Abstract
A method for estimating the dynamical statistical properties of the solutions of nonlinear Langevin-type stochastic differential equations is presented. The non-linear equation is linearized within a small interval of the independent variable and statistical properties are expressed analytically within the interval. The linearization procedure is optimal in the sense of the Chebyshev inequality. Long-term behavior of the solution process is obtained by appropriately matching the approximate solutions at the boundaries between intervals. The method is applied to a model nonlinear equation for which the exact time-dependent moments can be obtained by numerical methods. The calculations demonstrate that the method represents a significant improvement over the method of statistical linearization in time regimes far from equilibrium.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. G. Van Kampen,Phys. Rep. 24:171 (1976).
R. F. Fox,Phys. Rep. 48:179 (1978).
S. A. Adelman and J. D. Doll,J. Chem. Phys. 64:2375 (1976).
M. Shugard, J. C. Tully, and A. Nitzan,J. Chem. Phys. 69:336 (1978).
R. Kapral and K. J. Shin,J. Chem. Phys. 70:5623 (1979).
R. C. Desai and R. Zwanzig,J. Stat. Phys. 19:l (1978).
F. T. Arecchi, M. Asdente, and A. M. Ricca,Phys. Rev. A 14:383 (1976); F. T. Arecchi and A. M. Ricca,Phys. Rev. A 15:308 (1977).
P. A. Ruymgaart and T. T. Soong,J. Math. Anal. Appl. 34:325 (1971).
S. H. Crandall,J. Acoust. Soc. Am. 35:1700 (1963).
T. K. Caughey,J. Acoust. Soc. Am. 35:1706 (1963).
J. D. Doll and D. R. Dion,Chem. Phys. Lett. 37:386 (1976).
A. Budgor,J. Stat. Phys. 15:355 (1976).
A. Budgor, K. Lindenberg, and K. E. Shuler,J. Stat. Phys. 15:375 (1976).
B. J. West, K. Lindenberg, and K. E. Shuler,J. Stat. Phys. 18:217 (1978).
A. Budgor and B. J. West,Phys. Rev. A 17:370 (1978).
H. A. Kramers,Physica (The Hague) 7:284 (1940).
K. Matsuo,J. Stat. Phys. 18:535 (1978).
S. H. Northrup and J. T. Hynes,J. Chem. Phys. 69:5261 (1978).
R. Wilcox and R. Bellman,J. Math. Anal. Appl. 32:532 (1970).
W. Feller,An Introduction to Probability Theory and Its Applications, Vol. I (Wiley, New York, 1968), p. 233.
J. O. Eaves and W. P. Reinhardt, unpublished work.
Author information
Authors and Affiliations
Additional information
Supported in part by the National Science Foundation under Grants CHE77-16307 and PHY76-04761.
Rights and permissions
About this article
Cite this article
Eaves, J.O., Reinhardt, W.P. Piecewise optimal linearization method for nonlinear stochastic differential equations. J Stat Phys 25, 127–141 (1981). https://doi.org/10.1007/BF01008482
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF01008482