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Malakhov A. N., Pankratov A. L. Temporal scales of stochastic transitions in the piecewise-parabolic bistable systems with noise. Izvestiya VUZ. Applied Nonlinear Dynamics, 1995, vol. 3, iss. 3, pp. 70-79.

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
539.219.3:621.328

Temporal scales of stochastic transitions in the piecewise-parabolic bistable systems with noise

Autors: 
Malakhov Askold Nikolaevich, Lobachevsky State University of Nizhny Novgorod
Pankratov Andrei Leonidovich, Lobachevsky State University of Nizhny Novgorod
Abstract: 

The kinetics of stochastic transitions from one stable state to another in bistable systems with noise which are described by piecewise-parabolic potential profiles is considered. Using the method of the Laplace transform of the initial Fokker - Planck equation it has obtained the exact values of the mean transition times (i.e. relaxation time) of Brownian particles across the potential barriers, which separate the stable states. For the three concrete piecewise-parabolic profiles it has been made а comparative analysis of the obtained results with the well-known approximate Kramers’ results. The influence of the shape and width of the potential barriers оп е relaxation times is analyzed. There аге found the factors of the relaxation time structure, which represent the shape of stable state and the shape, height and width of а potential barrier.

Key words: 
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Acknowledgments: 
The authors express their gratitude to N.V. Agudov for his interest in the work and valuable comments. This work was supported by the Russian Foundation for Basic Research (project 94-02-04698-a).
Reference: 
  1. Kramers Н. Brownian motion in а field of force and the diffusion model of chemical reactions. Physica. 1940;7(4):284-304. DOI: 10.1016/S0031-8914(40)90098-2.
  2. Hanggi P, Talkner Р, Borkovec M. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 1990;62(2):251-341. DOI: 10.1103/RevModPhys.62.251.
  3. Risken H. The Fokker - Planck Equation. Berlin: Springer; 1996. 472 p. DOI: 10.1007/978-3-642-61544-3.
  4. Malakhov АN. Diffusion through sharp potential barriers I. Exact solution. Radiophys. Quantum Electron. 1991;34(5):451-460. DOI: 10.1007/BF01036839.
  5. Malakhov АN. Diffusion through sharp potential barriers. 2. Time characteristics of diffusion. Radiophys. Quantum Electron. 1991;34(6):571-580. DOI: 10.1007/BF01039582.
  6. Agudov NV, Malakhov АN. Nonstationary diffusion through arbitrary piecewise-linear potential profile. Exact solution and time characteristics. Radiophys. Quantum Electron. 1993;36(2):97-109.  DOI: 10.1007/BF01059491.
  7. Blomberg С. The Brownian motion theory of chemical transition rates. Physica А. 1977;86(1):49-66. DOI: 10.1016/0378-4371(77)90061-9.
  8. Edholm O, Leimar О. The accuracy of Kramers’ theory of chemical kinetics. Physica A. 1979;98(1-2):313-324. DOI: 10.1016/0378-4371(79)90182-1.
  9. Larson RS, Kostin MD. Kramers’ theory of chemical kinetics: Eigenvalue and eigenfunction analysis. J. Chem. Phys. 1978;69(11):4821-4829. DOI: 10.1063/1.436510.
  10. Larson RS. Thermally activated crossing of а sharp potential barrier. J. Chem. Phys. 1984;81(4):1731-1738. DOI: 10.1063/1.447899.
  11. Miller JCP. Tables of Weber Parabolic Cylinder Functions. London: Her Majesty’s Stationery Office; 1955. 233 p.
  12. Malakhov АN. Cumulant Analysis of Random Non-Gaussian Processes and Their Transformations. М.: Sovetskoe Radio; 1978. 376 p.
Received: 
12.11.1994
Accepted: 
18.09.1995
Published: 
05.04.1996
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