Journal of Statistical Physics

, Volume 158, Issue 1, pp 192–230 | Cite as

Exit Time Distribution in Spherically Symmetric Two-Dimensional Domains

  • J.-F. RupprechtEmail author
  • O. Bénichou
  • D. S. Grebenkov
  • R. Voituriez


The distribution of exit times is computed for a Brownian particle in spherically symmetric two-dimensional domains (disks, angular sectors, annuli) and in rectangles that contain an exit on their boundary. The governing partial differential equation of Helmholtz type with mixed Dirichlet–Neumann boundary conditions is solved analytically. We propose both an exact solution relying on a matrix inversion, and an approximate explicit solution. The approximate solution is shown to be exact for an exit of vanishing size and to be accurate even for large exits. For angular sectors, we also derive exact explicit formulas for the moments of the exit time. For annuli and rectangles, the approximate expression of the mean exit time is shown to be very accurate even for large exits. The analysis is also extended to biased diffusion. Since the Helmholtz equation with mixed boundary conditions is encountered in microfluidics, heat propagation, quantum billiards, and acoutics, the developed method can find numerous applications beyond exit processes.


Exit time Residence time Mixed boundary condition Helmholtz equation Active transport Microfluidic Heat transfer 



O.B. is supported by the ERC Starting Grant No. FPTOpt-277998. D.G. is supported by an ANR project “INADILIC”.


