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

Physical Review Letters

Stochastic Description of the Quantum Thermal Mixture

Patrizia Ruggiero, Marco Zannetti

Published Apr 12, 1982• 35 citations
Publisher

Abstract

In the framework of stochastic quantization the stochastic process is derived for the harmonic oscillator in equilibrium with a thermal bath. This allows one to treat the quantum and thermal fluctuations within the same scheme of stochastic differential equations. As an application of the theory the quantum-classical crossover is computed in the dynamical critical exponent of the Van Hove model.

DOI
10.1103/physrevlett.48.963

Related papers

Microscopic derivation of the stochastic process for the quantum Brownian oscillator

Patrizia Ruggiero, Marco Zannetti

The microscopic model of Ford, Kac, and Mazur for the interaction of an oscillator with a thermal reservoir is considered within the framework of stochastic quantization. The process executed by the Brownian particle is derived exactly and it is analyzed in detail. The phenomenological equations for the dynamics of the quantum oscillator in the equilibrium state, which we have previously derived, are recovered as the weak coupling limit of the exact theory.

Physical Review A•Aug 1, 1983•29 citations

Quantum-classical crossover in critical dynamics

Patrizia Ruggiero, Marco Zannetti

The stochastic quantization of Nelson is extended to the finite-temperature equilibrium states and is applied to the study of the quantum effects in the critical dynamics of the time-dependent Ginzburg-Landau model in the large-$n$ limit. A temperature-dependent (non-Markovian) stochastic process for the order parameter is derived which contains the crossover from the ground-state dynamics dominated by the quantum fluctuations, to classical critical dynamics in the neighborhood of the finite-temperature critical points. The crossover is exhibited by means of the effective dynamical critical exponent.

Physical review. B, Condensed matter•Mar 1, 1983•25 citations

Low-temperature properties in the stochastic quantisation of the Brownian oscillator

S. De Martino, Silvio De Siena, Patrizia Ruggiero +1 more

The microscopic model of Ullersma (1966) for a harmonic oscillator in contact with a thermal reservoir is quantised in the framework of Nelson's stochastic mechanics. Eliminating the degrees of freedom of the thermal reservoir the stochastic process of the quantum Brownian oscillator is obtained as the sum of the thermal contribution and the quantum (zero-temperature) contribution. The properties of the quantum fluctuations are studied in detail and the equation of motion is derived, obtaining the first example of a non-Markovian Nelson process.

Journal of Physics A Mathematical and General•Jul 7, 1989•3 citations

Thermal-mixture Kepler system in stochastic mechanics

Tetsuya Misawa

An extension of Nelson's stochastic mechanics to the finite-temperature equilibrium states allows one to describe the quantal harmonic oscillator in equilibrium with a thermal bath in terms of a temperature-dependent diffusion process. The stochastic process is called a ``thermal-mixture diffusion process'' for the harmonic oscillator. In this paper, by the use of a regularization of the classical Kepler problem, a thermal-mixture diffusion process associated with the Coulomb potential in ${\mathit{openR}}^{\mathit{n}}$ (n=2,3,5) is derived from one associated with the harmonic oscillator in ${\mathit{openR}}^{\mathit{N}}$ (N=2,4,8), which reduces to Nelson's ground-state process for the quantal Kepler system at zero temperature.

Physical Review A•Nov 1, 1991•1 citation

Relaxation of the quantum Brownian oscillator

Rodolfo Figari, Patrizia Ruggiero, Marco Zannetti

The description of a quantum Brownian oscillator based on stochastic quantisation is applied to the non-equilibrium relaxation process. The quantum fluctuation-dissipation theorem is derived in a form which relates only the thermal component of the fluctuations to the response function. In the zero-temperature limit, it is shown that the transient is described by processes related to the solution of the nonlinear Schrodinger equation.

Journal of Physics A Mathematical and General•Jun 1, 1984•3 citations

Stochastic simulation of quantum systems and critical dynamics

T. Schneider, Marco Zannetti, R. Badii

We propose transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy levels of one-particle systems and coupled harmonic oscillators. Another application is the relationship between the dynamics of a d-dimensional system evolving according to the Langevin equation, an associated d-dimensional quantum system, and its (d+1)-dimensional classical counterpart. On this basis we relate the critical dynamics to the static properties of an associated quantum system and its (d+1)-dimensional classical analog. This then leads to equalities and scaling relations between the static and dynamic critical exponents. As a specific example, we treat the time-dependent Ginzburg-Landau model without conservation laws. In the large-n limit, the critical dynamics is traced back to Gaussian (tricritical Lifshitz) behavior of the associated (d+1)-dimensional classical model. The static exponents describing the critical dynamics are determined, however, by the crossover from Lifshitz to Gaussian (tricritical Lifshitz) behavior.

Physical review. B, Condensed matter•Mar 1, 1985•37 citations
In the framework of stochastic quantization the stochastic process is derived for the harmonic oscillator in equilibrium with a thermal bath. This allows one to treat the quantum and thermal fluctuations within the same scheme of stochastic differential equations. As an application of the theory the quantum-classical crossover is computed in the dynamical critical exponent of the Van Hove model.
The stochastic quantization of Nelson is extended to the finite-temperature equilibrium states and is applied to the study of the quantum effects in the critical dynamics of the time-dependent Ginzburg-Landau model in the large-nnn limit. A temperature-dependent (non-Markovian) stochastic process for the order parameter is derived which contains the crossover from the ground-state dynamics dominated by the quantum fluctuations, to classical critical dynamics in the neighborhood of the finite-temperature critical points. The crossover is exhibited by means of the effective dynamical critical exponent.
The microscopic model of Ullersma (1966) for a harmonic oscillator in contact with a thermal reservoir is quantised in the framework of Nelson's stochastic mechanics. Eliminating the degrees of freedom of the thermal reservoir the stochastic process of the quantum Brownian oscillator is obtained as the sum of the thermal contribution and the quantum (zero-temperature) contribution. The properties of the quantum fluctuations are studied in detail and the equation of motion is derived, obtaining the first example of a non-Markovian Nelson process.
An extension of Nelson's stochastic mechanics to the finite-temperature equilibrium states allows one to describe the quantal harmonic oscillator in equilibrium with a thermal bath in terms of a temperature-dependent diffusion process. The stochastic process is called a ``thermal-mixture diffusion process'' for the harmonic oscillator. In this paper, by the use of a regularization of the classical Kepler problem, a thermal-mixture diffusion process associated with the Coulomb potential in openRn{\mathit{openR}}^{\mathit{n}}openRn (n=2,3,5) is derived from one associated with the harmonic oscillator in openRN{\mathit{openR}}^{\mathit{N}}openRN (N=2,4,8), which reduces to Nelson's ground-state process for the quantal Kepler system at zero temperature.
We propose transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy levels of one-particle systems and coupled harmonic oscillators. Another application is the relationship between the dynamics of a d-dimensional system evolving according to the Langevin equation, an associated d-dimensional quantum system, and its (d+1)-dimensional classical counterpart. On this basis we relate the critical dynamics to the static properties of an associated quantum system and its (d+1)-dimensional classical analog. This then leads to equalities and scaling relations between the static and dynamic critical exponents. As a specific example, we treat the time-dependent Ginzburg-Landau model without conservation laws. In the large-n limit, the critical dynamics is traced back to Gaussian (tricritical Lifshitz) behavior of the associated (d+1)-dimensional classical model. The static exponents describing the critical dynamics are determined, however, by the crossover from Lifshitz to Gaussian (tricritical Lifshitz) behavior.