The bulletin of mathematical biophysics

, Volume 15, Issue 3, pp 311–338

Random walk with persistence and external bias

  • Clifford S. Patlak

DOI: 10.1007/BF02476407

Cite this article as:
Patlak, C.S. Bulletin of Mathematical Biophysics (1953) 15: 311. doi:10.1007/BF02476407


The partial differential equation of the random walk problem with persistence of direction and external bias is derived. By persistence of direction or internal bias we mean that the probability a particle will travel in a given direction need not be the same for all directions, but depends solely upon the particle's previous direction of motion. The external bias arises from an anisotropy of the medium or an external force on the particle. The problem is treated by considering that the net displacement of a particle arises from two factors, namely, that neither the probability of the particle traveling in any direction after turning nor the distance the particle travels in a given direction need be the same for all directions. A modified Fokker-Planck equation is first obtained using the assumptions that the particles have a distribution of travel times and speeds and that the average time of travel between turns need not be zero. The fional equation incopporating the assumption of a persistence of direction and an external bias is then derived. Applications to the study of diffusion and to long-chain polymers are then made.

Copyright information

© University of Chicago 1953

Authors and Affiliations

  • Clifford S. Patlak
    • 1
  1. 1.Committee on Mathematical BiologyThe University of ChicagoChicagoUSA