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.

This study examined the role of feedback from cutaneous mechanoreceptors in the stability of human upright posture. A two-link, one degree of freedom, inverted pendulum model was constructed for the human body with ankle joint torque proportional to the delayed outputs from muscle receptors, joint receptors, and cutaneous mechanoreceptors in the foot. Theoretical analysis and numerical simulations indicated that the use of mechanoreceptive information reduced the frequency range and the maximum peak-peak value of the dynamic response of the system. However, without the use of muscle receptors, the mechanoreceptive feedback could not stabilize the system. In addition, body movement of human subjects was measured when their balanced upright posture was disturbed by a transient, forward/backward movement of a supporting platform. The loss of or change in cutaneous mechanoreceptive sense in their feet was induced by (a) having healthy subjects stand on a soft surface and (b) testing neuropathic patients with loss of vibratory sensation in their feet. The results showed significant increases in frequency range and maximum peak-peak value of ankle rotation and velocity for subjects standing on a soft (vs. hard) surface and for neuropathic patients (vs. age- and gender-matched healthy subjects).

Historical Background and Introductory Concepts Methods for Solving Langevin and Fokker-Planck Equations Matrix Continued Fractions Escape Rate Theory Linear and Nonlinear Response Theory Brownian Motion of a Free Particle and a Harmonic Oscillator Rotational Brownian Motion about a Fixed Axis in a Periodic Potential Brownian Motion in a Tilted Periodic Potential: Application to the Josephson Tunnelling Junction and Ring Lasers Brownian Motion in a Double-Well Potential Isotropic and Anisotropic Rotational Brownian Motion in Space in the Presence of an External Potential with Applications to Dielectric and Kerr Effect Relaxation in Fluids and Liquid Crystals Brownian Motion of Classical Spins with Applications to Superparamagnetism Magnetic Stochastic Resonance Dynamic Hysteresis Switching Field Surfaces Inertial Langevin Equations with Applications to Orientational Relaxation in Liquids Itinerant Oscillator Model Anomalous Diffusion Continuous Time Random Walks Methods for the Solution of Fractional Fokker-Planck Equations.

The topic of the present work are non-Brownian particles in shear flow. As reported in literature, the occurring phenomenon in this context is shear-induced diffusion which takes place in the absence of the well-known Brownian diffusion. Diffusive processes can be described stochastically in terms of a stochastic differential equation (Langevin equation or Langevin-like equation) or a differential equation for the probability density, in second order referred to as Fokker-Planck equation. It is known that in contrast to Brownian diffusion, the shear-induced diffusion is a long-time diffusion which poses a challenge to the stochastic description of this phenomenon. The present work analyzes the problem of non-Brownian particles in shear-induced diffusion with regard to the Markov property of the treated variables. This concludes that the Fokker-Planck equation so far derived in pure position space may not be sufficient. In order to ensure the Markov process property, a Fokker-Planck equation extended to coupled position colored-noise velocity space is derived. Throughout the extension, the colored-noise velocity is modeled as an Ornstein-Uhlenbeck process. These first two steps were also treated in the author’s Master thesis (Lukassen 2012). A detailed substantiation of this approach is published in (Lukassen & Oberlack 2014b) including a new multiple time scale analysis and a Gaussian solution. The multiple time scale analysis results in the dimensionless form of the equation of motion which serves as a starting point for the derivation of the new colored-noise Fokker-Planck equation. As a next step, this coupled Fokker-Planck equation is integrated over velocity space and approximated to yield a reduced position-space Fokker-Planck equation. It is shown that such a reduction as in the present work is only possible under certain conditions concerning the correlation time. The resulting position-space equation is analyzed and compared to the traditional position-space models. The reduced form exhibits additional correction terms. In an outlook, possible extensions of the presented model are discussed with exemplary simulation results. Chapter 5 and 6 as a whole are based on the author’s publication (Lukassen & Oberlack 2014b).

