Abstract
A general theory is derived for the moments of the first passage time of a one-dimensional Markov process in the presence of a weak time-dependent forcing. The linear corrections to the moments can be expressed by quadratures of the potential and of the time-dependent probability density of the unperturbed system or equivalently by its Laplace transform. If none of the latter functions is known, the derived formulas may still be useful for specific cases including a slow driving or a driving with power at only small or large times. In the second part of the paper, explicit expressions for the mean and variance of the first passage time are derived for the cases of a linear or a parabolic potential and an exponentially decaying driving force. The analytical results are found to be in excellent agreement with computer simulations of the respective first-passage processes. The particular examples furthermore demonstrate that already the effect of a simple exponential driving can be fairly involved implying a nontrivial nonmonotonic behavior of mean and variance as functions of the driving’s time scale.
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REFERENCES
L. Pontryagin, A. Andronov, and A. Witt, Zh.Eksp.Teor.Fiz., 3:172 (1933): Reprinted in Noise in Nonlinear Dynamical Systems, F. Moss and P. V. E. McClintock (eds) Vol. 1, (Cambridge University Press, Cambridge, 1989), p. 329.
A. J. F. Siegert, On the first passage time problem, Phys.Rev. 81:617 (1951).
M. Bier and R. D. Astumian, Matching a diffusive and a kinetic approach for escape over an fluctuating barrier, Phys.Rev.Lett. 71:1649 (1993).
C. R. Doering and J. C. Gadoua, Resonant activation over a fluctuating barrier, Phys.Rev.Lett. 16:2318 (1992).
P. Pechukas and P. H¨anggi, Rates of activated processes with fluctuating barriers, Phys.Rev.Lett. 73:2772 (1994).
P. Reimann, Thermally driven escape with fluctuating potentials: A new type of resonant activation, Phys.Rev.Lett. 74:4576 (1995).
L. Gammaitoni, P. H¨anggi, P. Jung, and F. Marchesoni, Stochastic resonance, Rev.Mod.Phys. 70:223 (1998).
J. E. Fletcher, S. Havlin, and G. H. Weiss, First passage time problems in time-dependent fields, J.Stat.Phys. 51:215 (1988).
M. Gitterman and G. H. Weiss, Coherent stochastic resonance in the presence of a field, Phys.Rev.E 52:5708 (1995).
J. Masoliver, A. Robinson, and G. H. Weiss, Coherent stochastic resonance, Phys.Rev.E 51:4021 (1995).
J. M. Porr´a, When coherent stochastic resonance appears, Phys.Rev.E 55:6533 (1997).
B. Lindner and A. Longtin, Nonrenewal spike trains generated by stochastic neuron models, in L. Schimansky-Geier, D. Abbott, A. Neiman, and Ch. Van den Broeck (eds) Noise in Complex Systems and Stochastic Dynamics, Vol 5114 (Bellingham, Washington, 2003), SPIE, p. 209.
M. J. Chacron, A. Longtin, M. St-Hilaire, and L. Maler, Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors, Phys.Rev.Lett. 85:1576 (2000).
M. J. Chacron, K. Pakdaman, and A. Longtin, Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue, Neural Comp. 15:253 (2003).
A. Bulsara, T. C. Elston, Ch. R. Doering, S. B. Lowen, and K. Lindenberg, Cooperative behavior in periodically driven noisy integrate-and-fire models of neuronal dynamics, Phys.Rev.E 53:3958 (1996).
A. Bulsara, S. B. Lowen, and C. D. Rees, Cooperative behavior in the periodically modulated Wiener process: Noise-induced complexity in a model neuron, Phys.Rev.E 49:4989 (1994).
P. L´ansk´y, Sources of periodical force in noisy integrate-and-fire models of neuronal dynamics, Phys.Rev.E 55:2040 (1997).
H. E. Plesser and T. Geisel, Stochastic resonance in neuron models: Endogenous stimulation revisited, Phys.Rev.E 63:031916 (2001).
