#### DMCA

## 4 Exponential approach to, and properties of, a non-equilibrium steady state in a dilute gas (2014)

### Citations

393 | The geometry of dissipative evolution equations: the porous medium equation
- Otto
(Show Context)
Citation Context ...pology of weak convergence of probability measures together with convergence of second moments, and we refer to the book Villani [21] for the definition and the proof of this fact.) As Otto has shown =-=[17]-=-, the evolution described by ∂f ∂t = ηT ∂ ∂v [ MT ∂ ∂v ( f MT )] is exponentially contractive in the 2-Wasserstein metric: If f and g are any two solutions of this equation dW2(f(·, t), g(·, t)) ≤ e −... |

118 | Rey-Bellet: Fourier’s law: a challenge to theorists
- Bonetto, Lebowitz, et al.
- 2000
(Show Context)
Citation Context ...ere are only a few models in which the isolated system evolves according to classical Hamiltonian mechanics or according to quantum mechanics for which we have even partial answers to these questions =-=[3, 9, 16]-=-. In addition to the rather unphysical models corresponding to harmonic crystals and ideal gases, existence and uniqueness was proven for systems interacting with soft potentials in contact with therm... |

112 |
Thermal conduction in classical lowdimensional lattices, Phys. Rep
- Lepri, Livi, et al.
- 2003
(Show Context)
Citation Context ...ere are only a few models in which the isolated system evolves according to classical Hamiltonian mechanics or according to quantum mechanics for which we have even partial answers to these questions =-=[3, 9, 16]-=-. In addition to the rather unphysical models corresponding to harmonic crystals and ideal gases, existence and uniqueness was proven for systems interacting with soft potentials in contact with therm... |

110 |
The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules
- Bobylev
- 1988
(Show Context)
Citation Context ...t and second moments as R, dGTW(Φ(f),Φ(g)) ≤ ( 1− γ [ 1 2 − 1 4 ∫ 1 −1 sb(s) ds ]) dGTW(f, g) . In particular, if b is even, dGTW(Φ(f),Φ(g)) ≤ ( 1− γ 2 ) dGTW(f, g) . Proof. Using the Bobylev formula =-=[1, 2]-=-, Q̂+(f, g) = ∫ S2 f(ξ+)g(ξ−)b ( σ · ξ |ξ| ) dσ (1.8) where ξ± = ξ ± |ξ|σ 2 . (1.9) Note that |ξ+| 2 + |ξ−| 2 = |ξ|2. Then we decompose Q̂+(f, f)− Q̂+(g, g) = Q̂+(f − g, f) + Q̂+(g, f − g) and we dedu... |

72 |
Topics in Optimal Transportation, in: Graduate
- Villani
- 2003
(Show Context)
Citation Context ...ic here, other than to say that like the GTW metric, it metrizes the topology of weak convergence of probability measures together with convergence of second moments, and we refer to the book Villani =-=[21]-=- for the definition and the proof of this fact.) As Otto has shown [17], the evolution described by ∂f ∂t = ηT ∂ ∂v [ MT ∂ ∂v ( f MT )] is exponentially contractive in the 2-Wasserstein metric: If f a... |

53 |
Probabilistic treatment of the Boltzmann equation of Maxwellian molecules
- Tanaka
- 1978
(Show Context)
Citation Context ...he contractive property proved in the previous section for (1.4), now using a different metric, but one that is equivalent to the GTW metric [11], namely the 2-Wasserstein metric. A theorem of Tanaka =-=[19, 20]-=- says that the evolution described by the spatially homogeneous Boltzmann equation for Maxwellian molecules is contractive in this metric. (We do not describe this metric here, other than to say that ... |

44 |
Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation.
- Gabetta, Toscani, et al.
- 1995
(Show Context)
Citation Context ...s on R3 into itself by Φ(f) = (1− γ)Q+(f, f) + γQ+(f, R) (1.6) so that the steady state equation is simply f = Φ(f) . (1.7) 6We shall show that Φ is contractive in the Gabetta-Toscani-Wennberg metric =-=[11]-=-, which is the metric defined as follows: Let f and g be two probability densities on R3 with finite second moments such that the first and second moments are identical. Let f̂ and ĝ denote their Fou... |

43 | On the (Boltzmann) entropy of non-equilibrium systems
- Goldstein, Lebowitz
(Show Context)
Citation Context ...s remain valid when the Boltzmann collision kernel Q(f, f) is replaced by the modified Enskog collision kernel which is generally considered to be a good approximation for a moderately dense gas; see =-=[12]-=- and references provided there. As noted above, in some physical situations it is more appropriate to model the interaction of a system with reservoirs by an Ornstein-Uhlenbeck continuous time diffusi... |

35 | Heat transport in low-dimensional systems
- Dhar
- 2008
(Show Context)
Citation Context ...ere are only a few models in which the isolated system evolves according to classical Hamiltonian mechanics or according to quantum mechanics for which we have even partial answers to these questions =-=[3, 9, 16]-=-. In addition to the rather unphysical models corresponding to harmonic crystals and ideal gases, existence and uniqueness was proven for systems interacting with soft potentials in contact with therm... |

32 | Central limit theorem for Maxwellian molecules and truncation of the Wild expansion.
- Carlen, Carvalho, et al.
- 2000
(Show Context)
Citation Context ...quation Because we have fixed the time scale so that the total loss term is simply f , the steady state equation can be written as f = (1− γ)Q+(f, f) + γQ+(f, R). We now follow a method introduced in =-=[8]-=- to solve this equation. Define the function Φ from the space of probability densities on R3 into itself by Φ(f) = (1− γ)Q+(f, f) + γQ+(f, R) (1.6) so that the steady state equation is simply f = Φ(f)... |