The transport characteristics of collective agent motion, governed by a combination of anomalous drift and Brownian motion, are investigated. The anomalous drift is modeled as a power-law function of the magnitude of the agents’ peculiar velocity. In the mean-field limit, the Fokker–Planck equation incorporates both anomalous drift and Brownian motion. Assuming that the velocity distribution function is weakly nonequilibrium, transport coefficients are derived by applying the first Maxwellian iteration to Grad’s 13-moment equations, which are obtained from the Fokker–Planck equation. However, when anomalous drift is included, the fluctuation amplitude–i.e. the diffusion coefficient–depends on the nonequilibrium state in order to conserve energy. Consequently, the expansion of the velocity distribution function using modified Hermite polynomials becomes inadequate under strongly nonequilibrium conditions, because the diffusion coefficient cannot be expressed solely in terms of temperature. As the drift becomes super-drift, the collective motion of agents asymptotically approaches a perfect fluid. Finally, the convergence of the velocity distribution function toward the equilibrium distribution, as governed by the Fokker–Planck equation, is confirmed, numerically. Transport coefficients obtained via their Green–Kubo expression using the direct simulation Monte Carlo method are compared with their corresponding analytical results.

This chapter extends the discussion of stochastic differential equations and Fokker–Planck equations on Euclidean space initiated in Chapter 4 to the case of processes that evolve on a Riemannian manifold. The manifold either can be embedded in ℝn or can be an abstract manifold with Riemannian metric defined in coordinates. Section 8.1 formulates SDEs and Fokker–Planck equations in a coordinate patch. Section 8.2 formulates SDEs for an implicitly defined embedded manifold using Cartesian coordinates in the ambient space. Section 8.3 focuses on Stratonovich SDEs on manifolds. The subtleties involved in the conversion between Ito and Stratonovich formulations are explained. Section 8.4 explores entropy inequalities on manifolds. In Section 8.5 the following examples are used to illustrate the general methodology: (1) Brownian motion on the sphere and (2) the stochastic kinematic cart described in Chapter 1. Section 8.6 discusses methods for solving Fokker–Planck equations on manifolds. Exercises involving numerical implementations are provided at the end of the chapter. The main points to take away from this chapter are: SDEs and Fokker–Planck equations can be formulated for stochastic processes in any coordinate patch of a manifold in a way that is very similar to the case of Rn; Stochastic processes on embedded manifolds can also be formulated extrinsically, i.e., using an implicit description of the manifold as a system of constraint equations; In some cases Fokker–Planck equations can be solved using separation of variables; Practical examples of this theory include Brownian motion on the sphere and the kinematic cart with noise.

When formulated properly, most geophysical transport-type process involving passive scalars or motile particles may be described by the same space–time nonlocal field equation which consists of a classical mass balance coupled with a space–time nonlocal convective/dispersive flux. Specific examples employed here include stretched and compressed Brownian motion, diffusion in slit-nanopores, subdiffusive continuous-time random walks (CTRW), super diffusion in the turbulent atmosphere and dispersion of motile and passive particles in fractal porous media. Stretched and compressed Brownian motion, which may be thought of as Brownian motions run with nonlinear clocks, are defined as the limit processes of a special class of random walks possessing nonstationary increments. The limit process has a mean square displacement that increases as tα+1 where α > −1 is a constant. If α = 0 the process is classical Brownian, if α < 0 we say the process is compressed Brownian while if α > 0 it is stretched. The Fokker–Planck equations for these processes are classical ade’s with dispersion coefficient proportional to tα. The Brownian-type walks have fixed time step, but nonstationary spatial increments that are Gaussian with power law variance. With the CTRW, both the time increment and the spatial increment are random. The subdiffusive Fokker–Planck equation is fractional in time for the CTRW’s considered in this article. The second moments for a Levy spatial trajectory are infinite while the Fokker–Planck equation is an advective–dispersive equation, ade, with constant diffusion coefficient and fractional spatial derivatives. If the Lagrangian velocity is assumed Levy rather than the position, then a similar Fokker–Planck equation is obtained, but the diffusion coefficient is a power law in time. All these Fokker–Planck equations are special cases of the general non-local balance law.

The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto $c$-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.

Brownian motion theory is always challenging to describe diffusion phenomena around a black hole, with the main issue is how to extend the classical theory of Brownian motion to the general relativity framework. In this study, we extended the Brownian motion theory in a curved space-time come from a strong gravitational field on the Schwarzschild black hole. The Brownian motion theory in Schwarzschild space-time was derived by using the Fokker-Planck equation, and the stationary solution was analyzed by Ito, Stratonovich-Fisk, and Hanggi-Klimontovich Approach. The numerical result was found that the Brownian motion in Schwarzschild space-time 1   was reduced to the standard Brownian motion in Newtonian classical theory. According to the Hanggi-Klimontovich approach for 1   the result showed a consistent with the relativistic Maxwell distribution. The Fokker-Planck equation in Schwarzschild space-time was also formulated as a generalization of relativistic Brownian motion theory. This work could open a promising interpretation to formulate the diffusion phenomena around a massive object in the general relativity framework.

