Home » Classical Statistical Mechanics

Pressure / log-partition density

Thermodynamic-limit pressure as volume-normalized log partition function; derivatives generate densities and correlations.
Pressure / log-partition density

Extension: pressure as a thermodynamic-limit object

For a finite system in volume Λ\Lambda with canonical partition function ZΛ(β,h)Z_\Lambda(\beta,\mathbf{h}) (possibly including external fields/couplings h\mathbf{h}), define the log-partition density

ψΛ(β,h)=1∣Λ∣log⁡ZΛ(β,h),ψ(β,h)=lim⁡Λ↑∞ψΛ(β,h), \psi_\Lambda(\beta,\mathbf{h})=\frac{1}{|\Lambda|}\log Z_\Lambda(\beta,\mathbf{h}), \qquad \psi(\beta,\mathbf{h})=\lim_{\Lambda\uparrow\infty}\psi_\Lambda(\beta,\mathbf{h}),

when the limit exists.

In thermodynamic language, the pressure is typically

p(β,h)=1β ψ(β,h),f(β,h)=−1β ψ(β,h), p(\beta,\mathbf{h})=\frac{1}{\beta}\,\psi(\beta,\mathbf{h}), \qquad f(\beta,\mathbf{h})=-\frac{1}{\beta}\,\psi(\beta,\mathbf{h}),

so that ff is the Helmholtz free energy density (compare and ).

Why it matters: generating thermodynamics by differentiation

When differentiation under the limit is justified, derivatives of ψ\psi produce macroscopic observables:

  • Energy density

    u(β,h)=−∂βψ(β,h). u(\beta,\mathbf{h})=-\partial_\beta \psi(\beta,\mathbf{h}).

  • Field-conjugate densities (schematically)

    ∂hiψ(β,h)=⟨Oi⟩β,h, \partial_{h_i}\psi(\beta,\mathbf{h})=\langle O_i\rangle_{\beta,\mathbf{h}},

    where the right-hand side is an .

  • Fluctuations / susceptibilities are second derivatives, linking ψ\psi to and two-point .

Convexity and phase transitions

  • As a log-moment generating function, ψ(β,h)\psi(\beta,\mathbf{h}) is convex in h\mathbf{h} and typically convex in β\beta after appropriate reparameterizations (a convex-analysis perspective connected to ).
  • Non-analyticity of ψ\psi (or pp or ff) in the thermodynamic limit is a standard signature among .

Prerequisites / cross-links