The Boltzmann distribution gives the probability that a system in thermal equilibrium at temperature \(T\) occupies a microstate with energy \(E\). It is the foundational result of equilibrium statistical mechanics.
Statement
For a system in contact with a heat bath at temperature \(T\), the probability of finding the system in a microstate \(i\) with energy \(E_i\) is:
\[ P_i = \frac{e^{-E_i / k_B T}}{Z} \]
where the partition function \(Z\) normalises the distribution:
\[ Z = \sum_i e^{-E_i / k_B T} \]
The sum runs over all microstates. For continuous systems, this becomes an integral over phase space.
Derivation from Maximum Entropy
Consider an ensemble of \(\mathcal{N}\) identical systems (the canonical ensemble). Let \(n_i\) be the number of systems in microstate \(i\). We have constraints:
\[ \sum_i n_i = \mathcal{N}, \qquad \sum_i n_i E_i = \mathcal{N} \langle E \rangle = U \]
where \(U\) is the total internal energy.
The number of ways to realise a distribution \(\{n_i\}\) is the multinomial coefficient:
\[ W = \frac{\mathcal{N}!}{\prod_i n_i!} \]
The Boltzmann entropy is \(S = k_B \ln W\). Using Stirling’s approximation \(\ln n! \approx n \ln n - n\):
\[ \ln W \approx \mathcal{N} \ln \mathcal{N} - \sum_i n_i \ln n_i \]
Introducing \(P_i = n_i / \mathcal{N}\):
\[ \frac{\ln W}{\mathcal{N}} \approx -\sum_i P_i \ln P_i \]
We recognise the Shannon entropy \(H = -\sum_i P_i \ln P_i\). The equilibrium distribution maximises \(H\) subject to the constraints \(\sum_i P_i = 1\) and \(\sum_i P_i E_i = \langle E \rangle\).
Using Lagrange multipliers \(\alpha\) and \(\beta\):
\[ \mathcal{L} = -\sum_i P_i \ln P_i - \alpha \left( \sum_i P_i - 1 \right) - \beta \left( \sum_i P_i E_i - \langle E \rangle \right) \]
Setting \(\partial \mathcal{L} / \partial P_i = 0\):
\[ -\ln P_i - 1 - \alpha - \beta E_i = 0 \]
Thus:
\[ P_i = e^{-(1 + \alpha)} e^{-\beta E_i} \]
The normalisation constraint determines \(e^{-(1+\alpha)} = 1/Z\), giving:
\[ P_i = \frac{e^{-\beta E_i}}{Z}, \qquad Z = \sum_i e^{-\beta E_i} \]
Identifying \(\beta\) with Temperature
To identify \(\beta = 1/k_B T\), consider two systems \(A\) and \(B\) in thermal contact. At equilibrium, the combined system maximises total entropy. If system \(A\) gains energy \(\delta E\) from \(B\):
\[ \delta S_{\text{total}} = \left( \frac{\partial S_A}{\partial E_A} - \frac{\partial S_B}{\partial E_B} \right) \delta E = 0 \]
This defines thermal equilibrium: \(\partial S / \partial E\) is equal for both systems. We define temperature via:
\[ \frac{1}{T} = \frac{\partial S}{\partial E} \]
From the canonical distribution, \(S = k_B \ln Z + k_B \beta \langle E \rangle\). Differentiating with respect to \(\langle E \rangle\) at fixed \(\beta\):
\[ \frac{\partial S}{\partial \langle E \rangle} = k_B \beta \]
Comparing with the thermodynamic definition gives \(\beta = 1/k_B T\).
Alternative Derivation: System Coupled to a Reservoir
Consider a small system \(S\) coupled to a large heat bath \(R\) with combined energy \(E_{\text{tot}}\). When \(S\) is in microstate \(i\) with energy \(E_i\), the reservoir has energy \(E_{\text{tot}} - E_i\).
The probability \(P_i\) is proportional to the number of reservoir microstates \(\Omega_R(E_{\text{tot}} - E_i)\):
\[ P_i \propto \Omega_R(E_{\text{tot}} - E_i) \]
Since \(E_i \ll E_{\text{tot}}\), expand the reservoir entropy \(S_R = k_B \ln \Omega_R\):
\[ S_R(E_{\text{tot}} - E_i) \approx S_R(E_{\text{tot}}) - E_i \frac{\partial S_R}{\partial E} = S_R(E_{\text{tot}}) - \frac{E_i}{T} \]
Therefore:
\[ \Omega_R(E_{\text{tot}} - E_i) = e^{S_R/k_B} \propto e^{-E_i / k_B T} \]
This directly yields the Boltzmann factor.
The Partition Function as Generating Function
The partition function encodes all thermodynamic information. Define \(\beta = 1/k_B T\). Then:
Mean energy: \[ \langle E \rangle = -\frac{\partial \ln Z}{\partial \beta} \]
Energy variance (heat capacity): \[ \langle E^2 \rangle - \langle E \rangle^2 = \frac{\partial^2 \ln Z}{\partial \beta^2} = k_B T^2 C_V \]
Helmholtz free energy: \[ F = -k_B T \ln Z \]
Entropy: \[ S = -\frac{\partial F}{\partial T} = k_B \ln Z + \frac{\langle E \rangle}{T} \]
Connection to the Gibbs Measure
From a more abstract viewpoint, the Boltzmann distribution is the unique probability measure on state space that maximises entropy given a constraint on mean energy. This is a special case of the exponential family in statistics, and the Gibbs measure in the theory of large deviations.
If we generalise to multiple conserved quantities \(\{Q_\alpha\}\) with conjugate “chemical potentials” \(\{\mu_\alpha\}\), we obtain the grand canonical distribution:
\[ P_i = \frac{1}{\Xi} \exp\left( -\beta E_i + \beta \sum_\alpha \mu_\alpha Q_{\alpha,i} \right) \]
The Boltzmann distribution is the special case with only energy conservation.