Skip to main content
SpringerLink
Log in

A Spectral Approach to Survival Probabilities in Porous Media

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider a diffusive process in a bounded domain with heterogeneously distributed traps, reactive regions or relaxing sinks. This is a mathematical model for chemical reactors with heterogeneous spatial distributions of catalytic germs, for biological cells with specific arrangements of organelles, and for mineral porous media with relaxing agents in NMR experiments. We propose a spectral approach for computing survival probabilities which are represented in the form of a spectral decomposition over the Laplace operator eigenfunctions. We illustrate the performances of the approach by considering diffusion inside the unit disk filled with reactive regions of various shapes and reactivities. The role of the spatial arrangement of these regions and its influence on the overall reaction rate are investigated in the long-time regime. When the reactivity is finite, a uniform filling of the disk is shown to provide the highest reaction rate. Although the heterogeneity tends to reduce the reaction rate, reactive regions can still be heterogeneously arranged to get nearly optimal performances.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Weiss, G.H.: Aspects and Applications of the Random Walk. North-Holland, Amsterdam (1994)

    MATH  Google Scholar 

  2. Redner, S.: A Guide to First-Passage Processes. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  3. Hughes, B.D.: Random Walks and Random Environments. Clarendon, Oxford (1995)

    MATH  Google Scholar 

  4. Weiss, G.H.: Overview of theoretical models for reaction rates. J. Stat. Phys. 42, 3 (1986)

    Article  ADS  Google Scholar 

  5. Coppens, M.-O.: The effect of fractal surface roughness on diffusion and reaction in porous catalysts: from fundamentals to practical application. Catalysis Today 53, 225–243 (1999)

    Article  Google Scholar 

  6. Zwanzig, R., Szabo, A.: Time dependent rate of diffusion-influenced ligand binding to receptors on cell surfaces. Biophys. J. 60, 671–678 (1991)

    Article  ADS  Google Scholar 

  7. Holcman, D., Marchewka, A., Schuss, Z.: Survival probability of diffusion with trapping in cellular neurobiology. Phys. Rev. E 72, 031910 (2005)

    Article  ADS  Google Scholar 

  8. Brownstein, K.R., Tarr, C.E.: Importance of classical diffusion in NMR studies of water in biological cells. Phys. Rev. A 19, 2446–2453 (1979)

    Article  ADS  Google Scholar 

  9. Smoluchowski, M.V.: Phys. Z. 17, 557 (1916)

    ADS  Google Scholar 

  10. Balagurov, B.Ya., Vaks, V.G.: Random walks of a particle on lattices with traps. J. Exp. Theor. Phys. 38, 968 (1974)

    ADS  Google Scholar 

  11. Grassberger, P., Procaccia, I.: The long time properties of diffusion in a medium with static traps. J. Chem. Phys. 77, 6281–6284 (1982)

    Article  ADS  Google Scholar 

  12. Kayser, R.F., Hubbard, J.B.: Diffusion in a medium with a random distribution of static traps. Phys. Rev. Lett. 51, 79 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  13. Kayser, R.F., Hubbard, J.B.: Reaction diffusion in a medium containing a random distribution of nonoverlapping traps. J. Chem. Phys. 80, 1127 (1984)

    Article  ADS  Google Scholar 

  14. Lee, S.B., Kim, I.C., Miller, C.A., Torquato, S.: Random-walk simulation of diffusion-controlled processes among static traps. Phys. Rev. B 39, 11833 (1989)

    Article  ADS  Google Scholar 

  15. Torquato, S., Kim, I.C.: Efficient simulation technique to compute effective properties of heterogeneous media. Appl. Phys. Lett. 55, 1847 (1989)

    Article  ADS  Google Scholar 

  16. Miller, C.A., Torquato, S.: Diffusion-controlled reactions among spherical traps: effect of polydispersity in trap size. Phys. Rev. B 40, 7101 (1989)

    Article  ADS  Google Scholar 

  17. Miller, C.A., Kim, I.C., Torquato, S.: Trapping and flow among random arrays of oriented spheroidal inclusions. J. Chem. Phys. 94, 5592 (1991)

    Article  ADS  Google Scholar 

  18. Kansal, A.R., Torquato, S.: Prediction of trapping rates in mixtures of partially absorbing spheres. J. Chem. Phys. 116, 10589 (2002)

