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Register a new userFractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries
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Fractional Brownian motion is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically fractional Brownian motion confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected fractional Brownian motion at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude $1/L$ in an interval of length $L$ found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement $langle X^2(t)ranglesimeq t^{alpha}$ with $1<alpha<2$, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion ($0<alpha<1$) this behaviour is reversed and the particle density is depleted close to the boundaries. The mean squared displacement in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent $alpha$. Our a priori surprising results may have interesting consequences for the application of fractional Brownian motion for processes such as molecule or tracer diffusion in the confined of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.
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Diffusive transport in many complex systems features a crossover between anomalous diffusion at short times and normal diffusion at long times. This behavior can be mathematically modeled by cutting off (tempering) beyond a mesoscopic correlation time the power-law correlations between the increments of fractional Brownian motion. Here, we investigate such tempered fractional Brownian motion confined to a finite interval by reflecting walls. Specifically, we explore how the tempering of the long-time correlations affects the strong accumulation and depletion of particles near reflecting boundaries recently discovered for untempered fractional Brownian motion. We find that exponential tempering introduces a characteristic size for the accumulation and depletion zones but does not affect the functional form of the probability density close to the wall. In contrast, power-law tempering leads to more complex behavior that differs between the superdiffusive and subdiffusive cases.
Complex systems display anomalous diffusion, whose signature is a space/time scaling $xsim t^delta$ with $delta e 1/2$ in the Probability Density Function (PDF). Anomalous diffusion can emerge jointly with both Gaussian, e.g., fractional Brownian motion, and power-law decaying distributions, e.g., Levy Flights (LFs) or Levy Walks (LWs). LFs get anomalous scaling, but also infinite position variance and also infinite energy and discontinuous velocity. LWs are based on random trapping events, resemble a Levy-type power-law distribution that is truncated in the large displacement range and have finite moments, finite energy and discontinuous velocity. However, both LFs and LWs cannot describe friction-diffusion processes. We propose and discuss a model describing a Heterogeneous Ensemble of Brownian Particles (HEBP) based on a linear Langevin equation. We show that, for proper distributions of relaxation time and velocity diffusivity, the HEBP displays features similar to LWs, in particular power-law decaying PDF, long-range correlations and anomalous diffusion, at the same time keeping finite position moments and finite energy. The main differences between the HEBP model and two LWs are investigated, finding that, even if the PDFs are similar, they differ in three main aspects: (i) LWs are biscaling, while HEBP is monoscaling; (ii) a transition from anomalous ($delta e 1/2$) to normal ($delta = 1/2$) diffusion in the long-time regime; (iii) the power-law index of the position PDF and the space/time diffusion scaling are independent in the HEBP, while they both depend on the scaling of the inter-event time PDF in LWs. The HEBP model is derived from a friction-diffusion process, it has finite energy and it satisfies the fluctuation-dissipation theorem.
We find the exact winding number distribution of Riemann-Liouville fractional Brownian motion for large times in two dimensions using the propagator of a free particle. The distribution is similar to the Brownian motion case and it is of Cauchy type. In addition we find the winding number distribution of fractal time process, i.e., time fractional Fokker-Planck equation, in the presence of finite size winding center.
Fractional Brownian motion is a non-Markovian Gaussian process indexed by the Hurst exponent $Hin [0,1]$, generalising standard Brownian motion to account for anomalous diffusion. Functionals of this process are important for practical applications as a standard reference point for non-equilibrium dynamics. We describe a perturbation expansion allowing us to evaluate many non-trivial observables analytically: We generalize the celebrated three arcsine-laws of standard Brownian motion. The functionals are: (i) the fraction of time the process remains positive, (ii) the time when the process last visits the origin, and (iii) the time when it achieves its maximum (or minimum). We derive expressions for the probability of these three functionals as an expansion in $epsilon = H-tfrac{1}{2}$, up to second order. We find that the three probabilities are different, except for $H=tfrac{1}{2}$ where they coincide. Our results are confirmed to high precision by numerical simulations.
Efficiency of search for randomly distributed targets is a prominent problem in many branches of the sciences. For the stochastic process of Levy walks, a specific range of optimal efficiencies was suggested under variation of search intrinsic and extrinsic environmental parameters. In this article, we study fractional Brownian motion as a search process, which under parameter variation generates all three basic types of diffusion, from sub- to normal to superdiffusion. In contrast to Levy walks, fractional Brownian motion defines a Gaussian stochastic process with power law memory yielding anti-persistent, respectively persistent motion. Computer simulations of search by time-discrete fractional Brownian motion in a uniformly random distribution of targets show that maximising search efficiencies sensitively depends on the definition of efficiency, the variation of both intrinsic and extrinsic parameters, the perception of targets, the type of targets, whether to detect only one or many of them, and the choice of boundary conditions. In our simulations we find that different search scenarios favour different modes of motion for optimising search success, defying a universality across all search situations. Some of our numerical results are explained by a simple analytical model. Having demonstrated that search by fractional Brownian motion is a truly complex process, we propose an over-arching conceptual framework based on classifying different search scenarios. This approach incorporates search optimisation by Levy walks as a special case.
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