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arXiv:cond-mat/0104084

A Quantum Description of Brownian Motion and the RC Circuit

Wayne H. Richardson

Published Apr 4, 2001• 0 citations
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Abstract

An equation for the reduced density matrix which describes a free particle, that is interacting with a linearly dissipative medium, is derived using the total Hamiltonian, and without resorting to any artificial model. A Master equation is also obtained for the dissipative medium. The theory erases the notion of the reservoir. It is shown that the dynamical interaction with the medium is not completely determined by the friction constant (or the corresponding fluctuation-dissipation relation). Unexpectedly, through the density-density correlation, the longitudinal dielectric function is shown to play a critical role.

DOI
10.48550/arxiv.cond-mat/0104084

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An equation for the reduced density matrix which describes a free particle, that is interacting with a linearly dissipative medium, is derived using the total Hamiltonian, and without resorting to any artificial model. A Master equation is also obtained for the dissipative medium. The theory erases the notion of the reservoir. It is shown that the dynamical interaction with the medium is not completely determined by the friction constant (or the corresponding fluctuation-dissipation relation). Unexpectedly, through the density-density correlation, the longitudinal dielectric function is shown to play a critical role.
An equation for the reduced density matrix which describes a capacitor, that is coupled to a resistor, is derived using the total Hamiltonian, and without resorting to any artificial model. A Master equation is also obtained for the dissipative medium. The theory erases the notion of the reservoir. It is shown that the dynamical interaction with the medium is not completely determined by the resistance (or the corresponding fluctuation–dissipation relation). Unexpectedly, through the charge density–density correlation, the longitudinal dielectric function is shown to play a critical role.
A fully quantum treatment of Einstein's Brownian motion is given, stressing in particular the role played by the two original requirements of translational invariance and connection between dynamics of the Brownian particle and atomic nature of the medium. The former leads to a clearcut relationship with a generator of translation-covariant quantum-dynamical semi-groups recently characterized by Holevo, the latter to a formulation of the fluctuation-dissipation theorem in terms of the dynamic structure factor, a two-point correlation function introduced in seminal work by Van Hove, directly related to density fluctuations in the medium and therefore to its atomistic, discrete nature. A microphysical expression for the generally temperature dependent friction coefficient is given in terms of the dynamic structure factor and of the interaction potential describing the single collisions. A comparison with the Caldeira-Leggett model is drawn, especially in view of the requirement of translational invariance, further characterizing general structures of reduced dynamics arising in the presence of symmetry under translations.
The quantum-mechanical motion of a free particle coupled to a linearly dissipative environment is analyzed in the Hamiltonian formalism. The total Hamiltonian is diagonalized and its eigenstates displayed. These results are used to discuss the role of initial conditions on the subsequent motion. A widely used initial condition---heat bath uncoupled from particle---is compared with another one in which the initial off-diagonal coherence of the reduced density matrix is comparable to that in the state of thermal equilibrium of the coupled system. Different transient behaviors, on time scales longer than the inverse cutoff frequency of the bath are found. The mean-square momentum of the particle in the steady state is found to depend on this highest bath frequency. An analysis of the correlated states of particle and heat reservoir shows that the latter behaves in some sense like a position-measuring apparatus.
A non-Markovian master equation is derived for the reduced probability density matrix of a subsystem interacting with a general reservoir of coupled anharmonic modes. Relaxation of the subsystem is mediated through resonant interactions with the reservoir. These interactions correspond to local vibrational motions with finite lifetimes due to diffusion/dephasing. The derivation assumes that the density of these interaction modes is very large so that they can be treated using statistical methods. The resulting master equation is shown to preserve probability, Hermiticity, and translational invariance, and conditions are stated which guarantee the positivity of the reduced density. At long time the master equation reduces to a Markovian equation of dynamical semigroup type. In the high temperature limit the theory is parameter free, and shows good agreement with the exact master equation of a spin-boson system.
Chandrasekhar's phenomenological theory of brownian motion is transcribed to the quantum regime. The statistics used is that of Boltzmann but the free particle was treated, in addition, with Fermi-Dirac statistics followed by some discussion relating to the measurements of low temperatures. Difficulties arising from the dissipative processes involved are by-passed with the aid of a new methodology enabling one to obtain a propagator in configuration space, which embraces single-particle dissipative forces. This sort of propagator is in turn used for the construction of a type of a nonequilibrium density matrix for the quantum brownian particle. Exact nonequilibrium density matrices for the free and the harmonically bound brownian particles under the influence of external time dependent forces are obtained. Finally, a quantum Fokker-Planck type equation for the general brownian particle is derived. The analysis proceeds via functional integration.
The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in the literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, focusing in particular on the correlations and the squeezing of the probability distribution induced by the environment.