  1. 1.
    Redner, S.: A Guide to First-Passage Processes. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Condamin, S., Bénichou, O., Tejedor, V., Voituriez, R., Klafter, J.: First-passage times in complex scale-invariant media. Nature 450(7166), 77–80 (2007)ADSCrossRefGoogle Scholar
  3. 3.
    Bénichou, O., Chevalier, C., Meyer, B., Voituriez, R.: Facilitated diffusion of proteins on chromatin. Phys. Rev. Lett. 106, 38102 (2011)CrossRefGoogle Scholar
  4. 4.
    Sheinman, M., Bénichou, O., Kafri, Y., Voituriez, R.: Classes of fast and specific search mechanisms for proteins on DNA. Rep. Prog. Phys. 75(2), 026601 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    Mazzolo, A.: Properties of diffusive random walks in bounded domains. Europhys. Lett. (EPL) 68(3), 350–355 (2004)ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Reingruber, J., Holcman, D.: Diffusion in narrow domains and application to phototransduction. Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) 79(3), 30904 (2009)ADSCrossRefGoogle Scholar
  7. 7.
    Meyer, B., Bénichou, O., Kafri, Y., Voituriez, R.: Geometry-induced bursting dynamics in gene expression. Biophys. J. 102(9), 2186–2191 (2012)ADSCrossRefGoogle Scholar
  8. 8.
    Singer, A., Schuss, Z., Holcman, D.: Narrow escape, Part II: the circular disk. J. Stat. Phys. 122(3), 465–489 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Pillay, S., Ward, M.J., Peirce, A., Kolokolnikov, T.: An asymptotic analysis of the mean first passage time for narrow escape problems: part I: two-dimensional domains. Multiscale Model. Simul. 8(3), 803–835 (2009)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Chevalier, C., Bénichou, O., Meyer, B., Voituriez, R.: First-passage quantities of Brownian motion in a bounded domain with multiple targets: a unified approach. J. Phys. A 44, 25002 (2011)CrossRefGoogle Scholar
  11. 11.
    Isaacson, Samuel A., Newby, Jay: Uniform asymptotic approximation of diffusion to a small target. Phys. Rev. E 88(1), 012820 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Bénichou, O., Chevalier, C., Klafter, J., Meyer, B., Voituriez, R.: Geometry-controlled kinetics. Nat. Chem. 2(6), 472–477 (2010)CrossRefGoogle Scholar
  13. 13.
    Meyer, B., Chevalier, C., Voituriez, R., Bénichou, O.: Universality classes of first-passage-time distribution in confined media. Phys. Rev. E 83(5), 51116 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    Nguyen, Binh T., Grebenkov, Denis S.: A spectral approach to survival probabilities in porous media. J. Stat. Phys. 141(3), 532–554 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Sneddon, I.N.: Mixed Boundary Value Problems in Potential Theory. North-Holland, Amsterdam (1966)zbMATHGoogle Scholar
  16. 16.
    Caginalp, C., Chen, X.: Analytical and numerical results for an escape problem. Arch. Ration. Mech. Anal. 203, 329–342 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Benichou, O., Voituriez, R.: Narrow-escape time problem: time needed for a particle to exit a confining domain through a small window. Phys. Rev. Lett. 100(16), 168104–168105 (2008)ADSCrossRefGoogle Scholar
  18. 18.
    Mattos, Thiago G., Mejía-Monasterio, Carlos, Metzler, Ralf, Oshanin, Gleb: First passages in bounded domains: when is the mean first passage time meaningful? Phys. Rev. E 86(3), 031143 (2012)ADSCrossRefGoogle Scholar
  19. 19.
    Singer, A., Schuss, Z., Holcman, D.: Narrow escape, Part III: non-smooth domains and Riemann surfaces. J. Stat. Phys. 122(3), 491–509 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Lagache, T., Holcman, D.: Effective motion of a virus trafficking inside a biological cell. SIAM J. Appl. Math. 68(4), 1146–1167 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Carslaw, H.: Conduction of Heat in Solids. Clarendon, Oxford (1959)Google Scholar
  22. 22.
    Crank, J.: The Mathematics of Diffusion. Oxford Science Publications, Oxford (1975)Google Scholar
  23. 23.
    Sbragaglia, M., Prosperetti, A.: A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces. Phys. Fluids 19(4), 043603 (2007)ADSCrossRefGoogle Scholar
  24. 24.
    Joseph, P., Cottin-Bizonne, C., Benoît, J.-M., Ybert, C., Journet, C., Tabeling, P., Bocquet, L.: Slippage of water past superhydrophobic carbon nanotube forests in microchannels. Phys. Rev. Lett. 97(15), 156104 (2006)ADSCrossRefGoogle Scholar
  25. 25.
    Cottin-Bizonne, C., Barentin, C., Charlaix, E., Bocquet, L., Barrat, J.-L.: Dynamics of simple liquids at heterogeneous surfaces: molecular-dynamics simulations and hydrodynamic description. Eur. Phys. J. E Soft Matter 15(4), 427–438 (2004)CrossRefGoogle Scholar
  26. 26.
    Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
  27. 27.
    Abramowitz, Milton: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Am. J. Phys. 56(10), 958 (1988)ADSCrossRefGoogle Scholar
  28. 28.
    Grebenkov, D.: Efficient Monte Carlo methods for simulating diffusion-reaction processes in complex systems. arXiv:1304.7807 (2013)
  29. 29.
    Berezhkovskii, A.M., Barzykin, A.V.: Extended narrow escape problem: boundary homogenization-based analysis. Phys. Rev. E 82(1), 011114 (2010)ADSCrossRefGoogle Scholar
  30. 30.
    Bénichou, O., Grebenkov, D., Levitz, P., Loverdo, C., Voituriez, R.: Optimal reaction time for surface-mediated diffusion. Phys. Rev. Lett. 105, 150606 (2010)CrossRefGoogle Scholar
  31. 31.
    Bénichou, O., Grebenkov, D., Levitz, P., Loverdo, C., Voituriez, R.: Mean first-passage time of surface-mediated diffusion in spherical domains. J. Stat. Phys. 142(4), 657–685 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Rupprecht, J.F., Bénichou, O., Grebenkov, D., Voituriez, R.: Kinetics of active surface-mediated diffusion in spherically symmetric domains. J. Stat. Phys. 147(5), 891–918 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Pryor, R.W.: Multiphysics Modeling Using COMSOL: A First Principles Approach. Jones & Bartlett Learning, Sudbury (2009)Google Scholar
  34. 34.
    Castro, L.P., Speck, F.O., Teixeira, F.S.: Mixed boundary value problems for the helmholtz equation in a quadrant. Integral Equ. Oper. Theory 56(1), 1–44 (2005)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Grebenkov, D.: Efficient Monte Carlo methods for simulating diffusion-reaction processes in complex systems. In: First-Passage Phenomena and their Applications. World Scientific Publishing Company, Singapore (2013)Google Scholar
  36. 36.
    Temkin, S.: Elements of Acoustics. American Institute of Physics, Woodbury (2001)Google Scholar
  37. 37.
    Duffy, D.G.: Mixed Boundary Value Problems. Chapman & Hall, London (2007)Google Scholar
  38. 38.
    Valsa, J.: Invlap package (from Matlabcentral/fileexchange), (2011)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • J.-F. Rupprecht
    • 1
    Email author
  • O. Bénichou
    • 1
  • D. S. Grebenkov
    • 2
  • R. Voituriez
    • 1
  1. 1.Sorbonne Universités, UPMC Univ Paris 06, UMR 7600Laboratoire de Physique Théorique de la Matière Condensée75005 ParisFrance
  2. 2.Laboratoire de Physique de la Matière Condensée (UMR 7643)CNRS – Ecole PolytechniquePalaiseau CedexFrance

Personalised recommendations