Advertisement

Study of Brownian functionals for a Brownian process model of snow melt dynamics with purely time dependent drift and diffusion

  • Ashutosh Dubey
  • Malay Bandyopadhyay
Regular Article
  • 13 Downloads

Abstract

In this paper, we investigate a Brownian motion (BM) with purely time dependent drift and diffusion by suggesting and examining several Brownian functionals, which characterize the stochastic model of water resources availability in snowmelt dominated regions with power law time dependent drift and diffusion. Snow melt process is modelled by a overdamped Langevin equation for a Brownian process with power law time dependent drift (μ(t) ~ qktα) and diffusion (D(t) ~ ktα) where they are proportional to each other. We introduce several probability distribution functions (PDFs) associated with such time dependent BMs. For instance, with initial starting value of snow amount H0, we derive analytical expressions for: (i) the PDF P(tf|H0) of the first passage time tf which specify the lifetime of such stochastic process, (ii) the PDF P(A|H0) of the area A till the first passage time and it provides us numerous valuable information about the average available water resources, (iii) the PDF P(M) associated with the maximum amount of available water M of the BM process before the complete melting of snow, and (iv) the joint PDF P(M;tm) of the maximum amount of available water M and its occurrence time tm before the first passage time. We further confirm our analytical predictions by computing the same PDFs with direct numerical simulations of the corresponding Langevin equation. We obtain a very good agreement of our theoretical predictions with the numerically simulated results. Finally, several nontrivial scaling behaviour in the asymptotic limits for the above mentioned PDFs are predicted, which can be verified further from experimental observation.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    D. Marks, J. Kimball, D. Tingey, T. Link, Hydrol. Process. 12, 1569 (1998) ADSCrossRefGoogle Scholar
  2. 2.
    A. Hamlet, D. Lettenmaier, J. Am. Water Resour. Assoc. 35, 1597 (1999) ADSCrossRefGoogle Scholar
  3. 3.
    M. Pascual, M. Bouma, A. Dobson, Microbes Infect. 4, 237 (2002) CrossRefGoogle Scholar
  4. 4.
    J. Patz, D. Campbell-Lendrum, T. Holloway, J. Foley, Nature 438, 310 (2005) ADSCrossRefGoogle Scholar
  5. 5.
    C. Barranguet, J. Kromkamp, J. Peene, Mar. Ecol. Prog. Ser. 173, 117 (1998) ADSCrossRefGoogle Scholar
  6. 6.
    M. Bertness, G. Leonard, Ecology 78, 1976 (1997) CrossRefGoogle Scholar
  7. 7.
    H. Charles, J.S. Dukes, Ecol. Appl. 19, 1758 (2009) CrossRefGoogle Scholar
  8. 8.
    A.R. Bulsara, S.B. Lowen, C.D. Rees, Phys. Rev. E 49, 4989 (1994) ADSCrossRefGoogle Scholar
  9. 9.
    A.R. Bulsara, T.C. Elston, C.R. Doering, S.B. Lowen, K. Lindenberg, Phys. Rev. E 53, 3958 (1996) ADSCrossRefGoogle Scholar
  10. 10.
    H.E. Plesser, S. Tanaka, Phys. Lett. A 225, 228 (1997) ADSCrossRefGoogle Scholar
  11. 11.
    J.R.R. Duarte, M.V.D. Vermelho, M.L. Lyra, Physica A 387, 1446 (2008) ADSCrossRefGoogle Scholar
  12. 12.
    R.J. Williams, Introduction to the Mathematics of Finance (AMS, Providence, RI, 2006) Google Scholar
  13. 13.
    M. Yor, Exponential Functionals of Brownian Motion and Related Topics (Springer, Berlin, 2000) Google Scholar
  14. 14.
    A. Comtet, C. Monthus, M. Yor, J. Appl. Probab. 35, 255 (1998) MathSciNetCrossRefGoogle Scholar
  15. 15.
    M.J. Kearney, J. Phys. A: Math. Gen. 37, 8421 (2004) ADSCrossRefGoogle Scholar
