Abstract.
We study the Brownian dynamics of a solid particle on a vibrating solid surface. Phenomenologically, the interaction between the two solid surfaces is modeled by solid friction, and the Gaussian white noise models the vibration of the solid surface. The solid friction force is proportional to the sign of relative velocity. We derive the Fokker-Planck (FP) equation for the time-dependent probability distribution to find the particle at a given location. We calculate analytically the steady state velocity distribution function, mean-square velocity and diffusion coefficient in d-dimensions. We present a generic method of calculating the autocorrelations in d-dimensions. This results in one dimension in an exact evaluation of the steady state velocity autocorrelation. In higher dimensions our exact general expression enables the analytic evaluation of the autocorrelation to any required approximation. We present approximate analytic expressions in two and three dimensions. Next, we numerically calculate the mean-square velocity and steady state velocity autocorrelation function up to d = 3 . Our numerical results are in good agreement with the analytically obtained results.
Graphical abstract
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
V. Balakrishnan, Elements of Nonequilibrium Statistical Mechanics (CRC Press, 2008)
L. Da Vinci, Static measurements of sliding and rolling friction, Codex Arundel, folios 40v, 41r, British Library
G. Amontons, Histoire de l’Academie Royale des Sciences avec les Memoires de Mathematique et de Physique, 1699-1708 (Chez Gerald Kuyper, Amsterdam, 1706-1709) p. 206
C.A. De Coulomb, Théorie des machines simples, en ayant égard au frottement de leurs parties et à la roideur des cordages (reprinted by Bachelier, Paris, 1821)
H. Olsson, K.J. Aström, C. Canudas de Wit, M. Gäfvert, P. Lischinsky, Eur. J. Control 4, 176 (1998)
B.N.J. Persson, Sliding Friction: Physical Principles and Applications (Springer, Berlin, 1998)
J.-C. Piedboeuf, J. de Carufel, R. Hurteau, Multibody Syst. Dyn. 4, 341 (2000)
E.J. Berger, Appl. Mech. Rev. 55, 535 (2002)
F.-J. Elmer, J. Phys. A: Math. Gen. 30, 6057 (1997)
R.I. Leine, D.H. van Campen, A. de Kraker, L. van den Steen, Nonlinear Dyn. 16, 41 (1998)
Q. Feng, Comput. Methods Appl. Mech. Eng. 192, 2339 (2003)
G.J. Stein, R. Zahoransky, P. Mucka, J. Sound Vib. 311, 74 (2008)
R. Blumenfeld, S.F. Edwards, M. Schwartz, Eur. Phys. J. E 32, 333 (2010)
M. Schwartz, R. Blumenfeld, Granular Matter 13, 241 (2011)
M. Schwartz, R. Blumenfeld, arXiv:1310.0983 [cond-mat.soft]
P. Das, S. Puri, M. Schwartz, Phys. Rev. E 94, 032907 (2016)
Y. Murayama, M. Sano, J. Phys. Soc. Jpn. 67, 1826 (1998)
A. Kawarada, H. Hayakawa, J. Phys. Soc. Jpn. 73, 2037 (2004)
G. Peng, H.J. Herrmann, Phys. Rev. E 49, R1796 (1994)
G. Peng, H.J. Herrmann, Phys. Rev. E 51, 1745 (1995)
O. Moriyama, N. Kuroiwa, M. Matsushita, H. Hayakawa, Phys. Rev. Lett. 80, 2833 (1998)
E. Azana, F. Chevoir, P. Moucheront, J. Fluid Mech. 400, 199 (1999)
A. Buguin, F. Brochard, P.G. de Gennes, Eur. Phys. J. E 19, 31 (2006)
D. Fleishman, Y. Asscher, M. Urbakh, J. Phys.: Condens. Matter 19, 096004 (2007)
P.S. Goohpattader, S. Mettu, M.K. Chaudhury, Langmuir 25, 9969 (2009)
A. Gnoli, A. Puglisi, H. Touchette, EPL 102, 014002 (2013)
A. Gnoli, A. Petri, F. Dalton, G. Pontuale, G. Gradenigo, A. Sarracino, A. Puglisi, Phys. Rev. Lett. 110, 120601 (2013)
A. Mauger, Physica A 367, 129 (2006)
Y. Chen, A. Baule, H. Touchette, W. Just, Phys. Rev. E 88, 052103 (2013)
Y. Chen, W. Just, Phys. Rev. E 90, 042102 (2014)
T.G. Sano, K. Kanazawa, H. Hayakawa, Phys. Rev. E 94, 032910 (2016)
A.A. Dubkov, P. Hänggi, I Goychuk, J. Stat. Mech. 2009, P01034 (2009)
J. Talbot, R.D. Wildman, P. Viot, Phys. Rev. Lett. 107, 138001 (2011)
A. Baule, P. Sollich, EPL 97, 020001 (2012)
T.G. Sano, H. Hayakawa, Phys. Rev. E 94, 032104 (2014)
K. kanazawa, T.G. Sano, T. Sagawa, H. Hayakawa, J. Stat. Phys. 160, 1294 (2015)
P.G. de Gennes, J. Stat. Phys. 119, 953 (2005)
H. Hayakawa, Physica D 205, 48 (2005)
A. Baule, E.G.D. Cohen, H. Touchette, J. Phys. A: Math. Theor. 43, 025003 (2010)
A. Baule, H. Touchette, E.G.D. Cohen, Nonlinearity 24, 351 (2011)
H. Touchette, E. van der Straeten, W. Just, J. Phys. A: Math. Theor. 43, 445002 (2010)
A.M. Menzel, N. Goldenfeld, Phys. Rev. E 84, 011122 (2011)
A.M. Menzel, Phys. Rev. E 92, 052308 (2015)
P.E. Kloedan, E. Plater, Numerical Solutions of Stochastic Differential Equations (Springer-Verlag, Berlin, 1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Das, P., Puri, S. & Schwartz, M. Single particle Brownian motion with solid friction. Eur. Phys. J. E 40, 60 (2017). https://doi.org/10.1140/epje/i2017-11549-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/i2017-11549-9