Brownian motion near a soft surface

The theory of Brownian motion as described by nonlinear Langevin equations and the corresponding Fokker–Planck equations is discussed. The general problems of the theory of nonlinear Brownian motion considered are: Brownian motion in a medium with nonlinear friction; the critical analysis of three forms of the relevant Langevin and Fokker–Planck equations (Ito's form, Stratonovich's form, and the kinetic form); the Smoluchowski equations and master equations for different cases; two methods of transition from master equation to Fokker–Planck equation; master equations for one-step processes; traditional and nontraditional definition of transition probabilities; evolution of free energy and entropy in Brownian motion; Lyapunov functionals. The following particular examples are considered: Brownian motion in self-oscillatory systems; H-theorem for the van der Pol oscillator; S-theorem; oscillator with inertial nonlinearity; bifurcation of energy of the limiting cycle; oscillator with multistable stationary states; oscillators in discrete time; bifurcations of energy of the limiting cycle and the period of oscillations; criterion of instability upon transition to discrete time, based on the H-theorem; Brownian motion of quantum atoms oscillators in the equilibrium electromagnetic field; Brownian motion in chemically reacting systems; partially ionised plasmas; the Malthus–Verhulst process.

Bed load transport is a highly complex process. The probability density function (PDF) of particle velocities results from the local particle momentum variability in response to fluid drag and interactions with the bed. Starting from the forces exerted on a single particle under low transport rates (i.e., rolling and sliding regimes), we derive here the nonlinear stochastic Langevin equation (LE) to describe the dynamics of a single particle, accounting for both the deterministic and the stochastic components of such forces. Then, the Fokker-Planck equation (FPE), which describes the evolution of the PDF of the ensemble particle velocities, is derived from the LE. We show that the theoretical PDFs of both streamwise and cross-stream velocities obtained by solving the FPE under equilibrium conditions have exponential form (PDFs of both positive and negative velocities decay exponentially), consistent with the experimental data by Roseberry et al. (2012). Moreover, we theoretically show how the exponential-like PDF of an ensemble of particle velocities results from the forces exerted on a single particle. We also show that the simulated particle motions using the proposed Langevin model exhibit an emergent nonlinear relationship between hop distances and travel times (power law with exponent 5/3), in agreement with the experimental data, providing a statistical description of the particles' random motion in the context of a stochastic transport process. Finally, our study emphasizes that the motion of individual particles, described by the LE, and the behavior of the ensemble, described by the FPE, are connected within a statistical mechanics framework.

Non-Markovian expansion in quantum Brownian motion

Dielectrophoresis (DEP) is a popular, noncontact electrokinetic method for separating and transporting nanosize biomolecules and colloids in microdevices. DEP is the movement of polarizable particles arising from the action of nonuniform electric fields. The spatial-temporal distribution of nanosize particles moving under the action of a deterministic DEP force and stochastic Brownian thermal motion can be described by the Fokker Planck equation (FPE). The application of DEP electrokinetics in micro-technologies means nanoscale particle movement needs to be modeled and measured quantitatively. Quantitative FPE prediction (using numerical values for relevant dielectric and fluid parameters) of DEP-driven particle transport is usually achieved numerically by using Finite Element methods (FEMs). The drawbacks of FEMs are inaccuracy where the electric field is extremely inhomogeneous and they offer little insight into the mathematical structure of the FPE solution. The latter is important, not only for prediction of particle movement, but also the 'reverse' process where parameter values are estimated from measurements of DEP experiments. In this paper, a Fourier-Bessel series solution to the FPE is derived that describes particle movement under the action of DEP in a simple chamber. The solution assumes the DEP force exhibits a hyperbolic spatial profile and can be extended to the case that assumes an exponential decay. This applies to planar arrays, such as, interdigitated electrodes. Time-dependent DEP particle collection and release (after the DEP is switched off) from a surface is evaluated for strong and weak DEP forces. Temporal DEP responses can be classified as state-transitions and perturbations, respectively.

