Langevin theory of anomalous Brownian motion made simple

Abstract

During the century from the publication of the work by Einstein (1905 Ann. Phys. 17 549) Brownian motion has become an important paradigm in many fields of modern science. An essential impulse for the development of Brownian motion theory was given by the work of Langevin (1908 C. R. Acad. Sci., Paris 146 530), in which he proposed an ‘infinitely more simple’ description of Brownian motion than that by Einstein. The original Langevin approach has however strong limitations, which were rigorously stated after the creation of the hydrodynamic theory of Brownian motion (1945). Hydrodynamic Brownian motion is a special case of ‘anomalous Brownian motion’, now intensively studied both theoretically and in experiments. We show how some general properties of anomalous Brownian motion can be easily derived using an effective method that allows one to convert the stochastic generalized Langevin equation into a deterministic Volterra-type integro-differential equation for the mean square displacement of the particle. Within the Gibbs statistics, the method is applicable to linear equations of motion with any kind of memory during the evolution of the system. We apply it to memoryless Brownian motion in a harmonic potential well and to Brownian motion in fluids, taking into account the effects of hydrodynamic memory. Exploring the mathematical analogy between Brownian motion and electric circuits, which are at nanoscales also described by the generalized Langevin equation, we calculate the fluctuations of charge and current in RLC circuits that are in contact with the thermal bath. Due to the simplicity of our approach it could be incorporated into graduate courses of statistical physics. Once the method is established, it allows bringing to the attention of students and effectively solving a number of attractive problems related to Brownian motion.

Erratum to: On the Correlation Properties of Thermal Noise in Fluids

“…11 use the above given equation and are correct. They agree with the previously obtained expressions for the VAF (see [2] and references therein). …”

Fractional Langevin Equation Model for Characterization of Anomalous Brownian Motion from NMR Signals

“…One can check that this equation gives the known results for particles in diffusion regime, when X(t) = Ct α (C is a temperature-dependent parameter, α = 1 corresponds to normal diffusion with C = 2D and the diffusion coefficient D = k B T/γ, α < 1 to sub-diffusion, and α > 1 to super-diffusion [15]. In our study the solution (3) was used to obtain the NMR signal decay from Eq.…”

“…(1) becomes the standard memoryless LE with the white noise force and a purely viscous Stokes friction force −γυ (t). To our opinion, the simplest method of solving linear GLE equations, which goes back to the old work [14], is as follows [15][16][17][18][19]. If we are interested in finding the MSD of the particle,…”

An Efficient Method to Study Nondiffusive Motion of Brownian Particles

“…So, in contrast to the commonly assumed overdamped motion, experimental access to short timescales has revealed a resonance peak in the spectrum of the particle fluctuations in a liquid [3], the transition from ballistic to diffusive BM has been observed [4] and even, as reported in [5], the instantaneous velocity of a Brownian particle has been measured for the first time in air. For the history and more references on the experiments and theory of the BM we refer to [1,[6][7][8]. The obtained results are well described by the hydrodynamic theory of the BM in harmonic potentials [9] and are interesting also because they can be considered as measurements of the "color" of the thermal noise driving the Brownian particles through collisions with the fluid molecules [3,10,11].…”

“…The solution of (4) requires knowledge of the system memory. However, independently of a concrete form of Γ(t), at short times the generalized Langevin equation describes the ballistic motion, X(t) ≈ k B T t 2 /M as t → 0 [7,18]. The longtime properties of the MSD can be related to the behavior of the Laplace transform of the kernel Γ, Γ = L{Γ(t)} at small s [19].…”