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

arXiv:cond-mat/9505037 · Journal of Physics A Mathematical and General

Variational schemes in the Fokker - Planck equation

Thomas Blum, Alan J. McKane

Published May 7, 1996• 10 citations
arXivHTMLPDF

Abstract

We investigate variational methods for finding approximate solutions to the Fokker - Planck equation, especially in cases lacking detailed balance. These schemes fall into two classes: those in which a Hermitian operator is constructed from the (non-Hermitian) Fokker - Planck operator, and those which are based on soluble approximations to this operator. The various approaches are tested on a simple quantum-mechanical problem, a toy Fokker - Planck equation and finally on the problem of a particle moving in a potential and subject to external non-white noise.

DOI
10.1088/0305-4470/29/9/003

Related papers

On the Quantum-Mechanical Fokker—Planck Equation

H. TED DAVIS, Kazuo Hiroike, Stuart A. Rice

Taking the Wigner function as the basic N-body distribution function, quantum-mechanical modifications to the Fokker—Planck equation are studied using the two different approaches introduced, respectively, by Kirkwood and Prigogine. In Sec. II the weak coupling limit is studied using the time-smoothing technique first introduced by Kirkwood; the results of this section can perhaps form the basis of an approximate treatment of the transport phenomena of pure liquids. In Sec. III we treat the problem of the Brownian motion of a heavy particle employing the perturbation techniques developed by Prigogine and co-workers; in this section it is found that the form of the quantum-mechanical Fokker—Planck equation is identical to the classical equation, but that the friction coefficient contains quantum-mechanical corrections. Finally, it is noted that the weak coupling result obtained by the time-smoothing technique is identical to the approximation of small momentum transfer (Sec. III), if in the latter case the scattering cross section is represented in the Born approximation.

The Journal of Chemical Physics•Oct 15, 1965•52 citations

Quantum simulation of the Fokker-Planck equation via Schrodingerization

Shi Jin, Nana Liu, Yue Yu

This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrodingerization method-it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrodinger-type equations. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrodingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrodingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrodingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique. We discuss the relationship between operator splitting in the Schrodingerization method and its application directly to the original problem, illustrating how the Schrodingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibits greater efficiency than the latter.

arXiv:2404.13585•Apr 21, 2024•1 citation

Langevin Formulation of Quantum Dynamics

M. Roncadelli

We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.

Europhysics Letters (EPL)•Oct 14, 1991•9 citations

Unified approach to the Bloch–Redfield theory and quantum Fokker–Planck equations

YiJing Yan, Feng Shuang, Rui‐Xue Xu +4 more

By using a rather simple algebraic approach, we revisit and further bridge between two most commonly used quantum dissipation theories, the Bloch–Redfield theory and a class of Fokker–Planck equations. The nature of the common approximation scheme involving in these two theories is analyzed in detail. While the Bloch–Redfield theory satisfies the detailed-balance relation, we also construct a class of Fokker–Planck equations that satisfy the detailed-balance relation up to the second moments in phase-space. Developed is also a generalized Fokker–Planck equation that preserves the general positivity of the reduced density operator. Both T1-relaxation and pure-T2 dephasing are considered, and their temperature dependence is shown to be very different. Provided is also an analogy between the quantum pure-T2 dephasing and the classical heat transport.

The Journal of Chemical Physics•Aug 8, 2000•53 citations

Variational orthogonalization

Farrokh Atai, Jens Hoppe, Mariusz Hynek +1 more

We introduce variational methods for finding approximate eigenfunctions and eigenvalues of quantum Hamiltonians by constructing a set of orthogonal wave functions which approximately solve the eigenvalue equation.

arXiv:1307.4010•Jul 15, 2013•1 citation

Stochastic variational approaches to non-Hermitian quantum-mechanical problems / vorgelegt von Daniel Kraft

Daniel Kraft

•Jan 1, 2013•0 citations
We investigate variational methods for finding approximate solutions to the Fokker - Planck equation, especially in cases lacking detailed balance. These schemes fall into two classes: those in which a Hermitian operator is constructed from the (non-Hermitian) Fokker - Planck operator, and those which are based on soluble approximations to this operator. The various approaches are tested on a simple quantum-mechanical problem, a toy Fokker - Planck equation and finally on the problem of a particle moving in a potential and subject to external non-white noise.
Taking the Wigner function as the basic N-body distribution function, quantum-mechanical modifications to the Fokker—Planck equation are studied using the two different approaches introduced, respectively, by Kirkwood and Prigogine. In Sec. II the weak coupling limit is studied using the time-smoothing technique first introduced by Kirkwood; the results of this section can perhaps form the basis of an approximate treatment of the transport phenomena of pure liquids. In Sec. III we treat the problem of the Brownian motion of a heavy particle employing the perturbation techniques developed by Prigogine and co-workers; in this section it is found that the form of the quantum-mechanical Fokker—Planck equation is identical to the classical equation, but that the friction coefficient contains quantum-mechanical corrections. Finally, it is noted that the weak coupling result obtained by the time-smoothing technique is identical to the approximation of small momentum transfer (Sec. III), if in the latter case the scattering cross section is represented in the Born approximation.
This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrodingerization method-it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrodinger-type equations. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrodingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrodingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrodingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique. We discuss the relationship between operator splitting in the Schrodingerization method and its application directly to the original problem, illustrating how the Schrodingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibits greater efficiency than the latter.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
By using a rather simple algebraic approach, we revisit and further bridge between two most commonly used quantum dissipation theories, the Bloch–Redfield theory and a class of Fokker–Planck equations. The nature of the common approximation scheme involving in these two theories is analyzed in detail. While the Bloch–Redfield theory satisfies the detailed-balance relation, we also construct a class of Fokker–Planck equations that satisfy the detailed-balance relation up to the second moments in phase-space. Developed is also a generalized Fokker–Planck equation that preserves the general positivity of the reduced density operator. Both T1-relaxation and pure-T2 dephasing are considered, and their temperature dependence is shown to be very different. Provided is also an analogy between the quantum pure-T2 dephasing and the classical heat transport.