Nonlinear friction forces are a problem in classical Brownian motion for a long time [1]. They are described either by Langevin or by Fokker-Planck equations [2-4]. However, the rigorous generalized Langevin equation is linear, which points out that nonlinear friction forces possess a macroscopic hydrodynamic origin [1]. For this reason, they are not present in the modern quantum theory of open systems [5]. The scope of the present paper is to investigate the nonlinear friction effect on quantum mechanics. The analysis is based on a dissipative Madelung quantum hydrodynamics. Thus, the nonlinear friction is a result of nonlinear non-equilibrium thermodynamics in contrast to the traditional mechanical nonlinear friction forces. A quantum particle in vacuum is described by the Schrodinger equation

Brownian motion theory is always challenging how to describe diffusion phenomena with the main issue is how to extend the classical theory of Brownian motion to the special relativity framework. In this study, we formulated dynamics and distribution of a Brownian particle in relativistic framework by using Langevin and Fokker-Planck equation. By representing Brownian particle dynamics by Langevin equation, the velocity curves were dependent on the presence of viscous friction coefficient (heat bath), and were used generalized in special relativity theory, A relativistic Langevin equation reduces to the classical theory at low velocities. Likewise, the distribution of Brownian particles is represented as a stationary solution of the relativistic Fokker-Planck equation. From numerical results, we found that the probability density in the relativistic Fokker-Planck equation for was reduced to the standard Fokker-Planck equation in Netownian classical theory. For the calculation result showed that the Hanggi-Klimontovich approach has a consistent result to the relativistic Maxwell distribution. This work could open a promising interpretation to formulate the diffusion phenomena into general relativity theory.

In both the kinetic theory and in the theory of Brownian motion we dealt with the transition from the kinetic description in six-dimensional phase space to a more coarse description in the space of three dimensions. In Ch. 11, for example, it was the transition from the Boltzmann equation to the comparatively simple equations of gas dynamics. Then, in the theory of Brownian motion, we went over from Fokker — Planck equation for the distribution function f(r,v,t) to the simpler Einstein — Smoluchowski equation in the spatial distribution f(r,t). As indicated in Sect. 15.10, this transition corresponds to the simplified hydrodynamic description of Brownian motion

The statistical mechanics of arbitrary holonomic scleronomous systems subjected to arbitrary external forces is described by specializing the Lagrange and Hamilton equations of motion to those of the Brownian motion on a manifold. In this context, the Klein‐Kramers and Smoluchowski equations are derived in covariant form, and it is demonstrated that these equations have equilibrium solutions corresponding to the Gibbs distribution, in agreement with standard thermodynamics. At last, the Langevin dynamics corresponding to the Smoluchowski limit is found to exactly correspond to the Brownian motion on a smooth manifold. These results find significant applications in the study of several statistical properties of constrained molecular assemblies (e.g. polymers) of interest in chemistry, physics and biology.

Under the conditions of developed fluidization there are intensive fluctuations both in the fluidizing medium and in the dispersed solid phase. These motions have a decisive effect on the rheologlcal properties of the fluidized bed, and on the chemical reactions and transport processes taking place in it [1], Thus, for example, in the experiments of Wicke and Fetting [2], who investigated the heat transfer between a fluidized bed and the walls of a heated container, the effective heat transfer coefficient was found to be higher by an order of magnitude than the corresponding result for a fluidized bed held down by a wire grid so that the random motion of the solid phase was reduced. It is clear that the initial stage of any study of the structure of the fluidized bed as a whole, and of the subsequent development of any model, must involve an investigation of local structural properties, including the above fluctuations. The time variation of the individual particle velocities is due to two different causes. First, there is the interaction between the particles both through direct collisions and through the medium of the liquid phase, and, secondly, there is the interaction with the viscous fluid. These two factors are not independent, so that the set of fluidized particles has certain features characteristic for both a dense gas, with a potential intramolecular interaction, and a set of particles executing Brownian motion in a continuous medium. Any detailed statistical theory of a system of fluidized particles must be based on a representation of the random particle motions in the medium by a stochastic process with some definite properties (see, for example, [3–4]). Ideally, this theory should lead to the formulation of a transport equation which, in view of the above properties of the system, should have some of the features of both the usual Boltzman transport equation and the Fokker-Planck equation. The solution of this final equation is, of course, more difficult than the solution of the Boltzman or Fokker-Planck equations. Moreover, there is also the problem of applying this equation to different special cases. An alternative approach is to develop an approximate, but still sufficiently effective, theory of the local properties of the fluidized bed, which would combine relative simplicity in application with sufficient rigor and generality. This kind of theory is put forward in the present paper. The conclusions to which it leads are in good qualitative agreement with experiment.

