[Paper Review] An old efficient approach to anomalous Brownian motion

V. Lisý, Jana Tóthová|arXiv (Cornell University)|Jun 5, 2010

Complex Systems and Time Series Analysis3 references3 citations

TL;DR

This paper revisits Vladimirsky's 1942 method for solving linear Langevin equations with memory, demonstrating its effectiveness in deriving anomalous Brownian motion (BM) behavior. By transforming the stochastic equation into a deterministic one via Gibbs statistics, the approach efficiently handles arbitrary memory kernels—particularly exponential decay—and yields exact solutions for mean square displacement and hydrodynamic BM in magnetic fields.

ABSTRACT

A number of random processes in various fields of science is described by phenomenological equations containing a stochastic force, the best known example being the Langevin equation (LE) for the Brownian motion (BM) of particles. Long ago Vladimirsky (1942) proposed a simple method for solving such equations. The method, based on the classical Gibbs statistics, consists in converting the stochastic LE into a deterministic one, and is applicable to linear equations with any kind of memory. When the memory effects are taken into account in the description of the BM, the mean square displacement of the particle at long times can exhibit an "anomalous" (different from that in the Einstein theory) time dependence. In the present paper we show how some general properties of such anomalous BM can be easily derived using the Vladimirsky approach. The method can be effectively used in solving many of the problems currently considered in the literature. We apply it to the description of the BM when the memory kernel in the Volterra-type integro-differential LE exponentially decreases with the time. The problem of the hydrodynamic BM of a charged particle in an external magnetic field is also solved.

Motivation & Objective

• To demonstrate the applicability of Vladimirsky's 1942 method to modern problems in anomalous Brownian motion.
• To derive general properties of anomalous BM with arbitrary memory kernels using a classical statistical mechanics approach.
• To solve the hydrodynamic BM of a charged particle in an external magnetic field using the same method.
• To show that the method provides exact analytical solutions for mean square displacement under non-Markovian dynamics.
• To establish the method as a powerful, overlooked tool for solving complex stochastic integro-differential equations in statistical mechanics.

Proposed method

• Transforms the stochastic Langevin equation with memory into a deterministic equation using classical Gibbs statistics.
• Applies the method to linear Volterra-type integro-differential equations with general memory kernels.
• Uses the assumption of Gaussian white noise in the Langevin equation to enable analytical treatment.
• Applies the method to the case where the memory kernel decays exponentially in time.
• Solves the hydrodynamic BM problem for a charged particle under a magnetic field by incorporating the Lorentz force into the deterministic formulation.
• Derives the mean square displacement analytically by solving the resulting deterministic equation in Fourier space.

Experimental results

Research questions

• RQ1 Can Vladimirsky's 1942 method be effectively applied to derive anomalous diffusion behavior in systems with memory?
• RQ2 What are the general analytical properties of mean square displacement in non-Markovian Brownian motion using this method?
• RQ3 How does the method handle the hydrodynamic BM of a charged particle in an external magnetic field?
• RQ4 What is the exact form of the mean square displacement when the memory kernel decays exponentially?
• RQ5 Can this classical method outperform modern numerical or perturbative techniques for such problems?

Key findings

• The method successfully transforms the stochastic Langevin equation with memory into a deterministic equation, enabling exact analytical solutions.
• For an exponential memory kernel, the mean square displacement exhibits subdiffusive behavior at long times, consistent with known results.
• The approach yields exact expressions for the time-dependent mean square displacement in both Markovian and non-Markovian regimes.
• The hydrodynamic BM of a charged particle in a magnetic field is solved analytically, showing modified diffusion characteristics due to the Lorentz force.
• The method is general and applicable to any linear Langevin equation with arbitrary memory, regardless of the kernel form.
• The solution for the exponential memory case confirms that anomalous diffusion arises naturally from the memory structure without additional assumptions.