Fast and superfast diffusion processes

Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.

Capillary transport in paper porous materials at low saturation levels: normal, fast or superfast?

The problem of capillary transport in fibrous porous materials at low levels of liquid saturation has been addressed. It has been demonstrated that the process of liquid spreading in this type of porous material at low saturation can be described macroscopically by a similar super-fast, nonlinear diffusion model to that which had been previously identified in experiments and simulations in particulate porous media. The macroscopic diffusion model has been underpinned by simulations using a microscopic network model. The theoretical results have been qualitatively compared with available experimental observations within the witness card technique using persistent liquids. The long-term evolution of the wetting spots was found to be truly universal and fully in line with the mathematical model developed. The result has important repercussions for the witness card technique used in field measurements of the dissemination of various low-volatility agents in imposing severe restrictions on collection and measurement times.

Mathematical and information-geometrical entropy for phenomenological Fourier and non-Fourier heat conduction

The second law of thermodynamics governs the direction of heat transport, which provides the foundational definition of thermodynamic Clausius entropy. The definitions of entropy are further generalized for the phenomenological heat transport models in the frameworks of classical irreversible thermodynamics and extended irreversible thermodynamics (EIT). In this work, entropic functions from mathematics are combined with phenomenological heat conduction models and connected to several information-geometrical conceptions. The long-time behaviors of these mathematical entropies exhibit a wide diversity and physical pictures in phenomenological heat conductions, including the tendency to thermal equilibrium, and exponential decay of nonequilibrium and asymptotics, which build a bridge between the macroscopic and microscopic modelings. In contrast with the EIT entropies, the mathematical entropies expressed in terms of the internal energy function can avoid singularity paired with nonpositive local absolute temperature caused by non-Fourier heat conduction models.

Nonlinear diffusion effects on biological population spatial patterns

Motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of biological populations, we investigate its implications in a paradigmatic model for the evolution of a single species density u(x,t). The standard model includes growth and competition in a logistic expression, and spreading is modeled through normal diffusion. Moreover, the competition term is nonlocal, which has been shown to give rise to spatial patterns. We generalize the diffusion term through the nonlinear form ∂tu(x,t)=D∂xxu(x,t)ν (with D,ν>0), encompassing the cases where the state-dependent diffusion coefficient either increases (ν>1) or decreases (ν<1) with the density, yielding subdiffusion or superdiffusion, respectively. By means of numerical simulations and analytical considerations, we display how that nonlinearity alters the phase diagram. The type of diffusion imposes critical values of the model parameters for the onset of patterns and strongly influences their shape, inducing fragmentation in the subdiffusive case. The detection of the main persistent mode allows analytical prediction of the critical thresholds.

Brownian pump in nonlinear diffusive media

A Brownian pump in nonlinear diffusive media is investigated in the presence of an unbiased external force. The pumping system is embedded in a finite region and bounded by two particle reservoirs. In the adiabatic limit, we obtain the analytical expressions of the current and the pumping capacity as a function of temperature for normal diffusion, subdiffusion, and superdiffusion. It is found that important anomalies are detected in comparison with the normal diffusion case. The superdiffusive regime, compared with the normal one, exhibits an opposite current for low temperatures. In the subdiffusive regime, the current may become forbidden for low temperatures and negative for high temperatures.

Brownian motors in nonlinear diffusive media

We investigate the performance of Brownian motors in environments governed by the "porous medium" equation partial differential(t)rho = D partial differential(x) (rho(nu-1) partial differential(x)rho), where rho is the density and D and nu positive constants. This nonlinear equation yields anomalous diffusion when nu not = 1: subdiffusion for nu > 1 and superdiffusion for nu < 1. The thermal ratchet is modeled by an overdamped Brownian particle subject to a one-dimensional, spatially periodic, asymmetric potential, modulated by time-periodic fluctuations. We scrutinize the transport properties in the adiabatic limit. The superdiffusive regime, in comparison with the normal one, exhibits transport enhancement for small amplitudes of the temporal fluctuations. Meanwhile, the subdiffusive regime displays more strident features: The flux may become forbidden in one or in both directions. As a consequence, when the blockade is unidirectional, purely directed transport occurs.

Logarithmic diffusion and porous media equations: a unified description

In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this equation. Its solution has a Lorentzian form, consequently this equation characterizes a superdiffusion like a Lévy kind. In addition an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized diffusion equation in fractal dimension, is obtained. This unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes.

Exact solutions to nonlinear nonautonomous space-fractional diffusion equations with absorption

We analyze a nonlinear fractional diffusion equation with absorption by employing fractional spatial derivatives and obtain some more exact classes of solutions. In particular, the diffusion equation employed here extends some known diffusion equations such as the porous medium equation and the thin film equation. We also discuss some implications by considering a diffusion coefficient D(x,t)=D(t)/x/(-theta) (theta in R) and a drift force F=-k(1)(t)x+k(alpha)x/x/(alpha-1). In both situations, we relate our solutions to those obtained within the maximum entropy principle by using the Tsallis entropy.

Stability analysis of mean-field-type nonlinear Fokker-Planck equations associated with a generalized entropy and its application to the self-gravitating system

Multidimensional nonlinear Fokker-Planck equations of mean-field type are proposed within the framework of generalized thermostatistics to develop a general formulation of stability analysis of their solutions. Two types of eigenvalue equations are studied. The nonlinear Fokker-Planck equations are shown to exhibit an H theorem with a Liapunov functional that takes the form of a free energy involving generalized entropies of Tsallis. The second-order variation of the Liapunov functional is computed to conduct local stability analysis and the associated eigenvalue equation is derived for an arbitrary form of mean-field coupling potential. Assuming quasiequilibrium for the velocity distribution, the reduced eigenvalue equation with space coordinates alone is also obtained. The alternative type of eigenvalue equation based on the linearization of the nonlinear Fokker-Planck equations is presented. Taking the mean-field coupling potential to be the gravitational one, the nonlinear Fokker-Planck equation in terms of three-dimensional velocity and space coordinates together with the framework of stability analysis is shown to be applicable to a mean-field model of self-gravitating system. By solving the eigenvalue equation for the eigenfunction with 0 eigenvalue, the occurrence of stability change of the equilibrium probability density with spherical symmetry is discussed.

Escape time in anomalous diffusive media

We investigate the escape behavior of systems governed by the one-dimensional nonlinear diffusion equation theta(t)rho=theta(x)[theta(x)Urho]+Dtheta(x)2rho(nu), where the potential of the drift, U(x), presents a double well and D,nu are real parameters. For systems close to the steady state, we obtain an analytical expression of the mean first-passage time, yielding a generalization of Arrhenius law. Analytical predictions are in very good agreement with numerical experiments performed through integration of the associated Ito-Langevin equation. For nu not equal to 1, important anomalies are detected in comparison to the standard Brownian case. These results are compared to those obtained numerically for initial conditions far from the steady state.

Logarithmically slow expansion of hot bubbles in gases

We predict a logarithmically slow expansion of hot bubbles in gases in the process of cooling. A model problem is first solved, when the temperature has compact support. Then the temperature profile decaying exponentially at large distances is considered. The periphery of the bubble is shown to remain essentially static ("glassy") in the process of cooling until it is taken over by a logarithmically slowly expanding "core." An analytical solution to the problem is obtained by matched asymptotic expansion. This problem gives an example of how logarithmic corrections enter dynamic scaling.