H. E. Plesser and S. Tanaka, Stochastic resonance in a model neuron with reset, Phys.Lett.A 225:228 (1997).
A. Longtin, Stochastic resonance in neuron models, J.Stat.Phys. 70:309 (1993).
V. Bezak, The first-passage-time problems with time-varying driving fields, Acta Phys.Slov. 39:337 (1989).
V. Balakrishnan, C. Van den Broeck, and P. H¨anggi, First-passage times of non-markovian processes:The case of a reflecting boundary, Phys.Rev.A 38:4213 (1988).
S. Redner, A Guide to First-Passage Processes. (Cambridge University Press, Cambridge, UK, 2001).
M. H. Choi and R. F. Fox, Evolution of escape processes with a time-varying load, Phys.Rev.E 66:031103 (2002).
S. V. G. Menon, First passage time distribution in an oscillating field, J.Stat.Phys. 66:1675 (1992).
I. Klik and Y. D. Yao, Resonant activation in a system with deterministic oscillations of barrier height, Phys.Rev.E 64:012101 (2001).
T. C. Elston and C. R. Doering, Numerical and analytical studies of nonequilibrium fluctuation-induced transport processes, J.Stat.Phys. 83:359 (1996).
R. Bartussek, P. Reimann, and P. H¨anggi, Precise numerics versus theory for correlation ratchets, Phys.Rev.Lett. 76:1176 (1996).
R. Guti´errez, L. M. Ricciardi, P. Rom´an, and F. Torres, First-passage-time densities for time-non-homogeneous diffusion processes, J.Appl.Prob. 34:623 (1997).
R. Guti´errez J´aimez, A. Juan Gonzalez, and P. TRom´an Rom´an, Construction of firstpassage-time densities for a diffusion process which is not necessarily time-homogeneous, J.Appl.Prob. 28:903 (1991).
R. Guti´errez J´aimez, P. Rom´an Rom´an, and F. Torres Ruiz, A note on the Volterra integral equation for the first-passage-time probability density, J.Appl.Prob. 32:635 (1995).
J. Lehmann, P. Reimann, and P. H¨anggi, Surmounting oscillating barriers, Phys.Rev.Lett. 84:1639 (2000).
V. N. Smelyanski, M. I. Dykman, and B. Golding, Time oscillations of escape rates in periodically driven systems, Phys.Rev.Lett. 82:3193 (1999).
P. Talkner and J. Luczka, Rate description of Fokker-Planck processes with time dependent parameters, cond-mat/0307498, (2003).
A. I. Shushin, Effect of external force on the kinetics of diffusion-controlled escaping from a one-dimensional potential well, Phys.Rev.E 62:4688 (2000).
P. H¨anggi, P. Talkner, and M. Borkovec, Reaction rate theory: Fifty years after kramers, Rev.Mod.Phys. 62:251 (1990).
H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7:284 (1940).
C. W. Gardiner, Handbook of Stochastic Methods, (Springer-Verlag, Berlin, 1985).
A. V. Holden, Models of the Stochastic Activity of Neurones, (Springer-Verlag, Berlin, 1976).
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1970).
N. G. Van Kampen, Short first-passage times, J.Stat.Phys. 70:15 (1993).
B. Lindner, L. Schimansky-Geier, and A. Longtin, Maximizing spike train coherence or incoherence in the leaky integrate-and-fire model, Phys.Rev.E 66:031916 (2002).
J. Honerkamp, Stochastic Dynamical Systems.Concepts, Numerical Methods, Data Analysis, (Wiley/VCH, Weinheim, 1993).
K. Pakdaman, S. Tanabe, and T. Shimokawa, Coherence resonance and discharge reliability in neurons and neuronal models, Neural Networks 14:895 (2001).
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Lindner, B. Moments of the First Passage Time Under External Driving. Journal of Statistical Physics 117, 703–737 (2004). https://doi.org/10.1007/s10955-004-2269-5
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DOI: https://doi.org/10.1007/s10955-004-2269-5