Abstract
We present a class of mean field model Hamiltonians consisting of a small, but macroscopic system S of N components interacting with a large reservoir R. The Dicke model of a laser is a particular example of S. Both R and S are fully quantum mechanical. The exact equations of motion are studied, and it is shown that it is possible to eliminate the reservoir variables and, in the limit N → ∞; to derive closed equations for the extensive variables of S. These equations are classical, and the effect of the reservoir is to provide damping and driving (or pumping) terms. As the parameters of the system are varied, S can undergo phase transitions in the sense that its equilibrium orbits bifurcate. To the next order, N 1/2, the reservoir drives the fluctuation observables of S linearly with Markovian, Gaussian random forces. Within the context of our class of models our results are rigorous and are obtained without any approximation.
On leave from the Department of Mathematics, M.I.T., Cambridge, Mass. 02139, U.S.A. Work partially supported by U.S. National Science Foundation Grant GP-31674 X and by a Guggenheim Memorial Foundation Fellowship.
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Hepp, K., Lieb, E.H. (1973). Phase Transitions in Reservoir-Driven Open Systems with Applications to Lasers and Superconductors. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds) Condensed Matter Physics and Exactly Soluble Models. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06390-3_13
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DOI: https://doi.org/10.1007/978-3-662-06390-3_13
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