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The Gibbs variational principle for inhomogeneous mean-field systems. (English) Zbl 0938.82500
A variational principle is established for inhomogeneous mean field models in the thermodynamic limit and is applied to the quasi-spin BCS model, the full BCS model, the Overhauser model and random mean-field models. Let $$\rho$$ be a modular state of the $$C^*$$-algebra $${\mathcal A}$$ (the KMS-state of the noninteracting single site Hamiltonian), $$\mu$$ be a probability measure on a compact (parameter) space $$X$$, $$\xi_n=(\xi_{n1},\cdots,\xi_{nn})$$, $$n\in{N}$$, be a sequence of $$n$$-tuples of points in $$X$$ with the limiting density $$\mu$$, $$n^{-1}\sum_i\delta(\xi_{ni})\to \mu$$ $$(\text{w}^*$$-topology), $${\mathcal A}^n$$ be the $$n$$-fold minimal $$C^*$$-tensor product, $$H_n(\xi_n)$$ be a sequence of symmetric continuous $${\mathcal A}^n$$-valued functions on $$X^n$$ (i.e. $$H_n\in C(X^n,{\mathcal A}^n)=C(X,{\mathcal A})^n)$$, $$nH_n$$ being considered as the interaction Hamiltonian, and assume the following approximate symmetry: for any $$\epsilon>0$$, there exists an $$m$$ such that for $$n>m$$, $$\|H_n-\text{sym}_n(H_m\otimes{1}_{n-m})\|\leq \epsilon$$, where $$\text{sym}_n$$ is the symmetrization projection. Let $$\rho^h$$ denote the perturbation of $$\rho$$ by $$h=h^*\in{\mathcal A}$$, let $$F(\rho,h)=-\log\rho^{-h}(1)$$ (the relative free energy), let $$S(\rho,\phi)$$ denote the relative entropy of a state $$\phi$$ with respect to $$\rho$$, let $$K(C(X,\mathcal A))$$ denote the set of states of $$C(X,\mathcal A)$$ and let $$K^\mu_{\text{s}}$$ denote the set of symmetric states of $$C(X,\mathcal A)^\infty$$ whose restrictions to $$C(X)^\infty$$ are $$\mu^\infty$$. The limit $$j(H)(\phi)=\lim_n\phi^n(H_n)$$ ($$\phi$$ a state of $$C(X,\mathcal A))$$ exists and defines a continuous function $$j(H)$$ over the state space $$K(C(X,\mathcal A))$$. The limit $$S_M(\psi^\infty,\phi)=\lim_n n^{-1}S(\psi^n,\phi|C(X,\mathcal A)^n)$$ exists for any $$\phi\in K^\mu_{\text{s}}$$ and defines the mean relative entropy $$S_M$$. The main result of the paper asserts the following variational principle: $\begin{split} \lim_{n\to\infty}n^{-1}F(\rho^n,nH_n(\xi_n))=\\ \inf\Big\{\lim_n\phi(H_n)+S_M((\mu\times\rho)^\infty,\phi)\colon \phi\in K^\mu_{\text{s}}\Big\}=\\ \text{inf}\{j(H)(\phi)+S(\mu\otimes\rho,\phi)\colon \phi\in K(C(X,\mathcal A)), \phi|C(X)=\mu\}.\end{split}$ The first infimum is attained at any $$\text{w}^*$$-cluster point of the sequence of states $$\psi_n\circ \Xi_n$$ of $$C(X,\mathcal A)^n\subset C(X,\mathcal A)^\infty$$, where $$\Xi_n$$ is defined by $$\Xi_nF=(\text{sym}_n F)(\xi_{n1},\cdots,\xi_{nn}) (F=F(\cdot)\in C(X^n,{\mathcal A}^n))$$ (symmetrized evaluation at $$\xi_n$$) and $$\Psi_n=\lambda^{-1}(\rho^n)^{-nH_n(\xi_n)}\in K({\mathcal A}^n)$$ $$(\lambda=(\rho^n)^{-nH_n(\xi_n)}(1))$$.
Reviewer: H.Araki (Kyoto)

##### MSC:
 82B10 Quantum equilibrium statistical mechanics (general) 46L60 Applications of selfadjoint operator algebras to physics 46N55 Applications of functional analysis in statistical physics