Four approaches for description of stochastic systems with small and finite inertia (original) (raw)

Small and finite inertia in stochastic systems: Moment and cumulant formalisms

28TH RUSSIAN CONFERENCE ON MATHEMATICAL MODELLING IN NATURAL SCIENCES, 2020

We analyze two approaches to elimination of a fast variable (velocity) in stochastic systems: moment and cumulant formalisms. With these approaches, we obtain the corresponding Smoluchovski-type equations, which contain only the coordinate/phase variable. The adiabatic elimination of velocity in terms of cumulants and moments requires the first three elements. However, for the case of small inertia, the corrected Smoluchowski equation in terms of moments requires five elements, while in terms of cumulants the same first three elements are sufficient. Compared to the method based on the expansion of the velocity distribution in Hermite functions, the considered approaches have comparable efficiency, but do not require individual mathematical preparation for the case of active Brownian particles, where one has to construct a new basis of eigenfunctions instead of the Hermite ones.

Some additional remarks on the cumulant expansion for linear stochastic differential equations

Physica A: Statistical Mechanics and its Applications, 1984

We summarize our previous results on cumulant expansions for linear stochastic differential equations with correlated multipliclative and additive noise. The application of the general formulas to equations with statistically independent multiplicative and additive noise is reconsidered in detail, because our earlier results for this case are only valid in special situations. Moreover, a second partially time ordered cumulant is introduced, which is appropriate for the calculation of multi-time averages of the solution process and which differs slightly from the one to be used for single-time averages. By the two supplements just mentioned we complete our earlier investigations on the subject.

An Outline of Stochastic Calculus

UNITEXT for Physics

For simplicity again, in this chapter we will only consider processes with just one component. We already remarked in the Sect. 6.3 that a white noise is a singular process whose main properties can be traced back to the non differentiability of some processes. As a first example we have shown indeed that the Poisson impulse process (6.63) and its associated compensated version (6.65) are white noises entailed by the formal derivation respectively of a simple Poisson process N (t) and of its compensated variant N (t). In the same vein we have shown then in the Example 6.22 that also the formal derivative of the Wiener process W (t)-not differentiable according to the Proposition 6.18-meets the conditions (6.69) to be a white noise, and in the Appendix H we also hinted that the role of the fluctuating force B(t) in the Langevin equation (6.78) for the Brownian motion is actually played by such a white noiseẆ (t). We can now give a mathematically more cogent justification for this identification in the framework of the Markovian diffusions. The Langevin equation (6.78) is a particular case of the more general equatioṅ X (t) = a(X (t), t) + b(X (t), t) Z (t) (8.1) where a(x, t) and b(x, t) are given functions and Z (t) is a process with E [Z (t)] = 0 and uncorrelated with X (t). From a formal integration of (8.1) we find X (t) = X (t 0) + t t 0 a(X (s), s) ds + t t 0 b(X (s), s)Z (s) ds so that, being X (t) assembled as a combination of Z (s) values with t 0 < s < t, to secure the non correlation of X (t) and Z (t) we should intuitively require also the non correlation of Z (s) and Z (t) for every pair s = t. Since moreover Z (t) is presumed

Theory of second- and higher-order stochastic processes

Physical Review E, 1995

This paper presents a general approach to linear stochastic processes driven by various random noises. Mathematically, such processes are described by linear stochastic differential equations of arbitrary order (the simplest non-trivial example is ddotx=R(t)\ddot x = R(t)ddotx=R(t), where R(t)R(t)R(t) is not a Gaussian white noise). The stochastic process is discretized into nnn time-steps, all possible realizations are summed up and the continuum limit is taken. This procedure often yields closed form formulas for the joint probability distributions. Completely worked out examples include all Gaussian random forces and a large class of Markovian (non-Gaussian) forces. This approach is also useful for deriving Fokker-Planck equations for the probability distribution functions. This is worked out for Gaussian noises and for the Markovian dichotomous noise.

Reduction of overdamped stochastic processes in the coherent state representation

Physical Review E, 1995

The adiabatic elimination of fast variables from overdamped stochastic processes by functional integration is demonstrated. Adiabatic elimination entails the evaluation of a reduced evolution operator from the full evolution equation. For Fokker-Planck processes, the reduced evolution operator may be expressed as a ground state expectation, and it is shown how this is represented as a coherent state path integral. The elimination is then achieved by functionally integrating out all reference to the fast variables. The end result is a decoupling of the full evolution equation into separate equations for the fast and slow variables. The method is demonstrated for Brownian motion and for a system with multiplicative colored noise.

Path-integral formulation of stochastic processes for exclusive particle systems

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000

We present a systematic formalism to derive a path-integral formulation for hard-core particle systems far from equilibrium. Writing the master equation for a stochastic process of the system in terms of the annihilation and creation operators with mixed commutation relations, we find the Kramers-Moyal coefficients for the corresponding Fokker-Planck equation (FPE), and the stochastic differential equation (SDE) is derived by connecting these coefficients in the FPE to those in the SDE. Finally, the SDE is mapped onto field theory using the path integral, giving the field-theoretic action, which may be analyzed by the renormalization group method. We apply this formalism to a two-species reaction-diffusion system with drift, finding a universal decay exponent for the long-time behavior of the average concentration of particles in arbitrary dimension.

Algebraic structure of stochastic expansions and efficient simulation

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012

We investigate the algebraic structure underlying the stochastic Taylor solution expansion for stochastic differential systems. Our motivation is to construct efficient integrators. These are approximations that generate strong numerical integration schemes that are more accurate than the corresponding stochastic Taylor approximation, independent of the governing vector fields and to all orders. The sinhlog integrator introduced by Malham & Wiese (2009) is one example. Herein, we show that the natural context to study stochastic integrators and their properties is the convolution shuffle algebra of endomorphisms; establish a new whole class of efficient integrators; and then prove that, within this class, the sinhlog integrator generates the optimal efficient stochastic integrator at all orders.