Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Chemical Physics

A quantum stochastic Fokker–Planck theory for adiabatic processes in condensed phases

William A. Wassam, Jack H. Freed

Published Jun 15, 1982• 9 citations
Publisher

Abstract

A maximum entropy approach to time-dependent phenomena is employed to construct a quantum stochastic theory for adiabatic processes in condensed phases. This theory is illustrated by considering the relative motion of a two-particle system interacting with a heat bath. We present a fully quantum mechanical phase space master equation, which is new. In the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion, our master equation reduces to a simpler form, and the principle of detailed balance emerges. This limit is shown to be equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: Δp̄Δq̄>>ℏ/2. The simpler master equation is utilized in the development of a quantum mechanical nonlinear Fokker–Planck equation. For cases in which correlations involving spatial and momentum fluxes are short ranged, the nonlinear Fokker–Planck equation reduces to the usual linear form, with ’’collision’’ and ’’streaming’’ terms that contain quantum corrections to all orders in ℏ. The relationship between the classical limit of the present theory and Brownian motion theory based on the Chapman–Kolmogorov equation is established.

DOI
10.1063/1.443017

Related papers

A quantum stochastic theory for nonadiabatic processes in condensed phases and on surfaces

William A. Wassam, Jack H. Freed

A quantum stochastic theory for nonadiabatic processes in condensed phases and on surfaces is given. The present theory is primarily concerned with processes involving the fragmentation of two-particle systems and/or the ’’collision’’ of two particles with internal structure, i.e., electronic, spin, and nuclear degrees of freedom. A fully quantum mechanical joint energy space–quantum phase space master equation for two-particle systems is presented. A simpler form of this equation emerges in the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion. (This limit is equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: Δp̄Δq̄>>ℏ/2.) The simpler master equation is converted into a nonlinear quantum Fokker–Planck equation for nonadiabatic systems. Introducing simplifying assumptions, we obtain a quantun analog of the stochastic Liouville equation augmented with irreversible kinetic terms, which describes nonadiabatic transitions and the interruption of coherence in energy space at different points in quantum phase space. The final results are appropriately modified to incorporate the influence of applied time-dependent electric and magnetic fields. The present work provides a formal theoretical framework for examining the role of quantum effects that are neglected in ’’semiclassical’’ kinetic and ’’diffusional’’ theories for nonadiabatic processes.

The Journal of Chemical Physics•Jun 15, 1982•11 citations

Brownian Motion of a Quantum Oscillator

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.

Physical Review A•Aug 1, 1971•208 citations

New Formulation of Stochastic Theory and Quantum Mechanics

L. de la Peña-Auerbach

The theory of stochastic motion is formulated from a new point of view. It is shown that the fundamental equations of the theory reduce to Schrödinger's equation for specific values of certain parameters. A generalized Fokker-Planck-Kolmogorov equation is obtained; with other values of the parameters, certain approximations reduce this to the Smoluchowski equation for Brownian movement. In particular, the potential function in the Schrödinger equation differs in the two cases. The usual uncertainty relations appear in a natural way in the theory, but in a broader context. A single theory thus covers both similarities and differences between quantum-mechanical and Brownian motion. Furthermore, possibilities for broadening nonrelativistic quantum mechanics are brought out and, as an example, the possible corrections due to non-Markoffian terms are briefly studied.

Journal of Mathematical Physics•Sep 1, 1969•99 citations

Fokker-Planck equation of the reduced Wigner function associated to an Ohmic quantum Langevin dynamics

Pedro J. Colmenares

This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.

arXiv:1712.02780•Physical review. E•May 18, 2018•16 citations

Wigner function approach to the quantum Brownian motion of a particle in a potential

W. T. Coffey, Yuri P. Kalmykov, С. В. Титов +1 more

Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.

Physical Chemistry Chemical Physics•Jan 1, 2007•59 citations

Brownian motion of a quantum particle

Deepak Kumar

Proposed here is an exactly soluble model for the field-free motion of a quantum particle interacting with a heat bath. The heat-bath interaction includes a fluctuating random force and a frictional force of Langevin type. The momentum and position distributions obtained in the model have a close relation to the corresponding solutions of the Fokker-Planck equation of a free classical particle.

Physical Review A•Mar 1, 1984•21 citations
A maximum entropy approach to time-dependent phenomena is employed to construct a quantum stochastic theory for adiabatic processes in condensed phases. This theory is illustrated by considering the relative motion of a two-particle system interacting with a heat bath. We present a fully quantum mechanical phase space master equation, which is new. In the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion, our master equation reduces to a simpler form, and the principle of detailed balance emerges. This limit is shown to be equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: Δp̄Δq̄>>ℏ/2. The simpler master equation is utilized in the development of a quantum mechanical nonlinear Fokker–Planck equation. For cases in which correlations involving spatial and momentum fluxes are short ranged, the nonlinear Fokker–Planck equation reduces to the usual linear form, with ’’collision’’ and ’’streaming’’ terms that contain quantum corrections to all orders in ℏ. The relationship between the classical limit of the present theory and Brownian motion theory based on the Chapman–Kolmogorov equation is established.
A quantum stochastic theory for nonadiabatic processes in condensed phases and on surfaces is given. The present theory is primarily concerned with processes involving the fragmentation of two-particle systems and/or the ’’collision’’ of two particles with internal structure, i.e., electronic, spin, and nuclear degrees of freedom. A fully quantum mechanical joint energy space–quantum phase space master equation for two-particle systems is presented. A simpler form of this equation emerges in the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion. (This limit is equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: Δp̄Δq̄>>ℏ/2.) The simpler master equation is converted into a nonlinear quantum Fokker–Planck equation for nonadiabatic systems. Introducing simplifying assumptions, we obtain a quantun analog of the stochastic Liouville equation augmented with irreversible kinetic terms, which describes nonadiabatic transitions and the interruption of coherence in energy space at different points in quantum phase space. The final results are appropriately modified to incorporate the influence of applied time-dependent electric and magnetic fields. The present work provides a formal theoretical framework for examining the role of quantum effects that are neglected in ’’semiclassical’’ kinetic and ’’diffusional’’ theories for nonadiabatic processes.
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 ccc-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.
The theory of stochastic motion is formulated from a new point of view. It is shown that the fundamental equations of the theory reduce to Schrödinger's equation for specific values of certain parameters. A generalized Fokker-Planck-Kolmogorov equation is obtained; with other values of the parameters, certain approximations reduce this to the Smoluchowski equation for Brownian movement. In particular, the potential function in the Schrödinger equation differs in the two cases. The usual uncertainty relations appear in a natural way in the theory, but in a broader context. A single theory thus covers both similarities and differences between quantum-mechanical and Brownian motion. Furthermore, possibilities for broadening nonrelativistic quantum mechanics are brought out and, as an example, the possible corrections due to non-Markoffian terms are briefly studied.
This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.
Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.