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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)

82B10 Quantum equilibrium statistical mechanics (general)
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