Continuous Time Random Walk in a velocity field: Role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixing
arXiv:2004.08605 · doi:10.1088/1367-2630/ab9ae2
Abstract
We consider the dynamics of a separable Continuous Time Random Walk (CTRW) when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding universe. The considered CTRW can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a Lévy flight. We first consider the case of zero velocity field. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of Lévy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: In one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.
arXiv comment35 pages, 8 figures
References in corpus (29)
- Fractional calculus and continuous-time finance
- In vivo anomalous diffusion and weak ergodicity breaking of lipid granules
- Models of traps and glass phenomenology
- Stochastic Ergodicity Breaking: a Random Walk Approach
- Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity
- The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times
- Exact Solution for the Influence of Spectral Diffusion on Single-Molecule Photon-Statistics
- On reaction-subdiffusion equations
- Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
- Distance matters: the impact of gene proximity in bacterial gene regulation
- Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary
- Towards deterministic equations for Levy walks: the fractional material derivative
- Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous time random walks
- Exploration and Trapping of Mortal Random Walkers
- Diffusion of Cosmic Rays in Expanding Universe. (I)
- The continuous time random walk formalism in financial markets
- Probabilistic approach to a proliferation and migration dichotomy in the tumor cell invasion
- Diffusion of Cosmic Rays in the Expanding Universe. II. Energy Spectra of Ultra-High Energy Cosmic Rays
- Continuous-time random-walk model for anomalous diffusion in expanding media
- Diffusion of cosmic rays at EeV energies in inhomogeneous extragalactic magnetic fields
- Diffusion in an expanding medium: Fokker-Planck equation, Green's function and first-passage properties
- Evanescent continuous time random walks
- Subdiffusion in an external force field
- Weak Galilean invariance as a selection principle for coarse-grained diffusive models
- Continuous time random walks and Fokker-Planck equation in expanding media
- Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains
- Generalised fractional diffusion equations for subdiffusion on arbitrarily growing domains
- Encounter-controlled coalescence and annihilation on a one-dimensional growing domain
- The standard and the fractional Ornstein-Uhlenbeck process on a growing domain