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SFD in a Semi-Infinite System with Absorbing Boundary

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Stochastic Dynamics and Energetics of Biomolecular Systems

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Abstract

In this chapter we consider diffusion of hard-core interacting Brownian particles in a semi-infinite one-dimensional interval with the absorbing boundary at the origin.

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Notes

  1. 1.

    Notice that the notation \(\mathbf {X}_{n}(t)\) has been used in the introductory Chap. 2 for noninteracting particles. Hence it seems to be more plausible to denote the position of the nth tracer and its absorption time as \(\mathbf {X}_{n:\infty }(t)\), \(\mathbf {T}_{n:\infty }\), respectively. However, we have decided not to include the symbol “\(\infty \)” into the notation. The precise meaning should be always clear from the context and we believe that no confusion may arise. From now on the position of a generic noninteracting particle and its absorption time will be denoted by unindexed random variables \(\mathbf {X}(t)\), \(\mathbf {T}\), respectively.

  2. 2.

    Everywhere in this chapter and Chap. 4 all PDFs that have originated in the one-dimensional single-particle problem are denoted by the letter “f”. Contrary to this, PDFs that occur in the many-particle problem with interaction will be designated by the letter “p”.

  3. 3.

    The vector notation used: \(\vec {x}_{N-k}=(x_{k+1},\ldots \,,x_{N})\), \(k=0,1,\ldots ,(N-1)\).

  4. 4.

    The idea behind the last term on the right-hand side of Eq. (3.32) is essentially the same as that behind equation (3.12).

  5. 5.

    Main steps of the underlying calculation are following. We define the auxiliary function B(x, t), \(F(x,t)=1-B(x,t)\). Then we expand the term \([F(x,t)]^{n-1}\) in the tracer PDF (3.39) according to the binomial theorem into the sum of powers of B. Properties of the auxiliary function B (\(f(x,t) = - \partial B/\partial x\), \(B(0,t)=S(t)\), \(B(\infty ,t)=0\)) allows for a direct integration of each term of the emerging sum which yields Eq. (3.43).

References

  1. A. Ryabov, P. Chvosta, Survival of interacting Brownian particles in crowded one-dimensional environment. J. Chem. Phys. 136, 064114 (2012). doi:10.1063/1.3684954

    Article  ADS  Google Scholar 

  2. A. Ryabov, P. Chvosta, Tracer dynamics in semi-infinite system with absorbing boundary. Phys. Rev. E 89, 022132 (2014). doi:10.1103/PhysRevE.89.022132

    Article  ADS  Google Scholar 

  3. S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, 2001). ISBN 0-521-65248-0

    Google Scholar 

  4. P. Pollett, Quasi-stationary distributions: a bibliography (2012). http://www.maths.uq.edu.au/~pkp/papers/qsds/qsds.pdf

  5. D. Steinsaltz, S.N. Evans, Markov mortality models: implications of quasistationarity and varying initial distributions. Theor. Popul. Biol. 65, 319 (2004). doi:10.1016/j.tpb.2003.10.007

    Article  MATH  Google Scholar 

  6. A.M. Yaglom, Certain limit theorems of the theory of branching random processes. Dokl. Akad. Nauk SSSR 56, 795 (1947) MR0022045 (in Russian)

    Google Scholar 

  7. I. Nåsell, The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Probab. 28, 895 (1996). doi:10.2307/1428186

    Article  MATH  Google Scholar 

  8. P.A. Ferrari, S. Martínez, J.S. Martín, Phase transition for absorbed Brownian motion with drift. J. Stat. Phys. 86, 213 (1997). doi:10.1007/BF02180205

    Article  ADS  MATH  Google Scholar 

  9. P. Cattiaux, P. Collet, A. Lambert, S. Martínez, S. Méléard, J.S. Martín, Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab. 37, 1647 (2009). doi:10.1214/09-AOP451

    Article  MathSciNet  Google Scholar 

  10. S. Méléard, D. Villemonais, Quasi-stationary distributions and population processes. Probab. Surv. 9, 340 (2012). doi:10.1214/11-PS191

    Article  MathSciNet  MATH  Google Scholar 

  11. E.A. DiMarzio, Proper accounting of conformations of a polymer near a surface. J. Chem. Phys. 42, 2101 (1965). doi:10.1063/1.1696251

    Article  ADS  Google Scholar 

  12. R.-J. Roe, Conformation of an isolated polymer molecule at an interface. II. Dependence on molecular weight. J. Chem. Phys. 43, 1591 (1965). doi:10.1063/1.1696976

