Sinai Billiards under Small External Forces II

Chernov, N.

doi:10.1007/s00023-007-0351-7

Cited by 28 publications

(53 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The change in velocity can be thought of as a kick or twist while a change in position can model a slip along the boundary at collision, or even reflection by a soft billiard potential [BT]. In [Ch2,Ch4], Chernov considered billiards under small external forces F with G = 0, and F to be stationary. In [Z] a twist force was considered assuming F = 0 and G depending on and affecting only the velocity, not the position.…”

Section: Description Of Model and Main Resultsmentioning

confidence: 99%

“…In this paper, we prove the steady state fluctuation relations for the periodic Lorentz gas with an external electric field and an iso-energetic thermostat [CELS1,CELS2] as well as several classes of related models with different forcing mechanisms [Ch2,Ch4,CZZ,Z]. While the models at hand are uniformly hyperbolic, the singularities of the billiard dynamics (due to grazing collisions) preclude the use of Markov partitions to study the fluctuation properties of ergodic averages.…”

Section: Introductionmentioning

confidence: 92%

See 1 more Smart Citation

Fluctuation of the Entropy Production for the Lorentz Gas Under Small External Forces

Demers

Rey-Bellet

Zhang

2018

Commun. Math. Phys.

View full text Add to dashboard Cite

In this paper we study the physical and statistical properties of the periodic Lorentz gas with finite horizon driven to a non-equilibrium steady state by the combination of non-conservative external forces and deterministic thermostats. A version of this model was introduced by Chernov, Eyink, Lebowitz, and Sinai and subsequently generalized by Chernov and the third author. Non-equilibrium steady states for these models are SRB measures and they are characterized by the positivity of the steady state entropy production rate. Our main result is to establish that the entropy production, in this context equal to the phase space contraction, satisfies the Gallavotti-Cohen fluctuation relation. The main tool needed in the proof is the family of anisotropic Banach spaces introduced by the first and third authors to study the ergodic and statistical properties of billiards using transfer operator techniques.

show abstract

Section: Description Of Model and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 92%

Fluctuation of the Entropy Production for the Lorentz Gas Under Small External Forces

Demers

Rey-Bellet

Zhang

2018

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…Our mathematical constructions and arguments are also unusual in many ways. In all the previous studies [9,11,13,15,18] of billiard-like particles moving under external forces, the dynamics was time-reversible, or at least invertible. That is, the flow Φ t was well defined for all −∞ < t < ∞, and the collision map F was invertible, i.e., F −1 existed.…”

Section: Super-diffusionmentioning

confidence: 99%

“…The last theorem can be extended to multiple correlations, and it implies, via a standard argument, Central Limit Theorem for the map F ε , see [17,Chapter 7] and [11].…”

mentioning

confidence: 99%

Lorentz Gas with Thermostatted Walls

Chernov

Dolgopyat

2010

Ann. Henri Poincaré

Self Cite

View full text Add to dashboard Cite

In a planar periodic Lorentz gas, a point particle (electron) moves freely and collides with fixed round obstacles (ions). If a constant force (induced by an electric field) acts on the particle, the latter will accelerate, and its speed will approach infinity (Chernov and Dolgopyat in J Am Math Soc 22: 821-858, 2009; Phys Rev Lett 99, paper 030601, 2007). To keep the kinetic energy bounded one can apply a Gaussian thermostat, which forces the particle's speed to be constant. Then an electric current sets in and one can prove Ohm's law and the Einstein relation (Chernov and Dolgopyat in Russian Math Surv 64:73-124, 2009; Chernov et al. Comm Math Phys 154:569-601, 1993; Phys Rev Lett 70:2209-2212. However, the Gaussian thermostat has been criticized as unrealistic, because it acts all the time, even during the free flights between collisions. We propose a new model, where during the free flights the electron accelerates, but at the collisions with ions its total energy is reset to a fixed level; thus our thermostat is restricted to the surface of the scatterers (the 'walls'). We rederive all physically interesting facts proven for the Gaussian thermostat in Chernov, Dolgopyat (Russian Math Surv 64:73-124, 2009) and Chernov et al. (Comm Math Phys 154:569-601, 1993; Phys Rev Lett 70:2209-2212, including Ohm's law and the Einstein relation. In addition, we investigate the superconductivity phenomenon in the infinite horizon case.

show abstract

“…Since the pioneering work by Sinai [47] on dispersing billiards, the dynamical structures and stochastic properties have been extensively studied for chaotic billiards [31,6,7,8,10,11,41,54,18,19,12,1,17,13,20,2,53], and also for abstract hyperbolic systems with or without singularities [48,42,37,43,46,57,58,49,24,14,21,25,26,27]. Among all the physical measures, the SRB measures -named after Sinai [48], Ruelle [45] and Bowen [4,5] -are shown to display several levels of ergodic properties, including the decay rate of correlations, the large deviation principles and the central limit theorem, etc.…”

Section: Introductionmentioning

confidence: 99%

Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities

2018

View full text Add to dashboard Cite

Dedicated to the memory of Nikolai Chernov.Abstract. We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically Hölder observables. The observables could be unbounded, and the process may be non-stationary and need not have linearly growing variances. Our results apply to Anosov diffeomorphisms, Sinai dispersing billiards and their perturbations. The random processes under consideration are related to the fluctuation of Lyapunov exponents, the shrinking target problem, etc.2010 Mathematics Subject Classification. 37D50, 37A25, 60F17.

show abstract

Sinai Billiards under Small External Forces II

Cited by 28 publications

References 44 publications

Fluctuation of the Entropy Production for the Lorentz Gas Under Small External Forces

Fluctuation of the Entropy Production for the Lorentz Gas Under Small External Forces

Lorentz Gas with Thermostatted Walls

Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities

Contact Info

Product

Resources

About