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

Molecular Physics

On irreversible processes in quantum mechanics

I. Prigogine, Morikazu Toda

Published Jan 1, 1958• 9 citations
Publisher

Abstract

The methods developed by Prigogine and his colleagues are applied to the study of quantum mechanical systems. The density matrix is divided into a slowly varying and an oscillatory part, and asymptotic equations are established for the elements of this matrix. In the representation in which the unperturbed hamiltonian is diagonal, these equations take a particularly simple form for weakly coupled systems: the matrix elements lying on diagonals parallel to the principal diagonal undergo separate transformations. A particular case is that of the diagonal elements themselves; for these the well-known Pauli equations are obtained. As an illustrative example the problem of the frictional forces on an oscillator immersed in a thermostat is considered in detail. The frictional forces introduced by Langevin and Lorentz are special cases of this.

DOI
10.1080/00268975800100071

Related papers

Quantum Langevin Dynamics

Mohammad Attrash, Roi Baer

Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of $E_{gs}+k_BT$. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.

arXiv:2509.09743•Sep 11, 2025•0 citations

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

Approach to Equilibrium in Quantum Systems

Pierre M. V. Résibois

Prigogine and his co-workers developed a genera; theory of approach to equilibrium in classical mechanics. A inethod is presented which indicates the possibility for obtaining similar results for quantum systems. This method makes use of a representation for the density matrix which stresses the formal analogy between the von Neumann equation and the classical Liouville equation. This representation was used in the particular cases of weakly-coupled systems and of quantum plasmas. (B.O.G.)

Physical Review Letters•Nov 1, 1960•6 citations

Coherent states and stochastic formulation of quantum mechanics

A. Lonke

For physical systems with Hamiltonians analytic in the position and momentum operators the diagonal matrix elements of the density operator in the coherent state representation may be considered as a phase-space probability measure in the stochastic formulation of quantum mechanics. The correspondence rule relating quantum-mechanical observables to ordinary functions in phase space is obtained. It is shown that the dynamics of the time-dependent phase-space probability densities is again representable in the form of a stochastic process.

Journal of Mathematical Physics•May 1, 1978•7 citations

A general theory of matrix transformation and its application to quantum mechanical problems

Shoon K. Kim

A general formalism is given to construct a transformation matrix TAB which connects two matrices A and B of order n × n satisfying any given polynomial equations of degree r(≤n). The matrix TAB is given explicitly by a polynomial of degree (r − 1) in A and B. A special case where B is a diagonal matrix equivalent to A leads to a general theory of matrix diagonalization. The formalism contains the projection operator method as a special case. Illustrative examples are given on the Dirac theory of the electron and the basic 2 × 2 spinor representation of the proper orthochronous Lorentz group; the latter is completely prameterized by the spatial rotation and the velocity of the parallel translation.

International Journal of Quantum Chemistry•Jun 19, 2009•0 citations

Note on the Quantum Theory of Irreversible Processes

U. Fano

Perturbation treatments of irreversible processes are shown to rest on the assumption that the rate of dissipation of disturbances in a system exceeds their rate of input from an external source. This hypothesis is implicit in the concept of a dissipative system and is formulated mathematically in terms of the relaxation time of quantum-mechanical correlation functions or in terms of equivalent properties of the matrices that represent interaction operators in the energy scheme.

Physical Review•Nov 15, 1954•27 citations
The methods developed by Prigogine and his colleagues are applied to the study of quantum mechanical systems. The density matrix is divided into a slowly varying and an oscillatory part, and asymptotic equations are established for the elements of this matrix. In the representation in which the unperturbed hamiltonian is diagonal, these equations take a particularly simple form for weakly coupled systems: the matrix elements lying on diagonals parallel to the principal diagonal undergo separate transformations. A particular case is that of the diagonal elements themselves; for these the well-known Pauli equations are obtained. As an illustrative example the problem of the frictional forces on an oscillator immersed in a thermostat is considered in detail. The frictional forces introduced by Langevin and Lorentz are special cases of this.
Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of Egs+kBTE_{gs}+k_BTEgs​+kB​T. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.
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.
For physical systems with Hamiltonians analytic in the position and momentum operators the diagonal matrix elements of the density operator in the coherent state representation may be considered as a phase-space probability measure in the stochastic formulation of quantum mechanics. The correspondence rule relating quantum-mechanical observables to ordinary functions in phase space is obtained. It is shown that the dynamics of the time-dependent phase-space probability densities is again representable in the form of a stochastic process.
A general formalism is given to construct a transformation matrix TAB which connects two matrices A and B of order n × n satisfying any given polynomial equations of degree r(≤n). The matrix TAB is given explicitly by a polynomial of degree (r − 1) in A and B. A special case where B is a diagonal matrix equivalent to A leads to a general theory of matrix diagonalization. The formalism contains the projection operator method as a special case. Illustrative examples are given on the Dirac theory of the electron and the basic 2 × 2 spinor representation of the proper orthochronous Lorentz group; the latter is completely prameterized by the spatial rotation and the velocity of the parallel translation.