non-Markovian Stochastic Equation Research Articles

Correlation functions of non-Markovian systems out of equilibrium: analytical expressions beyond single-exponential memory

This paper is concerned with correlation functions of stochastic systems with memory, a prominent example being a molecule or colloid moving through a complex (e.g. viscoelastic) fluid environment. Analytical investigations of such systems based on non-Markovian stochastic equations are notoriously difficult. A common approximation is that of a single-exponential memory, corresponding to the introduction of one auxiliary variable coupled to the Markovian dynamics of the main variable. As a generalization, we here investigate a class of ‘toy’ models with altogether three degrees of freedom, giving rise to more complex forms of memory. Specifically, we consider, mainly on an analytical basis, the under- and overdamped motion of a colloidal particle coupled linearly to two auxiliary variables, where the coupling between variables can be either reciprocal or non-reciprocal. Projecting out the auxiliary variables, we obtain non-Markovian Langevin equations with friction kernels and colored noise, whose structure is similar to that of a generalized Langevin equation. For the present systems, however, the non-Markovian equations may violate the fluctuation–dissipation relation as well as detailed balance, indicating that the systems are out of equilibrium. We then study systematically the connection between the coupling topology of the underlying Markovian system and various autocorrelation functions. We demonstrate that already two auxiliary variables can generate surprisingly complex (e.g. non-monotonic or oscillatory) memory and correlation functions. Finally, we show that a minimal overdamped model with two auxiliary variables and suitable non-reciprocal coupling yields correlation functions resembling those describing hydrodynamic backflow in an optical trap.

The Quadrature Master Equations

In this paper, we derive the non-Markovian stochastic equation of motion (SEM) and master equations (MEs) for the open quantum system by using the non-Markovian stochastic Schrodinger equations (SSEs) for the quadrature unraveling in linear and nonlinear cases. The SSEs for quadrature unraveling arise as a special case of a quantum system. Also we derive the Markovian SEM and ME by using linear and nonlinear Ito SSEs for the measurement probabilities. In linear non-Markovian case, we calculate the convolutionless linear quadrature non-Markovian SEM and ME. We take advantage from example and show that corresponding theory.

Numerical solutions for non-Markovian stochastic equations of motion

The reliability and precision of numerically solving stochastic non-Markovian equations by standard numerical codes, more specifically, with the fourth-order Runge–Kutta routine for solving differential equations, is gauged by comparing the results obtained from analytical solutions for the equations. The results for different prescriptions for transforming the non-Markovian equations in a system of Markovian ones are compared so to check the reliability of the numerical method.

Quantum Non-Markovian Stochastic Equations

We derive quantum diffusion equations with transport coefficients explicitly depending on time from generalized non-Markovian Langevin equations and obtain generalized fluctuation-dissipation relations. We substantiate the axiomatic Lindblad approach in the microscopic model. For non-Markovian dynamics, we find sets of diffusion coefficients that ensure the purity of states at any instant.

Memory Function Approach to Chaos and Turbulence and the Continued Fraction Expansion

The chaotic orbits of dynamical systems are deterministic and predictable on short timescales τ γ , but they become stochastic and random on long timescales T M (» τ γ ) due to the orbital instability of chaos. This randomization of chaotic orbits has been formulated recently by deriving a non-Markovian stochastic equation for macrovariables in terms of a fluctuating force and a memory function. In order to develop this memory function approach to chaos and turbulence, we explore the following problems by studying the Duffing oscillator and the Navier-Stokes equation for an incompressible fluid: 1) the physical meaning of the projection of macrovariables A(t) onto A(0); 2) the method of calculating the short-lived motion with short timescale τ γ , which determines the memory functions and the macroscopic transport coefficients due to chaos and turbulence; 3) the continued fraction expansion of the memory function, and the order estimation of short timescales τ γ and long timescales τM; 4) the relation between the memory function and the time correlation function of a nonlinear force, which gives computable theoretical expressions for the macroscopic transport coefficients.

