General Solution of the Linear Biased Correlated Walk

抄録

The arrival probabilities with directions for a linear biased correlated walk, satisfy coupled linear difference equations. An exact analytic solution to these equations with a periodic boundary and a general initial condition, is obtained. The solution for an inhomogeneous initial condition can be expressed in terms of the Chevyshev polynomials of the second kind, and is quite complicated. For a homogeneous initial condition, the analytic expressions for the arrival probabilities can be written down in single lines. With a finite bias, the system approaches a stationary state in which the ar-rival probabilities are different for different directions, and a finite current exists. As an application of the theory, a dilute solution of CO in liquid Ar under an electric field is discussed.