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Motion of a particle in a velocity-dependent random force

Published online by Cambridge University Press:  14 July 2016

Karmeshu
Karmeshu*
Affiliation:
University of Delhi
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Abstract

The motion of a particle is investigated in the presence of a velocity-dependent random force assumed to be proportional to velocity. Two different possibilities are considered, namely, the presence and absence of random driving force. In the absence of random driving force, the velocity and displacement auto-correlation function are calculated. The probability distribution in velocity space is also evaluated. It is found that in the absence of intrinsic damping, the energy of the particle increases without limit. The condition for the energetic stability of the particle in the presence of random driving force is obtained. The Fokker-Planck equation for the probability distribution in velocity space is derived from the stochastic Liouville equation for delta-correlated velocity-dependent random force.

Keywords

EXACT SOLUTIONSLANGEVIN EQUATIONSTOCHASTIC DAMPINGSTOCHASTIC DRIVING FORCEDELTA-CORRELATEDSTATIONARY GAUSSIAN PROCESSENERGETIC STABILITYFOKKER-PLANCK EQUATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 4 , December 1976 , pp. 684 - 695
Copyright
Copyright © Applied Probability Trust 1976 

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References

Brissaud, A. and Frisch, U. (1974) Solving linear stochastic differential equations. J. Math. Phys. 15, 524–534.CrossRefGoogle Scholar
Bourret, R. C. (1971) Energetic stability of the harmonic oscillator with random parametric driving. Physica 54, 623–629.CrossRefGoogle Scholar
Bourret, R. C., Frisch, U. and Pouquet, A. (1973) Brownian motion of harmonic oscillator with stochastic frequency. Physica 65, 303–320.Google Scholar
Chandrasekhar, S. (1943) Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89.Google Scholar
Fox, R. F. (1974) Multiplicative stochastic processes, Fokker-Planck equation, and possible dynamical mechanism for critical behaviour. J. Math. Phys. 15, 1918–1929.CrossRefGoogle Scholar
Hakim, R. (1968) Relativistic stochastic processes. J. Math. Phys. 9, 1805–1818.Google Scholar
Hoel, P. G. (1954) Introduction to Mathematical Statistics. Wiley, New York.Google Scholar
Kubo, R. (1963) Stochastic Liouville equations. J. Math. Phys. 4, 174–183.CrossRefGoogle Scholar
Kubo, R. (1974) Response, relaxation and fluctuation. In Lecture Notes in Physics 31, ed. Kirczenow, G. and Marro, J., Springer-Verlag, Berlin.Google Scholar
Landau, L. D. and Lifshitz, E. M. (1960) Mechanics. Pergamon Press, New York.Google Scholar
Papanicolaou, G. C. (1971) Motion of a particle in a random field. J. Math. Phys. 12, 1494–1496.Google Scholar
Papanicolaou, G. C. and Keller, J. B. (1971) Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media. SIAM J. Appl. Math. 21, 287–305.CrossRefGoogle Scholar
Puri, S. (1966) Plasma-heating and diffusion in stochastic fields. Phys. Fluids 9, 2043–2046;Google Scholar
Puri, S. (1968) Statistical particle acceleration in random field. Phys. Fluids 11, 1745–1753.Google Scholar
Soong, T. T. (1973) Random Differential Equations in Science and Engineering. Academic Press, New York and London.Google Scholar
Sturrock, P. A. (1966) Stochastic acceleration. Phys. Rev. 141, 186–191.Google Scholar
Wong, E. (1964) The construction of a class of stationary Markoff processes. In Proceedings of the Symposia on Applied Mathematics, ed. Bellman, R. American Mathematical Society, Philadelphia.Google Scholar