Stochastic treatment of disoriented chiral condensates within a Langevin description

Applying a microscopically motivated semi-classical Langevin description of the linear sigma model we investigate for various different scenarios the stochastic evolution of a disoriented chiral condensate (DCC) in a rapidly expanding system. Some particular emphasize is put on the numerical realisation of colored noise in order to treat the underlying dissipative and nonmarkovian stochastic equations of motion. A comparison with an approximate markovian (i.e. instantaneous) treatment of dissipation and noise will be made in order to identify the possible influence of memory effects in the evolution of the chiral order parameter. Assuming a standard Rayleigh cooling term to simulate a D-dimensional scaling expansion we present the probability distribution in the low momentum pion number stemming from the relaxing zero mode component of the chiral field. The best DCC signal is expected for initial conditions centered around $<\sigma > \approx 0 $ as would be the case of effective light `pions' close to the phase transition. By choosing appropriate idealized global parameters for the expansion our findings show that an experimentally feasible DCC, if it does exist in nature, has to be a rare event with some finite probability following a nontrivial and nonpoissonian distribution on an event by event basis. DCCs might then be identified experimentally by inspecting higher order factorial cumulants $\theta_m$ ($m\ge 3$) in the sampled distribution.

Relaxation to a bistable state in a quantum cell

We analyze a simple model of a quantum cell composed of two Coulomb-coupled double dot systems with one electron in each double dot. The interaction of electrons with acoustic phonons is considered as the principal mechanism that relaxes the cell to the ground state. Non-Markovian stochastic equations for population differences and dipole moments of the constituent double dots are derived. It is found that in order for the bistable state of the quantum cell to exist, the effective energy of Coulomb repulsion between the double dot systems must exceed the individual tunnel splitting energy, as well as the phonon temperature. The behavior of the cell polarization near the critical temperature is described analytically. We calculate the damping rates determining cell relaxation to its minimum-energy state and discuss limitations on the speed of cell response to external fields caused by the finite relaxation time.

Integral expansion often reducing to the density-gradient expansion, extended to non-Markov stochastic processes: Consequent non-Markovian stochastic equation whose leading terms coincide with Schrödinger's.

An integral expansion is obtained that reduces under explicitly given conditions to the density-gradient expansion for the number density of stochastic particles. Explicit coefficients in terms of moments are calculated up to and including the fourth order, corresponding to the super-Burnett approximation. The expansion is proven to be valid for both Markov and non-Markov processes, including those with infinite memory, typical of a stochastic motion with inertia. An application of this expansion is the relation between local average velocities in Markovian stochastic processes and the drift velocity of a probability cloud. As a further application we obtain a non-Markovian stochastic equation whose leading terms coincide with the Schr\"odinger equation. The additional terms could be interpreted as corrections that produce contributions to atomic spectra of the fine-structure type. In the case of scattering, where the density gradients can be locally high, the correction terms become relevant and they could perhaps explain the differences between the calculated and measured rotational and/or vibrational cross sections for hydrogen molecules.

Non-Markovian theory of activated rate processes. I. Formalism

The escape of a particle from a potential well is treated using a generalized Langevin equation (GLE) in the low friction limit. The friction is represented by a memory kernel and the random noise is characterized by a finite correlation time. This non-Markovian stochastic equation is reduced to a Smoluchowski diffusion equation for the action variable of the particle and explicit expressions are obtained for the drift and diffusion terms in this equation in terms of the Fourier coefficients of the deterministic trajectory (associated with the motion without coupling to the heat bath) and of the Fourier transform of the friction kernel. The latter (frequency dependent friction) determines the rate of energy exchange with the heat bath. The resulting energy (or action) diffusion equation is used to determine the rate of achieving the critical (escape) energy. Explicit expressions are obtained for a Morse potential. These results for the escape rate agree with those from stochastic trajectories based on the original GLE. Non-Markovian effects are shown to have large effects on the rate of energy accumulation and relaxation within the well.