Functional Distribution Of Tempered Anomalous Motion Particles: Modelling, Simulations,and Physical Applications

Brownian motion generally corresponds to a large number of random small fluctu-ations. The (rare) big fluctuations in many cases lead to non-Brownian motion, anoma-lous diffusion. The tempered anomalous diffusion describes the slow transition from anomalous to normal diffusion, even though the transition sometimes can not be de-tected in the observation time because of the finite lifespan of particles. The continuous time random walk (CTRW), a natural generalization of the Brownian random walk, al-lows the incorporation of the waiting time distribution ψ(t) and the general jump length distribution η(x). For the normal diffusion, ψ(t) has bounded first moment and η(x) bounded second moment; for the anomalous diffusion, ψ(t) has divergent first momen-t and/or η(x) has divergent second moment. Functionals of Brownian/non-Brownian motions have diverse applications and attracted a lot of interest of scientists. This paper focuses on the modelling and simulations of functional distribution of tempered anomalous moving particles. Then we derive the equations, called fractional Feynman-Kac equation, describing the distribution of the functionals of tempered anomalous d-iffusion, belonging to the continuous time random walk class. Several examples of the functionals are explicitly treated, including the occupation time in half-space, the first passage time, the maximal displacement, the fluctuations of the occupation fraction, and the fluctuations of the time-averaged position.