Abstract
Solutions of the Fokker-Planck (Kramers) equation in position-velocity space for the double-well potentiald 2x2/2+d4x4/4 in terms of matrix continued fractions are derived. It is shown that the method is also applicable to a Boltzmann equation with a BGK collision operator. Results of eigenvalues and of the Fourier transform of correlation functions are presented explicitly. The lowest nonzero eigenvalue is compared with the escape rate in the weak noise limit for various damping constants and the susceptibility is compared with the zero-friction-limit result.
Similar content being viewed by others
References
P. Fulde, L. Pietronero, W. R. Schneider, and S. Strässler.Phys. Rev. Lett. 35:1776 (1975).
T. Geisel, inPhysics of Superionic Conductors, M. B. Salamon, ed.,Topics in Current Physics, Vol. 15 (Springer, Berlin, 1979), p. 201.
W. Dieterich, P. Fulde, and I. Peschel,Adv. Phys. 29:527 (1980).
B. D. Josephson,Phys. Lett. 1:251 (1962).
L. Solymar,Superconductive Tunneling and Applications (Chapman and Hall, London, 1972).
A. Barone, G. Paterno,Physics and Applications of the Josephson Effect (Wiley, New York 1982).
J. McConnel,Rotational Brownian Motion and Dielectric Theory (Academic, London, 1980).
W. T. Coffey,Adv. Mol. Relaxation Interaction Processes 17:169 (1982); M. Evans, G. J. Evans, W. T. Coffey, and P. Grigolini,Molecular Dynamics (Wiley, New York, 1982).
E. Praestgard and N. G. van Kampen,Molec. Phys. 43:33 (1981).
A. J. Viterbi,Principles of Coherent Communication (McGraw-Hill, New York, 1966).
W. C. Lindsey,Synchronization Systems in Communication and Control (Prentice-Hall, Englewood Cliffs, New Jersey, 1972).
G. A. Toombs,Phys. Rev. 40C:181 (1978).
W. Wonneberger and H. J. Breymayer,Z. Phys. B43:329 (1981).
D. Chandler,J. Chem. Phys. 68:2959 (1978).
B. J. Berne, J. L. Skinner, and P. G. Wolynes,J. Chem. Phys. 73:4314 (1980).
D. L. Hasha, T. Eguchi, and J. Jonas,J. Chem. Phys. 73:1571 (1981);J. Am. Chem. Soc. 104:2290 (1982).
D. K. Garrity and J. L. Skinner,Chem. Phys. Lett. 95:46 (1983).
B. Carmeli and A. Nitzan,J. Chem. Phys. 80:3596 (1984).
H. A. Staab,Einführung in die theoretische organische Chemie, 4th ed., 2nd printing (Verlag Chemie, Weinheim, 1970).
H. A. Kramers, Physica7:284 (1940).
H. C. Brinkman,Physica 22:29 (1956);Physica 22:149 (1956).
C. Blomberg,Physica 86A:49 (1977).
P. B. Visscher,Phys. Rev. B 14:347 (1976).
J. L. Skinner and P. G. Wolynes,J. Chem. Phys. 69:2143 (1978);
J. L. Skinner and P. G. Wolynes,J. Chem. Phys. 72:4913 (1980).
R. S. Larson and M. D. Kostin,J. Chem. Phys. 69:4821 (1978);
R. S. Larson and M. D. Kostin,J. Chem. Phys. 72:1392 (1980).
S. C. Northrup and J. T. Hynes,J. Chem. Phys. 69:5246 (1978);J. Chem. Phys. 69:5261 (1978);J. Chem. Phys. 73:2700 (1980); R. F. Grote and J. T. Hynes,J. Chem. Phys. 73:2715 (1980).