    Article  ADS  Google Scholar 

  19. Richards, P.M.: Diffusion to finite-size traps. Phys. Rev. Lett. 56, 1838 (1986)

    Article  ADS  Google Scholar 

  20. Richards, P.M.: Diffusion to nonoverlapping or spatially correlated traps. Phys. Rev. B 35, 248 (1987)

    Article  ADS  Google Scholar 

  21. Richards, P.M., Torquato, S.: Upper and lower bounds for the rate of diffusion-controlled reactions. J. Chem. Phys. 87, 4612 (1987)

    Article  ADS  Google Scholar 

  22. Rubinstein, J., Torquato, S.: Diffusion-controlled reactions: mathematical formulation, variational principles, and rigorous bounds. J. Chem. Phys. 88, 6372 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  23. Torquato, S., Avellaneda, M.: Diffusion and reaction in heterogeneous media: pore-size distribution, relaxation times, and mean survival time. J. Chem. Phys. 95, 6477 (1991)

    Article  ADS  Google Scholar 

  24. Torquato, S.: Diffusion and reaction among traps: some theoretical and simulation results. J. Stat. Phys. 65, 1173 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Torquato, S., Yeong, C.L.Y.: Universal scaling for diffusion-controlled reactions among traps. J. Chem. Phys. 106, 8814 (1997)

    Article  ADS  Google Scholar 

  26. Riley, M.R., Muzzio, F.J., Buettner, H.M., Reyes, S.C.: The effect of structure on diffusion and reaction in immobilized cell systems. Chem. Eng. Sci. 50, 3357 (1995)

    Article  Google Scholar 

  27. Singer, A., Schuss, Z., Holcman, D., Eisenberg, R.S.: Narrow escape. Part I. J. Stat. Phys. 122, 437 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Singer, A., Schuss, Z., Holcman, D.: Narrow escape. Part II: the circular disk. J. Stat. Phys. 122, 465 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. Bénichou, 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, 168105 (2008)

    Article  ADS  Google Scholar 

  30. Kolokolnikov, T., Titcombe, M.S., Ward, M.J.: Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps. Eur. J. Appl. Math. 16, 161 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  31. Ward, M.J.: Asymptotic methods for reaction-diffusion systems: past and present. Bull. Math. Biol. 68, 1151 (2006)

    Article  Google Scholar 

  32. 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. SIAM Multiscale Model. Simul. 8, 803–835 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  33. Cheviakov, A.F., Ward, M.J., Straube, R.: An asymptotic analysis of the mean first passage time for narrow escape problems: part II. The sphere. SIAM Multiscale Model. Simul. 8, 836–870 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. Cheviakov, A., Ward, M.J.: Optimizing the principal eigenvalue of the Laplacian in a sphere with interior traps. Math. Comput. Model. (2010). doi:10.1016/j.mcm.2010.02.025

    Google Scholar 

  35. Ryu, S.: Effects of inhomogeneous partial absorption and the geometry of the boundary on population evolution of molecules diffusing in general porous media. Phys. Rev. E 80, 026109 (2009)

    Article  ADS  Google Scholar 

  36. Ryu, S., Johnson, D.L.: Aspects of diffusive-relaxation dynamics with a nonuniform, partially absorbing boundary in general porous media. Phys. Rev. Lett. 103, 118701 (2009)

    Article  ADS  Google Scholar 

  37. Condamin, S., Bénichou, O., Tejedor, V., Voituriez, R., Klafter, J.: First-passage time in complex scale-invariant media. Nature 450, 77 (2007)

    Article  ADS  Google Scholar 

  38. Condamin, S., Bénichou, O., Moreau, M.: First-passage times for random walks in bounded domains. Phys. Rev. Lett. 95, 260601 (2005)

    Article  ADS  Google Scholar 

  39. Condamin, S., Bénichou, O., Moreau, M.: First-exit times and residence times for discrete random walks on finite lattices. Phys. Rev. E 72, 016127 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  40. Condamin, S., Bénichou, O., Moreau, M.: Random walks and Brownian motion: a method of computation for first-passage times and related quantities in confined geometries. Phys. Rev. E 75, 021111 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  41. Condamin, S., Tejedor, V., Bénichou, O.: Occupation times of random walks in confined geometries: from random trap model to diffusion-limited reactions. Phys. Rev. E 76, 050102R (2007)

    Article  ADS  Google Scholar 

  42. Yuste, S.B., Oshanin, G., Lindenberg, K., Bénichou, O., Klafter, J.: Survival probability of a particle in a sea of mobile traps: a tale of tails. Phys. Rev. E 78, 021105 (2008)

