By generalizing Bogolyubov’s reduced description method, we suggest a formalism to derive kinetic equations for many-body dissipative systems in external stochastic field. As a starting point, we use a stochastic Liouville equation obtained from Hamilton’s equations taking dissipation and stochastic perturbations into account. The Liouville equation is then averaged over realizations of the stochastic field by an extension of the Furutsu-Novikov formula to the case of a non-Gaussian field. As the result, a generalization of the classical Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy is derived. In order to get a kinetic equation for the single-particle distribution function, we use a regular cutoff procedure of the BBGKY hierarchy by assuming weak interaction between the particles and weak intensity of the field. Within this approximation, we get the corresponding Fokker-Planck equation for the system in a non-Gaussian stochastic field. Two particular cases are discussed by assuming either Gaussian statistics of external perturbation or homogeneity of the system.
REFERENCES
1.
N.
Bogolyubov
, Problems of Dynamical Theory in Statistical Physics
(Providence College
, Providence, RI
, 1959
).2.
A. I.
Akhiezer
and S. V.
Peletminskii
, Methods of Statistical Physics
(Pergamon
, Oxford
, 1981
).3.
G. E.
Uhlenbeck
and G. W.
Ford
, Lectures in Statistical Mechanics
(American Mathematical Society
, Providence
, 1963
).4.
M. H.
Ernst
, inProgress in Statistical Physics
, edited by W.
Sung
, I.
Chang
, K.
Byungnam
, et al (World Scientific
, Singapore
, 1998
), pp. 3
-28
.5.
E. G. D.
Cohen
, Physica A
194
, 229
(1993
).6.
Yu. L.
Klimontovich
, The Kinetic Theory of Electromagnetic Processes
(Springer-Verlag
, Berlin
, 1983
).7.
W. B.
Russel
, D. A.
Saville
, and W. R.
Schowalter
, Colloidal Dispersions
(Cambridge University Press
, Cambridge
, 1989
).8.
M.
Doi
and S. F.
Edwards
, The Theory of Polymer Dynamics
(Clarendon Press
, Oxford
, 1986
).9.
V. N.
Pokrovsky
, The Mesoscopic Theory of Polymer Dynamics
, Springer Series in Chemical Physics
(Springer Science and Business Media
, 2010
), Vol. 95
.10.
M.
Mayorga
, L.
Romero-Salazar
, and J. M.
Rubi
, Physica A
307
, 297
(2002
).11.
A. J.
Archer
and M.
Rauscher
, J. Phys. A: Math. Gen.
37
, 9325
(2004
).12.
A.
Basu
and S.
Ramaswamy
, J. Stat. Mech.: Theory Exp.
11
, 11003
(2007
).13.
H.
Xia
, K.
Ishii
, T.
Iwaii
, H.
Li
, and B.
Yang
, Appl. Opt.
47
, 1257
(2008
).14.
P.-H.
Chavanis
, Physica A
390
, 1546
(2011
).15.
N. V.
Brilliantov
and T.
Poschel
, Kinetic Theory of Granular Gases
(Oxford University Press
, Oxford
, 2004
).16.
A.
Mehta
, Granular Physics
(Cambridge University Press
, Cambridge
, 2007
).17.
D.
Chowdhury
, L.
Santen
, and A.
Schadschneider
, Phys. Rep.
329
, 199
(2000
).18.
D.
Helbing
, Rev. Mod. Phys.
73
, 1067
(2001
).19.
A.
Barrat
, E.
Trizac
, and M. H.
Ernst
, J. Phys.: Condens. Matter
17
, S2429
(2005
).20.
T. P. C.
Van Noije
, M. H.
Ernst
, and R.
Brito
, Physica A
251
, 266
(1998
).21.
T. P. C.
Van Noije
, M. H.
Ernst
, E.
Trizac
, and I.
Pagonabarraga
, Phys. Rev. E
59
, 4326
(1999
).22.
M. H.
Ernst
, Dynamics: Models and Kinetic Methods for Non-Equilibrium Many Body Systems
, NATO Advanced Science Institutes Series, Series E, Applied Sciences
edited by J.
Karkheck
(Springer Science & Business Media
, 2000
), Vol. 371
, p. 239
.23.
T. P. C.
Noije
and M. H.
Ernst
, Granular Gases
, Lecture Notes in Physics
, edited byT.
Pöschel
and S.
Luding
(Springer Science & Business Media
, 2001
), Vol. 564
, p. 3
.24.