Invalidity of the spectral Fokker–Planck equation for Cauchy noise driven Langevin equation

Brownian motion is a universal characteristic of colloidal particles embedded in a host medium, and it is the fingerprint of molecular transport or diffusion, a generic feature of relevance not only in physics but also in several branches of science and engineering. Since its discovery, Brownian motion, also known as colloidal dynamics, has been important in elucidating the connection between the molecular details of the diffusing macromolecule and the macroscopic information on the host medium. However, colloidal dynamics is far from being completely understood. For instance, the diffusion of non-spherical colloids and the effects of the underlying geometry of the host medium on the dynamics of either passive or active particles are a few representative cases that are part of the current challenges in soft matter physics. In this contribution, we take a step forward to introduce a covariant description of the colloidal dynamics in curved spaces. Without the loss of generality, we consider the case where hydrodynamic interactions are neglected. This formalism will allow us to understand several phenomena, for instance, the curvature effects on the kinetics during spinodal decomposition and the thermodynamic properties of colloidal dispersion, to mention a few examples. This theoretical framework will also serve as the starting point to highlight the role of geometry on colloidal dynamics, an aspect that is of paramount importance to understanding more complex transport phenomena, such as the diffusive mechanisms of proteins embedded in cell membranes.

Using the quantum-mechanical Boltzmann equation we study the motion of a heavy ion (mass M) moving through a Fermi fluid of light particles [mass m, γ=(m/M)½≪1] under the influence of an electric field. We find, contrary to the classical case, that the condition γ≪1 is not sufficient to ensure that the ion undergoes Brownian motion (i.e., its motion is characterized by a Fokker—Planck equation). The condition γξ≪1 must hold, where ξ2 is the ratio of the Fermi energy to the temperature. When ξ is too large the ion will suffer large momentum changes upon colliding with the fermions and hence its motion will not be describable by a Fokker—Planck equation. In liquid 3He there is a temperature region (4°&amp;gt;T&amp;gt;1°K) in which a heavy ion experiences what we call quantum-mechanical Brownian motion; the Fokker—Planck equation still holds but the friction coefficient contains quantum-mechanical effects. Using our formula for mobility and the mobility data of Meyer we estimate the effective mass of an ion in liquid 3He to be about 20 times the mass of an 3He atom. This is in agreement with the recent measurements of Dahm and Sanders in liquid 4He. They have found the effective mass to be between 20 to 40 4He atoms. However, since only binary scattering events are accounted for in our mobility formula, its application to liquid 3He should be taken with reservation.

The Brownian motion in a harmonic trap is studied by magnetic tweezers experiment and computer simulation. The results of the experiment and simulation validate the theory. Then the theory is used to analyze the experimental results including the effect of persistent length of DNA on the displacement distribution of the bead and the error in force measurements. It can be concluded that the variation of the persistent length affects more the Brownian motion along the DNA chain than in the other direction; under a small force, a considerable error of the force measurement will occur.

Fractional Brownian motion with complex variance via random walk in the complex plane and applications

Differential equations governing the time evolution of distribution functions for Brownian motion in the full phase space were first derived independently by Klein and Kramers. From these so-called Fokker-Planck equations one may derive the reduced differential equations in coordinate space known as Smoluchowski equations. Many such derivations have previously been reported, but these either involved unnecessary assumptions or approximations, or were performed incompletely. We employ an iterative reduction scheme, free of assumptions, and calculate formally exact corrections to the Smoluchowski equations for many-particle systems with and without hydrodynamic interaction, and for a single particle in an external field. In the absence of hydrodynamic interaction, the lowest order corrections have been expressed explicitly in terms of the coordinate space distribution function. An additional application of the method is made to the reduction of the stress tensor used in evaluating the intrinsic viscosity of particles in solution. Most of the present work is based on classical Brownian motion theory, but brief consideration is given in an appendix to some recent developments regarding non-Markovian equations for Brownian motion.