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Il Nuovo Cimento B

Friction and noise in quantum mechanics: A model for the interactions between a system and a thermal bath( ∗ )

Bruno Bianco, Elsa Moggia, Stefano Giordano, Walter Rocchia, A. Chiabrera

Published Feb 1, 2001• 5 citations
OpenAlex

Abstract

densitymatrixofaquan-tumsystemcomposedofoneormanyparticlessubjectedtoaconservativeforcefieldand interacting with a thermal bath has been derived. This result is achieved byanalogy with classical models based upon the Langevin and Fokker-Planck equa-tions. According to the Langevin approach, the interaction is modelled by meansof a random force field and a viscous friction term. We generalise this classicalapproach introducing suitable operators that describe the quantum evolution of asystemweaklycoupledtoathermalbath. Inparticular,wedefineakindoffrictionoperator thatcan be thoughtas the quantalcounterpartof the classical Langevintermcorrespondingtotheviscousforce. Theproposedapproachhastheinvaluableadvantageofyieldingahandydifferentialequationtomodelthequantuminteractionbetweenasystemandathermalbath.PACS42.50.Lc–Quantumfluctuations,quantumnoise,andquantumjumps.PACS05.30–Quantumstatisticalmechanics.PACS05.70.Ln–Nonequilibriumandirreversiblethermodynamics.PACS05.10.Gg–Stochasticanalysismethods(Fokker-Planck,Langevin,etc.).

Related papers

Brownian motion of a quantum particle

Deepak Kumar

Proposed here is an exactly soluble model for the field-free motion of a quantum particle interacting with a heat bath. The heat-bath interaction includes a fluctuating random force and a frictional force of Langevin type. The momentum and position distributions obtained in the model have a close relation to the corresponding solutions of the Fokker-Planck equation of a free classical particle.

Physical Review A•Mar 1, 1984•21 citations

Classical stochastic systems

Alex Kamenev

This chapter is devoted to the classical limit of the quantum dissipative action obtained in Chapter 3. We show how it yields Langevin, Fokker–Planck and optimal path descriptions of classical stochastic systems. These approaches are used to discuss activation escape, fluctuation relation, reaction models and other examples. Classical dissipative action In Section 3.2 we derived the Keldysh action for a quantum particle coupled to an Ohmic environment, Eq. (3.20). If only linear terms in the quantum coordinate X q (t) are kept in this action, it leads to a classical Newtonian equation with a viscous friction force, Eq. (3.21). Such an approximation completely disregards any fluctuations: both quantum and classical . Our goal now is to do better than that and to keep classical thermal fluctuations, while still neglecting quantum effects. To this end it is convenient to restore the Planck constant ħ in the action and then take the limit ħ → 0. For dimensional reasons, the factor ħ -1 should stay in front of the entire action. To keep the part of the action responsible for the classical equation of motion (3.21) free from the Planck constant it is convenient to rescale the quantum component as X q → ħ X q . Indeed, when this is done all terms linear in X q do not contain ħ. Finally, to have the temperature in energy units, one needs to substitute T with T /ħ.

Cambridge University Press eBooks•Sep 8, 2011•0 citations

Classical Langevin dynamics derived from quantum mechanics

Håkon Hoel, Anders Szepessy

The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.

arXiv:1906.09858•Discrete and Continuous Dynamical Systems - B•Jan 1, 2020•5 citations

Quantum and classical Fokker-Planck equations for a Gaussian-Markovian noise bath

Yoshitaka Tanimura, Peter Guy Wolynes

The quantum Fokker-Planck equation for a Gaussian-Markovian bath is deduced by applying a method proposed by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1989)]. The results are expressed in the form of simultaneous differential equations in terms of density operators and can treat strong system-bath interactions where the correlated effects of the noise play an important role. The classical Fokker-Planck equation for a Gaussian-Markovian noise is obtained by performing the Wigner transformation, and its equilibrium state is shown to be the Maxwell-Boltzmann distribution. The method is convenient for numerical studies. Calculations for quantum-system harmonic oscillators and the double-well potential problems are demonstrated for cases of Gaussian-white noise and Gaussian-Markovian noise.

Physical Review A•Apr 1, 1991•224 citations

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

Explicit noise and dissipation operators for quantum stochastic thermodynamics

Stefano Giordano, Fabrizio Cleri, Ralf Blossey

Abstract The theory of open quantum systems plays a fundamental role in several scientific and technological disciplines, from quantum computing and information science to molecular electronics and quantum thermodynamics. Despite its widespread relevance, a rigorous formulation of quantum dissipation in conjunction with thermal noise remains a topic of active research. In this work, we establish a formal correspondence between classical stochastic thermodynamics, in particular the Fokker–Planck and Klein–Kramers equations, and the quantum master equation. Building on prior studies of multiplicative noise in classical stochastic differential equations, we demonstrate that thermal noise at the quantum level manifests as a multidimensional geometric stochastic process. By applying canonical quantization, we introduce a novel Hermitian dissipation operator that serves as a quantum analogue of classical viscous friction. This operator allows for a well-defined expression of heat exchange between a system and its environment, enabling the formulation of an alternative quantum equipartition theorem. Our framework ensures a precise energy balance that aligns with the first law of thermodynamics and an entropy balance consistent with the second law. The theoretical formalism is applied to two prototypical quantum systems, the harmonic oscillator and a particle in an infinite potential well, for which it provides new insights into nonequilibrium thermodynamics at the quantum scale. Our results advance the understanding of dissipation in quantum systems and establish a foundation for future studies on stochastic thermodynamics in the quantum domain.

