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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Master Langevin equations: Origin of asymptotic diffusion

Claudio Oscar Dorso, E. Susana Hernández, José Luis Vega

Published Jan 1, 1993• 6 citations
Publisher

Abstract

We extend the master-equation treatment of dynamical evolution of a system-plus-reservoir configuration including the propagation of initial correlations as a noise source. Specializing into the quantum harmonic oscillator coupled to a fermionic heat bath, we develop a model for the diffusion matrix in the space of diagonal density operators. It can be shown that mean values of observables undergo Langevin-like motion and, in particular, that the mean value and dispersion of the oscillator quanta approach the canonical equilibrium values. A final interpretation of the characteristics and role of the noise source is given.

DOI
10.1103/physreve.47.300

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We extend the master-equation treatment of dynamical evolution of a system-plus-reservoir configuration including the propagation of initial correlations as a noise source. Specializing into the quantum harmonic oscillator coupled to a fermionic heat bath, we develop a model for the diffusion matrix in the space of diagonal density operators. It can be shown that mean values of observables undergo Langevin-like motion and, in particular, that the mean value and dispersion of the oscillator quanta approach the canonical equilibrium values. A final interpretation of the characteristics and role of the noise source is given.
For a quantum system describable in a Markoffian way, via a set of variables $\mathrm{a}={{a}_{1}, {a}_{2}, \ensuremath{\cdots}}$, we show that the Langevin noise sources ${F}_{\ensuremath{\mu}}$ in the operator equations of motion $\frac{d{a}_{\ensuremath{\mu}}}{\mathrm{dt}}={A}_{\ensuremath{\mu}}(\mathrm{a})+{F}_{\ensuremath{\mu}}$ possess second moments $〈{F}_{\ensuremath{\mu}}(t){F}_{\ensuremath{\nu}}(u)〉=2〈{D}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathrm{a}, t)〉\ensuremath{\delta}(t\ensuremath{-}u)$. The diffusion coefficients ${D}_{\ensuremath{\mu}\ensuremath{\nu}}$ can be determined from a knowledge of the mean equations of motion via the (exact) time-dependent Einstein relation $2〈{D}_{\ensuremath{\mu}\ensuremath{\nu}}〉=\ensuremath{-}〈{A}_{\ensuremath{\mu}}{a}_{\ensuremath{\nu}}〉\ensuremath{-}〈{a}_{\ensuremath{\mu}}{A}_{\ensuremath{\nu}}〉+\frac{d〈{a}_{\ensuremath{\mu}}(t){a}_{\ensuremath{\nu}}(t)〉}{\mathrm{dt}}$, where 〈〉〈 〉〈〉 represents a reservoir average. The sources ${F}_{\ensuremath{\mu}}$, ${F}_{\ensuremath{\nu}}$ do not commute with one another, and as a result the commutation rules of the ${a}_{\ensuremath{\mu}}$ are shown to be preserved in time. The mean motion and diffusion coefficients are calculated for a harmonic oscillator, and for a set of atomic levels. We prove that two dynamically coupled systems (e.g., field and atoms) have uncorrelated Langevin forces if they are coupled to independent reservoirs. Radiation-field-atom coupling adds no new noise sources. We thus obtain simply the maser model including noise sources used in Quantum Noise V. Direct calculations of the mean motion and fluctuations in a system coupled to a reservoir yield relationships in agreement with the Einstein relation. For reservoirs violating time reversal, anomalous frequency shifts are found possible that violate the Ritz combination principle since $\ensuremath{\Delta}{\ensuremath{\omega}}_{12}+\ensuremath{\Delta}{\ensuremath{\omega}}_{23}+\ensuremath{\Delta}{\ensuremath{\omega}}_{31}$ need not vanish.
UNLABELLED: The Langevin equation is universally applied to describe noise in physical systems of all kinds. To derive it, rather than postulate it, one has to introduce a noise source such as an environment or "reservoir". The solution requires that at some initial time the reservoir is in equilibrium and uncorrelated with the system. This is unphysical and, moreover, in the quantum mechanical case leads to inconsistencies. The way to overcome this anomaly is to consider the equilibrium of system and noise source combined. Then the resulting time-dependent equations for the correlation of various operators can be given. To find out whether they can be reproduced by a Langevin equation, the result is applied to the explicitly solvable system of a harmonic oscillator in a harmonic bath. CONCLUSION: Langevin cannot describe quantum noise but belongs to the realm of mesoscopic physics.
The Langevin equation is universally applied to describe noise in physical systems of all kinds. In order to derive it, rather than postulating it, one has to introduce a noise source such as an environment or “reservoir”. The solution requires that at some initial time the reservoir is in equilibrium and uncorrelated with the system. This is unphysical and moreover, in the quantum mechanical case leads to inconsistencies. The way to overcome this anomaly is by considering the equilibrium of system and noise source combined. The resulting equations for the time dependent correlations between various operators can be computed. In order to find out whether the Langevin equation is able to reproduce the result we consider the explicitly solvable system of a harmonic oscillator in a harmonic bath. Conclusion: Langevin cannot describe quantum noise but belongs to the realm of mesoscopic physics.
Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed.
Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.
We analyze the quantum evolution of a particle moving in a potential in interaction with an environment of harmonic oscillators in a thermal state, using the quantum state diffusion (QSD) picture of Gisin and Percival. The QSD picture exploits mathematical connection between the usual Markovian master equation for the evolution of the density operator and a class of stochastic nonlinear Schr\"odinger equations (Ito equations) for a pure state \ensuremath{\Vert}\ensuremath{\psi}〉, and appears to supply a good description of individual systems and processes. We find approximate stationary solutions to the Ito equation (exact, for the case of quadratic potentials). The solutions are Gaussians, localized around a point in phase space undergoing classical Brownian motion. We show, for quadratic potentials, that every initial state approaches these stationary solutions in the long time limit. We recover the density operator corresponding to these solutions, and thus show, for this particular model, that the QSD picture effectively supplies a prescription for approximately diagonalizing the density operator in a basis of phase space localized states. We show that the rate of localization is related to the decoherence time, and also to the time scale on which thermal and quantum fluctuations become comparable. We use these results to exemplify the general connection between the QSD picture and the decoherent histories approach to quantum mechanics, discussed previously by Di\'osi, Gisin, Halliwell, and Percival. \textcopyright{} 1995 The American Physical Society.