Abstract
We study the motion of N = 2 overdamped Brownian particles in gravitational interaction in a space of dimension d = 2. This is equivalent to the simplified motion of two biological entities interacting via chemotaxis when time delay and degradation of the chemical are ignored. This problem also bears similarities with the stochastic motion of two point vortices in viscous hydrodynamics [O. Agullo, A. Verga, Phys. Rev. E 63, 056304 (2001)]. We analytically obtain the probability density of finding the particles at a distance r from each other at time t. We also determine the probability that the particles have coalesced and formed a Dirac peak at time t (i.e. the probability that the reduced particle has reached r = 0 at time t). Finally, we investigate the mean square separation \(\langle\) r 2 \(\rangle\) and discuss the proper form of the virial theorem for this system. The reduced particle has a normal diffusion behavior for small times with a gravity-modified diffusion coefficient \(\langle\) r 2 \(\rangle\) = r 0 2 + (4k B /ξ μ)(T–\(T_{*}\))t, where k B \(T_{*}\) = Gm 1 m 2/2 is a critical temperature, and an anomalous diffusion for large times \(\langle\) r 2 \(\rangle\) \(\propto\) \(t^{1-T_*/T}\). As a by-product, our solution also describes the growth of the Dirac peak (condensate) that forms at large time in the post collapse regime of the Smoluchowski-Poisson system (or Keller-Segel model in biology) for T < T c = GMm/(4k B ). We find that the saturation of the mass of the condensate to the total mass is algebraic in an infinite domain and exponential in a bounded domain. Finally, we provide the general form of the virial theorem for Brownian particles with power law interactions.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
A. Campa, T. Dauxois, S. Ruffo, Phys. Rep. 480, 57 (2009)
P.H. Chavanis, Physica A 361, 81 (2006)
B. Perthame, Appl. Math. 49, 539 (2004)
P.H. Chavanis, C. Sire, Physica A 384, 199 (2007)
P.H. Chavanis, M. Ribot, C. Rosier, C. Sire, Banach Center Publ. 66, 103 (2004)
P.H. Chavanis, Physica A 384, 392 (2007)
P.H. Chavanis, C. Sire, Physica A 387, 4033 (2008)
P.H. Chavanis, C. Sire, Phys. Rev. E 73, 066103 (2006)
P.H. Chavanis, C. Sire, Phys. Rev. E 73, 066104 (2006)
E.F. Keller, L.A. Segel, J. Theor. Biol. 30, 225 (1971)
C. Sire, P.H. Chavanis, Collapse and evaporation of a canonical self-gravitating gas, in Proceedings of the 12th Marcel Grossmann Meeting (World Scientific, Singapore, 2010) [arXiv:1003.1118]
M.A. Herrero, J.J.L. Velazquez, J. Math. Biol. 35, 177 (1996)
C. Sire, P.H. Chavanis, Phys. Rev. E 66, 046133 (2002)
C. Sire, P.H. Chavanis, Phys. Rev. E 78, 061111 (2008)
P.M. Lushnikov, Phys. Lett. A 374, 1678 (2010)
R. Mannella, P.H. Chavanis, in preparation
P.H. Chavanis, Eur. Phys. J. B 57, 391 (2007)
C. Sire, P.H. Chavanis, Phys. Rev. E 69, 066109 (2004)
O. Agullo, A. Verga, Phys. Rev. E 63, 056304 (2001)
T. Padmanabhan, Phys. Rep. 188, 285 (1990)
A.M. Salzberg, J. Math. Phys. 6, 158 (1965)
J. Katz, D. Lynden-Bell, Mon. Not. R. Astron. Soc. 184, 709 (1978)
T. Padmanabhan, Mon. Not. R. Astron. Soc. 253, 445 (1991)
M. Kiessling, Comm. Pure Appl. Math. 46, 27 (1993)
E. Abdalla, M.R. Tabar, Phys. Lett. B 440, 339 (1998)
J.J. Aly, J. Perez, Phys. Rev. E 60, 5185 (1999)
P.H. Chavanis, Eur. Phys. J. B 70, 413 (2009)
J. Sopik, C. Sire, P.H. Chavanis, Phys. Rev. E 72, 026105 (2005)
H. Risken, The Fokker-Planck equation (Springer, 1989)
C. Sire, S. Majumdar, A. Rödinger, Phys. Rev. E 61, 1258 (2000)
N.G. Van Kampen, Stochastic Processes in Physics and Chemistry (Amsterdam, North-Holland, 1992)
P.H. Chavanis, Int. J. Mod. Phys. B 20, 3113 (2006)
D.A. Kessler, E. Barkai, [arXiv:1005.4737]
L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory (Pergamon Press, 1977)
J. Sopik, C. Sire, P.H. Chavanis, Phys. Rev. E 74, 011112 (2006)
K. Gawedzki, M. Vergassola, Physica D 138, 63 (2000)
E. Weinan, E. Vanden-Eijnden, Proc. Natl. Acad. Sci. USA 97, 8200 (2000)
K. Gawedzki, P. Horvai, J. Stat. Phys. 116, 1247 (2004)
A. Gabrielli, F. Cecconi, J. Phys. A: Math. Theor. 41, 235003 (2008)
W. Feller, Commun. Pure Appl. Math. 8, 203 (1955)
S. Marksteiner, K. Ellinger, P. Zoller, Phys. Rev. A 53, 3409 (1996)
J. Farago, Europhys. Lett. 52, 379 (2000)
F. Lillo, S. Miccichè, R.N. Mantegna, arXiv:cond-mat/0203442
E. Lutz, Phys. Rev. Lett. 93, 1906021 (2004)
F. Bouchet, T. Dauxois, Phys. Rev. E 72, 5103 (2005)
P.H. Chavanis, M. Lemou, Eur. Phys. J. B 59, 217 (2007)
S. Miccichè, Phys. Rev. E 79, 031116 (2009)
P.H. Chavanis, M. Lemou, Phys. Rev. E 72, 061106 (2005)
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Oxford University Press, 1961)
J. Binney, S. Tremaine, Galactic Dynamics (Princeton Series in Astrophysics, 1987)
P.H. Chavanis, C.R. Phys. 7, 331 (2006)
F. Calogero, J. Math. Phys. 10, 2191 (1969)
B. Sutherland, J. Math. Phys. 12, 2191 (1971)
P.H. Chavanis, Eur. Phys. J. B 62, 179 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chavanis, P., Mannella, R. Self-gravitating Brownian particles in two dimensions: the case of N = 2 particles. Eur. Phys. J. B 78, 139–165 (2010). https://doi.org/10.1140/epjb/e2010-90839-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2010-90839-3