Classical discrete system

For a canonical ensemble that is classical and discrete, the canonical partition function is defined as

$${\displaystyle Z=\sum _{i}e^{-\beta E_{i}},}$$

where

• ${\displaystyle i}$ is the index for the microstates of the system;
• ${\displaystyle e}$ is Euler's number;
• ${\displaystyle \beta }$ is the thermodynamic beta, defined as ${\displaystyle {\tfrac {1}{k_{\text{B}}T}}}$ where ${\displaystyle k_{\text{B}}}$ is Boltzmann's constant;
• ${\displaystyle E_{i}}$ is the total energy of the system in the respective microstate.

The exponential factor ${\displaystyle e^{-\beta E_{i}}}$ is otherwise known as the Boltzmann factor.

Derivation of canonical partition function (classical, discrete)

There are multiple approaches to deriving the partition function. The following derivation follows the more powerful and general information-theoretic Jaynesian maximum entropy approach.

According to the second law of thermodynamics, a system assumes a configuration of maximum entropy at thermodynamic equilibrium. We seek a probability distribution of states ${\displaystyle \rho _{i}}$ that maximizes the discrete Gibbs entropy

$${\displaystyle S=-k_{\text{B}}\sum _{i}\rho _{i}\ln \rho _{i}}$$

subject to two physical constraints:

1. The probabilities of all states add to unity (second axiom of probability):
   $${\displaystyle \sum _{i}\rho _{i}=1.}$$

   
2. In the canonical ensemble, the average energy is fixed (conservation of energy):
   $${\displaystyle \langle E\rangle =\sum _{i}\rho _{i}E_{i}\equiv U.}$$

   

Applying variational calculus with constraints (analogous in some sense to the method of Lagrange multipliers), we write the Lagrangian (or Lagrange function) ${\displaystyle {\mathcal {L}}}$ as

$${\displaystyle {\mathcal {L}}=\left(-k_{\text{B}}\sum _{i}\rho _{i}\ln \rho _{i}\right)+\lambda _{1}\left(1-\sum _{i}\rho _{i}\right)+\lambda _{2}\left(U-\sum _{i}\rho _{i}E_{i}\right).}$$

Varying and extremizing ${\displaystyle {\mathcal {L}}}$ with respect to ${\displaystyle \rho _{i}}$ leads to

$${\displaystyle {\begin{aligned}0&\equiv \delta {\mathcal {L}}\\&=\delta \left(-\sum _{i}k_{\text{B}}\rho _{i}\ln \rho _{i}\right)+\delta \left(\lambda _{1}-\sum _{i}\lambda _{1}\rho _{i}\right)+\delta \left(\lambda _{2}U-\sum _{i}\lambda _{2}\rho _{i}E_{i}\right)\\&=\sum _{i}{\bigg [}\delta {\Big (}-k_{\text{B}}\rho _{i}\ln \rho _{i}{\Big )}+\delta {\Big (}\lambda _{1}\rho _{i}{\Big )}+\delta {\Big (}\lambda _{2}E_{i}\rho _{i}{\Big )}{\bigg ]}\\&=\sum _{i}\left[{\frac {\partial }{\partial \rho _{i}}}{\Big (}-k_{\text{B}}\rho _{i}\ln \rho _{i}{\Big )}\,\delta (\rho _{i})+{\frac {\partial }{\partial \rho _{i}}}{\Big (}\lambda _{1}\rho _{i}{\Big )}\,\delta (\rho _{i})+{\frac {\partial }{\partial \rho _{i}}}{\Big (}\lambda _{2}E_{i}\rho _{i}{\Big )}\,\delta (\rho _{i})\right]\\&=\sum _{i}{\bigg [}-k_{\text{B}}\ln \rho _{i}-k_{\text{B}}+\lambda _{1}+\lambda _{2}E_{i}{\bigg ]}\,\delta (\rho _{i}).\end{aligned}}}$$

Since this equation should hold for any variation ${\displaystyle \delta (\rho _{i})}$, it implies that

$${\displaystyle 0\equiv -k_{\text{B}}\ln \rho _{i}-k_{\text{B}}+\lambda _{1}+\lambda _{2}E_{i}.}$$

