Quantum Brownian Motion: Recent Developments (With 3 Figures)

A. O. Caldeira

Fabián H. Gaioli, Edgardo T. Garcia Alvarez, Diego G. Arbo

G. S. Agarwal

The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto $c$-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.

Aniello Lampo, Miguel Ángel García March, Maciej Lewenstein

Aurélien Drezet

The pilot wave interpretation proposed by de Broglie and later by Bohm contains not only a dynamical ontology but also relies on a statistical assumption known as quantum equilibrium. In this work which follows our recent article [1] we develop a Langevin force description of the relaxation process which leads to quantum equilibrium. Based on a application of the Caldera-Leggett model for a thermal bath we show how a Brownian motion leads naturally to quantum relaxation.

Fabrice Debbasch, C. Chevalier

We review all recent contributions to the literature on stochastic processes. In particular, the Relativistic Ornstein‐Uhlenbeck Process is presented in detail, as is the intrinsic Brownian motion studied by Franchi and Le Jan. The Relativistic Brownian Motion of Dunkel and Hänggi is also reviewed, together with a model introduced by Oron and Horwitz. We finally suggest some possible future developements of the current research.