On the quantum langevin equation

The quantum Langevin equation is the Heisenberg equation of motion for the (operator) coordinate of a Brownian particle coupled to a heat bath. We give an elementary derivation of this equation for a simple coupled-oscillator model of the heat bath.Deceased.     

Exact c-number representation of non-Markovian quantum dissipation

The reduced dynamics of a quantum system interacting with a linear heat bath finds an exact representation in terms of a stochastic Schr?dinger equation. All memory effects of the reservoir are transformed into noise correlations and mean-field friction. The classical limit of the resulting stochastic dynamics is shown to be a generalized Langevin equation, and conventional quantum state diffusion is recovered in the Born-Markov approximation. The non-Markovian exact dynamics, valid at arbitrary temperature and damping strength, is exemplified by an application to the dissipative two-state system.     

Equilibration and relaxation times at the chiral phase transition including reheating

We investigate the relaxational dynamics of the order parameter of chiral symmetry breaking, the sigma mean-field, with a heat bath consisting of quarks and antiquarks. A semiclassical stochastic Langevin equation of motion is obtained from the linear sigma model with constituent quarks. The equilibration of the system is studied for a first order phase transition and a critical point, where a different behavior is found. At the first order phase transition we observe the phase coexistence and at a critical point the phenomenon of critical slowing down with large relaxation times. We go beyond existing Langevin studies and include reheating of the heat bath by determining the energy dissipation during the relaxational process. The energy of the entire system is conserved. In a critical point scenario we again observe critical slowing down.     

Thermal relaxation of systems with quadratic heat bath coupling: I. Classical systems

We consider the evolution of systems whose coupling to the heat bath is quadratic in the bath coordinates. Performing an explicit elimination of the bath variables we arrive at an equation of evolution for the system variables alone. In the weak coupling limit we show that the equation is of the generalized Langevin form, with fluctuations that are Gaussian and that obey a fluctuation-dissipation relation. If the system-bath coupling is linear in the system coordinates the resulting fluctuations are additive and the dissipation is linear. If the coupling is nonlinear in the system coordinates, the resulting fluctuations are multiplicative and the dissipation is nonlinear.     

基于分数阶环境热浴的非各态历经判据研究

本文从分数阶谱形式的气固耦合模型出发, 理论推导出具有幂律记忆核的广义朗之万方程. 研究气体分子在自由场和简谐势场中的动力学演化和长时渐进行为, 着重分析三种各态历经判据: Khinchin判据、Lee判据以及内在判据和外在表现的适用性. 研究结果表明: Khinchin判据适用于广义朗之万方程描述的所有扩散和输运过程; Lee判据并不适用于布朗运动, 只能用来区分不同类型的扩散过程; 而内在判据和外在表现不仅能够把非各态历经分为两类, 同时可以揭示非各态历经的物理内在根源.     

简谐速度噪声与简谐噪声环境中谐振子的动力学共振

通过求解简谐势场中的广义量子朗之万方程，得到平均能量的精确表达式.由于简谐速度噪声与简谐噪声功率谱的不同特点，两种内部噪声驱动的谐振子在简谐外力的作用下具有不同的共振特征.这些特征可用来检验两种噪声.     

瓮模型中多粒子动力学的研究

研究了处于热库中的多颗粒在两个和多个瓮中的运动.通过求解含噪声项的一维朗之万方程,获得颗粒的位置和速度,并分析了其运动状态.研究发现，在高温下系统处于对称态的时间较长,反之系统将会出现多个定态.所有运动颗粒的速率分布都满足Gauss分布,非对称态的有效温度T₂与弹性恢复系数r有良好的指数关系. 关键词： 多颗粒 多瓮 速率分布 有效温度     

Langevin dynamics simulation of a one-dimensional linear spin chain with long-range interactions

In this paper we study the critical behavior of a simple one-dimensional rotor spin in the form of a linear chain with long-range interactions, using the mean field Langevin dynamics approach and in the presence of fluctuations added by a heat bath. We have computed the specific heat, the magnetic susceptibility, the Binder fourth-order cumulant, and the magnetization, and then we have calculated the critical exponents using finite-size scaling. In addition, we provide a relation between the thermal bath temperature and the temperature of the system. Our results confirm the existence of a second-order critical temperature in the one-dimensional chain of spins with long-range interaction.     

Fractional Kinetics in Kac–Zwanzig Heat Bath Models

We study a variant of the Kac–Zwanzig model of a particle in a heat bath. The heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data. As n → ∞ the trajectories of the distinguished particle weakly converge to the solution of a stochastic integro-differential equation—a generalized Langevin equation (GLE) with power-law memory kernel and driven by 1/f ^α -noise. The limiting process exhibits fractional sub-diffusive behaviour. We further consider the approximation of non-Markovian processes by higher-dimensional Markovian processes via the introduction of auxiliary variables and use this method to approximate the limiting GLE. In contrast, we show the inadequacy of a so-called fractional Fokker–Planck equation in the present context. All results are supported by direct numerical experiments.     

