Abstract
We complement the literature on the statistical mechanics of point vortices in two-dimensional hydrodynamics. Using a maximum entropy principle, we determine the multi-species Boltzmann-Poisson equation and establish a form of Virial theorem. Using a maximum entropy production principle (MEPP), we derive a set of relaxation equations towards statistical equilibrium. These relaxation equations can be used as a numerical algorithm to compute the maximum entropy state. We mention the analogies with the Fokker-Planck equations derived by Debye and Hückel for electrolytes. We then consider the limit of strong mixing (or low energy). To leading order, the relationship between the vorticity and the stream function at equilibrium is linear and the maximization of the entropy becomes equivalent to the minimization of the enstrophy. This expansion is similar to the Debye-Hückel approximation for electrolytes, except that the temperature is negative instead of positive so that the effective interaction between like-sign vortices is attractive instead of repulsive. This leads to an organization at large scales presenting geometry-induced phase transitions, instead of Debye shielding. We compare the results obtained with point vortices to those obtained in the context of the statistical mechanics of continuous vorticity fields described by the Miller-Robert-Sommeria (MRS) theory. At linear order, we get the same results but differences appear at the next order. In particular, the MRS theory predicts a transition between sinh and tanh-like ω − ψ relationships depending on the sign of Ku − 3 (where Ku is the Kurtosis) while there is no such transition for point vortices which always show a sinh-like ω − ψ relationship. We derive the form of the relaxation equations in the strong mixing limit and show that the enstrophy plays the role of a Lyapunov functional.
Similar content being viewed by others
References
P.H. Chavanis, Statistical Mechanics of Two-dimensional vortices and stellar systems, in Dynamics and Thermodynamics of Systems with Long Range Interactions, edited by T. Dauxois, S. Ruffo, E. Arimondo, and M. Wilkens, Lecture Notes in Physics (Springer, 2002), Vol. 602
R. Balescu, Statistical Mechanics of Charged Particles (Wiley, 1963)
J. Binney, S. Tremaine, Galactic Dynamics (Princeton Series in Astrophysics, 1987)
A. Campa, T. Dauxois, S. Ruffo, Phys. Rep. 480, 57 (2009)
L. Onsager, Nuovo Cimento, Suppl. 6, 279 (1949)
P. Debye, E. Hückel, Phys. Z. 24, 185 (1923)
P. Debye, E. Hückel, Phys. Z. 24, 305 (1923)
P.H. Chavanis, Physica A 391, 3657 (2012)
C. Marchioro, M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids (Springer, New York, 1994)
J. Miller, Phys. Rev. Lett. 65, 2137 (1990)
R. Robert, J. Sommeria, J. Fluid Mech. 229, 291 (1991)
D. Lynden-Bell, Mon. Not. R. Astron. Soc. 136, 101 (1967)
P.H. Chavanis, J. Sommeria, R. Robert, ApJ 471, 385 (1996)
G. Joyce, D. Montgomery, J. Plasma Phys. 10, 107 (1973)
P.H. Chavanis, Phys. Rev. E 64, 026309 (2001)
R. Kawahara, H. Nakanishi, J. Phys. Soc. Jpn 75, 054001 (2006)
R. Kawahara, H. Nakanishi, J. Phys. Soc. Jpn 76, 074001 (2007)
T. Padmanabhan, Phys. Rep. 188, 285 (1990)
D. Montgomery, G. Joyce, Phys. Fluids 17, 1139 (1974)
S. Kida, J. Phys. Soc. Jpn 39, 1395 (1975)
