zbMATH — the first resource for mathematics

Return to equilibrium. (English) Zbl 0257.46091
Summary: The problem of return to equilibrium is phrased in terms of a \(C^*\)-algebra \(\mathcal A\), and two one-parameter groups of automorphisms \(\tau, \tau^p\) corresponding to the unperturbed and locally perturbed evolutions. The asymptotic evolution, under \(\tau\), of \(\tau^p\)-invariant, and \(\tau^p\)-K.M.S., states is considered. This study is a generalization of scattering theory and results concerning the existence of limit states are obtained by techniques similar to those used to prove the existence, and intertwining properties, of wave-operators. Conditions of asymptotic abelianness of \((\mathcal A, \tau)\) provide the necessary dispersive properties for the return to equilibrium. It is demonstrated that the \(\tau^p\)-equilibrium states and their limit states are coupled by automorphisms with a quasi-local property; they are not necessarily normal with respect to one another. An application to the \(X-Y\) model is given which extends previously known results and other applications, and examples, are given for the Fermi gas.

46L05 General theory of \(C^*\)-algebras
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
46N55 Applications of functional analysis in statistical physics
Full Text: DOI
[1] Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer 1966. · Zbl 0148.12601
[2] Ruelle, D.: Statistical Mechanics. New York: Benjamin 1969. · Zbl 0177.57301
[3] Haag, R., Hugenholtz, N. M., Winnink, M.: Commun. math. Phys.5, 215 (1967). · Zbl 0171.47102 · doi:10.1007/BF01646342
[4] Kastler, D., Pool, J., Poulsen, E-Thue: Commun. math. Phys.12, 175 (1969). · Zbl 0179.30301 · doi:10.1007/BF01661572
[5] Winnink, M.: Thesis, Univ. of Groningen (1968).
[6] Abraham, D. B., Barouch, E., Gallavotti, G., Martin-Löf, A.: Phys. Rev. Letters25(II), 1449 (1970). · doi:10.1103/PhysRevLett.25.1449
[7] Radin, C.: Commun. math. Phys.23, 189 (1971). · Zbl 0223.46053 · doi:10.1007/BF01877741
[8] Robinson, D. W.: Commun. math. Phys.7, 337 (1968). · Zbl 0162.29304 · doi:10.1007/BF01646665
[9] Lanford, O. E.: Cargèse Lectures. New York: Gordon and Breach 1970.
[10] Rocca, F., Sirugue, M., Testard, D.: Commun. math. Phys.13, 317 (1969). · doi:10.1007/BF01645416
[11] Hepp, K.: Preprint, Zürich (1972).
[12] Emch, G., Radin, C.: J. Math. Phys.12, 2043 (1971). · Zbl 0231.46098 · doi:10.1063/1.1665497
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.