Fractional Brownian Motions, Fractional Noises and Applications

We investigate the evolutionary dynamics of a Moran model which consists of subdivided populations of individuals confined to a set of islands. In subdivided populations, migration acts with selection and genetic drift to determine the evolutionary dynamics. The individuals are assumed to be haploid with two types. They reproduce according to their fitness values, die at random, and migrate between the islands. The evolutionary dynamics of an individual based model is formulated in terms of a master equation and is approximated as the multidimensional Fokker-Planck equation (FPE) and the coupled non-linear stochastic differential equations (SDEs) with multiplicative noise. We first analyze the deterministic part of the SDEs to obtain the fixed points and determine the stability of each fixed point. We find that there is only one stable fixed point to which all populations started from any initial distribution other than the unstable absorbing points will evolve in both the symmetric and antisymmetric selection schemes. Next, we take demographic stochasticity into account and analyze the FPE by eliminating the fast variable to reduce the coupled two-variable FPE to the single-variable FPE. We derive a quasi-stationary distribution of the reduced FPE and predict the fixation probabilities and the mean fixation times to absorbing states. We also carry out numerical simulations in the form of the Gillespie algorithm and find that the results of simulations are consistent with the analytic predictions.

We investigate the stochastic dynamics of a model which consists of subdivided populations of individuals confined to a set of islands. In subdivided populations, migration acts with selection and genetic drift to determine the evolutionary dynamics. The individuals are assumed to be haploid with two types. They reproduce according to their fitness values, die at random, and migrate between the islands. The evolutionary dynamics of an individual-based model is formulated in terms of a master equation and is approximated as the multidimensional Fokker–Planck equation (FPE) and the coupled non-linear stochastic differential equations (SDEs) with multiplicative noise. We first analyze the deterministic part of the SDEs, to obtain the fixed points and determine the stability of each fixed point. We find that there is a continuous phase transition in the population distribution when the migration rate is equal to the selection strength in the antisymmetric selection scheme. Next, we take demographic stochasticity into account, and analyze the FPE by eliminating the fast variable to reduce the coupled two-variable FPE to the single-variable FPE. We derive a quasi-stationary distribution of the reduced FPE, and predict the fixation probabilities and the mean fixation times to absorbing states. We also carry out numerical simulations, in the form of the Gillespie algorithm, and find that the results of simulations are consistent with the analytic predictions.

This chapter is devoted to the study of Langevin equations, first order in time differential equations, which depend on a random noise, and which belong to a class of stochastic differential equations that describe diffusion processes, or random motion. From a Langevin equation, a Fokker–Planck (FP) equation for the probability distribution of the solutions, at given time, of the Langevin equation can be derived. It is shown that observables averaged over the noise can also be calculated from path integrals, whose integrands define automatically positive measures. The path integrals involve dynamic actions that have automatically a Becchi–Rouet–Stora–Tyutin (BRST) symmetry and, when the driving force derives from a potential, exhibit the simplest form of supersymmetry. In some cases, like Brownian motion on Riemannian manifolds, difficulties appear in the precise definition of stochastic equations, quite similar to the quantization problem encountered in quantum mechanics (QM). Time discretization provides one possible solution to the problem.

We prove that stochastic replicator dynamics can be interpreted as intrinsic Brownian motion on the simplex equipped with the Aitchison geometry. As an immediate consequence, we derive three approximation results in the spirit of Wong–Zakai approximation, Donsker’s invariance principle and a JKO-scheme. Using the Fokker–Planck equation and Wasserstein-contraction estimates, we also study the long time behavior of the stochastic replicator equation, as an example of a nongradient drift diffusion on the Aitchison simplex.

Transient shear flow data of untreated multiwalled carbon nanotubes (MWCNTs) dispersed in a Newtonian epoxy matrix are analyzed. A sequence of shearing and rest steps was applied to characterize the transient responses of the suspensions. Stress overshoots appeared at very small deformation during forward and reverse flow experiments and their intensity increased with rest time between two consecutive flows, during which the suspension structure was reconstructed. The transient behavior of the MWCNT suspensions is explained with the help of a recently proposed model [G. Natale et al., AIChE J. 60(4), 1476–1487 (2014)]. The MWCNTs are described as rigid rods dispersed in a Newtonian matrix, and the evolution of the system is controlled by hydrodynamics, rod-rod interactions, and Brownian motion. The force due to the interactions is modeled as a nonlinear lubrication force and the total stress tensor is evaluated introducing a fourth-order interaction tensor. The Fokker–Planck equation is numerically solved for transient simple shear flow using a finite volume method, avoiding the need of closure approximations. The model predictions show that interactions slow down the orientation evolution of the rods. For the first time, the effect of shear rate is directly accounted by the model, which predicts that a critical shear rate is necessary to break down the structure and let the rods orient in the flow direction. In addition, we confronted the model predictions with the rheological data of a glass fiber-filled polybutene [M. Sepehr et al., 48(5), 1023–1048 (2004)], demonstrating its ability to describe the behavior of micro and nano-scale particle suspensions.