dorsal/arxiv
View SchemaWhite noise approach to the low density limit of a quantum particle in a gas
| Authors | Alexander Pechen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0607134 |
| URL | https://arxiv.org/abs/quant-ph/0607134 |
| Journal | QP-PQ: Quantum Probability and White Noise Analysis, Vol. 18, Eds. M. Sch\"urmann and U. Franz, (2005) 428--447 |
Abstract
The white noise approach to the investigation of the dynamics of a quantum particle interacting with a dilute and in general non-equilibrium gaseous environment in the low density limit is outlined. The low density limit is the kinetic Markovian regime when only pair collisions (i.e., collisions of the test particle with one particle of the gas at one time moment) contribute to the dynamics. In the white noise approach one first proves that the appropriate operators describing the gas converge in the sense of appropriate matrix elements to certain operators of quantum white noise. Then these white noise operators are used to derive quantum white noise and quantum stochastic equations describing the approximate dynamics of the total system consisting of the particle and the gas. The derivation is given ab initio, starting from the exact microscopic quantum dynamics. The limiting dynamics is described by a quantum stochastic equation driven by a quantum Poisson process. This equation then applied to the derivation of quantum Langevin equation and linear Boltzmann equation for the reduced density matrix of the test particle. The first part of the paper describes the approach which was developed by L. Accardi, I.V. Volovich and the author and uses the Fock-antiFock (or GNS) representation for the CCR algebra of the gas. The second part presents the approach to the derivation of the limiting equations directly in terms of the correlation functions, without use of the Fock-antiFock representation. This approach simplifies the derivation and allows to express the strength of the quantum number process directly in terms of the one-particle $S$-matrix.
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"abstract": "The white noise approach to the investigation of the dynamics of a quantum\nparticle interacting with a dilute and in general non-equilibrium gaseous\nenvironment in the low density limit is outlined. The low density limit is the\nkinetic Markovian regime when only pair collisions (i.e., collisions of the\ntest particle with one particle of the gas at one time moment) contribute to\nthe dynamics. In the white noise approach one first proves that the appropriate\noperators describing the gas converge in the sense of appropriate matrix\nelements to certain operators of quantum white noise. Then these white noise\noperators are used to derive quantum white noise and quantum stochastic\nequations describing the approximate dynamics of the total system consisting of\nthe particle and the gas. The derivation is given ab initio, starting from the\nexact microscopic quantum dynamics. The limiting dynamics is described by a\nquantum stochastic equation driven by a quantum Poisson process. This equation\nthen applied to the derivation of quantum Langevin equation and linear\nBoltzmann equation for the reduced density matrix of the test particle. The\nfirst part of the paper describes the approach which was developed by L.\nAccardi, I.V. Volovich and the author and uses the Fock-antiFock (or GNS)\nrepresentation for the CCR algebra of the gas. The second part presents the\napproach to the derivation of the limiting equations directly in terms of the\ncorrelation functions, without use of the Fock-antiFock representation. This\napproach simplifies the derivation and allows to express the strength of the\nquantum number process directly in terms of the one-particle $S$-matrix.",
"arxiv_id": "quant-ph/0607134",
"authors": [
"Alexander Pechen"
],
"categories": [
"quant-ph",
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],
"journal_ref": "QP-PQ: Quantum Probability and White Noise Analysis, Vol. 18, Eds.\n M. Sch\\\"urmann and U. Franz, (2005) 428--447",
"title": "White noise approach to the low density limit of a quantum particle in a gas",
"url": "https://arxiv.org/abs/quant-ph/0607134"
},
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