I.
Pagonabarraga
, E.
Trizac
, T. P. C.
van Noije
, and M. H.
Ernst
, Phys. Rev. E
65
, 011303
(2001
).25.
P.
Maynar
, M. I. G.
de Soria
, and E.
Trizac
, Eur. Phys. J.: Spec. Top.
179
, 123
(2009
).26.
P.
Maynar
and M. I. G.
de Soria
, Math. Modell. Nat. Phenom.
6
, 87
(2011
).27.
N.
Khalil
and V.
Garzó
, Phys. Rev. E
88
, 052201
(2013
).28.
M. I. G.
De Soria
, P.
Maynar
, and E.
Trizac
, Phys. Rev. E
87
, 022201
(2013
).29.
A.
Prados
and E.
Trizac
, Phys. Rev. Lett.
112
, 198001
(2014
).30.
S.
Ramaswamy
, Annu. Rev. Condens. Matter Phys.
1
, 323
(2010
).31.
P.
Romanczuk
, M.
Bar
, W.
Ebeling
, B.
Lindner
, and L.
Schimansky-Geier
, Eur. Phys. J.: Spec. Top.
202
, 1
–162
(2012
).32.
M. C.
Marchetti
, J. F.
Joanny
, S.
Ramaswamy
, T. B.
Liverpool
, J.
Prost
, M.
Rao
, and R. A.
Simha
, Rev. Mod. Phys.
85
, 1143
(2013
).33.
R.
Grossman
, L.
Schimansky-Geier
, and P.
Romanczuk
, New J. Phys.
15
, 085014
(2013
).34.
T.
Ihle
, Eur. Phys. J.: Spec. Top.
223
, 1293
(2014
).35.
A.
Peshkov
, E.
Bertin
, F.
Ginelli
, and H.
Chaté
, Eur. Phys. J.: Spec. Top.
223
, 1315
(2014
).36.
E.
Bertin
, H.
Chaté
, F.
Ginelli
, G.
Grégoire
, S.
Léonardo
, and A.
Peshkov
, Eur. Phys. J.: Spec. Top.
223
, 1419
(2014
).37.
T.
Ihle
, Eur. Phys. J.: Spec. Top.
223
, 1423
(2014
).38.
N. V.
Laskin
, S. V.
Peletminskii
, and V. I.
Prikhod’ko
, J. Phys. Stud.
2
, 3
(1998
).39.
I.
Goldhirsch
, S. V.
Peletminskii
, A. S.
Peletminskii
, and A. I.
Sokolovskii
, “Application of Bogolyubov’s approach to the derivation of kinetic equations for dissipative systems
,” e-print arXiv:1307.3466v1 [cond-mat.stat-mech] (2013).40.
41.
42.
S. V.
Peletminskii
, Yu. V.
Slyusarenko
, and A. I.
Sokolovskii
, Physica A
326
, 412
(2003
).43.
L. D.
Landau
and E. M.
Lifshitz
, Statistical Physics
, Course of Theoretical Physics
(Pergamon
, London, Oxford
, 1980
), Vol.5
.44.
J. J.
Brey
, M. J.
Ruiz-Montero
, and D.
Cubero
, Phys. Rev. E
54
, 3664
(1996
).45.
J. J.
Brey
, J. W.
Dufty
, and A.
Santos
, J. Stat. Phys.
87
, 1051
(1997
).46.
J. J.
Brey
, M. I. G.
de Soria
, P.
Maynar
, and M. J.
Ruiz-Montero
, Phys. Rev. E
70
, 011302
(2004
).47.
A.
Baldassarri
, A.
Barrat
, G.
D’Anna
, V.
Loreto
, P.
Mayor
, and A.
Puglisi
, J. Phys.: Condens. Matter
17
, S2405
(2005
).48.
I.
Goldhirsch
, Powder Technol.
182
, 130
(2008
).49.
D.
Serero
, C.
Goldenberg
, S. H.
Noskowicz
, and I.
Goldhirsch
, Powder Technol.
182
, 257
(2008
).50.
S.
Peletminsky
and Y.
Slusarenko
, Physica A
210
, 165
(1994
).51.
J. A.
Krommes
and C.
Oberman
, J. Plasma Phys.
16
, 193
(1976
).52.
Yu. L.
Klimontovich
, Statistical Theory of Open Systems
(Kluwer Academic Publishers
, Dordrecht
, 1995
), Vol.1
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