Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas
We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat wh...
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| Published in | Journal of statistical physics Vol. 146; no. 6; pp. 1221 - 1243 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Boston
Springer US
01.03.2012
Springer |
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| Abstract | We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field
E
as well as a Gaussian thermostat which keeps the speed |
v
| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small
E
. Some of these projections are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions. Similarities and differences are noted. |
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| AbstractList | We investigate analytically and numerically the spatial structure of the nonequilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed [absolute value of v] constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of these projections are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions. Similarities and differences are noted. Keywords Billiard * Green-Kubo * SRB measure * Thermostat We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed | v | constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E . Some of these projections are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions. Similarities and differences are noted. |
| Audience | Academic |
| Author | Korepanov, Alexey Bonetto, Federico Chernov, Nikolai Lebowitz, Joel L. |
| Author_xml | – sequence: 1 givenname: Federico surname: Bonetto fullname: Bonetto, Federico email: bonetto@math.gatech.edu organization: School of Mathematics, Georgia Institute of Technology – sequence: 2 givenname: Nikolai surname: Chernov fullname: Chernov, Nikolai organization: Department of Mathematics, University of Alabama at Birmingham – sequence: 3 givenname: Alexey surname: Korepanov fullname: Korepanov, Alexey organization: Department of Mathematics, University of Alabama at Birmingham – sequence: 4 givenname: Joel L. surname: Lebowitz fullname: Lebowitz, Joel L. organization: Departments of Mathematics and Physics, Rutgers University |
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| Cites_doi | 10.1007/BF02102109 10.1007/PL00001034 10.1103/PhysRevE.65.051204 10.1016/0022-0396(83)90011-6 10.1143/PTP.38.1031 10.1023/A:1026414222092 10.1103/RevModPhys.57.617 10.1007/s00220-011-1262-5 10.1007/BF01019693 10.1007/978-3-642-84371-6 10.1142/5999 10.1090/surv/127 10.1007/978-3-662-00547-7 |
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| Keywords | Thermostat Billiard SRB measure Green-Kubo |
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| References | Bonetto, Daem, Lebowitz (CR1) 2000; 101 Spohn (CR13) 1991 Zhang (CR15) 2011; 306 Deimling (CR7) 1985 Frederickson, Kaplan, Yorke, Yorke (CR10) 1983; 49 Eckmann, Ruelle (CR12) 1985; 57 Chernov, Eyink, Lebowitz, Sinai (CR5) 1993; 154 Evans, Morris (CR9) 1990 Du (CR8) 2006 Chernov, Markarian (CR6) 2006 Bonetto, Lebowitz, Macias, Dagdug (CR3) 2009 Moran, Hoover (CR11) 1987; 48 Bonetto, Daem, Lebowitz, Ricci (CR2) 2002; 65 Chernov (CR4) 2001; 2 Yamada, Kawasaki (CR14) 1967; 38 N. Chernov (444_CR5) 1993; 154 N. Chernov (444_CR6) 2006 J.P. Eckmann (444_CR12) 1985; 57 K. Deimling (444_CR7) 1985 T. Yamada (444_CR14) 1967; 38 N. Chernov (444_CR4) 2001; 2 Y.H. Du (444_CR8) 2006 F. Bonetto (444_CR1) 2000; 101 D. Evans (444_CR9) 1990 B. Moran (444_CR11) 1987; 48 H.K. Zhang (444_CR15) 2011; 306 P. Frederickson (444_CR10) 1983; 49 F. Bonetto (444_CR3) 2009 F. Bonetto (444_CR2) 2002; 65 H. Spohn (444_CR13) 1991 |
