Thermodynamic limit convention

Standard convention for taking particle number and volume to infinity while keeping density fixed.

Thermodynamic limit convention

The thermodynamic limit is the standard scaling limit used to define bulk (infinite-system) thermodynamic quantities from finite systems.

Continuum convention (particles in a volume)

For a system of $N$ particles in volume $V$ , the thermodynamic limit is taken as

N \to \infty,\qquad V \to \infty,\qquad \frac{N}{V} \to \rho,

where $\rho$ is the fixed particle density. Intensive parameters such as temperature $T$ (equivalently $\beta=1/(k_BT)$ ) and external fields are held fixed.

One typically defines bulk quantities as limits of densities, for example the free energy density (or free energy per particle) when the limit exists:

f(\beta,\rho) ={} \lim_{N,V\to\infty\atop N/V\to\rho}\frac{F_{N,V}(\beta)}{V} ={} -\frac{1}{\beta}\lim_{N,V\to\infty\atop N/V\to\rho}\frac{1}{V}\ln Z_{N,V}(\beta).

By convention, $\ln$ is the natural logarithm; see natural logarithm convention .

Lattice convention (finite regions growing to infinity)

For lattice models on $\mathbb{Z}^d$ , one takes a sequence of finite regions $\Lambda_1\subset \Lambda_2\subset \cdots$ with $|\Lambda_n|\to\infty$ . A common regularity requirement is that boundary effects vanish in proportion to volume, e.g.

\frac{|\partial \Lambda_n|}{|\Lambda_n|} \to 0,

for an appropriate notion of boundary $\partial\Lambda_n$ (a “van Hove” type condition).

Bulk free energy per site is then defined by

f(\beta) = -\frac{1}{\beta}\lim_{n\to\infty}\frac{1}{|\Lambda_n|}\ln Z_{\Lambda_n}.

Boundary conditions and the limit

Finite-volume quantities depend on the choice of boundary condition, but for many short-range models the thermodynamic limit of bulk densities (when it exists) is independent of boundary conditions along regular sequences of regions. See boundary condition convention for lattice systems .