Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces

We derive a fractional Fokker-Planck equation for subdiffusion in a general space- and time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.     

On the Generalized Langevin Equation for a Rouse Bead in a Nonequilibrium Bath

We present the reduced dynamics of a bead in a Rouse chain which is submerged in a bath containing a driving agent that renders it out-of-equilibrium. We first review the generalized Langevin equation of the middle bead in an equilibrated bath. Thereafter, we introduce two driving forces. Firstly, we add a constant force that is applied to the first bead of the chain. We investigate how the generalized Langevin equation changes due to this perturbation for which the system evolves towards a steady state after some time. Secondly, we consider the case of stochastic active forces which will drive the system to a nonequilibrium state. Including these active forces results in an extra contribution to the second fluctuation–dissipation relation. The form of this active contribution is analysed for the specific case of Gaussian, exponentially correlated active forces. We also discuss the resulting rich dynamics of the middle bead in which various regimes of normal diffusion, subdiffusion and superdiffusion can be present.     

Grounds for the application of fractional-order diffusion equations in channeling theory using the Langevin approach

Transverse-energy distribution densities in the case of ultrarelativistic proton channeling in the (100) planes of diamond-like crystals have been numerically calculated within the framework of the Langevin approach. Based on the self-similarity principle, the subdiffusion character of the motion under consideration has been established, which can be regarded as grounds for using fractional-order diffusion equations to adequately describe channeling and dechanneling processes mathematically.     

Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule

By introducing fractional Gaussian noise into the generalized Langevin equation, the subdiffusion of a particle can be described as a stationary Gaussian process with analytical tractability. This model is capable of explaining the equilibrium fluctuation of the distance between an electron transfer donor and acceptor pair within a protein that spans a broad range of time scales, and is in excellent agreement with a single-molecule experiment.     

Mesoscale Simulation of Bacterial Chromosome and Cytoplasmic Nanoparticles in Confinement

In this study we investigated, using a simple polymer model of bacterial chromosome, the subdiffusive behaviors of both cytoplasmic particles and various loci in different cell wall confinements. Non-Gaussian subdiffusion of cytoplasmic particles as well as loci were obtained in our Langevin dynamic simulations, which agrees with fluorescence microscope observations. The effects of cytoplasmic particle size, locus position, confinement geometry, and density on motions of particles and loci were examined systematically. It is demonstrated that the cytoplasmic subdiffusion can largely be attributed to the mechanical properties of bacterial chromosomes rather than the viscoelasticity of cytoplasm. Due to the randomly positioned bacterial chromosome segments, the surrounding environment for both particle and loci is heterogeneous. Therefore, the exponent characterizing the subdiffusion of cytoplasmic particle/loci as well as Laplace displacement distributions of particle/loci can be reproduced by this simple model. Nevertheless, this bacterial chromosome model cannot explain the different responses of cytoplasmic particles and loci to external compression exerted on the bacterial cell wall, which suggests that the nonequilibrium activity, e.g., metabolic reactions, play an important role in cytoplasmic subdiffusion.     

Langevin equation for a free particle driven by power law type of noises

We study a generalized Langevin equation for a free particle driven by N internal noises. The mean square displacement and velocity autocorrelation function are derived in case of a mixture of Dirac delta, power law and Mittag-Leffler noises. Additionally, a frictional memory kernel of distributed order is considered. The long time limit and short time limit are analyzed, and the dominant contributions of noises on particle dynamics is discussed. Various different diffusive behaviors (subdiffusion, superdiffusion, normal diffusion, ultraslow diffusion) are obtained. The considered problem may be used in the theory of anomalous diffusion in complex environment.     

The rich phenomenology of Brownian particles in nonlinear potential landscapes

Non-interacting Brownian particles obey Langevin equations fulfilling a fluctuation–dissipation relation between friction and thermal noise. Under a linear potential (constant force) Einstein found a relation between diffusion and transport through mobility. In nonlinear potentials this prediction is only satisfied within the limits of very small and large constant external forces. Moreover, other more interesting behaviors do appear, such as: dispersionless transport, sorting, giant diffusion, subdiffusion, superdiffusion, subtransport, etc. All these phenomena depend on the characteristics of the nonlinear potential landscape: periodic or random, the symmetries and boundary conditions. Moreover, the presence of transport is the keystone of most of this phenomenology. In this review, we present numerical simulations illustrating these facts and theoretical analysis when possible.     

The effects of viscoelastic fluid on kinesin transport

Kinesins are molecular motors which transport various cargoes in the cytoplasm of cells and are involved in cell division. Previous models for kinesins have only targeted their in vitro motion. Thus, their applicability is limited to kinesin moving in a fluid with low viscosity. However, highly viscoelastic fluids have considerable effects on the movement of kinesin. For example, the high viscosity modifies the relation between the load and the speed of kinesin. While the velocity of kinesin has a nonlinear dependence with respect to the load in environments with low viscosity, highly viscous forces change that behavior. Also, the elastic nature of the fluid changes the velocity of kinesin. The new mechanistic model described in this paper considers the viscoelasticity of the fluid using subdiffusion. The approach is based on a generalized Langevin equation and fractional Brownian motion. Results show that a single kinesin has a maximum velocity when the ratio between the viscosity and elasticity is about 0.5. Additionally, the new model is able to capture the transient dynamics, which allows the prediction of the motion of kinesin under time varying loads.     

A multidimensional subdiffusion model： An arbitrage-free market

<正>To capture the subdiffusive characteristics of financial markets,the subordinated process,directed by the inverse Q-stale subordinator S_α(t) for 0 <α< 1,has been employed as the model of asset prices.In this article,we introduce a multidimensional subdiffusion model that has a bond and K correlated stocks.The stock price process is a multidimensional subdiffusion process directed by the inverse Q-stable subordinator.This model describes the period of stagnation for each stock and the behavior of the dependency between multiple stocks.Moreover,we derive the multidimensional fractional backward Kolmogorov equation for the subordinated process using the Laplace transform technique.Finally, using a martingale approach,we prove that the multidimensional subdiffusion model is arbitrage-free,and also gives an arbitrage-free pricing rule for contingent claims associated with the martingale measure.