  16. 16.
    S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943) ADSCrossRefGoogle Scholar
  17. 17.
    H. Risken, The Fokker-Planck Equation: Methods of Solutions and Applications, 2nd ed. (Springer-Verlag, Berlin, 1989) Google Scholar
  18. 18.
    C.W. Gardiner, Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences, 2nd edn. (Springer-Verlag, Berlin, 1985) Google Scholar
  19. 19.
    A. Siegert, Phys. Rev. 81, 617 (1951) ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    G.L. Gerstein, B. Mandelbrot, Biophys. J. 4, 41 (1964) CrossRefGoogle Scholar
  21. 21.
    M. Bandyopadhyay, S. Gupta, D. Segal, Phys. Rev. E 83, 031905 (2011) ADSCrossRefGoogle Scholar
  22. 22.
    A.M. Jayannavar, Chem. Phys. Lett. 199, 149 (1992) ADSCrossRefGoogle Scholar
  23. 23.
    E. Urdapilleta, Phys. Rev. E 83, 021102 (2011) ADSCrossRefGoogle Scholar
  24. 24.
    J. Benda, L. Maler, A. Longtin, J. Neurophysiol. 104, 2806 (2010) CrossRefGoogle Scholar
  25. 25.
    B. Lindner, A. Longtin, J. Theor. Biol. 232, 505 (2005) CrossRefGoogle Scholar
  26. 26.
    S. Kumar, G. Mishra, Phys. Rev. Lett. 110, 258102 (2013) Google Scholar
  27. 27.
    S. Kumar, R. Kumar, W. Janke, Phys. Rev. E 93, 010402(R) (2016) ADSCrossRefGoogle Scholar
  28. 28.
    B.S. Alexandrov, V. Gelev, A.R. Bishop, A. Usheva, K. Rasmussen, Phys. Lett. A 374, 1214 (2010) ADSCrossRefGoogle Scholar
  29. 29.
    E.S. Swanson, Phys. Rev. E 83, 040901(R) (2011) ADSCrossRefGoogle Scholar
  30. 30.
    A. Molini, P. Talkner, G.G. Katul, A. Porporatoa, Physica A 390, 1841 (2011) ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    S.N. Majumdar, Curr. Sci. 89, 2076 (2005) Google Scholar
  32. 32.
    J. Randon-Furling, S.N. Majumdar, J. Stat. Mech.: Theory Exp. 2007, P10008 (2007) CrossRefGoogle Scholar
  33. 33.
    M. Kac, Trans. Am. Math. Soc. 65, 1 (1949) CrossRefGoogle Scholar
  34. 34.
    S.N. Majumdar, M.J. Kearney, Phys. Rev. E 76, 031130 (2007) ADSCrossRefGoogle Scholar
  35. 35.
    P.L. Krapivsky, S.N. Majumdar, A. Rosso, J. Phys. A 43, 315001 (2010) ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    A. Hanke, R. Metzler, J. Phys. A 36, L473 (2003) ADSCrossRefGoogle Scholar
  37. 37.
    A. Bar, Y. Kafri, D. Mukamel, Phys. Rev. Lett. 98, 038103 (2007) ADSCrossRefGoogle Scholar
  38. 38.
    A. Bar, Y. Kafri, D. Mukamel, J. Phys.: Condens. Matter 21, 034110 (2009) Google Scholar
  39. 39.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 2007) Google Scholar
  40. 40.
    O. Krichevsky, G. Bonnet, Rep. Prog. Phys. 65, 251 (2002) ADSCrossRefGoogle Scholar
  41. 41.
    G. Altan-Bonnet, A. Libchaber, O. Krichevsky, Phys. Rev. Lett. 90, 138101 (2003) ADSCrossRefGoogle Scholar
  42. 42.
    A.C. Branka, D.M. Heyes, Phys. Rev. E 58, 2611 (1998) ADSCrossRefGoogle Scholar
  43. 43.
    T. Barnett, R. Malone, W. Pennell, D. Stammer, B. Semtner, W. Washington, Clim. Change 62, 1 (2004) CrossRefGoogle Scholar
  44. 44.
    T.P. Barnett, J.C. Adam, D.P. Lettenmaier, Nature 438, 303 (2005) ADSCrossRefGoogle Scholar
  45. 45.
    D. De Walle, A. Rango, Principles of Snow Hydrology (Cambridge University Press, Cambridge, UK, 2008) Google Scholar
  46. 46.
    R.L. Bras, Hydrology: An Introduction to Hydrological Science (Addison-Wesley, Reading, MA, 1990) Google Scholar
  47. 47.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1973) Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Basic Sciences, Indian Institute of Technology BhubaneswarBhubaneswarIndia

Personalised recommendations