Equilibrium States

Abstract

It is well known that an equilibrium state at inverse temperature $\beta = \frac{1}{kT}$ can be determined by the so-called variational principle of statistical mechanics. Let k be the Boltzmann constant and T the absolute temperature of a homogeneous boson system determined by the local Hamiltonians H _V , with one Hamiltonian for each finite volume V. The principle is defined as follows: Consider the real map f, called the grand canonical free energy density functional, defined on the set of homogeneous or periodic states by the following. For any state ω of the system, f is defined by

$$ f:\, \omega \rightarrow f(\omega) =\lim_V \frac{1}{V}( \beta\omega(H_V - \mu N_V)- S(\omega_V))$$

(1)

where μ is the chemical potential, N _V =∫ _V  dx a ^∗(x)a(x) the observable standing for the number of particles, and S(ω _V ) the entropy of the restriction of the state ω to the finite volume V of ℝⁿ. We indicate by ω _V this restriction of the state ω to the algebra $\mathfrak{A}_{V}$ of observables measurable within the volume V. This means that the set $\mathfrak{A}_{V}$ is generated by all creation and annihilation operators a ^♯(f) with test functions f having their support in V.