Equilibrium States of Long Range Interacting Quantum Lattice Systems
At this place some thermodynamic aspects of a quantum lattice in thermic equilibrium will be handled. At the center of the discussion stand different variational principles and their connection by generalized Legendre transformation. To convey this program, one restricts to a completely homogeneous lattice that will be realized by choosing the set of permutation invariant states S P (A) as an adequate thermodynamic state space . The generalization of polynomial interactions [2, 3, 1] to equivalence classes of approximately symmetric sequences  becomes apparent as a fruitful concept. This large class of possible thermodynamic systems is to be seen as a natural class of mean field interactions. To clarify the behavior of the thermodynamic functional internal energy, entropy and free energy in the thermodynamic limit, the full generality of  is not exhausted, especially only finite-dimensional algebras on a lattice point are treated and the absolute entropy is used in contrast to the relative one. The absolute properties allow a direct relation to the measurable expectation values of a macroscopic thermodynamic system. Starting with the principle of minimal free energy density for equilibrium states, the convex duality between free energy and entropy density is proved (Theorem 3.1). With a suited restriction of the variation to extremely permutation invariant states, the duality properties are modified in a way which distinguishes between an interaction in the lattice and an external field (Theorem 3.3). The rigorous treatment uses mainly results of convex analysis [5, 6]. Some ideas can be found in the investigations, presented in [7, 8] for short range interactions on translation invariant systems and the case of quadratic mean field interactions on a permutation invariant lattice in . In contrast to , the homogeneity simplifies the calculations. The introduction of the so-called sub gradient allows a geometrical interpretation of states, minimizing the free energy functional. The formalism is suited to treat non-trivial phase transitions of first and second order, [2, 3]. A discussion of models presented, e.g. in  seems to be possible and will be worked out in future.
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- Th. Gerisch and A. Rieckers, The Quantum Statistical Free Energy Minimum Principle for Multi-Lattice Mean Field Theories, Preprint Tübingen 1990.Google Scholar
- A. Rieckers and H. - J. Volkert, Variational Principles and Equilibrium States for a Class of Long-Range Interacting Quantum Lattice Systems, Preprint Tübingen 1988Google Scholar
- H. - J. Volkert and A. Rieckers, Equilibrium States and Phase Transitions of some FCC- Multi-Lattice Systems, Preprint Tübingen 1988Google Scholar
- G. A. Raggio and R. F. Werner, Quantum statistical mechanics of general mean field systems, Helvetica Physica Acta, Vol. 62 (1989) 980–1003Google Scholar
- R. T. Rockafellar, Conjugate Duality and Optimization, Society for Industrial and Applied Mathematics, Philadelphia (1974)Google Scholar
- R. T. Rockafellar, Convex Analysis, Princeton University Press (1970)Google Scholar
- R. B. Israel, Convexity in the Theory of Lattice Gases, Princeton University Press (1979)Google Scholar
- G. L. Sewell, Quantum Theory of Collective Phenomena, Clarendon Press, Oxford, (1986)Google Scholar
- J.-C. Toledano and P. Toledano, The Landau Theory of Phase Transitions, World Scientific Publishing, Singapore - New Jersey - Hong Kong, (1987)Google Scholar
- S. Sakai, C*-Algebras and W*-Algebras, Springer - Verlag, Berlin, (1971)Google Scholar
- O. Bratteli and D. W. Robinson Operator Algebras and Quantum Statistical Mechanics II, Springer Verlag, New York - Heidelberg - Berlin (1981)Google Scholar