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arXiv:math-ph/0301025 · Journal of Statistical Physics

Some Considerations on the Derivation of the Nonlinear Quantum Boltzmann Equation

Dario Benedetto, François Castella, R. Esposito, Mario Pulvirenti

Published Aug 1, 2004• 66 citations
arXivHTMLPDF
DOI
10.1023/b:joss.0000037205.09518.3f

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We formulate the classical and quantum Boltzmann equations for the one particle distribution function and density operator corresponding to the motion of several indistinguisable point charges in motion with mutual electromagnetic interaction attached to central potentials and also interacting with an external electromagnetic field. These equations are used to derive formulas for the scattered electromagnetic field and analyze their properties like frequency content. Using the Wigner distribution to draw an analogy between classical probability distribution functions in phase space and corresponding quantum analogues, we compare the classical and quantum Boltzmann equations showing explicitly how the quantum Boltzmann equation for the Wigner distribution is the same as the classical Boltzmann equation but with extra quantum corrections expressible as a power series in Planck’c constant. We then proceed to formulate the quantum Boltzmann equation for open quantum systems, ie, systems connected to a noisy bath and explain how to estimate the Lindblad noisy coupling parameters from the nonlinear quantum Boltzmann state evolution between successive POV measurements. We then proceed to formulate the quantum Boltzmann equation for the one particle density operator derived from the quantum Belavkin filter for N-particles taking into account non-demolition measurement noise. Partial tracing of the Belavkin filter with the molecular chaos hypothesis for the N-particle state yields this nonlinear one particle filtered quantum Boltzmann equation. We then apply Lec-Bouten’s method of control unitaries based on the measurement process to cancel out part of Lindblad noise from the filtered quantum Boltzmann equation. Finally, we explain how to perform a second order perturbative analysis of the classical Boltzmann equation in order to arrive at the frequency spectrum of the scattered field from the frequency spectrum of the incident field. The last section is devoted to the derivation of the two particle quantum Boltzmann equation for the Belavkin quantum filter. During the process of derivation, we explain how to derive the general rthr^{th}rth order quantum Boltzmann equation for the Belavkin quantum filter on NNN particles when the bath acts symmetrically, ie, in a permutation invariant way on all the particles.