M. Mangel,J. Chem. Phys. 72:6606 (1980).
K. Schulten, Z. Schulten, and A. Szabo,J. Chem. Phys. 74:4426 (1981).
M. Bixon and R. Zwanzig,J. Stat. Phys. 3:245 (1971).
M. I. Dykman and M. A. Krivoglaz,Synergetics, Proceedings of the International Symposium (Tallinn, 1983), p. 33;
M. I. Dykman and M. A. KrivoglazSov. Phys. 50:30 (1979).
P. Hänggi,Phys. Lett. 78A:304 (1980).
J. A. Krumhansl and J. R. Schrieffer,Phys. Rev. B 11:3535 (1975).
P. Hänggi,J. Stat. Phys. 30:401 (1983).
T. Shimizu,Phys. Lett. 71A:319 (1979); K. Takeyama,Progr. Theor. Phys. 60:613 (1978);Progr. Theor. Phys. 63:91 (1980).
H. Risken and H. D. Vollmer,Z. Phys. B33:297 (1979).
H. D. Vollmer and H. Risken,Z. Phys. B34:313 (1979).
H. Risken and H. D. Vollmer,Z. Phys. B39:339 (1980).
H. Risken, H. D. Vollmer, and M. Mörsch,Z. Phys. B40:343 (1981).
H. D. Vollmer and H. Risken,Physica 110A:106 (1982).
H. Risken and H. D. Vollmer,Molec. Phys. 46:555 (1982).
H. Risken,The Fokker-Planck Equation, Springer Series in Synergetics, Vol. 18 (Springer, Berlin, 1984).
L. D. Landau and E. M. Lifschitz,Lehrbuch der Theoretischen Physik, Vol. V,Statistische Physik (Akademie-Verlag, Berlin, 1975); L. Landau and E. M. Lifschitz,Statistical Physics (Pergamon, London, 1968).
S. Chandrasekar,Rev. Mod. Phys. 15:1 (1943).
M. C. Wang and G. E. Uhlenbeck,Rev. Mod. Phys. 17:323 (1945).
P. L. Bhatnagar, E. P. Gross, and M. Krook,Phys. Rev. 94:511 (1954).
L. de Haro,Physica 111A:65 (1982).
H. Dekker,Proefschrift (Physisch Laboratorium TNO, 's-Gravenhage, 1980).
H. Dekker and N. G. van Kampen,Phys. Lett. 76A:101 (1980).
B. J. Matkowsky, Z. Schuss, and E. Ben-Jacob,J. Appl. Math. 42:835 (1982).
C. W. Gardiner,J. Stat. Phys. 30:157 (1983).
J. T. Hynes,Chem. Phys. Lett. 79:344 (1981).
J. B. Morton and S. Corrsin,J. Stat. Phys. 2:153 (1970).
K. Matsuo,J. Stat. Phys. 6:535 (1978).
W. Magnus, F. Oberhettinger, and R. P. Soni,Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966).
H. Risken and K. Voigtlaender, in preparation.
H. Risken and H. D. Vollmer,Z. Phys. B35:177 (1979).
H. D. Vollmer and H. Risken,Z. Phys. B37:343 (1980).
P. Jung and H. Risken,Z. Phys. B54:357 (1984).
K. Voigtlaender and H. Risken,Chem. Phys. Lett. 105:506 (1984).
I. S. Gradstheyn and I. M. Ryzhik,Tables of Integrals, Series and Products (Academic, New York, 1965).
H. D. Vollmer and H. Risken,Z. Phys. B52:259 (1983).
M. Mörsch, H. Risken, and H. D. Vollmer,Z. Phys. B49:47 (1982).
P. Hänggi and H. Thomas,Phys. Rep. 88:207 (1982).
M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions (Dover, New York, 1965).
Y. Onodera,Progr. Theor. Phys. 44:1477 (1970).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Voigtlaender, K., Risken, H. Solutions of the Fokker-Planck equation for a double-well potential in terms of matrix continued fractions. J Stat Phys 40, 397–429 (1985). https://doi.org/10.1007/BF01017181
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01017181