Статья
2017

The general approach to the theory of the first passage problem for electrochemical stochastic diffusion in equilibrium


B. M. Grafov B. M. Grafov
Российский электрохимический журнал
https://doi.org/10.1134/S1023193517090075
Abstract / Full Text

I developed a general approach to the analyzing of the first passage problem for stochastic diffusion in equilibrium electrochemical systems. I found Fokker–Planck equation that controls the process of stochastic diffusion in any electrochemical system at the equilibrium. I found the link between the above-mentioned Fokker–Planck equation and electrochemical impedance. On the basis of the Fokker–Planck equation, I derived analytical expression for the characteristic function of random time passage by the process of electrochemical stochastic diffusion. The developed theory is useful for the working out of a method for electrochemical noise diagnostics based on the information on the process of electrochemical stochastic diffusion in the bounded limits.

Author information
  • Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Leninskii pr. 31, Moscow, 119071, Russia

    B. M. Grafov

References
  1. Iverson, W.P., Transient voltage changes produced in corroding metals and alloys, J. Electrochem. Soc., 1968, vol. 115, p. 617.
  2. Grafov, B.M. and Levich, V.G., On the fluctuationdissipation theorem in a stationary state, Soviet J. Expression Theor. Physics, 1968, vol. 27, p. 507.
  3. Fleischmann, M. and Oldfield, J.W., Generationrecombination noise in weak electrolytes, J. Electroanal. Chem., 1970, vol. 27, p. 207.
  4. Tyagai, V.A., Faradaic noise of complex electrochemical reactions, Electrochim. Acta, 1971, vol. 16, p. 1647.
  5. Barker, G.C., Flicker noise connected with the hydrogen evolution reaction on mercury, J. Electroanal. Chem., 1972, vol. 39, p. 484.
  6. Grafov, B.M. and Levich, V.G., Fluctuation-dissipation theorem for electrochemical networks, Soviet Electrochem., 1972, vol. 8, p. 478.
  7. Blanc, G., Gabrielli, C., and Keddam, M., Measurement of the electrochemical noise by a cross correlation method, Electrochim. Acta, 1975, vol. 20, p. 687.
  8. Epelboin, I., Gabrielli, C., Keddam, M., and Raillon, L., Measurement of the power spectral density of electrochemical noise: direct two-channel method, J. Electroanal. Chem., 1979, vol. 105, p. 389. ψ
  9. Uruchurtu, J.C. and Dawson, J.L., Noise analysis of pure aluminum under different pitting conditions, Corrosion, 1987, vol. 43, p. 19.
  10. Searson, P.C. and Dawson, J.L., Analysis of electrochemical noise generated by corroding electrodes under open circuit conditions, J. Electrochem. Soc., 1988, vol. 135, p. 1908.
  11. Gabrielli, C., Huet, F., and Keddam, M., Fluctuations in electrochemical systems. i. general theory on diffusion limited electrochemical reactions, J. Chem. Phys., 1993, vol. 99, p. 7232.
  12. Dawson, J.L., Electrochemical noise measurement: the definitive in-situ technique for corrosion applications? In: Electrochemical Noise Measurement for Corrosion Applications. ASTM Int., 1996.
  13. Martinet, S., Durand, R., Ozil, P., Leblanc, P., and Blanchard, P., Application of electrochemical noise analysis to the study of batteries: state-of-charge determination and overcharge detection, J. Power Sources, 1999, vol. 83, p. 93.
  14. Aballe, A., Bethencourt, M., Botana, F.J., and Marcos, M., Using wavelets transform in the analysis of electrochemical noise data. Electrochim. Acta, 1999, vol. 44, p. 4805.
  15. Mansfeld, F., Sun, Z., Hsu, C.H., and Nagiub, A., Concerning trend removal in electrochemical noise measurements, Corros. Sci., 2001, vol. 43, p. 341.
  16. Hassibi, A., Navid, R., Dutton, R.W., and Lee, T.H., Comprehensive study of noise processes in electrode electrolyte interfaces, J. Appl. Phys., 2004, vol. 96, p. 1074.
  17. Cottis, R.A., Sources of electrochemical noise in corroding systems, Russ. J. Electrochem., 2006, vol. 42, p. 497.
  18. Timashev, S.F. and Polyakov, Y.S., Review of flicker noise spectroscopy in electrochemistry, Fluctuation Noise Lett., 2007, vol. 7, p. R15.
  19. Loto, C.A., Electrochemical noise measurement technique in corrosion research, Int. J. Electrochem. Sci., 2012, vol. 7, p. 9248.
  20. Grafov, B.M., Electrochemical Symmetrical Stochastic Diffusion, Russ. J. Electrochem., 2012, vol. 48, p. 144.
  21. Coffey, W.T., Kalmykov, Yu.T., and Waldron, J.T., The Langevin Equation. With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, New Jersey: World Scientific, 2004.
  22. Lindenberg, K., West, B.J., and Masoliver, J., First passage time problems for non-Markovian processes. In: Noise in Nonlinear Dynamic Systems, vol. 1. Theory of Continuous Fokker-Planck Systems, Moss, F. and McClintock, P.V.E., Eds., Cambridge: Cambridge Univ. Press, 1989, p. 110.
  23. Grafov, B.M., Fokker–Planck equation for stochastic diffusion associated with Markovian electrochemical noise, Russ. J. Electrochem., 2015, vol. 51, p. 278.
  24. Risken, H., The Fokker-Planck Equation. Methods of Solution and Applications. Berlin: Springer, 1984.
  25. Mainardi, F. and Pironi, P., The fractional Langevin equation: brownian motion revisited, Extracta Mathematicae, 1996, vol. 10, p. 140.
  26. Grafov, B.M. Theory of the first encounter with the boundary by a stochastic diffusion process in an equilibrium electrochemical RC-circuit. Russ. J. Electrochem., 2016, vol. 52, p. 885.
  27. Metzler, R. and Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Reports, 2000, vol. 339, p. 1.
  28. Rangarajan, G. and Ding, M., Anomalous diffusion and the first passage time problem, Phys. Rev. E, 2000, vol. 62, p. 120.
  29. Rangarajan, G. and Ding, M., First passage time distribution for anomalous diffusion, Phys. Lett. A, 2000, vol. 273, p. 322.
  30. Sokolov, I.M., Thermodynamics and fractional Fokker–Planck equation, Phys. Rev. E, 2001, vol. 63, p. 056111.
  31. Sokolov, I.M., Solutions of a class of non-markovian Fokker–Planck equations, Phys. Rev., 2002, vol. 66, p. 041101.
  32. Metzler, R. and Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. Gen., 2004, vol. 37, p. R161.
  33. Burnecki, K., Magdziarz, M., and Weron, A., Identification and validation of fractional subdiffusion dynamics, Fractional Dynamics: Recent Advances, Klafter, J., Lim, S.C., and Metzler, R., Eds., New Jersey: World Scientific, 2012, p. 329.