Complex Langevin equations and Schwinger–Dyson equations

Stationary distributions of complex Langevin equations are shown to be the complexified path integral solutions of the Schwinger–Dyson equations of the associated quantum field theory. Specific examples in zero dimensions and on a lattice are given. The relevance to the study of quantum field theory solution space is discussed.     

Higher-order quasilinear approximations for the propagator of the Kramers equation

We present a systematic procedure for constructing higher-order quasilinear approximations for the propagator of the Klein-Kramers equation describing the motion of a Brownian particle in a general force field. Its key points are splitting the full force field into a linear contribution and an anharmonic correction, replacing the underlying Langevin equations by difference equations and solving these equations iteratively. An accurate single step propagator is then derived in terms of known statistical properties of the noise terms. Its use in a path integral shows this approach to be advantageous over a Taylor series expansion for the propagator recently derived employing standard techniques.     

Supersymmetry in stochastic processes with higher-order time derivatives

A supersymmetric path-integral representation is developed for stochastic processes whose Langevin equation contains any number N of time derivatives, thus generalizing the presently available treatment of first-order Langevin equations by Parisi and Sourlas [Phys. Rev. Lett. 43 (1979) 744; Nucl. Phys. B 206 (1982) 321] to systems with inertia (Kramers' process) and beyond. The supersymmetric action contains N fermion fields with first-order time derivatives whose path integral is evaluated for fermionless asymptotic states.     

Path integral representation of fractional harmonic oscillator

Fractional oscillator process can be obtained as the solution to the fractional Langevin equation. There exist two types of fractional oscillator processes, based on the choice of fractional integro-differential operators (namely Weyl and Riemann-Liouville). An operator identity for the fractional differential operators associated with the fractional oscillators is derived; it allows the solution of fractional Langevin equations to be obtained by simple inversion. The relationship between these two fractional oscillator processes is studied. The operator identity also plays an important role in the derivation of the path integral representation of the fractional oscillator processes. Relevant quantities such as two-point and n-point functions can be calculated from the generating functions.     

Nonlinear transport and dynamics of fluctuations

The time evolution of the macroscopic variables of a system initially in a state far from thermal equilibrium is studied from a statistical mechanical point of view. Exact nonlinear transport equations for the mean values and linear nonstationary Langevin equations for the fluctuations around the mean path are derived. Connections between the dynamics of fluctuations and the transport equations are discussed. The Langevin random forces depend on the macroscopic state and they are related to the transport kernels by a fluctuation-dissipation formula.     

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     

Complex fermionic and spin path integrals: An alternative parametrization

For a previously developed complex Langevin method we propose an alternative parametrization. This results in a simpler structure of the stochastic differential equation and a faster performance of the simulation algorithm. The equivalence of both complex stochastic processes is demonstrated for a simple path integral.     

Stability and fluctuation in chemical systems

Fluctuations around the steady state in chemical reactions are discussed. The connection between the two approaches in the literature, the Langevin equation approach and the master equation approach, is shown in terms of a path integral.     

Langevin Dynamics,Large Deviations and Instantons for the Quasi-Geostrophic Model and Two-Dimensional Euler Equations

We investigate a class of simple models for Langevin dynamics of turbulent flows, including the one-layer quasi-geostrophic equation and the two-dimensional Euler equations. Starting from a path integral representation of the transition probability, we compute the most probable fluctuation paths from one attractor to any state within its basin of attraction. We prove that such fluctuation paths are the time reversed trajectories of the relaxation paths for a corresponding dual dynamics, which are also within the framework of quasi-geostrophic Langevin dynamics. Cases with or without detailed balance are studied. We discuss a specific example for which the stationary measure displays either a second order (continuous) or a first order (discontinuous) phase transition and a tricritical point. In situations where a first order phase transition is observed, the dynamics are bistable. Then, the transition paths between two coexisting attractors are instantons (fluctuation paths from an attractor to a saddle), which are related to the relaxation paths of the corresponding dual dynamics. For this example, we show how one can analytically determine the instantons and compute the transition probabilities for rare transitions between two attractors.     

Generalized Langevin Equations for Systems with Local Interactions

Journal of Statistical Physics - We present a new method to approximate the Mori–Zwanzig (MZ) memory integral in generalized Langevin equations describing the evolution of smooth observables...     

Fundamental aspects of quantum Brownian motion

With this work we elaborate on the physics of quantum noise in thermal equilibrium and in stationary nonequilibrium. Starting out from the celebrated quantum fluctuation-dissipation theorem we discuss some important consequences that must hold for open, dissipative quantum systems in thermal equilibrium. The issue of quantum dissipation is exemplified with the fundamental problem of a damped harmonic quantum oscillator. The role of quantum fluctuations is discussed in the context of both, the nonlinear generalized quantum Langevin equation and the path integral approach. We discuss the consequences of the time-reversal symmetry for an open dissipative quantum dynamics and, furthermore, point to a series of subtleties and possible pitfalls. The path integral methodology is applied to the decay of metastable states assisted by quantum Brownian noise.     

Injected Power Fluctuations in Langevin Equation

In this paper, we consider the Langevin equation from an unusual point of view, that is as an archetype for a dissipative system driven out of equilibrium by an external excitation. Using path integral method, we compute exactly the probability density function of the power (averaged over a time interval of length ) injected (and dissipated) by the random force into a Brownian particle driven by a Langevin equation. The resulting distribution, as well as the associated large deviation function, display strong asymmetry, whose origin is explained. Connections with the so-called Fluctuation Theorem are thereafter discussed. Finally, considering Langevin equations with a pinning potential, we show that the large deviation function associated with the injected power is completely insensitive to the presence of a potential.     

非平衡统计场论与临界动力学(Ⅰ)——广义朗之万方程

本文从闭路格林函数顶角方程出发,推导出临界动力学中序参量和守恒量所满足的广义朗之万方程。根据Ward-Takahashi恒等式和线性响应理论,确定守恒量方程应具有的形式,它自动包含了模-模耦合项。考察不同的对称群,得到临界动力学的各种模型。整个理论框架也可用于描述远离平衡的稳态附近的行为。 关键词：     

WKB-type expansion for Langevin equations: II. Covariant expansion

We show how to compute in a covariant way the WKB-type expansion for the transition probability density of the Markovian processes generated by general multiplicative noise Langevin equations. The method uses phase space functional integrals and normal coordinates around the classical path. We compare with our earlier non-covariant methods and compute explicitly the first order correction to the WKB-approximation.     

Presentation of solutions of impulsive fractional Langevin equations and existence results

In this paper, a class of impulsive fractional Langevin equations is firstly offered. Formula of solutions involving Mittag-Leffler functions and impulsive terms of such equations are successively derived by studying the corresponding linear Langevin equations with two different fractional derivatives. Meanwhile, existence results of solutions are established by utilizing boundedness, continuity, monotonicity and nonnegative of Mittag-Leffler functions and fixed point methods. Further, other existence results of nonlinear impulsive problems are also presented. Finally, an example is given to illustrate our theoretical results.