    Article  ADS  Google Scholar 

  43. Levitz, P.E., Grebenkov, D.S., Zinsmeister, M., Kolwankar, K., Sapoval, B.: Brownian flights over a fractal nest and first passage statistics on irregular surfaces. Phys. Rev. Lett. 96, 180601 (2006)

    Article  ADS  Google Scholar 

  44. Callaghan, P.T.: A simple matrix formalism for spin echo analysis of restricted diffusion under generalized gradient waveforms. J. Magn. Reson. 129, 74–84 (1997)

    Article  ADS  Google Scholar 

  45. Barzykin, A.V.: Theory of spin echo in restricted geometries under a step-wise gradient pulse sequence. J. Magn. Reson. 139, 342–353 (1999)

    Article  ADS  Google Scholar 

  46. Axelrod, S., Sen, P.N.: Nuclear magnetic resonance spin echoes for restricted diffusion in an inhomogeneous field: methods and asymptotic regimes. J. Chem. Phys. 114, 6878–6895 (2001)

    Article  ADS  Google Scholar 

  47. Grebenkov, D.S.: NMR survey of reflected Brownian motion. Rev. Mod. Phys. 79, 1077–1137 (2007)

    Article  ADS  Google Scholar 

  48. Grebenkov, D.S.: Laplacian eigenfunctions in NMR I. A numerical tool. Concepts Magn. Reson. A 32, 277–301 (2008)

    Google Scholar 

  49. Grebenkov, D.S.: Laplacian eigenfunctions in NMR II. Theorical advances. Concepts Magn. Reson. A 34, 264–296 (2009)

    Google Scholar 

  50. Grebenkov, D.S.: Residence times and other functionals of reflected Brownian motion. Phys. Rev. E 76, 041139 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  51. Majumdar, S.N.: Brownian functionals in Physics and Computer Science. Curr. Sci. 89, 2076 (2005)

    Google Scholar 

  52. Truman, A., Williams, D.: In: Pinsky, M.A. (ed.) Diffusion Processes and Related Problems in Analysis, vol. 1. Birkhauser, Basel (1990)

    Google Scholar 

  53. Grebenkov, D.S.: Analytical solution for restricted diffusion in circular and spherical layers under inhomogeneous magnetic fields. J. Chem. Phys. 128, 134702 (2008)

    Article  ADS  Google Scholar 

  54. Grebenkov, D.S.: Subdiffusion in a bounded domain with a partially absorbing/reflecting boundary. Phys. Rev. E 81, 021128 (2010)

    Article  ADS  Google Scholar 

  55. Grebenkov, D.S.: Multiple correlation function approach: rigorous results for simples geometries. Diffus. Fundam. 5, 1 (2007)

    Google Scholar 

  56. Crank, J.: The Mathematics of Diffusion, 2nd edn. Clarendon, Oxford (1975)

    Google Scholar 

  57. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, 2nd edn. Clarendon, Oxford (1959)

    Google Scholar 

  58. Bowman, F.: Introduction to Bessel Functions, 1st edn. Dover, New York (1958)

    MATH  Google Scholar 

  59. Lapidus, M.: Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture. Trans. Am. Math. Soc. 325, 465 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  60. Collins, F.C., Kimball, G.E.: Diffusion-controlled reaction rates. J. Coll. Sci. 4, 425–437 (1949)

    Article  Google Scholar 

  61. Sapoval, B.: General formulation of Laplacian transfer across irregular surfaces. Phys. Rev. Lett. 73, 3314–3317 (1994)

    Article  ADS  Google Scholar 

  62. Grebenkov, D.S.: Partially reflected Brownian motion: a stochastic approach to transport phenomena. In: Velle, L.R. (ed.) Focus on Probability Theory, pp. 135–169. Nova Science Publishers, New York (2006)

    Google Scholar 

  63. Singer, A., Schuss, Z., Osipov, A., Holcman, D.: Partially reflected diffusion. SIAM J. Appl. Math. 68, 844–868 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique de la Matière Condensée, CNRS – Ecole Polytechnique, 91128, Palaiseau, France

    Binh T. Nguyen & Denis S. Grebenkov

Authors
  1. Binh T. Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Denis S. Grebenkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Denis S. Grebenkov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nguyen, B.T., Grebenkov, D.S. A Spectral Approach to Survival Probabilities in Porous Media. J Stat Phys 141, 532–554 (2010). https://doi.org/10.1007/s10955-010-0054-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-010-0054-1

Keywords

Search

Navigation