The bulletin of mathematical biophysics

, Volume 15, Issue 3, pp 311–338 | Cite as

Random walk with persistence and external bias

  • Clifford S. Patlak


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.


Brownian Motion Random Walk External Bias Soret Effect Random Walk Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Benoit, M. H. 1947. “Sur la statistique des chaines avec interactions et empêchements stériques.”Jour de Chimie Physique,44, 18–21.Google Scholar
  2. Chandrasekhar, S. 1943. “Stochastic Problems in Physics and Astronomy.”Rev. Mod. Physics,15, 1–89.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chapman, S. 1928. “On the Brownian Displacements and Thermal Diffusion of Grains Suspended in a Non-Uniform Fluid.”Proc. Roy. Soc. London (A),119, 34–54.Google Scholar
  4. — and T. C. Cowling. 1939.The Mathematical Theory of Non-Uniform Gases. Cambridge: Cambridge University Press.Google Scholar
  5. Coulson, C. A. 1947. “Note on the Random-Walk Problem”.Proc. Cambridge Philos. Soc.,43, 583–86.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Cramér, H. 1946.Mathematical Methods of Statistics. Princeton: Princeton University Press.zbMATHGoogle Scholar
  7. Einstein, A. 1926.Investigations on the Theory of the Brownian Motion. (Ed. R. Furth) London: Methuen and Co., Ltd.Google Scholar
  8. Eyring, H. 1932. “The Resultant Electric Moment of Complex Molecules.”Phys. Rev.,39, 746–48.CrossRefGoogle Scholar
  9. Feller, W. 1949. “On the Theory of Stochastic Processes, with Particular Reference to Applications.”Proc. Berkeley Symp. on Math. Statistics and Probability, 403–32. Berkeley, Calif.: University of California Press.Google Scholar
  10. — 1950. “Some Recent Trends in the Mathematical Theory of Diffusion.”Proc. Internat. Congress of Mathematicians,2, 322–39.Google Scholar
  11. — 1951. “Diffusion Processes in Genetics.”Proc. Second Berkeley Symp. on Math. Statics and Probability, 227–46. Berkeley, Calif.: University of California Press.Google Scholar
  12. Fraenkel, G. S. and D. L. Gunn. 1940.The Orientation of Animals. Oxford: Oxford University Press.Google Scholar
  13. Furry, W. H. 1948. “On the Elementary Explanation of Diffusion Phenomena in Gases.”Am. Jour. Physics,16, 63–78.CrossRefGoogle Scholar
  14. — and P. H. Pitkanen. 1951. “Gaseous Diffusion as a Random Process.”Jour. Chem. Physics,19, 729–38.CrossRefGoogle Scholar
  15. Furth, R. 1920. “Die Brownsche Bewegung bei Berücksichtigung einer Persistenz der Bewegungsrichtung. Mit. Anwendungen auf die Bewegung lebender Infusorien.”Z. für Physik,2, 244–56.CrossRefGoogle Scholar
  16. Guth, E. and H. Mark. 1934. “Zur innermolekularen Statistik, inbesondere bei Kettermolekülen I.”Monatshefte Chem.,65, 93–121.CrossRefGoogle Scholar
  17. Hartley, G. S. 1931a. “Theories of the Soret Effect.”Trans. Farad. Soc.,27, 1–10.CrossRefGoogle Scholar
  18. — 1931b. “Diffusion and Distribution in a Solvent of Graded Composition.”Ibid.,,27, 10–29.CrossRefGoogle Scholar
  19. Infeld, L. 1940. “On the Theory of Brownian Motion.”Univ. Toronto Studies, Applied Math., Series #4. Toronto: University of Toronto Press.Google Scholar
  20. Jeans, J. H. 1940.An Introduction to the Kinetic Theory of Gases. Cambridge: Cambridge University Press.Google Scholar
  21. Jost, W. 1952.Diffusion in Solids, Liquids, Gases. New York: Academic Press Inc.Google Scholar
  22. King, G. W. 1951. “Biassed Random Walks in Classical, Statistical, and Quantum Mechanics.”O.N.R. Project N.R. 033243, Tech. Report #5.Google Scholar