| References_xml | – year: 1991 ident: CR13 publication-title: Large Scale Dynamics of Interacting Particles – volume: 154 start-page: 569 year: 1993 end-page: 601 ident: CR5 article-title: Steady state electric conductivity in the periodic Lorentz gas publication-title: Commun. Math. Phys. doi: 10.1007/BF02102109 – year: 2006 ident: CR8 publication-title: Order Structure and Topological Methods in Nonlinear Partial Differential Equations – volume: 2 start-page: 197 year: 2001 end-page: 236 ident: CR4 article-title: Sinai billiards under small external forces publication-title: Ann. Henri Poincaré doi: 10.1007/PL00001034 – volume: 65 year: 2002 ident: CR2 article-title: Properties of stationary nonequilibrium states in the thermostatted periodic Lorentz gas: the multiparticle system publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.65.051204 – year: 1990 ident: CR9 publication-title: Statistical Mechanics of Nonequilibrium Fluids – volume: 49 start-page: 185 year: 1983 end-page: 207 ident: CR10 article-title: The Lyapunov dimension of strange attractors publication-title: J. Differ. Equ. doi: 10.1016/0022-0396(83)90011-6 – volume: 38 start-page: 1031 year: 1967 end-page: 1051 ident: CR14 article-title: Nonlinear effects in the shear viscosity of a critical mixture publication-title: Prog. Theor. Phys. doi: 10.1143/PTP.38.1031 – volume: 101 start-page: 35 year: 2000 end-page: 60 ident: CR1 article-title: Properties of stationary nonequilibrium states in the thermostatted periodic Lorentz gas I: The one particle system publication-title: J. Stat. Phys. doi: 10.1023/A:1026414222092 – year: 2006 ident: CR6 publication-title: Chaotic Billiards – volume: 57 start-page: 617 year: 1985 end-page: 656 ident: CR12 article-title: Ergodic theory of chaos and strange attractors publication-title: Rev. Mod. Phys. doi: 10.1103/RevModPhys.57.617 – start-page: 27 year: 2009 end-page: 36 ident: CR3 article-title: Nonequilibrium stationary solution of thermostatted Boltzmann equation in a field publication-title: New Trends in Statistical Physics: Festschrift in Honor of Leopoldo Garcia-Colin’s 80th Birthday – volume: 306 start-page: 747 year: 2011 end-page: 776 ident: CR15 article-title: Current in periodic Lorentz gases with twists publication-title: Commun. Math. Phys. doi: 10.1007/s00220-011-1262-5 – year: 1985 ident: CR7 publication-title: Nonlinear Functional Analysis – volume: 48 start-page: 709 year: 1987 end-page: 726 ident: CR11 article-title: Diffusion in the periodic Lorentz billiard publication-title: J. Stat. Phys. doi: 10.1007/BF01019693 – volume: 154 start-page: 569 year: 1993 ident: 444_CR5 publication-title: Commun. Math. Phys. doi: 10.1007/BF02102109 – volume-title: Large Scale Dynamics of Interacting Particles year: 1991 ident: 444_CR13 doi: 10.1007/978-3-642-84371-6 – volume: 2 start-page: 197 year: 2001 ident: 444_CR4 publication-title: Ann. Henri Poincaré doi: 10.1007/PL00001034 – start-page: 27 volume-title: New Trends in Statistical Physics: Festschrift in Honor of Leopoldo Garcia-Colin’s 80th Birthday year: 2009 ident: 444_CR3 – volume: 49 start-page: 185 year: 1983 ident: 444_CR10 publication-title: J. Differ. Equ. doi: 10.1016/0022-0396(83)90011-6 – volume: 57 start-page: 617 year: 1985 ident: 444_CR12 publication-title: Rev. Mod. Phys. doi: 10.1103/RevModPhys.57.617 – volume: 38 start-page: 1031 year: 1967 ident: 444_CR14 publication-title: Prog. Theor. Phys. doi: 10.1143/PTP.38.1031 – volume: 48 start-page: 709 year: 1987 ident: 444_CR11 publication-title: J. Stat. Phys. doi: 10.1007/BF01019693 – volume-title: Order Structure and Topological Methods in Nonlinear Partial Differential Equations year: 2006 ident: 444_CR8 doi: 10.1142/5999 – volume-title: Statistical Mechanics of Nonequilibrium Fluids year: 1990 ident: 444_CR9 – volume: 306 start-page: 747 year: 2011 ident: 444_CR15 publication-title: Commun. Math. Phys. doi: 10.1007/s00220-011-1262-5 – volume: 65 year: 2002 ident: 444_CR2 publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.65.051204 – volume-title: Chaotic Billiards year: 2006 ident: 444_CR6 doi: 10.1090/surv/127 – volume-title: Nonlinear Functional Analysis year: 1985 ident: 444_CR7 doi: 10.1007/978-3-662-00547-7 – volume: 101 start-page: 35 year: 2000 ident: 444_CR1 publication-title: J. Stat. Phys. doi: 10.1023/A:1026414222092 |
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| Snippet | We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two... We investigate analytically and numerically the spatial structure of the nonequilibrium stationary states (NESS) of a point particle moving in a two... |
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| Title | Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas |
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