Energy dissipation and violation of the fluctuation-response relation for harmonic oscillators described by the generalized Langevin equation

The generalized Langevin equation (GLE) satisfied by harmonic oscillators coupled to a heat bath is transformed into an equality between the rate of energy dissipation and an extent of violation of the fluctuation-response relation. Its significance is discussed. When the system reaches a stationary state and a single harmonic oscillator’s frequency is set to zero, the equality reduces to a fluctuation-dissipation relation, which is slightly different from the usual Kubo’s formalism.     

Stochastic thermodynamics: principles and perspectives

Stochastic thermodynamics provides a framework for describing small systems like colloids or biomolecules driven out of equilibrium but still in contact with a heat bath. Both, a first-law like energy balance involving exchanged heat and entropy production entering refinements of the second law can consistently be defined along single stochastic trajectories. Various exact relations involving the distribution of such quantities like integral and detailed fluctuation theorems for total entropy production and the Jarzynski relation follow from such an approach based on Langevin dynamics. Analogues of these relations can be proven for any system obeying a stochastic master equation like, in particular, (bio)chemically driven enzyms or whole reaction networks. The perspective of investigating such relations for stochastic field equations like the Kardar-Parisi-Zhang equation is sketched as well.     

Effects of temperature on the relaxation to equilibrium and stationary nonequilibrium states of some Langevin systems

We investigate the effects of temperature on the properties of the time relaxation to equilibrium and nonequilibrium steady states of correlation functions of some Langevin harmonic systems. We consider commonly used dissipative and conservative Langevin dynamics, and show that the time relaxation rate depends on the temperature in the case of thermal reservoirs at different temperatures connected to the system, but it does not happen in the case of relaxation to equilibrium, i.e., if all the heat bath are at the same temperature. Our formalism maps the initial stochastic problem on a noncanonical quantum field theory, and the calculations of the relaxation rates are based on a perturbative analysis. We argue to show the reliability of the perturbative computation.     

The generalized Schrödinger–Langevin equation

In this work, for a Brownian particle interacting with a heat bath, we derive a generalization of the so-called Schrödinger–Langevin or Kostin equation. This generalization is based on a nonlinear interaction model providing a state-dependent dissipation process exhibiting multiplicative noise. Two straightforward applications to the measurement process are then analyzed, continuous and weak measurements in terms of the quantum Bohmian trajectory formalism. Finally, it is also shown that the generalized uncertainty principle, which appears in some approaches to quantum gravity, can be expressed in terms of this generalized equation.     

Fourier's Law for a Harmonic Crystal with Self-Consistent Stochastic Reservoirs

Journal of Statistical Physics - We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the “exterior” left...     

Master and Langevin equations for electromagnetic dissipation and decoherence of density matrices

We set up a forward - backward path integral for a point particle in a bath of photons to derive a master equation for the density matrix which describes electromagnetic dissipation and decoherence. We also derive the associated Langevin equation. As an application, we recalculate the Wigner-Weisskopf formula for the natural line width of an atomic state at zero temperature and find, in addition, the temperature broadening caused by the decoherence term. Our master equation also yields the correct Lamb shift of atomic levels. The two equations may have applications to dilute interstellar gases or to few-particle systems in cavities. Received 29 November 2000 and Received in final form 11 February 2001     

Nonlinear momentum relaxation of an impurity in a harmonic chain

A microscopic derivation of the generalized Langevin equation for arbitrary powers of the momentum of an impurity in a harmonic chain is presented. As a direct consequence of the Gaussian character of the conditional momentum distribution function, nonlinear momentum coupling effects are absent for this system and the Langevin equation takes on a particularly simple form. The kernels which characterize the decay of higher powers of the impurity momentum depend on the ratio of the masses of the impurity and bath particles, in contrast to the situation for the momentum Langevin equation for this system. The simplicity of the harmonic chain dynamics is exploited in order to investigate several features of the relaxation, such as the factorization approximation for time-dependent correlation functions and the decay of the kinetic energy autocorrelation function.     

Quantum Dynamics of a Harmonic Oscillator in a Defomed Bath in the Presence of Lamb Shift

In this paper, we investigate the dissipative quantum dynamics of a harmonic oscillator in the presence a deformed bath by considering the Lamb shift term. The deformed bath is modelled by a collection of deformed quantum harmonic oscillators as a generalization of Hopfield model. The Langevin equation for both the photon number and the fluctuation spectrum under the Weisskopf–Winger approximation are obtained and discussed.