  23. Klein, G. 1950–51. “A Generalization of the Classical Random-Walk Problem and a Simple Model of Brownian Motion Based Thereon.”Proc. Roy. Soc. Edinburgh,63, 268–79.Google Scholar
  24. Kolmogoroff, A. 1931. “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung.”Mathematische Ann.,104, 415–58.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Kuhn, H. 1947. “Restricted Bond Rotation and Shape of Unbranched Saturated Hydrocarbon Chain Molecules.”Jour. Chem. Physics,15, 843–44.CrossRefGoogle Scholar
  26. Loeb, L. B. 1934.The Kinetic Theory of Gases. Sec. Ed. New York: McGraw-Hill Book Co. Inc.Google Scholar
  27. Maxwell, J. C. 1927.Collected Scientific Papers (Ed. W. D. Niven) Vol. 1, p. 392. Paris: Library Scientifique.Google Scholar
  28. Meyer, O. E. 1899.Kinetic Theory of Gases. Sec. Rev. Ed. (Trans. R. E. Baynes) New York: Longmans, Green and Co.Google Scholar
  29. Moran, P. A. P. 1948. “The Statistical Distribution of the Length of a Rubber Molecule.”Proc. Cambridge Philos. Soc.,44, 342–44.MathSciNetzbMATHGoogle Scholar
  30. Patlak, C. S. 1953. In press.Google Scholar
  31. Pearson, K. 1905. “The Problem of the Random Walk.”Nature,72, 294.Google Scholar
  32. Pearson, K. 1906. “Mathematical Contributions to the Theory of Evolution. XV. A Mathematical Theory of Random Migration.”Draper's Company Research Memoirs, Biometric Series III. 54 pp.Google Scholar
  33. Rayleigh, Lord. 1945.The Theory of Sound. Sec. Ed. Vol. I, paragraph 42a. New York: Dover Publications.zbMATHGoogle Scholar
  34. Skellam, J. G. 1951. “Random Dispersal in Theoretical Populations.”Biometrika,38, 196–218.MathSciNetCrossRefzbMATHGoogle Scholar
  35. Taylor, G. I. 1922. “Diffusion by Continuous Movements.”Proc. London Math. Soc., Series 2,20, 196–212.Google Scholar
  36. Taylor, W. J. 1947. “Average Square Length and Radius of Unbranched Long-Chain Molecules with Restricted Internal Rotation.”Jour. Chem. Physics,15, 412–13.CrossRefGoogle Scholar
  37. — 1948. “Average Length and Radius of Normal Paraffin Hydrocarbon Molecules.”Ibid.,,16, 257–67.CrossRefGoogle Scholar
  38. Tchen, C. M. 1952. “Random Flight with Multiple Partial Correlations.”Jour. Chem. Physics,20, 214–17.CrossRefGoogle Scholar
  39. Treloar, L. R. G. 1942–43. “The Structure and Elasticity of Rubber.”Reprots on Progress in Physics, IX, 113–36.Google Scholar
  40. Uhlenbeck, G. E. and L. S. Ornstein. 1930. “On the Theory of the Brownian Motion.”Phys. Rev.,36, 823–41.CrossRefGoogle Scholar
  41. Wang, M. C. and G. E. Uhlenbeck. 1945. “On the Theory of the Brownian Motion II.”Rev. Mod. Physics,17, 323–42.MathSciNetCrossRefzbMATHGoogle Scholar
  42. Wilkinson, D. H. 1952. “The Random Element in Bird Navigation.”Jour. Exp. Biol.,29, 532–60.Google Scholar
  43. Wright, S. 1945. “The Differential Equation of the Distribution of Gene Frequencies.”Proc. Nat. Acad. Sci.,31, 382–89.MathSciNetCrossRefzbMATHGoogle Scholar
  44. Yang, L. M. 1949. “Kinetic Theory of Diffusion in Gases and Liquids I. Diffusion and the Brownian Motion.”Proc. Roy. Soc. London (A),198, 94–116.zbMATHCrossRefGoogle Scholar
  45. Zernike, F. 1928. “Wahrscheinlichkeitsrechnung und mathematische Statistik.”Hand. der Physik,3, 481–82. Berlin: Julius Springer.Google Scholar

Copyright information

© University of Chicago 1953

Authors and Affiliations

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

Personalised recommendations