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

Journal of Mathematical Physics

Physical applications of multiplicative stochastic processes. II. Derivation of the Bloch equations for magnetic relaxation

Ronald F. Fox

Published Feb 1, 1974• 10 citations
Publisher

Abstract

The multiplicative stochastic process treatment of the time development of the density matrix for a subsystem in contact with a heat reservoir is applied to the specific problem of the relaxation of a nuclear magnetic moment which is interacting with a fluctuating magnetic environment. A model for the fluctuating interaction Hamiltonian, appropriate for the magnetic moment case, is presented, and the Bloch equations for nuclear magnetic relaxation are constructed as a consequence. Agreement with empirical observations is noted.

DOI
10.1063/1.1666625

Related papers

Physical applications of multiplicative stochastic processes

Ronald F. Fox

The theory of multiplicative stochastic processes has been shown to lead to a density matrix description of nonequilibrium quantum mechanical phenomena. In the present paper a detailed treatment of the approach to the uniform, microcanonical, and canonical equilibrium density matrices is presented. The canonical equilibrium density matrix is approached by the density matrix which represents a subsystem in contact with a constant temperature heat reservoir.

Journal of Mathematical Physics•Jan 1, 1973•14 citations

Physical applications of multiplicative stochastic processes. III. Nonequilibrium entropy

Ronald F. Fox

It is argued that the expression − KB Trace [〈ρ (t) 〉 ln〈ρ (t) 〉], which appears in a stochastic treatment of the dynamics of the density matrix is indeed the nonequilibrium entropy. The reasoning involves consideration of the time evolution of the free energy for a relaxing magnetic moment in a fluctuating magnetic environment. It is shown that the H theorem, and the monotonic decrease of the free energy, as described by Pauli’s master equation can be generalized to the full density matrix, at least for the case of magnetic relaxation, which requires the presence of off-diagonal density matrix elements.

Journal of Mathematical Physics•Jun 1, 1975•1 citation

Simple Derivation of the Bloch Equation

D. ter Haar

It is shown how one can derive Bloch's induction equation from the equation of motion of the density matrix by coarse-graining in time. The relaxation of the magnetization is assumed to be due to a fluctuating magnetic field acting upon the spins. The derivation does not provide any new information but is sufficiently simple that it may be used as a lecture application of the density matrix.

American Journal of Physics•Dec 1, 1966•2 citations

A stochastic approach to open quantum systems

Robert Biele, Roberto D’Agosta

Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.

arXiv:1112.2694•Journal of Physics Condensed Matter•Jun 20, 2012•56 citations

Quantum relaxation dynamics of magnetic moments in a radiative reservoir

Yuh‐Jen Cheng

A quantum Langevin equation is derived to describe the relaxation dynamics of a magnetic moment in a static magnetic field and radiative reservoir. The damping and fluctuation forces are derived from the radiative interaction between magnetic moment and surrounding reservoir. Through the use of a symmetrized interaction Hamiltonian, the damping force is identified due to the combination of radiation self-reaction and reservoir fluctuations. The radiation self-reaction is a quantum version of its classical counterpart, whereas the reservoir fluctuations are solely a quantum effect resulting from the quantization of the electromagnetic field. The relative magnitude between these two effects changes during the relaxation process. This equation shows that the relative magnitude of the quantum correction to the classical Landau-Lifshitz description is inversely proportional to the system angular momentum quantum number.

Physical Review B•Dec 29, 2004•1 citation

Stochastic Boltzmann equation for magnetic relaxation in high-spin molecules

Daniel M. Packwood, Helmut G. Katzgraber, Winfried Teizer

We introduce the stochastic Boltzmann equation (SBE) as an approach for exploring the spin dynamics of magnetic molecules coupled to a stochastic environment. The SBE is a time-evolution equation for the probability density of the spin density matrix of the system. This probability density is relevant to experiments which take measurements on single molecules, in which probabilities of observing particular spin states (rather than ensemble averages) are of interest. By analogy with standard treatments of the regular Boltzmann equation, we propose a relaxation-time approximation for the SBE and show that solutions to the SBE under the relaxation-time approximation can be obtained by performing simple trajectory simulations for the case of a boson gas environment. Cases where the relaxation-time approximation are satisfied can therefore be investigated by careful choice of the parameters for the boson gas environment, even if the actual environment is quite different from a boson gas. The application of the SBE approach is demonstrated through an illustrative example.

arXiv:1603.01320•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Mar 1, 2016•1 citation
The multiplicative stochastic process treatment of the time development of the density matrix for a subsystem in contact with a heat reservoir is applied to the specific problem of the relaxation of a nuclear magnetic moment which is interacting with a fluctuating magnetic environment. A model for the fluctuating interaction Hamiltonian, appropriate for the magnetic moment case, is presented, and the Bloch equations for nuclear magnetic relaxation are constructed as a consequence. Agreement with empirical observations is noted.
It is argued that the expression − KB Trace [〈ρ (t) 〉 ln〈ρ (t) 〉], which appears in a stochastic treatment of the dynamics of the density matrix is indeed the nonequilibrium entropy. The reasoning involves consideration of the time evolution of the free energy for a relaxing magnetic moment in a fluctuating magnetic environment. It is shown that the H theorem, and the monotonic decrease of the free energy, as described by Pauli’s master equation can be generalized to the full density matrix, at least for the case of magnetic relaxation, which requires the presence of off-diagonal density matrix elements.
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.
A quantum Langevin equation is derived to describe the relaxation dynamics of a magnetic moment in a static magnetic field and radiative reservoir. The damping and fluctuation forces are derived from the radiative interaction between magnetic moment and surrounding reservoir. Through the use of a symmetrized interaction Hamiltonian, the damping force is identified due to the combination of radiation self-reaction and reservoir fluctuations. The radiation self-reaction is a quantum version of its classical counterpart, whereas the reservoir fluctuations are solely a quantum effect resulting from the quantization of the electromagnetic field. The relative magnitude between these two effects changes during the relaxation process. This equation shows that the relative magnitude of the quantum correction to the classical Landau-Lifshitz description is inversely proportional to the system angular momentum quantum number.
We introduce the stochastic Boltzmann equation (SBE) as an approach for exploring the spin dynamics of magnetic molecules coupled to a stochastic environment. The SBE is a time-evolution equation for the probability density of the spin density matrix of the system. This probability density is relevant to experiments which take measurements on single molecules, in which probabilities of observing particular spin states (rather than ensemble averages) are of interest. By analogy with standard treatments of the regular Boltzmann equation, we propose a relaxation-time approximation for the SBE and show that solutions to the SBE under the relaxation-time approximation can be obtained by performing simple trajectory simulations for the case of a boson gas environment. Cases where the relaxation-time approximation are satisfied can therefore be investigated by careful choice of the parameters for the boson gas environment, even if the actual environment is quite different from a boson gas. The application of the SBE approach is demonstrated through an illustrative example.