Equilibrium States

Part of the Theoretical and Mathematical Physics book series (TMP)


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
$$ f:\, \omega \rightarrow f(\omega) =\lim_V \frac{1}{V}( \beta\omega(H_V - \mu N_V)- S(\omega_V))$$
where μ is the chemical potential, N V =∫ V dxa (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.


Density Matrix Variational Principle Solvable Model Thermodynamic Limit Free Energy Density 


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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsK.U.LeuvenLeuvenBelgium

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