    Article  ADS  Google Scholar 

  13. A. Silberberg, Adsorption of flexible macromolecules. III. Generalized treatment of the isolated macromolecule; the effect of selfexclusion. J. Chem. Phys. 46, 1105 (1967). doi:10.1063/1.1840775

    Article  ADS  Google Scholar 

  14. E.A. DiMarzio, R.J. Rubin, Adsorption of a chain polymer between two plates. J. Chem. Phys. 55, 4318 (1971). doi:10.1063/1.1676755

    Article  ADS  Google Scholar 

  15. H. Taitelbaum, S. Havlin, G.H. Weiss, Effects of hard-core interaction on nearest-neighbor distances at a single trap for random walks and for diffusion processes. Chem. Phys. 146, 351 (1990). doi:10.1016/0301-0104(90)80055-3

    Article  ADS  Google Scholar 

  16. G.H. Weiss, K.E. Shuler, K. Lindenberg, Order statistics for first passage times in diffusion processes. J. Stat. Phys. 31, 255 (1983). doi:10.1007/BF01011582

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. S.B. Yuste, K. Lindenberg, Order statistics for first passage times in one-dimensional diffusion processes. J. Stat. Phys. 85, 501 (1996). doi:10.1007/BF02174217

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. K. Lindenberg, V. Seshadri, K.E. Shuler, G.H. Weiss, Lattice random walks for sets of random walkers. First passage times. J. Stat. Phys. 23, 11 (1980). doi:10.1007/BF01014427

    Article  MathSciNet  ADS  Google Scholar 

  19. S.B. Yuste, L. Acedo, K. Lindenberg, Order statistics for d-dimensional diffusion processes. Phys. Rev. E 64, 052102 (2001). doi:10.1103/PhysRevE.64.052102

    Article  ADS  Google Scholar 

  20. L. Acedo, S.B. Yuste, Survival probability and order statistics of diffusion on disordered media. Phys. Rev. E 66, 011110 (2002). doi:10.1103/PhysRevE.66.011110

    Article  ADS  Google Scholar 

  21. S.B. Yuste, L. Acedo, Order statistics of the trapping problem. Phys. Rev. E 64, 061107 (2001). doi:10.1103/PhysRevE.64.061107

    Article  ADS  Google Scholar 

  22. S.B. Yuste, Escape times of \(j\) random walkers from a fractal labyrinth. Phys. Rev. Lett. 79, 3565 (1997). doi:10.1103/PhysRevLett.79.3565

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. S.B. Yuste, Order statistics of diffusion on fractals. Phys. Rev. E 57, 6327 (1998). doi:10.1103/PhysRevE.57.6327

    Article  ADS  Google Scholar 

  24. S.B. Yuste, L. Acedo, Multiparticle trapping problem in the half-line. Physica A 297, 321 (2001). doi:10.1016/S0378-4371(01)00244-8

    Article  ADS  MATH  Google Scholar 

  25. E. Ben-Naim, Mixing of diffusing particles. Phys. Rev. E 82, 061103 (2010). doi:10.1103/PhysRevE.82.061103

    Article  ADS  Google Scholar 

  26. E. Ben-Naim, P.L. Krapivsky, First-passage exponents of multiple random walks. J. Phys. A: Math. Theor. 43, 495008 (2010). doi:10.1088/1751-8113/43/49/495008

    Article  MathSciNet  Google Scholar 

  27. W. Feller, An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. (Wiley, New York, 1968). ISBN 13 978-0471257080

    Google Scholar 

  28. N. Leibovich, E. Barkai, Everlasting effect of initial conditions on single-file diffusion. Phys. Rev. E 88, 032107 (2013). doi:10.1103/PhysRevE.88.032107

    Article  ADS  Google Scholar 

  29. L. Lizana, T. Ambjörnsson, A. Taloni, E. Barkai, M.A. Lomholt, Foundation of fractional Langevin equation: harmonization of a many-body problem. Phys. Rev. E 81, 051118 (2010). doi:10.1103/PhysRevE.81.051118

    Article  ADS  Google Scholar 

  30. L.P. Sanders, T. Ambjörnsson, First passage times for a tracer particle in single file diffusion and fractional Brownian motion. J. Chem. Phys. 136, 175103 (2012). doi:10.1063/1.4707349

    Article  ADS  Google Scholar 

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  1. Department of Macromolecular Physics, Charles University in Prague, Prague, Czech Republic

    Artem Ryabov

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Ryabov, A. (2016). SFD in a Semi-Infinite System with Absorbing Boundary. In: Stochastic Dynamics and Energetics of Biomolecular Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-27188-0_3

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