Y.B. Pointin, T.S. Lundgren, Phys. Fluids 19, 1459 (1976)
T.S. Lundgren, Y.B. Pointin, J. Stat. Phys. 17, 323 (1977)
E. Caglioti, P.L. Lions, C. Marchioro, M. Pulvirenti, Commun. Math. Phys. 143, 501 (1992)
M. Kiessling, Commun. Pure Appl. Math. 47, 27 (1993)
G.L. Eyink, H. Spohn, J. Stat. Phys. 70, 833 (1993)
E. Caglioti, P.L. Lions, C. Marchioro, M. Pulvirenti, Commun. Math. Phys. 174, 229 (1995)
M. Kiessling, J. Lebowitz, Lett. Math. Phys. 42, 43 (1997)
K. Sawada, T. Suzuki, Theoret. Appl. Mech. Jpn 56, 285 (2008)
R. Robert, J. Sommeria, Phys. Rev. Lett. 69, 2776 (1992)
J. Sopik, C. Sire, P.H. Chavanis, Phys. Rev. E 72, 026105 (2005)
E. Keller, L.A. Segel, J. Theor. Biol. 26, 399 (1970)
C.E. Leith, Phys. Fluids 27, 1388 (1984)
R.H. Kraichnan, J. Fluid Mech. 67, 155 (1975)
P.H. Chavanis, J. Sommeria, J. Fluid Mech. 314, 267 (1996)
P.K. Newton, The N-Vortex Problem: Analytical Techniques, in Applied Mathematical Sciences (Springer-Verlag, Berlin, 2001), Vol. 145
G. Kirchhoff, in Lectures in Mathematical Physics, Mechanics (Teubner, Leipzig, 1877)
J. Fröhlich, D. Ruelle, Commun. Math. Phys. 87, 1 (1982)
D. Ruelle, J. Stat. Phys. 61, 865 (1990)
S.F. Edwards, J.B. Taylor, Proc. R. Soc. Lond. A 336, 257 (1974)
P.H. Chavanis, Physica A 387, 6917 (2008)
P.H. Chavanis, Eur. Phys. J. Plus 127, 159 (2012)
J.G. Esler, T.L. Ashbee, N.R. McDonald, Phys. Rev. E 88, 012109 (2013)
P.H. Chavanis, J. Sommeria, J. Fluid Mech. 356, 259 (1998)
P.H. Chavanis, Eur. Phys. J. B 70, 73 (2009)
R. Ellis, K. Haven, B. Turkington, J. Stat. Phys. 101, 999 (2000)
P.H. Chavanis, Int. J. Mod. Phys. B 20, 3113 (2006)
M. Kiessling, Lett. Math. Phys. 34, 49 (1995)
L. Onsager, Phys. Rev. 37, 405 (1931)
P.H. Chavanis, Eur. Phys. J. B 62, 179 (2008)
P.H. Chavanis, Phys. Rev. E 58, R1199 (1998)
P.H. Chavanis, Int. J. Mod. Phys. B 26, 1241002 (2012)
S. Chandrasekhar, ApJ 97, 255 (1943)
W. Nernst, Z. Phys. Chem. 2, 613 (1888)
W. Nernst, Z. Phys. Chem. 4, 129 (1889)
M. Planck, Ann. Phys. 39, 161 (1890)
A. Naso, P.H. Chavanis, B. Dubrulle, Eur. Phys. J. B 77, 187 (2010)
J.B. Taylor, M. Borchardt, P. Helander, Phys. Rev. Lett. 102, 124505 (2009)
H.J.H. Clercx, S.R. Maassen, G.J.F. van Heijst, Phys. Rev. Lett. 80, 5129 (1998)
A. Naso, S. Thalabard, G. Collette, P.H. Chavanis, B. Dubrulle, J. Stat. Mech 6, 06019 (2010)
S. Thalabard, B. Dubrulle, F. Bouchet, arXiv:1306.1081 (2013)
A. Venaille, F. Bouchet, J. Stat. Phys. 143, 346 (2011)
A. Naso, P.H. Chavanis, B. Dubrulle, Eur. Phys. J. B 80, 493 (2011)
C. Herbert, B. Dubrulle, P.H. Chavanis, D. Paillard, Phys. Rev. E 85, 056304 (2012)
C. Herbert, B. Dubrulle, P.H. Chavanis, D. Paillard, J. Stat. Mech. 5, 05023 (2012)
A. Venaille, F. Bouchet, Phys. Rev. Lett. 102, 104501 (2009)
P.H. Chavanis, Phys. Rev. E 68, 036108 (2003)
P.H. Chavanis, A. Naso, B. Dubrulle, Eur. Phys. J. B 77, 167 (2010)
F. Bouchet, A. Venaille, Phys. Rep. 515, 227 (2012)
R.S. Ellis, Large Deviations and Statistical Mechanics (Springer, New York, 1985)
H. Touchette, Phys. Rep. 478, 1 (2009)
T. Ashbee, Ph.D. Thesis, University College London, 2014
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chavanis, PH. Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit. Eur. Phys. J. B 87, 81 (2014). https://doi.org/10.1140/epjb/e2014-40869-x
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2014-40869-x