Random-walk propagator

Reaction-Biased Random Walks. Propagation Failure 175... 

Polymer molecules in a solution undergo random thermal motions, which give rise to space and time fluctuations of the polymer concentration. If the concentration of the polymer solution is dilute enough, the interaction between individual polymer molecules is negligible. Then the random motions of the polymer can be described as a three dimensional random walk, which is characterized by the diffusion coefficient D. Light is scattered by the density fluctuations of the polymer solution. The propagation of phonons is overdamped in water and becomes a simple diffusion process. In the case of polymer networks, however, such a situation can never be attained because the interaction between chains (in... 

The work of DiMarzio and Rubin (DiMarzio, 1965 Rubin, 1965 DiMarzio and Rubin, 1971) began the development of a related but more powerful approach. Rather than calculating microstructural details from a presumed architecture, Rubin s matrix method concentrates on the effect of local interactions on the propagation of the chain, thereby deriving the statistical properties of the random walk and the structure of the entire chain. This formalism is the foundation for several subsequent models, so some details are reviewed here. The notation is transposed into a form consistent with the contemporary models discussed below. 

In all substances, at high temperatures, the electrical resistivity is dominated by inelastic scattering of the electrons by phonons, and other electrons. As classical particles, the electrons travel on trajectories that resemble random walks, but their apparent motion is diffusive over large-length scales because there is enough constructive interference to allow propagation to continue. Ohm s law holds and with increasing numbers of inelastic... 

Molecular dynamics Propagator Random walk Surface barriers... 

As p 0", we have v oo, and as p oo, we find that v grows with p. In consequence, a minimum velocity exists, but no maximum velocity. In the previous section we considered the cases when the microscopic transport processes are described by Markovian random walks. The great advantage of the Hamilton-Jacobi formulation of the front propagation problem in general, and the formulas (4.46) in particular, is that they allow us to study quite complicated transport operators for the evolution of the scalar field p and the underlying random walk model, including non-Markovian processes [118,121,125]. 

In previous sections we showed that the macroscopic dynamics of propagating fronts depend on the statistical characteristics of the underlying random walk model for the mesoscopic transport process. Since the dynamics of fronts are nonuniversal, it is an important problem to find universal rules relating both levels of description. The goal of this section is to address this problem. We are interested in exploring the physical properties of systems of particles that disperse according to a general CTRW. 

Front Propagation in Persistent Random Walks with Reactions 169... 

As discussed in Sect. 2.2, persistent random walks provide a mesoscopic description of reaction-transport systems with inertia. This approach provides another opportunity to explore the effects of a finite velocity in the transport mechanism on propagating fronts. We consider two cases. The first corresponds to reaction walks where the kinetic terms do not depend on the direction of the particles. This corresponds to choosing /f = 1/2 in (2.38), and persistent random walks with such kinetics are called direction-independent reaction walks (DIRW). The second case corresponds to walks where reactions occur only between particles with opposite velocities. We call such systems direction-dependent reaction walks (DDRWs). 

An interesting question arises when the dispersal is biased in a direction away from the region occupied by the unstable state [286]. What are the conditions on the reaction rate and bias that will result in a stalled front Or phrased differently, what is the critical (minimal) value of the reaction rate to sustain front propagation when the underlying random walk has a bias in the opposite direction The goal of this section is to show the following (i) The standard diffusion approximation of the transport process always provides an inaccurate value for the critical reaction rate, (ii) If the reaction rate exceeds the jump frequency of the random walk, then the front cannot stall and will always propagate into the unstable state, independently of the values of the other statistical parameters of the random walk. 

We choose the initial conditions to be /o(x, 0) = 1 for jc <0 and /o(Jt, 0) = 0 for X > 0. This initial condition describes, for example, a territory divided into an invaded zone, x < 0, and a noninvaded zone, x > 0, separated by a frontier at X = 0. If particles disperse according to an isotropic random walk with KPP kinetics, this initial condition turns into a front propagating from left to right, i.e., the invasion starts. Since the particle jumps are isotropic, the reaction is responsible for the motion of the front from left to right. It is the reaction process that starts and maintains a successful invasion. A bias to the left in the random walk will hinder the invasion. Therefore we expect that the critical reaction rate is given by a balance between the factor favoring the invasion, the reaction process, and the factor opposing the invasion, the bias in the transport process. 

Lattice models and other discrete models are used to describe a wide variety of dynamical systems [270, 77]. In this section we study the propagation of reaction-random walk wavefronts on heterogeneous lattices that consist of a main backbone with a regular distribution of secondary branches. An example is comb-like structures, see Fig. 6.3 [53]. 

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