Brownian motion in a quantizing magnetic field

A model previously discussed by the author to study Brownian motion of charged carriers in a quantizing magnetic field is extended to include a Landau level-dependent friction parameter. A phase-space Fokker-Planck equation is used to derive a generalized diffusion equation describing spatial diffusion of the carriers, coupled with random jumps between adjacent Landau levels. This partial differential-difference equation is solved analytically. The longitudinal “global” diffusion coefficient is calculated and shown to be enhanced over the value in the extreme quantum limit.     

限流扩散理论的局限性及其改进

对几种典型的限流扩散理论的局限性与含时间吸收项的渐近扩散理论的长处作了分析和数字计算比较。结论是,理论上所导出的限流扩散方程,虽然形式上解决了中子迁移的限流问题,但由于理论推导中所作的假定带有一定的局限性,往往使限流过度;而带有时间吸收项的渐近扩散理论避免了上述的局限性,因而它对中子迁移问题的描述优于限流扩散理论。     

Fluctuation-dissipation theorems for systems in non-thermal equilibrium and applications

The linear response theory of systems obeying Fokker-Planck equations is discussed under the assumption that the principle of detailed balance is satisfied. This theory is used to obtain fluctuation-dissipation theorems for systems in a non-thermal equilibrium state corresponding to the steady state solution of the Fokker-Planck equation. Some of the aspects of the threshold region of a single mode laser (without detuning) are discussed. In particular it is shown that, contrary to the results established recently by others, the “linear susceptibility” χ(Ω) is acontinuous function of the pump parameterp. The results are then specialized to Gaussian Markov process in which case an alternate form of the fluctuation-dissipation theorems is also given. Finally few other generalizations of the fluctuation-dissipation theorems are also briefly discussed.     

Kinetic and hydrodynamic models of chemotactic aggregation

We derive general kinetic and hydrodynamic models of chemotactic aggregation that describe certain features of the morphogenesis of biological colonies (like bacteria, amoebae, endothelial cells or social insects). Starting from a stochastic model defined in terms of N coupled Langevin equations, we derive a nonlinear mean-field Fokker-Planck equation governing the evolution of the distribution function of the system in phase space. By taking the successive moments of this kinetic equation and using a local thermodynamic equilibrium condition, we derive a set of hydrodynamic equations involving a damping term. In the limit of small frictions, we obtain a hyperbolic model describing the formation of network patterns (filaments) and in the limit of strong frictions we obtain a parabolic model which is a generalization of the standard Keller-Segel model describing the formation of clusters (clumps). Our approach connects and generalizes several models introduced in the chemotactic literature. We discuss the analogy between bacterial colonies and self-gravitating systems and between the chemotactic collapse and the gravitational collapse (Jeans instability). We also show that the basic equations of chemotaxis are similar to nonlinear mean-field Fokker-Planck equations so that a notion of effective generalized thermodynamics can be developed.     

Fokker-Planck equation for interacting waves: Dispersion law renormalization and wave turbulence

The multiple timescale method for removing secularities is used to generate the Fokker-Planck (“FP”) equation for a system of interacting waves. This FP equation describes diffusion in the phase space of the angle, as well as the action, variables of all underlying modes. The first moment of the FP equation gives a kinetic (or Boltzmann-type) equation governing the averaged actions, and describing the diffusion of action in time. Angle diffusion leads to a renormalization of the dispersion law. Stationary solutions for the average action (or so-called spectral intensity) are derived for equilibrium and for the driven off-equilibrium state corresponding to a cascade of wave energy from low to high frequencies (wave turbulence). The reduced distribution function for these states is derived.The derivation of the FP equation from the Liouville equation, as well as the derivation of the kinetic equation from the FP equation, requires that the distribution of modes be sufficiently dense. In this limit, cumulants that are initially zero increase at a rate that is thermodynamically sm all. A Langevin equation, governing the evolution of a distinguished oscillator, that is applicable even in off-equilibrium conditions, is derived. The concept of winding numbers is extended to the general phase space motion of action-angle variables through the introduction of a multiple-valued probability density.     

