Unbiased Diffusion to Escape through Small Windows: Assessing the Applicability of the Reduction to Effective One-Dimension Description in a Spherical Cavity

Marco-Vinicio Vazquez, Leonardo Dagdug

.

DOI: 10.4236/jmp.2011.24037                

Abstract

This study is devoted to unbiased motion of a point Brownian particle that escapes from a spherical cavity through a round hole. Effective one-dimensional description in terms of the generalized Fick-Jacobs equation is used to derive a formula which gives the mean first-passage time as a function of the geometric parameters for any value of a, where a is the hole’s radius. This is our main result and is given in equation (19). This result is a generalization of the Hill’s formula, which is restricted to small values of a.

Keywords

Diffusion, Brownian Particle, Fick-Jacobs Equation, Narrow-Escape Time

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. Redner, “A Guide to First Passage Time Processes,” Cambridge University Press, 2001. doi:10.1017/CBO9780511606014
[2]	P. H?nggi, P. Talkner and M. Borkovec, “Reaction-rate Theory: Fifty Years after Kramers,” Reviews of Modern Physics, Vol. 62, No. 2, pp. 251-341.
[3]	M. Coppey, O. Bénichou, R. Voituriez and M. Moreau, “Kinetics of Target Site Localization of a Protein on DNA: A Stochastic Approach,” Biophysical Journal, Vol. 87, No. 3, pp. 1640-1649.
[4]	O. Bénichou, M. Coppey, M. Moreau, P. H. Suet and R. Voituriez, “Optimal Search Strategies for Hidden Targets,” Physical Review Letters, Vol. 94, No. 19, pp. 198101 (1-4).
[5]	L. Gallos, C. Song, S. Havlin and H. A. Makse, “Scaling Theory of Transport in Complex Biological Networks,” Proceedings of the National Academy of Sciences U. S. A., Vol. 104, No. 19, pp. 7746-7751.
[6]	D. Holcman and Z. Schuss, “Escape Through a Small Opening: Receptor Trafficking in a Synaptic Membrane,” Journal of Statistical Physics, Vol. 117, No. 5--6, pp. 975-1014.
[7]	O. Bénichou, and R. Voituriez, “Narrow-Escape Time Problem: Time Needed for a Particle to Exit a Confining Domain through a Small Window,” Physical Review Letters, Vol. 100, pp. 168105(1-4).
[8]	Z. Schuss, A. Singer and D. Holcman, “The Narrow Escape Problem for Diffusion in Cellular Microdomains,” Proceedings of the National Academy of Sciences U. S. A., Vol. 104, No. 41, pp. 16098-16103.
[9]	S. W. Cowan, T. Schirmer, G. Rummel, M. Steiert, R. Ghosh, R. A. Pauptit, J. N. Jansonius, and J. P. Rosenbusch, Nature, Vol. 358, pp. 727. doi:10.1038/358727a0
[10]	L. Z. Song, M. R. Hobaugh, C. Shustak, S. Cheley, H. Bayley and J. E. Gouaux, Science, Vol. 274, 1996, pp. 1859. doi:10.1126/science.274.5294.1859
[11]	M. Gershow and J. A. Golovchenko, “Recapturing and Trapping Single Molecules with a Solid-state Nanopore,” Nature Nanotechnology, Vol. 2, pp. 775-779. doi:10.1038/nnano.2007.381
[12]	L. T. Sexton, L. P. Horne, S. A. Sherrill, G. W. Bishop, L. A. Baker and C. R. Martin, “Resistive-Pulse Studies of Proteins and Protein/Antibody Complexes Using a Conical Nanotube Sensor,” Journal of the American Chemical Society, Vol. 129, No. 43, pp. 13144-13152.
[13]	I. D. Kosinska, I. Goychuk, M. Kostur, G. Schmidt and P. H?nggi, “Rectification in Synthetic Conical Nanopores: A One-dimensional Poisson-Nernst-Planck Model,” Physical Review E, Vol. 77, No. 3, pp. 031131.
[14]	J. K?rger and D. M. Ruthven, “Diffusion in Zeolites and Other Microporous Solids,” Wiley, New York, 1992.
[15]	R. A. Siegel, “Theoretical Analysis of Inward Hemispheric Release above and below Drug Solubility,” Journal of Controlled Release, Vol. 69, No. 1, pp. 109-126.
[16]	N. F. Sheppard, D. J. Mears, and S. W. Straka, “Micromachined Silicon Structures for Modelling Polymer Matrix Controlled Release Systems ,” Journal of Controlled Release, Vol. 42, No. 1, pp. 15-24.
[17]	M. H. Jacobs, “Diffusion Processes,” Springer, New York, 1967.
[18]	R. Zwanzig, “Diffusion past an Entropy Barrier,” Journal of Physical Chemistry, Vol. 96, No. 10, pp. 3926-3930.
[19]	D. Reguera and J. M. Rub, “Kinetic Equations for Diffusion in the Presence of Entropic Barriers,” Physical Review E, Vol. 64, No. 6, pp. 061106(1-8).
[20]	A. M. Berezhkovskii, M. A. Pustovoit and S. M. Bezrukov, “Diffusion in a Tube of Varying Cross Section: Numerical Study of Reduction to Effective One-Dimensional Description,” Journal of Chemical Physics, Vol. 126, No. 13, pp. 134706(1-5).
[21]	P. Kalinay, “Mapping of Forced Diffusion in Quasi-One- Dimensional Systems,” Physical Review E, Vol. 80, No. 3, pp. 031106(1-10).
[22]	R. M. Bradley, “Diffusion in a Two-Dimensional Channel with Curved Midline and Varying Width: Reduction to an Effective One-Dimensional Description,” Physical Review E, Vol. 80, No. 6, pp. 061142(1-7).
[23]	P. S. Burada and G. Schmid, “Steering the Potential Barriers: Entropic to Energetic,” Physical Review E, Vol. 82, No. 5, pp. 051128(1-6).
[24]	I. V. Grigoriev, Yu. A. Makhnovskii, A. M. Berezhkovskii and V. Yu. Zitserman, “Kinetics of Escape through a Small Hole,” Journal of Chemical Physics, Vol. 116, No. 22, pp. 9574-9577.