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

Stochastic quantization of a dissipative dynamical system and its hydrodynamical interpretation

K. Hajra

Published Jun 1, 1991• 5 citations
Publisher

Abstract

In the framework of stochastic quantization the quantum theory of an open system is envisaged here. A hydrodynamical model has been suggested for the motion of a quantum particle in the presence of frictional dissipative force. Finally, a comparison is made between quantum conservative and nonconservative systems.
DOI
10.1063/1.529257

Related papers

Hydrodynamic quantization of mechanical systems

R. J. Wysocki

In the present paper we propose a method of quantization of mechanical systems that is applicable, inter alia, to systems acted on by nonconservative forces, such as dissipative systems. First, the correspondence between classical and quantum mechanics is considered and we make some important observations that can be used in defining the rules of quantization. Next, the quantization rules are presented and we give a few examples of the application of these rules to mechanical systems that are difficult to quantize within any other systemetic approach. Finally, it is shown that for any quantum-mechanical system there are stationary solutions of the evolution equations, and that the existence of these solutions is directly related to the proposed quantization rules.

Physical Review A•Sep 30, 2005•1 citation

Dynamical Invariant for Dissipative Systems via Complex Quantum Hydrodynamics

Dieter Schuch, Moise Bonilla–Licea

For Hamiltonian systems with time-dependent potential, the Hamiltonian, and thus the energy, is no longer a constant of motion. However, for such systems as the parametric oscillator, i.e., an oscillator with time-dependent frequency ω(t), still, a dynamical invariant can be found that now has the dimension of action. The question, if such an invariant still exists after the addition of a dissipative friction force is analyzed for the classical as well as for the quantum mechanical case from different perspectives, particularly from that of a complex hydrodynamic formulation of quantum mechanics.

Dynamics•Jan 16, 2023•0 citations

Nonlinear Friction in Quantum Mechanics

Roumen Tsekov

Nonlinear friction forces are a problem in classical Brownian motion for a long time [1]. They are described either by Langevin or by Fokker-Planck equations [2-4]. However, the rigorous generalized Langevin equation is linear, which points out that nonlinear friction forces possess a macroscopic hydrodynamic origin [1]. For this reason, they are not present in the modern quantum theory of open systems [5]. The scope of the present paper is to investigate the nonlinear friction effect on quantum mechanics. The analysis is based on a dissipative Madelung quantum hydrodynamics. Thus, the nonlinear friction is a result of nonlinear non-equilibrium thermodynamics in contrast to the traditional mechanical nonlinear friction forces. A quantum particle in vacuum is described by the Schrodinger equation

•Jan 1, 2012•3 citations

Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics

Michel Bauer, Denis Bernard, Tony Jin

Motivated by the search for a quantum analogue of the macroscopic fluctuation theory, we study quantum spin chains dissipatively coupled to quantum noise. The dynamical processes are encoded in quantum stochastic differential equations. They induce dissipative friction on the spin chain currents. We show that, as the friction becomes stronger, the noise induced dissipative effects localize the spin chain states on a slow mode manifold, and we determine the effective stochastic quantum dynamics of these slow modes. We illustrate this approach by studying the quantum stochastic Heisenberg spin chain.

arXiv:1706.03984•SciPost Physics•Nov 4, 2017•71 citations

Dissipative and Quantum Mechanics

Roumen Tsekov

Three existing interpretations of quantum mechanics, given by Heisenberg, Bohm and Madelung, are examined to describe dissipative quantum systems as well. It is found that the Madelung quantum hydrodynamics is the only correct approach. A new stochastic reinterpretation of the quantum mechanics is proposed, which represents the microscopic face of the Madelung hydrodynamics. The main idea is that the vacuum fluctuates permanently, which explains the probabilistic character of the quantum mechanics. Thus, it is an objective theory independent of the human beings and their measurements. The effect of the thermal fluctuations in the surrounding is also accounted for via a heuristic Langevin equation with two random forces. Some statistical characteristics of these quantum and thermal noises are determined by reproducing known results for the system phase-space dynamics.