A hydrodynamic approach to multidimensional dissipation-based Schrödinger models from quantum Fokker-Planck dynamics

In this paper we are concerned with the modeling of quantum dissipation and diffusion effects at the level of the multidimensional Schrödinger equation. Our starting point is the quantum Fokker-Planck master equation describing dissipative interactions (of mass and energy) of the particle ensemble with a thermal bath in thermodynamic equilibrium. When considering its associated hydrodynamic system, which rules the temporal evolution of the local density and the mean fluid-flow velocity, and imposing physically admissible closure relations, these equations can be seen as describing the fluid-mechanical evolution of the macroscopic amplitude and phase of an envelope wavefunction, thus giving rise to a family of dissipative Schrödinger equations of logarithmic type whose steady state and radial dynamics are analyzed. Also, numerical comparison with the exactly solvable models for the free particle and the damped harmonic oscillator is performed.     

A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims made in the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T. D. Frank [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker-Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker-Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was offered that a Chapman-Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a “non-linear Markov process”, nor a “non-linear Fokker-Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker-Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the Appendix A we present the theory of Fokker-Planck pdes and Chapman-Kolmogorov equations for stochastic processes with finite memory.     

Expansion of the kinetic hierarchy for a massive particle: The repeated ring and Fokker-Planck equations

The tagged particle BBGKY hierarchy is systematically expanded in inverse powers of the square root of the particle mass. In the Brownian limit, for fixed Knudsen number, the hierarchy reduces to the Brownian limit of the repeated ring equation which itself reduces to the Fokker-Planck equation. The friction coefficient of the Fokker-Planck equation is found to be a functional of the solution of Dorfman, van Beijeren, and McClure's extended Boltzmann equation for a fixed object in a flowing gas. As a consequence, the tagged particle diffusion coefficient calculated in the Brownian limit of the repeated ring equation is valid for all particle sizes.     

The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

Relaxation dynamics of a quantum Brownian particle in an ideal gas

We show how the quantum analog of the Fokker-Planck equation for describing Brownian motion can be obtained as the diffusive limit of the quantum linear Boltzmann equation. The latter describes the quantum dynamics of a tracer particle in a dilute, ideal gas by means of a translation-covariant master equation. We discuss the type of approximations required to obtain the generalized form of the Caldeira-Leggett master equation, along with their physical justification. Microscopic expressions for the diffusion and relaxation coefficients are obtained by analyzing the limiting form of the equation in both the Schr?dinger and the Heisenberg picture.     

Statistical mechanics of nonlinear hydrodynamic fluctuations

The paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics. The generalized Fokker-Planck equation for the distribution function of coarse-grained densities of conserved quantities is derived from the Liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix. We have obtained the functional-differential Fokker-Planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse-grained fluctuating fields. An outline of the derivation of Fokker-Planck equations containing the Burnett terms is also given. The explicit coordinate representation for the hydrodynamic Fokker-Planck equation is discussed in the case of one-component simple fluid. The general scheme of a change of coarse-grained functional variables is developed for hydrodynamic Fokker-Planck equations. The corresponding transformation rules are found for “drift” terms, “diffusion coefficients” and thermodynamic forces. The dynamical equations and stationary conditions for averages of functions (functionals) of hydrodynamic fields are discussed by using the Fokker-Planck operators acting on such functions. The explicit form of these operators are found for various sets of fluctuating fields. As an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state.     

Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N→+∞, it reduces to the Vlasov equation governing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasistationary state (QSS) on the coarse-grained scale. We interpret the physical nature of the QSS in relation to Lynden-Bell’s statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-Markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interesting factor in our approach is the development of a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.     

A generalised model of hydrogen diffusion in metals with multiple trap types

Continuum modelling of hydrogen diffusion in metals, which accounts for both trapping and an imposed force field, is revisited. A generalised model of hydrogen diffusion and trapping is developed as a continuous interpretation of the discrete random-walk theory. A system of nonlinear equations describing the phenomenon of diffusion with multiple types of traps is derived without the assumption of a local equilibrium among hydrogen populations in dissimilar positions. Lattice-trap interchange kinetics can degenerate into local equilibrium as a limit case. Moreover, certain terms in general equations may be negligible in specific situations. By removing these terms, known particularised models of hydrogen diffusion and trapping are recovered. Determining the terms, which are disregarded in reduced models, enables a straightforward assessment of the applicability of these models. The advantages and limitations of particularised models applied to hydrogen embrittlement analyses are discussed.     

