Research ArticleOpen AccessGoogle Scholar indexed
Unbiased Diffusion to Escape through Small Windows: Assessing the Applicability of the Reduction to Effective One-Dimension Description in a Spherical Cavity
- 1
- 2
Journal of Modern Physics·Volume 02 (2011)·Pages 284–288·Published 8 April 2011·DOI10.4236/jmp.2011.24037
Copy link · social · email
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.
KeywordsDiffusionBrownian ParticleFick-Jacobs EquationNarrow-Escape Time
- S. Redner, “A Guide to First Passage Time Processes,” Cambridge University Press, 2001. doi:10.1017/CBO9780511606014
- 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.
- 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.
- 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).
- 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.
- 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.
- 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).
- 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.
- 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
- 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
- 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
- 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.
- 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.