In this paper we analyze a system of N identical quantum particles in a weak-coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermi's Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension d≥2.

Some considerations on the derivation of the nonlinear quantum Boltzmann equation / Benedetto, Dario; F., Castella; R., Esposito; Pulvirenti, Mario. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - STAMPA. - 116:1-4(2004), pp. 381-410. [10.1023/b:joss.0000037205.09518.3f]

Some considerations on the derivation of the nonlinear quantum Boltzmann equation

BENEDETTO, Dario;PULVIRENTI, Mario
2004

Abstract

In this paper we analyze a system of N identical quantum particles in a weak-coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermi's Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension d≥2.
2004
kinetic equations; quantum systems; weak coupling
01 Pubblicazione su rivista::01a Articolo in rivista
Some considerations on the derivation of the nonlinear quantum Boltzmann equation / Benedetto, Dario; F., Castella; R., Esposito; Pulvirenti, Mario. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - STAMPA. - 116:1-4(2004), pp. 381-410. [10.1023/b:joss.0000037205.09518.3f]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/236775
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