arXiv:2504.11938•Journal of Statistical Mechanics Theory and Experiment•Aug 1, 2025•1 citation
densitymatrixofaquan-tumsystemcomposedofoneormanyparticlessubjectedtoaconservativeforcefieldand interacting with a thermal bath has been derived. This result is achieved byanalogy with classical models based upon the Langevin and Fokker-Planck equa-tions. According to the Langevin approach, the interaction is modelled by meansof a random force field and a viscous friction term. We generalise this classicalapproach introducing suitable operators that describe the quantum evolution of asystemweaklycoupledtoathermalbath. Inparticular,wedefineakindoffrictionoperator thatcan be thoughtas the quantalcounterpartof the classical Langevintermcorrespondingtotheviscousforce. Theproposedapproachhastheinvaluableadvantageofyieldingahandydifferentialequationtomodelthequantuminteractionbetweenasystemandathermalbath.PACS42.50.Lc–Quantumfluctuations,quantumnoise,andquantumjumps.PACS05.30–Quantumstatisticalmechanics.PACS05.70.Ln–Nonequilibriumandirreversiblethermodynamics.PACS05.10.Gg–Stochasticanalysismethods(Fokker-Planck,Langevin,etc.).
This chapter is devoted to the classical limit of the quantum dissipative action obtained in Chapter 3. We show how it yields Langevin, Fokker–Planck and optimal path descriptions of classical stochastic systems. These approaches are used to discuss activation escape, fluctuation relation, reaction models and other examples. Classical dissipative action In Section 3.2 we derived the Keldysh action for a quantum particle coupled to an Ohmic environment, Eq. (3.20). If only linear terms in the quantum coordinate X q (t) are kept in this action, it leads to a classical Newtonian equation with a viscous friction force, Eq. (3.21). Such an approximation completely disregards any fluctuations: both quantum and classical . Our goal now is to do better than that and to keep classical thermal fluctuations, while still neglecting quantum effects. To this end it is convenient to restore the Planck constant ħ in the action and then take the limit ħ → 0. For dimensional reasons, the factor ħ -1 should stay in front of the entire action. To keep the part of the action responsible for the classical equation of motion (3.21) free from the Planck constant it is convenient to rescale the quantum component as X q → ħ X q . Indeed, when this is done all terms linear in X q do not contain ħ. Finally, to have the temperature in energy units, one needs to substitute T with T /ħ.
The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.
The quantum Fokker-Planck equation for a Gaussian-Markovian bath is deduced by applying a method proposed by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1989)]. The results are expressed in the form of simultaneous differential equations in terms of density operators and can treat strong system-bath interactions where the correlated effects of the noise play an important role. The classical Fokker-Planck equation for a Gaussian-Markovian noise is obtained by performing the Wigner transformation, and its equilibrium state is shown to be the Maxwell-Boltzmann distribution. The method is convenient for numerical studies. Calculations for quantum-system harmonic oscillators and the double-well potential problems are demonstrated for cases of Gaussian-white noise and Gaussian-Markovian noise.
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.
Abstract The theory of open quantum systems plays a fundamental role in several scientific and technological disciplines, from quantum computing and information science to molecular electronics and quantum thermodynamics. Despite its widespread relevance, a rigorous formulation of quantum dissipation in conjunction with thermal noise remains a topic of active research. In this work, we establish a formal correspondence between classical stochastic thermodynamics, in particular the Fokker–Planck and Klein–Kramers equations, and the quantum master equation. Building on prior studies of multiplicative noise in classical stochastic differential equations, we demonstrate that thermal noise at the quantum level manifests as a multidimensional geometric stochastic process. By applying canonical quantization, we introduce a novel Hermitian dissipation operator that serves as a quantum analogue of classical viscous friction. This operator allows for a well-defined expression of heat exchange between a system and its environment, enabling the formulation of an alternative quantum equipartition theorem. Our framework ensures a precise energy balance that aligns with the first law of thermodynamics and an entropy balance consistent with the second law. The theoretical formalism is applied to two prototypical quantum systems, the harmonic oscillator and a particle in an infinite potential well, for which it provides new insights into nonequilibrium thermodynamics at the quantum scale. Our results advance the understanding of dissipation in quantum systems and establish a foundation for future studies on stochastic thermodynamics in the quantum domain.