含义Meaning
一旦知道正则配分函数 Q(N, V, T)，所有热力学量都可通过求导得到——连接微观能级与宏观热力学的桥梁。
Once Q(N, V, T) is known, every thermodynamic quantity follows by differentiation — the bridge between molecular levels and bulk thermodynamics.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(Q\) | 正则配分函数canonical partition function | 1 |
| \(U_0, A_0\) | T = 0 时的值values at T = 0 | J |
推导Derivation
平均能 \(U = \sum_i E_ie^{-\beta E_i}/Q = -\partial\ln Q/\partial\beta\)，\(\beta = 1/k_\mathrm{B}T\)，换成 T 得上式。
Mean energy \(U = -\partial\ln Q/\partial\beta\).
由 \(A = U - TS\) 及 \(U = -T^2\partial(A/T)/\partial T\)（吉布斯–亥姆霍兹）积分得 \(A = -k_\mathrm{B}T\ln Q\)。
Integrating Gibbs–Helmholtz gives \(A = -k_\mathrm{B}T\ln Q\).
其余量由热力学关系 \(p = -(\partial A/\partial V)_T\)、\(S = -(\partial A/\partial T)_V\) 得到。
The rest follow from \(p = -(\partial A/\partial V)_T\), \(S = -(\partial A/\partial T)_V\).
理想气体 \(\ln Q = N\ln q - \ln N!\)，\(q_\text{trans} \propto V\) ⇒ \(p = Nk_\mathrm{B}T/V\)，统计力学推出理想气体状态方程。
For an ideal gas \(q \propto V\) gives \(p = Nk_\mathrm{B}T/V\) — the ideal-gas law from statistics.
适用条件与局限Assumptions & limitations
要求体系处于平衡态且能级已知；凝聚相的 Q 难以精确计算（需模拟）。
Requires equilibrium and known levels; condensed-phase Q needs simulation.