Abstract
The Ventsel'-Freidlin probability estimates for small random perturbations of dynamical systems are used to generalize and justify the Onsager-Machlup irreversible thermodynamic variational description of Gaussian statistical distributions in the limit where Boltzmann's constant tends to zero for non-Gaussian diffusion processes. A Hamiltonian formulation is used to determine the maximum likelihood paths for the growth and decay of nonequilibrium fluctuations, in the same limit, subject to the imposed constraints. The paths of maximum likelihood manifest a symmetry in past and future and are the stationary conditions of the constrained thermodynamic variational principle of least dissipation of energy. The power balance equations supply the required constraints and the most likely path for the growth of a fluctuation is characterized by a negative entropy production. The entropy plays the role of the quasipotential of Ventsel' and Freidlin and exit from a bounded domain containing a deterministically stable steady state is made at that state on the boundary with maximum entropy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Onsager and S. Machlup,Phys. Rev. 91:1505 (1953); S. Machlup and L. Onsager,Phys. Rev. 91:1512(1931).
B. H. Lavenda,Found Phys. 9:405 (1979).
A. D. Ventsel' and M. I. Freidlin,Russ. Math. Surv. 25:1 (1970);Theory Probab. Appl. 17:269 (1972).
A. Friedmann,Stochastic Differential Equations and Applications, Vol. 2, Academic Press, New York (1976), Chap. 14.
D. Ludwig,SIAM (Soc. Ind. Appl. Math.) Rev. 17:605 (1975).
R. F. Anderson and S. Orey,Nagoya Math. J. 60:189 (1976).
B. J. Matkowsky and Z. Schuss,Bull. Am. Math. Soc. 82:321 (1976);SIAM (Soc. Ind. Appl. Math.)J. Appl. Math. 33:365 (1977); see also Z. Schuss,Theory and Applications of Stochastic Differential Equations, Wiley, New York (1980), Chap. 7.
R. G. Williams,SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 40:208 (1981).
B. H. Lavenda,Thermodynamics of Irreversible Processes, Macmillan, London/Halsted, New York (1978).
B. H. Lavenda, in preparation.
E. Wong and M. Zakai,Int. J. Eng. Sci. 3:213 (1965).
R. L. Stratonovich,SIAM (Soc. Ind. Appl. Math.) J. Control 4:362 (1966); see also R. L. Stratonovich,Conditional Markov Processes and Their Application to the Theory of Optimal Control, American Elsevier, New York (1968), Chap. 2.
K. Itô,Memoir Am. Math. Soc. 4:1 (1951).
I. V. Girsanov,Theory Prob. Appl. 5:285 (1960).
B. H. Lavenda and E. Santamato,J. Math. Phys. 22:2926 (1981).
L. D. Landau and E. M. Lifshitz,Statistical Physics, Pergamon Press, Oxford (1959), p. 377.
B. H. Lavenda,Rivista Nuovo Cimento 7:229 (1977).
Ref. 9, Chap. V.
Ref. 9, Chap. VI.
B. H. Lavenda and E. Santamato,Lett. Nuovo Cimento 26:27 (1979).
E. Santamato and B. H. Lavenda,The stochastic H theorem, submitted toJ. Math. Phys.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lavenda, B.H., Santamato, E. Thermodynamic criteria governing irreversible processes under the influence of small thermal fluctuations. J Stat Phys 29, 345–361 (1982). https://doi.org/10.1007/BF01020791
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01020791