Schr�dinger-type equation with damping for a dynamical system in a thermal bath

1970 ◽  
Vol 5 (2) ◽  
pp. 1150-1158 ◽  
Author(s):  
K. Valyasek ◽  
D. N. Zubarev ◽  
A. L. Kuzemskii
Keyword(s):  
Dynamical System ◽  
Type Equation ◽  
Thermal Bath

scholarly journals Existence of Random Attractors for ap-Laplacian-Type Equation with Additive Noise

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Wenqiang Zhao ◽  
Yangrong Li
Keyword(s):  
Dynamical System ◽  
Additive Noise ◽  
Single Point ◽  
Type Equation ◽  
Random Attractor ◽  
Approximation Procedure ◽  
Restrictive Assumption ◽  
Stochastic Dynamical System ◽  
Additive White Noise ◽  
Uniqueness Of A Solution

We first establish the existence and uniqueness of a solution for a stochasticp-Laplacian-type equation with additive white noise and show that the unique solution generates a stochastic dynamical system. By using the Dirichlet forms of Laplacian and an approximation procedure, the nonlinear obstacle, arising from the additive noise is overcome when we make energy estimate. Then, we obtain a random attractor for this stochastic dynamical system. Finally, under a restrictive assumption on the monotonicity coefficient, we find that the random attractor consists of a single point, and therefore the system possesses a unique stationary solution.

ON THE STOCHASTIC EVOLUTION OF THE INFLATON FIELD

2004 ◽  
Vol 13 (07) ◽  
pp. 1315-1320 ◽  
Author(s):  
JOÃO MARIA SILVA ◽  
JOSÉ A. S. LIMA
Keyword(s):  
Thermal Contact ◽  
Initial Density ◽  
Type Equation ◽  
Thermal Bath ◽  
Large Set ◽  
Stochastic Evolution ◽  
Thermally Induced ◽  
Inflaton Field ◽  
Warm Inflation ◽  
Density Perturbations

During the inflationary regime, the expansion of the Universe is driven by a scalar field ϕ(t) which may be in thermal contact with the radiation fluid. In this work, we study the influence of the thermal bath assuming that it is responsible for the stochastic evolution of the inflaton field. Assuming that the fluctuation dynamics is described by a Langevin-type equation of motion, a large set of analytical solutions including white and colored noises are derived. It is found that even in the case of white noise the field experience an anomalous diffusion. Such results may be important for studying thermally induced initial density perturbations in inflationary cosmologies, mainly in the framework of warm inflation.

scholarly journals GENERALIZED KINETIC AND EVOLUTION EQUATIONS IN THE APPROACH OF THE NONEQUILIBRIUM STATISTICAL OPERATOR

2005 ◽  
Vol 19 (06) ◽  
pp. 1029-1059 ◽  
Author(s):  
A. L. KUZEMSKY
Keyword(s):  
Kinetic Equation ◽  
Evolution Equations ◽  
Kinetic Equations ◽  
Type Equation ◽  
Nonequilibrium State ◽  
Nonequilibrium Statistical Operator ◽  
Thermal Bath ◽  
Relaxation Processes ◽  
Collision Term ◽  
Statistical Operator

The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.

scholarly journals Hidden Dynamical Symmetry and Quantum Thermodynamics from the First Principles: Quantized Small Environment

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1546
Author(s):  
Ashot S. Gevorkyan ◽  
Alexander V. Bogdanov ◽  
Vladimir V. Mareev
Keyword(s):  
First Principles ◽  
Type Equation ◽  
Dynamical Symmetry ◽  
Thermal Bath ◽  
Quantum Thermodynamics ◽  
Joint System ◽  
Hidden Symmetry ◽  
Noise Type ◽  
Self Consistent ◽  
Time Continuum

Evolution of a self-consistent joint system (JS), i.e., a quantum system (QS) + thermal bath (TB), is considered within the framework of the Langevin–Schrödinger (L-Sch) type equation. As a tested QS, we considered two linearly coupled quantum oscillators that interact with TB. The influence of TB on QS is described by the white noise type autocorrelation function. Using the reference differential equation, the original L-Sch equation is reduced to an autonomous form on a random space–time continuum, which reflects the fact of the existence of a hidden symmetry of JS. It is proven that, as a result of JS relaxation, a two-dimensional quantized small environment is formed, which is an integral part of QS. The possibility of constructing quantum thermodynamics from the first principles of non-Hermitian quantum mechanics without using any additional axioms has been proven. A numerical algorithm has been developed for modeling various properties and parameters of the QS and its environment.

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