Quantum tunneling at zero temperature in the strong friction regime

In the large damping limit we derive a Fokker-Planck equation in configuration space (the so-called Smoluchowski equation) describing a Brownian particle immersed into a thermal environment and subjected to a nonlinear external force. We quantize this stochastic system and survey the problem of escape over a double-well potential barrier. Our finding is that the quantum Kramers rate does not depend on the friction coefficient at low temperatures; i.e., we predict a superfluidity phenomenon in overdamped open systems. Moreover, at zero temperature we show that the quantum escape rate does not vanish in the strong friction regime. This result, therefore, is in contrast with the work by Ankerhold et al. [Phys. Rev. Lett. 87, 086802 (2001)]] in which no quantum tunneling is predicted at zero temperature.     

The diffusion in the quantum Smoluchowski equation

A novel quantum Smoluchowski dynamics in an external, nonlinear potential has been derived recently. In its original form, this overdamped quantum dynamics is not compatible with the second law of thermodynamics if applied to periodic, but asymmetric ratchet potentials. An improved version of the quantum Smoluchowski equation with a modified diffusion function has been put forward in L. Machura et al. (Phys. Rev. E 70 (2004) 031107) and applied to study quantum Brownian motors in overdamped, arbitrarily shaped ratchet potentials. With this work we prove that the proposed diffusion function, which is assumed to depend (in the limit of strong friction) on the second-order derivative of the potential, is uniquely determined from the validity of the second law of thermodynamics in thermal, undriven equilibrium. Put differently, no approximation-induced quantum Maxwell demon is operating in thermal equilibrium. Furthermore, the leading quantum corrections correctly render the dissipative quantum equilibrium state, which distinctly differs from the corresponding Gibbs state that characterizes the weak (vanishing) coupling limit.     

Chapman-Enskog development of the multivariate master equation

A systematic development for a Multivariate Master Equation (MME), describing reaction diffusion systems, is presented along the same lines as the Chapman-Enskog development of kinetic gas theory. Diffusion, which occurs at a very fast rate, brings the system near a state of diffusional equilibrium, the analogue of local equilibrium for gases. In diffusional equilibrium, the global Master Equation (ME) is shown to be an exact consequence of the MME. For finite systems, corrections to the global ME, resulting from the finiteness of the diffusion times, are calculated. The development is verified on an exactly solvable model and illustrated on the Schlögl model. The difficulties encountered in the thermodynamic limit are discussed, and possible outcomes suggested.     

Structure and size distribution of percolating clusters. Comparison with gelling systems

Dynamic properties of Brownian particles immersed in a periodic potential with two barriers V₁ and V₂ (symmetric bistable potential) are studied by using the Fokker-Planck equation which we solve numerically by the matrix continued fraction method. This study will therefore serve to demonstrate the influence of this form of potential, which is of great interest for superionic conductors and for many other solid systems, on the diffusion process. Thus, we have calculated the full width at half maximum (FWHM) ) of the quasi-elastic line of the dynamic structure factor, for a large range of values of the wave-vectors q. Our results show clearly that, by varying the ratio of the barriers strictly between and 1, the Fokker-Planck equation describes a diffusive process which has some characteristic of jump and liquid-like regimes. While in the limit cases, i.e. when tends to or 1, the diffusion process can be described only by a simple jump motion. However, the jump-lengths corresponding to each limit case are not equal. In general the change of the ratio is found to have a significant effect on the character of the diffusive motion. We have also performed Fokker-Planck dynamics calculations of the diffusion coefficient in a bistable potential. We have found a good agreement between numerical calculations and analytical approximation results obtained in the high friction limit. Received 25 May 1998 and Received in final form 15 November 1998