Isolating for ${\displaystyle \rho _{i}}$ yields

$${\displaystyle \rho _{i}=\exp \left({\frac {-k_{\text{B}}+\lambda _{1}+\lambda _{2}E_{i}}{k_{\text{B}}}}\right).}$$

To obtain ${\displaystyle \lambda _{1}}$, one substitutes the probability into the first constraint:

$${\displaystyle {\begin{aligned}1&=\sum _{i}\rho _{i}\\&=\exp \left({\frac {-k_{\text{B}}+\lambda _{1}}{k_{\text{B}}}}\right)Z,\end{aligned}}}$$

where ${\displaystyle Z}$ is a constant number defined as the canonical ensemble partition function:

$${\displaystyle Z\equiv \sum _{i}\exp \left({\frac {\lambda _{2}}{k_{\text{B}}}}E_{i}\right).}$$

Isolating for ${\displaystyle \lambda _{1}}$ yields ${\displaystyle \lambda _{1}=-k_{\text{B}}\ln(Z)+k_{\text{B}}}$.

Rewriting ${\displaystyle \rho _{i}}$ in terms of ${\displaystyle Z}$ gives

$${\displaystyle \rho _{i}={\frac {1}{Z}}\exp \left({\frac {\lambda _{2}}{k_{\text{B}}}}E_{i}\right).}$$

Rewriting ${\displaystyle S}$ in terms of ${\displaystyle Z}$ gives

$${\displaystyle {\begin{aligned}S&=-k_{\text{B}}\sum _{i}\rho _{i}\ln \rho _{i}\\&=-k_{\text{B}}\sum _{i}\rho _{i}\left({\frac {\lambda _{2}}{k_{\text{B}}}}E_{i}-\ln(Z)\right)\\&=-\lambda _{2}\sum _{i}\rho _{i}E_{i}+k_{\text{B}}\ln(Z)\sum _{i}\rho _{i}\\&=-\lambda _{2}U+k_{\text{B}}\ln(Z).\end{aligned}}}$$

To obtain ${\displaystyle \lambda _{2}}$, we differentiate ${\displaystyle S}$ with respect to the average energy ${\displaystyle U}$ and apply the first law of thermodynamics, ${\displaystyle dU=TdS-PdV}$:

$${\displaystyle {\frac {dS}{dU}}=-\lambda _{2}\equiv {\frac {1}{T}}.}$$

Thus the canonical partition function ${\displaystyle Z}$ becomes

$${\displaystyle Z\equiv \sum _{i}e^{-\beta E_{i}},}$$

where ${\displaystyle \beta \equiv 1/(k_{\text{B}}T)}$ is defined as the thermodynamic beta. Finally, the probability distribution ${\displaystyle \rho _{i}}$ and entropy ${\displaystyle S}$ are respectively

$${\displaystyle {\begin{aligned}\rho _{i}&={\frac {1}{Z}}e^{-\beta E_{i}},\\S&={\frac {U}{T}}+k_{\text{B}}\ln Z.\end{aligned}}}$$

Classical continuous system

In classical mechanics, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable. In classical statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms. In this case we must describe the partition function using an integral rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as

$${\displaystyle Z={\frac {1}{h^{3}}}\int e^{-\beta H(q,p)}\,\mathrm {d} ^{3}q\,\mathrm {d} ^{3}p,}$$

where

• ${\displaystyle h}$ is the Planck constant;
• ${\displaystyle \beta }$ is the thermodynamic beta, defined as ${\displaystyle {\tfrac {1}{k_{\text{B}}T}}}$;
• ${\displaystyle H(q,p)}$ is the Hamiltonian of the system;
• ${\displaystyle q}$ is the canonical position;
• ${\displaystyle p}$ is the canonical momentum.

To make it into a dimensionless quantity, we must divide it by h, which is some quantity with units of action (usually taken to be Planck's constant).

