Quantum Brownian motion and generalizations of Hermite polynomials

Abstract

We introduce new families of orthogonal polynomials H D , n , motivated by the non-equilibrium evolution of a quantum Brownian particle (qBp). The H D , n ’s generalize non-trivially the standard Hermite polynomials, employed for classical Brownian motion. We treat several models (labelled by D ) for a non-equilibrium qBp, by means of the Wigner function W , in the presence of a “heat bath” at thermal equilibrium, with and without ab initio friction. For long times (for a suitable class of initial conditions), the non-equilibrium Wigner function W should approach, in some sense, the (time-independent) equilibrium Wigner function W e q , D , which describes the thermal equilibrium of the qBp with the “heat bath” and plays a central role. W e q , D is chosen to be the weight function which orthogonalizes the H D , n ’s. New results on W e q , D and on the H D , n ’s are reported. We justify the key role of the H D , n ’s as follows. Using the H D , n ’s, moments W e q , D , n and W n are introduced for W e q , D and W , respectively. At equilibrium, all moments W e q , D , n except the lowest one ( W e q , D , 0 ) vanish identically. Off-equilibrium, one expects that, for long times (for suitable initial conditions): (i) all non-equilibrium moments W n (except the lowest moment W 0 ), will approach zero, while (ii) the lowest non-equilibrium moment W 0 will tend to W e q , D , 0 ( ≠ 0 ) . To complete the justification, we outline how the approximate long-time non-equilibrium theories determined by W 0 for the different models ( D ) yield Smoluchowski equations and irreversible evolutions of the qBp towards thermal equilibrium.