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In order to describe non-Gaussian kinetics in weakly damped systems, the concept of continuous time random walks is extended to particles with finite inertia. One thus obtains a generalized Kramers-Fokker-Planck equation, which retains retardation effects, i.e., nonlocal couplings in time and space. It is shown that despite this complexity, exact solutions(More)
Muller's ratchet is a paradigmatic model for the accumulation of deleterious mutations in a population of finite size. A click of the ratchet occurs when all individuals with the least number of deleterious mutations are lost irreversibly due to a stochastic fluctuation. In spite of the simplicity of the model, a quantitative understanding of the process(More)
The anomalous (i.e., non-Gaussian) dynamics of particles subject to a deterministic acceleration and a series of "random kicks" is studied. Based on an extension of the concept of continuous time random walks to position-velocity space, a new fractional equation of the Kramers-Fokker-Planck type is derived. The associated collision operator necessarily(More)
We formulate the generalized master equation for a class of continuous-time random walks in the presence of a prescribed deterministic evolution between successive transitions. This formulation is exemplified by means of an advection-diffusion and a jump-diffusion scheme. Based on this master equation, we also derive reaction-diffusion equations for(More)
We introduce a population model for species under cyclic competition. This model allows individuals to coexist and interact on single cells while migration takes place between adjacent cells. In contrast to the model introduced by Reichenbach, Mobilia, and Frey [Reichenbach, Mobilia, and Frey, Nature (London) 448, 1046 (2007)], we find that the emergence of(More)
We obtain the exact solution for the Burgers equation with a time dependent forcing, which depends linearly on the spatial coordinate. For the case of a stochastic time dependence an exact expression for the joint probability distribution for the velocity fields at multiple spatial points is obtained. A connection with stretched vortices in hydrodynamic(More)
The description of diffusion processes is possible in different frameworks such as random walks or Fokker-Planck or Langevin equations. Whereas for classical diffusion the equivalence of these methods is well established, in the case of anomalous diffusion it often remains an open problem. In this paper we aim to bring three approaches describing anomalous(More)
The role of external forces in systems exhibiting anomalous diffusion is discussed on the basis of the describing Langevin equations. Since there exist different possibilities to include the effect of an external field the concept of biasing and decoupled external fields is introduced. Complementary to the recently established Langevin equations for(More)
We present a model of stochastic molecular transport in spiny dendrites. In this model the molecules perform a random walk between the spines that trap the walkers. If the molecules interact with each other inside the spines the trapping time in each spine depends on the number of molecules in the respective trap. The corresponding mathematical problem has(More)