Classical continuous system (multiple identical particles)

For a gas of ${\displaystyle N}$ identical classical particles in three dimensions, the partition function is

$${\displaystyle Z={\frac {1}{N!h^{3N}}}\int \,\exp \left(-\beta \sum _{i=1}^{N}H({\textbf {q}}_{i},{\textbf {p}}_{i})\right)\;\mathrm {d} ^{3}q_{1}\cdots \mathrm {d} ^{3}q_{N}\,\mathrm {d} ^{3}p_{1}\cdots \mathrm {d} ^{3}p_{N}}$$

where

• ${\displaystyle h}$ is the Planck constant;
• ${\displaystyle \beta }$ is the thermodynamic beta, defined as ${\displaystyle {\tfrac {1}{k_{\text{B}}T}}}$;
• ${\displaystyle i}$ is the index for the particles of the system;
• ${\displaystyle H}$ is the Hamiltonian of a respective particle;
• ${\displaystyle q_{i}}$ is the canonical position of the respective particle;
• ${\displaystyle p_{i}}$ is the canonical momentum of the respective particle;
• ${\displaystyle \mathrm {d} ^{3}}$ is shorthand notation to indicate that ${\displaystyle q_{i}}$ and ${\displaystyle p_{i}}$ are vectors in three-dimensional space.

The reason for the factorial factor N! is discussed below. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not dimensionless. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by h^3N (where h is usually taken to be Planck's constant).

Quantum mechanical discrete system

For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the trace of the Boltzmann factor:

$${\displaystyle Z=\operatorname {tr} (e^{-\beta {\hat {H}}}),}$$

where:

• ${\displaystyle \operatorname {tr} (\circ )}$ is the trace of a matrix;
• ${\displaystyle \beta }$ is the thermodynamic beta, defined as ${\displaystyle {\tfrac {1}{k_{\text{B}}T}}}$;
• ${\displaystyle {\hat {H}}}$ is the Hamiltonian operator.

The dimension of ${\displaystyle e^{-\beta {\hat {H}}}}$ is the number of energy eigenstates of the system.

Quantum mechanical continuous system

For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as

$${\displaystyle Z={\frac {1}{h}}\int \langle q,p|e^{-\beta {\hat {H}}}|q,p\rangle \,\mathrm {d} q\,\mathrm {d} p,}$$

where:

• ${\displaystyle h}$ is the Planck constant;
• ${\displaystyle \beta }$ is the thermodynamic beta, defined as ${\displaystyle {\tfrac {1}{k_{\text{B}}T}}}$;
• ${\displaystyle {\hat {H}}}$ is the Hamiltonian operator;
• ${\displaystyle q}$ is the canonical position;
• ${\displaystyle p}$ is the canonical momentum.

In systems with multiple quantum states s sharing the same energy E_s, it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j) as follows:

$${\displaystyle Z=\sum _{j}g_{j}\cdot e^{-\beta E_{j}},}$$

where g_j is the degeneracy factor, or number of quantum states s that have the same energy level defined by E_j = E_s.

The above treatment applies to quantum statistical mechanics, where a physical system inside a finite-sized box will typically have a discrete set of energy eigenstates, which we can use as the states s above. In quantum mechanics, the partition function can be more formally written as a trace over the state space (which is independent of the choice of basis):

$${\displaystyle Z=\operatorname {tr} (e^{-\beta {\hat {H}}}),}$$

where Ĥ is the quantum Hamiltonian operator. The exponential of an operator can be defined using the exponential power series.

The classical form of Z is recovered when the trace is expressed in terms of coherent states^[1] and when quantum-mechanical uncertainties in the position and momentum of a particle are regarded as negligible. Formally, using bra–ket notation, one inserts under the trace for each degree of freedom the identity:

$${\displaystyle {\boldsymbol {1}}=\int |x,p\rangle \langle x,p|{\frac {dx\,dp}{h}},}$$

where |x, p⟩ is a normalised Gaussian wavepacket centered at position x and momentum p. Thus

$${\displaystyle Z=\int \operatorname {tr} \left(e^{-\beta {\hat {H}}}|x,p\rangle \langle x,p|\right){\frac {dx\,dp}{h}}=\int \langle x,p|e^{-\beta {\hat {H}}}|x,p\rangle {\frac {dx\,dp}{h}}.}$$

A coherent state is an approximate eigenstate of both operators ${\displaystyle {\hat {x}}}$ and ${\displaystyle {\hat {p}}}$, hence also of the Hamiltonian Ĥ, with errors of the size of the uncertainties. If Δx and Δp can be regarded as zero, the action of Ĥ reduces to multiplication by the classical Hamiltonian, and Z reduces to the classical configuration integral.