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On the derivation of a quantum Boltzmann equation from the periodic von-Neumann equation. (English) Zbl 0954.82023

Author’s summary: We present the semi-conductor Boltzmann equation, which is time-reversible, and indicate that it can be formally derived by considering the large time and small perturbing potential limit in the van Neumann equation (time-reversible). We then rigorously compute the corresponding asymptotics in the case of the von Neumann equation on the torus. We show that the limiting equation we obtain does not coincide with the physically realistic model. The former is indeed an equation of Boltzmann type, yet with memory in time, so that it appears to be time-reversible. We comment on this point, and further describe the properties of the limiting equation.

MSC:

82C70 Transport processes in time-dependent statistical mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
35Q40 PDEs in connection with quantum mechanics
70F99 Dynamics of a system of particles, including celestial mechanics
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