20 |
On the derivation of the Boltzmann equation
- Lanford
- 1976
(Show Context)
Citation Context ...Hamiltonian microscopic dynamics, and it yields, in some suitable limit, a Boltzmann equation for the single particle distribution; e.g., hard-sphere collisions under the Boltzmann-Grad scaling limit =-=[15, 18]-=-. Adding stochastic interactions with thermal reservoirs should then lead, in a suitable limit, to (4.1). However, this is beyond our current reach, even for the case in which the system is in contact... |

19 |
Fourier transform method in the theory of the Boltzmann equation for Maxwell molecules
- Bobylev
- 1975
(Show Context)
Citation Context ...t and second moments as R, dGTW(Φ(f),Φ(g)) ≤ ( 1− γ [ 1 2 − 1 4 ∫ 1 −1 sb(s) ds ]) dGTW(f, g) . In particular, if b is even, dGTW(Φ(f),Φ(g)) ≤ ( 1− γ 2 ) dGTW(f, g) . Proof. Using the Bobylev formula =-=[1, 2]-=-, Q̂+(f, g) = ∫ S2 f(ξ+)g(ξ−)b ( σ · ξ |ξ| ) dσ (1.8) where ξ± = ξ ± |ξ|σ 2 . (1.9) Note that |ξ+| 2 + |ξ−| 2 = |ξ|2. Then we decompose Q̂+(f, f)− Q̂+(g, g) = Q̂+(f − g, f) + Q̂+(g, f − g) and we dedu... |

16 | A hydrodynamic limit of the stationary Boltzmann equation in a slab - Esposito, Lebowitz, et al. - 1994 |

16 |
An inequality for a functional of probability distributions and its application to Kac’s one-dimensional model of a Maxwellian gas. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 27
- Tanaka
- 1973
(Show Context)
Citation Context ...he contractive property proved in the previous section for (1.4), now using a different metric, but one that is equivalent to the GTW metric [11], namely the 2-Wasserstein metric. A theorem of Tanaka =-=[19, 20]-=- says that the evolution described by the spatially homogeneous Boltzmann equation for Maxwellian molecules is contractive in this metric. (We do not describe this metric here, other than to say that ... |

11 |
Stationary states for a mechanical system with stochastic boundary conditions
- Goldstein, Kipnis, et al.
- 1985
(Show Context)
Citation Context ...tion to the rather unphysical models corresponding to harmonic crystals and ideal gases, existence and uniqueness was proven for systems interacting with soft potentials in contact with thermal walls =-=[14]-=-. The resulting NESS is spatially non-uniform, and we have little information about its structure. This is true even for cases in which the system is described mesoscopically by a one particle distrib... |

8 | On the validity of the Boltzmann equation for short range potentials
- Pulvirenti, Saffirio, et al.
(Show Context)
Citation Context ...Hamiltonian microscopic dynamics, and it yields, in some suitable limit, a Boltzmann equation for the single particle distribution; e.g., hard-sphere collisions under the Boltzmann-Grad scaling limit =-=[15, 18]-=-. Adding stochastic interactions with thermal reservoirs should then lead, in a suitable limit, to (4.1). However, this is beyond our current reach, even for the case in which the system is in contact... |

3 | Mechanical systems with stochastic boundaries - Goldstein, Lebowitz, et al. - 1979 |

1 |
The Kac Model Coupled to a Thermostat, eprint arXiv:1309.2715
- Bonetto, Loss, et al.
- 2013
(Show Context)
Citation Context ...ervoir. If one drops the requirement 16 that the microscopic dynamics be Hamiltonian, the situation is much better. Such a microscopic derivation was proven recently by Bonetto, Loss and Vaidyanathan =-=[4]-=- when the isolated system dynamics is given by the Kac stochastic collision model and there is a single thermal reservoir. One may expect a similar result to be valid for the Kac system in contact wit... |

1 |
Nonunique stationary states in driven kinetic systems with applications to plasmas, Phys
- Carlen, Esposito, et al.
- 1995
(Show Context)
Citation Context ...en particles are neglible. More is possible to prove for NESS of kinetic systems that are spatially uniform. Such a system, with one reservoir, but acted upon by an electric field, is investigated in =-=[5, 6]-=- and will be discussed later in this paper. Here we extend this investigation to the case in which the system is coupled to several thermal reservoirs at different temperatures. Remarkably, we find, f... |

1 |
Rokhlenko,Hydrodynamic Limit of a driven kinetic system with non-unique stationary states, Archive for Rational Mechanics and Analysis
- Carlen, Esposito, et al.
- 1998
(Show Context)
Citation Context ...en particles are neglible. More is possible to prove for NESS of kinetic systems that are spatially uniform. Such a system, with one reservoir, but acted upon by an electric field, is investigated in =-=[5, 6]-=- and will be discussed later in this paper. Here we extend this investigation to the case in which the system is coupled to several thermal reservoirs at different temperatures. Remarkably, we find, f... |

1 |
Propagation of Smoothness and the Rate of Exponential decay to Equilibrium for a spatially Homogeneous Maxwellian Gas
- Carlen, Gabetta, et al.
- 1999
(Show Context)
Citation Context ..., f∞) decreases to zero exponentially fast. 9We can dispense with the requirement that the initial data f(0) has the same first and second moments as R by using the correction technique introduced in =-=[7]-=-. Let us describe briefly this argument. Let us denote λ0 := 1 2 [ 1− 1 2 ∫ +1 −1 sb(s) ds ] > 0, λ1 := [ 1− ( (1− γ) + γ [ 1 4 ∫ 1 −1 (1 + s) b(s) ds ])] > 0 . We define (in Fourier variables) M̂[f ]... |