•Jan 1, 2009•4 citations

Dissipative and Quantum Mechanics

Roumen Tsekov

Three existing interpretations of quantum mechanics, given by Heisenberg, Bohm and Madelung, are examined to describe dissipative quantum systems as well. It is found that the Madelung quantum hydrodynamics is the only correct approach. A new stochastic reinterpretation of the quantum mechanics is proposed, which represents the microscopic face of the Madelung hydrodynamics. The main idea is that the vacuum fluctuates permanently, which explains the probabilistic character of the quantum mechanics. Thus, it is an objective theory independent of the human beings and their measurements. The effect of the thermal fluctuations in the surrounding is also accounted for via a heuristic Langevin equation with two random forces. Some statistical characteristics of these quantum and thermal noises are determined by reproducing known results for the system phase-space dynamics.

arXiv:0903.0283•Jan 1, 2009•5 citations
In the present paper we propose a method of quantization of mechanical systems that is applicable, inter alia, to systems acted on by nonconservative forces, such as dissipative systems. First, the correspondence between classical and quantum mechanics is considered and we make some important observations that can be used in defining the rules of quantization. Next, the quantization rules are presented and we give a few examples of the application of these rules to mechanical systems that are difficult to quantize within any other systemetic approach. Finally, it is shown that for any quantum-mechanical system there are stationary solutions of the evolution equations, and that the existence of these solutions is directly related to the proposed quantization rules.
For Hamiltonian systems with time-dependent potential, the Hamiltonian, and thus the energy, is no longer a constant of motion. However, for such systems as the parametric oscillator, i.e., an oscillator with time-dependent frequency ω(t), still, a dynamical invariant can be found that now has the dimension of action. The question, if such an invariant still exists after the addition of a dissipative friction force is analyzed for the classical as well as for the quantum mechanical case from different perspectives, particularly from that of a complex hydrodynamic formulation of quantum mechanics.
Nonlinear friction forces are a problem in classical Brownian motion for a long time [1]. They are described either by Langevin or by Fokker-Planck equations [2-4]. However, the rigorous generalized Langevin equation is linear, which points out that nonlinear friction forces possess a macroscopic hydrodynamic origin [1]. For this reason, they are not present in the modern quantum theory of open systems [5]. The scope of the present paper is to investigate the nonlinear friction effect on quantum mechanics. The analysis is based on a dissipative Madelung quantum hydrodynamics. Thus, the nonlinear friction is a result of nonlinear non-equilibrium thermodynamics in contrast to the traditional mechanical nonlinear friction forces. A quantum particle in vacuum is described by the Schrodinger equation
Motivated by the search for a quantum analogue of the macroscopic fluctuation theory, we study quantum spin chains dissipatively coupled to quantum noise. The dynamical processes are encoded in quantum stochastic differential equations. They induce dissipative friction on the spin chain currents. We show that, as the friction becomes stronger, the noise induced dissipative effects localize the spin chain states on a slow mode manifold, and we determine the effective stochastic quantum dynamics of these slow modes. We illustrate this approach by studying the quantum stochastic Heisenberg spin chain.
Three existing interpretations of quantum mechanics, given by Heisenberg, Bohm and Madelung, are examined to describe dissipative quantum systems as well. It is found that the Madelung quantum hydrodynamics is the only correct approach. A new stochastic reinterpretation of the quantum mechanics is proposed, which represents the microscopic face of the Madelung hydrodynamics. The main idea is that the vacuum fluctuates permanently, which explains the probabilistic character of the quantum mechanics. Thus, it is an objective theory independent of the human beings and their measurements. The effect of the thermal fluctuations in the surrounding is also accounted for via a heuristic Langevin equation with two random forces. Some statistical characteristics of these quantum and thermal noises are determined by reproducing known results for the system phase-space dynamics.
Three existing interpretations of quantum mechanics, given by Heisenberg, Bohm and Madelung, are examined to describe dissipative quantum systems as well. It is found that the Madelung quantum hydrodynamics is the only correct approach. A new stochastic reinterpretation of the quantum mechanics is proposed, which represents the microscopic face of the Madelung hydrodynamics. The main idea is that the vacuum fluctuates permanently, which explains the probabilistic character of the quantum mechanics. Thus, it is an objective theory independent of the human beings and their measurements. The effect of the thermal fluctuations in the surrounding is also accounted for via a heuristic Langevin equation with two random forces. Some statistical characteristics of these quantum and thermal noises are determined by reproducing known results for the system phase-space dynamics.