Statistical physics

A chain of $n$ links of length $a$, each pointing independently right or left, has extensionand multiplicity $n!/(n_{+}!n_{-}!)$.

Weighting each configuration by $e^{\tau l}$ gives

The physical tension is $f=k_{B}T\tau$. At small extension,At fixed tension the extension is $L=na\tanh (fa/(k_{B}T))$ and decreases as temperature rises.

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

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

A microstate is a complete microscopic specification of a system.

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

An ideal quantum gas has mean occupationwith one sign for bosons and the other for fermions.

When $E-\mu \gg k_{B}T$, either quantum occupation law reduces to the classical factor

The mean occupation of a bosonic one-particle state of energy $E$ is

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

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.

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

For two polarizations in area $A$ and $\omega =C|k{|}^{\alpha }$, counting wavevector states in a disk gives

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.

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.

The Debye temperature is $T_D=\hslash {\omega }_D/k_B$.

For two polarizations, $n$ atoms in area $A$, and $\omega =C|k{|}^{\alpha }$, mode counting givesAt low temperature its energy scales as $T^{1+2/\alpha }$ and its constant-volume heat capacity as $T^{2/\alpha }$.

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 $g(E)$.

The mean occupation of a fermionic one-particle state of energy $E$ isAt zero temperature this becomes the step function $1_{E<E_F}$.

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

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.

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.

For spin-one-half particles in three dimensions with dispersion $\epsilon =pc$, the density of states isAt zero temperature,

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 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

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

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

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 $e^{-\beta E_F}$.

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 $T^2$.

At zero temperature, a two-dimensional free Fermi gas satisfies

When $E-\mu \gg k_{B}T$, the additive $\mp 1$ is negligible and both quantum distributions reduce to the Maxwell--Boltzmann factor $e^{-(E-\mu )/(k_{B}T)}$.

For particles of mass $m$, spin degeneracy $g_s$, and energy $m{c}^2+p^2/(2m)$,

For photons in thermal equilibrium at temperature $T$, the number density per frequency interval is

Changing variables to $x=h\nu /(k_{B}T)$ in the Planck distribution giveswhere $\alpha$ and $\xi$ are temperature-independent constants.

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

The Boltzmann constant converts temperature into an energy scale.

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

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

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

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.

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

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

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.

Absolute thermodynamic temperature can be defined so that a reversible engine exchanging heats $Q_h$ and $Q_c$ with reservoirs at $T_h$ and $T_c$ satisfiesA choice of one reference temperature fixes the scale.

Heat is energy transferred because of a temperature difference.

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

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

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

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

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

An isotropic ideal gas whose particles obey the ultrarelativistic dispersion $\epsilon =pc$ satisfies

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

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

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

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

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

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

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

An ideal gas obeys $pV=N{k}_{B}T$ and neglects intermolecular interactions except during elastic collisions.

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

A gas is nondegenerate when its occupation numbers are low enough that quantum statistics reduce to the Maxwell-Boltzmann distribution. A common criterion is $n{\lambda }_{\mathrm{th}}^3\ll 1$, where ${\lambda }_{\mathrm{th}}$ is the thermal de Broglie wavelength.

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

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

The Dieterici equation of statemodels excluded volume through $b$ and attraction through $a$. Its critical point has $v_c=2b$ and $T_c=a/(4b{k}_B)$.

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.

If $f(\lambda x)={\lambda }^{k}f(x)$, differentiation with respect to $\lambda$ at one gives

Extensivity and $dE=TdS-pdV+\mu dN$ give

The Gibbs free energy is $G=E-TS+pV$. For a simple system of fixed composition,

For a reaction with stoichiometric changes ${\nu }_i$, varying the reaction extent at fixed temperature and pressure givesEquilibrium requires ${\sum }_i{\nu }_i{\mu }_i=0$.

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.

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 $L=T\Delta S$.

The latent heat from phase $\alpha$ to phase $\beta$ isper mole or per particle, according to the normalization used.

Along a first-order coexistence curve,At a critical point the two phases merge, their entropy jump vanishes, and the latent heat tends to zero.

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.

For a simple compressible system with fixed particle number, $dE=TdS-pdV$.

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

For a fixed amount of ideal gas with constant heat capacities, the first law and $\delta Q=0$ givewhere $\gamma =C_p/C_V$.

Enthalpy is $H=E+pV$ and satisfies $dH=TdS+Vdp$ for a simple compressible system.

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

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

With translational and rotational modes active but vibration frozen, a diatomic ideal gas has $E=5N{k}_{B}T/2$ and $H=7N{k}_{B}T/2$.

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.

At fixed pressure, $Q=\Delta H$ and the expansion work is $p\Delta V=N{k}_B\Delta T$.

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.

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

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 $\gamma$,

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

For a discrete probability distribution over microstates, the Gibbs entropy is

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

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

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

The grand canonical partition function is

The grand potential isFor a homogeneous extensive system, $\Phi =-pV$.

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.

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

The canonical partition functionnormalizes Boltzmann probabilities and generates equilibrium thermodynamic quantities.

Canonical differentiation gives

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

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.

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

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

For $N$ independent spins with $s_z\in \{-1,0,1\}$ and one-spin Hamiltonian $-\mu B{s}_z$,Writing $x=\beta \mu B$, its heat capacity is

For $N$ noninteracting units with energies $-ϵ,0,ϵ$, put $x=\beta ϵ$. Thenand

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.

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

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

Each independent quadratic term in a classical Hamiltonian contributes $k_{B}T/2$ to the mean energy.

In the dilute Maxwell--Boltzmann regime, a species has number density proportional to $g(m{k}_{B}T{)}^{3/2}{e}^{(\mu -m{c}^2)/(k_{B}T)}$.

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

An isothermal noninteracting gas in potential $mgz$ has an exponential vertical density and scale height $k_{B}T/(mg)$.

For constant gravitational acceleration and temperature,

The pressure and density decay length in an isothermal atmosphere is $H=k_{B}T/(mg)$.

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

Up to volume-independent terms, $F=-N{k}_{B}T\log V$, so $-\partial F/\partial V=N{k}_{B}T/V$.