Random walks master equations

Exercise. The random walk with continuous time is defined as follows. The states are all integers n (— oo < n < oo). A particle can jump between neighboring states. In a short time dt has probability dt to jump to the right, and the same probability to jump to the left. Construct the master equation for pn(t) (compare VI.2). 

A fundamental property of the master equation is As t -> oo all solutions tend to the stationary solution or - in the case of decomposable or splitting W - to one of the stationary solutions. Again this statement is strictly true only for a finite number of discrete states. For an infinite number of states, and a fortiori for a continuous state space, there are exceptions, e.g., the random walk (2.11). Yet it is a useful rule of thumb for a physicist who knows that many systems tend to equilibrium. We shall therefore not attempt to give a general proof covering all possible cases, but restrict ourselves to a finite state space. There exist several ways of proving the theorem. Of course, they all rely on the property (2.5), which defines the class of W-matrices. 

Consider the unbounded symmetrical random walk rn and g constant and equal to one another. This constant may then be absorbed into the time unit, so that the master equation is as in (V.2.11)... 

Exercise. The asymmetric random walk on an infinite lattice is governed by the master equation... 

This is the master equation for a random walk with an absorbing n = 0. Exercise. Solve by the same method the M-equation boundary at... 

Exercise. Solve the following master equation for a random walk between two reflecting boundaries... 

To make the Fokker-Planck equation exact, rather than an approximation, one has to allow the coefficients in W to depend on a parameter e in such a way that the assumptions made are exact in the limit ->0.t) We demonstrate this approach for the asymmetric random walk, whose master equation (VI.2.13) is... 

THE CONTINUOUS-TIME RANDOM WALK VERSUS THE GENERALIZED MASTER EQUATION... 

Equation (20) is the central result of the Zwanzig projection method, and it is one of the two theoretical tools under scrutiny in this chapter, the first being the Generalized Master Equation (GME), of which Eq. (20) is a remarkable example, and the second being the Continuous Time Random Walk (CTRW) [17]. It must be pointed out that to make Eq. (20) look like a master equation, it is necessary to make the third term on the right-hand side of it vanish. To do so, the easiest way is to make the following two assumptions ... 

Equation (1) may be derived using a variety of microscopic models of the relaxation process. In the derivation of Eq. (1), Debye [1] used the theory of the Brownian motion developed by Einstein and Smoluchowski. Einstein s theory of Brownian motion [2] is based on the notion of a discrete time walk. The walk may be described in simple schematic terms as follows. Consider a two-dimensional lattice then, in discrete time steps of length At, the random walker is assumed to jump to one of its nearest-neighbor sites, displayed, for example [7], on a square lattice with lattice constant Ax, the direction being random. Such a process, which is local both in space and time, can be modeled [7] in the one-dimensional analogue by the master equation... 

Continuous time random walk processes with decoupled A,(x) and v /(t) can be rephrased in terms of a generalized master equation [49]. This is also true for a general external force F(x), where we obtain a relation of the type... 

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