Model and Statistical Analysis of the Motion of a Tired Random Walker in Continuum

ABSTRACT

The model of a tired random walker, whose jump-length decays exponentially in time, is proposed and the motion of such a tired random walker is studied systematically in one-, two- and three-dimensional continuum. In all cases, the diffusive nature of walker breaks down due to tiring which is quite obvious. In one-dimension, the distribution of the displacement of a tired walker remains Gaussian (as observed in normal walker) with reduced width. In two and three dimensions, the probability distribution of displacement becomes nonmonotonic and unimodal. The most probable displacement and the deviation reduce as the tiring factor increases. The probability of return of a tired walker decreases as the tiring factor increases in one and two dimensions. However, in three dimensions, it is found that the probability of return is almost insensitive to the tiring factor. The probability distributions of first return time of a tired random walker do not show the scale invariance as observed for a normal walker in continuum. The exponents, of such power law distributions of first return time, in all three dimensions are estimated for normal walker. The exit probability and the probability distribution of first passage time are found in all three dimensions. A few results are compared with available analytical calculations for normal walker.

The model of a tired random walker, whose jump-length decays exponentially in time, is proposed and the motion of such a tired random walker is studied systematically in one-, two- and three-dimensional continuum. In all cases, the diffusive nature of walker breaks down due to tiring which is quite obvious. In one-dimension, the distribution of the displacement of a tired walker remains Gaussian (as observed in normal walker) with reduced width. In two and three dimensions, the probability distribution of displacement becomes nonmonotonic and unimodal. The most probable displacement and the deviation reduce as the tiring factor increases. The probability of return of a tired walker decreases as the tiring factor increases in one and two dimensions. However, in three dimensions, it is found that the probability of return is almost insensitive to the tiring factor. The probability distributions of first return time of a tired random walker do not show the scale invariance as observed for a normal walker in continuum. The exponents, of such power law distributions of first return time, in all three dimensions are estimated for normal walker. The exit probability and the probability distribution of first passage time are found in all three dimensions. A few results are compared with available analytical calculations for normal walker.

Cite this paper

Acharyya, M. (2015) Model and Statistical Analysis of the Motion of a Tired Random Walker in Continuum.*Journal of Modern Physics*, **6**, 2021-2034. doi: 10.4236/jmp.2015.614208.

Acharyya, M. (2015) Model and Statistical Analysis of the Motion of a Tired Random Walker in Continuum.

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[1] Bhattacherjee, S.M., Giacometti, A. and Maritan, A. (2013) Journal of Physics: Condensed Matter, 25, Article ID: 503101.

[2] Hsu, H.-P. and Grassberger, P. (2011) Journal of Statistical Physics, 144, 597.

http://dx.doi.org/10.1007/s10955-011-0268-x

[3] Bhattacharjee, J.K. (1996) Physical Review Letters, 77, 1524.

http://dx.doi.org/10.1103/PhysRevLett.77.1524

[4] Tejedor, V. (2012) Ph.D. Thesis, Universite Pierre and Marie Curie, France and Technische Universitat, Munchen, Germany.

[5] Lubeck, S. and Hucht, F. (2001) Journal of Physics A: Mathematical and Theoretical, 34, L577.

http://dx.doi.org/10.1088/0305-4470/34/42/103

[6] Sumedha and Dhar, D. (2006) Journal of Statistical Physics, 115, 55.

[7] Kapri, R. and Dhar, D. (2009) Physical Review E, 80, Article ID: 1051118.

http://dx.doi.org/10.1103/PhysRevE.80.051118

[8] Manna, S.S. and Stella, A.L. (2002) Physica A, 316, 135.

http://dx.doi.org/10.1016/S0378-4371(02)01497-8

[9] Goswami, S. and Sen, P. (2012) Physical Review A, 86, Article ID: 022314.

http://dx.doi.org/10.1103/PhysRevA.86.022314

[10] Dhar, D. and Stauffer, D. (1998) International Journal of Modern Physics C, 9, 349.

http://dx.doi.org/10.1142/S0129183198000273

[11] De Bacco, C., Majumdar, S.N. and Sollich, P. (2015) Journal of Physics A: Mathematical and Theoretical, 48, Article ID: 205004.

http://dx.doi.org/10.1088/1751-8113/48/20/205004

[12] Kearney, M.J. and Majumdar, S.N. (2014) Journal of Statistical Physics, 47, Article ID: 465001.

[13] Condamin, S., Benichou, O., Tejedor, V., Voituriez, R. and Klafter, J. (2007) Nature, 450, 77-80.

http://dx.doi.org/10.1038/nature06201

[14] Tejedor, V., Benichou, O., Metzler, R. and Voituriez, R. (2011) Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 255003.

http://dx.doi.org/10.1088/1751-8113/44/25/255003

[15] Majumdar, S.N., Mounaix, P. and Schehr, G. (2014) Journal of Statistical Mechanics: Theory and Experiment, 2014, Article ID: P09013.

[16] Godreche, C., Majumdar, S.N. and Schehr, G. (2014) Journal of Physics A: Mathematical and Theoretical, 47, Article ID: 255001.

http://dx.doi.org/10.1088/1751-8113/47/25/255001

[17] Baule, A., Vijay Kumar, K. and Ramaswamy, S. (2008) Journal of Statistical Mechanics: Theory and Experiment, 2008, Article ID: P11008.

http://dx.doi.org/10.1088/1742-5468/2008/11/P11008

[18] Schnitzer, M.J. (1993) Physical Review E, 48, 2553-2568.

http://dx.doi.org/10.1103/PhysRevE.48.2553

[19] Drewry, H.P.G. and Seaton, N.A. (1995) AIChE Journal, 41, 880-893.

http://dx.doi.org/10.1002/aic.690410415

[20] Acharyya, A.B. (2015) Return Probability of a Random Walker in Continuum with Uniformly Distributed Jump-Length. arxiv.org:1506.00269[cond-mat,stat-mech].

[21] Serino, C.A. and Redner, S. (2010) Journal of Statistical Mechanics: Theory and Experiment, 2010, Article ID: P01006.

http://dx.doi.org/10.1088/1742-5468/2010/01/P01006

[22] Krapivsky, P.L. and Redner, S. (2004) American Journal of Physics, 72, 591.

http://dx.doi.org/10.1119/1.1632487

[23] Redner, S. (2001) A Guide to First-Passage Process. Cambridge University Press, Cambridge, UK.

http://dx.doi.org/10.1017/CBO9780511606014

[24] Poyla, G. (1921) Mathematische Annalen, 84, 149-160.

http://dx.doi.org/10.1007/BF01458701