Many-Body Boson Systems pp 27-42 | Cite as

# Equilibrium States

Chapter

## 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 where

*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)

*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 ℝ^{ n }. 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*.## Keywords

Density Matrix Variational Principle Solvable Model Thermodynamic Limit Free Energy Density## Preview

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© Springer-Verlag London Limited 2011