The Kac model with a thermostat. F . Bonetto, School of Mathematics, GeorgiaTech GGI 30/5/2014 F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
Work in collaboration with Michael Loss Ranjini Vaidyanathan Hagop Tossounian F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
General Setting. I have N particle in a box. They may interact with several different things to make the dynamics more interesting. Among others, we have considered: Binary elastic collisions. 1 Elastic ollisions with scatterers. 2 Interaction with an external electric field E plus some 3 mechanism to keep the energy finite. Normally this is given by a Gausssian thermostat. Thermal reservoir at the boundary of the system. This can 4 be modeled in different ways. F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
E T T + 1 - 2 F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
The questions. The system is described by the collection of the positions q i and velocity v i of the particle. We set V = ( v 1 , . . . , v N ) Q = ( q 1 , . . . , q N ) The state of the system is a probability distribution F N ( Q , V ; t ) and the evolution is in general given by a linear operator L N , that is ˙ F N ( t ) = L F N ( t ) F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
Calling ¯ F N = lim t →∞ F N ( t ) the Steady State of the system, the typical question one can ask are: Existence of the limit for N → ∞ of ¯ F N and its behavior with respect to the parameters of the system. Rate of convergence to the steady state of a generic initial state. Call f ( v ; t ) the 1-particle marginal of F N ( V ; t ) for N very large. Can we write a closed evolution equation for f ( v ; t ) in the style of the Boltzmann equation. F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
A rigorous answer to the above question in the original deterministic models is way too difficult for me. The deterministic collisionS make the problem extremely difficult. One way out is to simplify the model by replacing the deterministic collisison with random collision. This idea was introduced first by Mark Kac in 1956. F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
The Kac model. The particle are in 1 dimension and they are initially unifrmly distributed in space so that one can forget their positions. The collision are described by a Poisson process whose intensity will be chosen later. Every time a collision take place we select at random and uniformly two particles i and j with incoming velocities v i and v j . The outgoing velocoties v ∗ i and v ∗ j of the two particle are � v 2 i + v 2 chosen uniformly on the circle of radius j . This rule is very similar to that used in the KMP model. F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
If the state of the system if F N ( V ) we can describe the effect of a collision by the operator R i , j given by R 1 , 2 F N ( V ) = 1 � F N ( v 1 cos ( θ ) − v 2 sin ( θ ) , v 1 sin ( θ )+ v 2 cos ( θ ) , V ( 2 ) ) 2 π where V ( k ) = ( v k + 1 , . . . , v N ) . The evolution is thus given by ˙ F N = L N F N with L N = λ N � ( R i , j − I ) � N � 2 i < j The scaling factor in front of the sum assures that the average number of collision a particle suffers in a given time is independent of N . F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
This evolution preserve the total kinetic energy. It is thus natural to look at F ( V ) as defined of the sphere √ √ S N − 1 ( N in R N . In this way the evarage kinetic N ) of radius energy per particle is 1/2. √ Let d σ ( V ) the normalized volume measure on S N − 1 ( N ) . It is easy to see that there is a unique steady state given by F N ( V ) = 1. F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
Known facts. The spectral gap of L N can be computed exactly (Carlen-Carvalho-Loss (2000)): = − 1 N + 1 Λ ( 1 ) N 2 N − 2 It is clearly unifrom in N . This is only useful very close to the stedy state. Indeed if the initial state is of the form N � F ( V ) = f ( v i ) restriced on the sphere i = 1 then � F − 1 � 2 ≃ C N So that if we start far from the steady state, it takes a time of order N to get close. F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
We can define the entropy with respect to the steady state as � F ( V ) � � S ( F | ¯ F ) = F ( V ) log d σ ( V ) ¯ F ( V ) where in this case ¯ F ≡ 1. It is easy to show that ˙ S ( F | ¯ S ( F ( t ) | ¯ F ) ≤ e − c N t S ( F ( 0 ) | ¯ F ) ≥ 0 F ) The constant c N is not uniform in N . Indeed for every δ there exists C δ such that: 1 C δ N ≤ c N ≤ N 1 − δ (Villani (2003), Einav (2011)) F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
Boltzmann Property Given a symmetric disfribution F N ( V ) we define the k particle marginal as � f k F N ( V ) dV ( k ) N ( v 1 , . . . v k ) = A sequence of distributions F N ( V ) has the Boltzamnn Property if k � N →∞ f k lim N ( v 1 , . . . , v k ) = f ( v i ) i = 1 where N →∞ f 1 f ( v ) = lim N ( v ) F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
Propagation of Chaos Let F N ( t ) be the state of the system at tine t with initial condition F N ( 0 ) . Kac (1956) (see also McKean (1966)) proved that if F N ( 0 ) has the Boltzman property that F N ( t ) also has the Boltzmann Property. His result is not uniform in t . Form the above if follows, rather easily, that the limiting 1-particle marginal satisfy � � ˙ d θ ( f ( v ∗ ) f ( w ∗ ) − f ( v ) f ( w )) f ( v ; t ) = 2 dw − where v ∗ = v cos ( θ ) − w sin ( θ ) w ∗ = v sin ( θ ) + w cos ( θ ) F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
Electric Conduction. E 1 2 F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
In B., Chernov, Korepanov, Lebowitz we studied a system of point like particles colliding with “virtual” obstacles under the influence of an electric field and a Gaussian thermostat. In B., Carlen, Esposito, Lebowitz, Marra (2013) we proved validity of a self-consistent Boltzmann Equation with a technique completely different from that used by Kac or McKean. This result is being extended to colliding particle by Carlen, Mustafa, Wennberg (2014). We could also analize in detail the steady state for small electric field (B. Loss(2013)). F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
Thermostated Kac System. We add to the system a second collision process. Again at Poisson distributed times a particle collides with a termostated wall. We can represent this wall in two ways. A strong thermostat described by the operator: � T s F ( v 1 , V ( 1 ) ) dv 1 1 F ( V ) = γ β ( v 1 ) where � β 2 π e − β v 2 2 , γ β ( v ) = or a weak thermostat � T w 1 , V ( 1 ) ) d θ dw 1 F ( V ) = γ β ( w ∗ ) F ( v ∗ F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
The generator of the evolution becomes 2 λ � � L N = µ ( T i − I ) + ( R i , j − I ) N − 1 i i < j The evolution now take place on the full R N since the energy is not conserved. It is easy to see that there is a unique steady state given by the Maxwellian at inverse temperature β � Γ β ( V ) = γ β ( v i ) . i We can ask the same question we asked for the original Kac model. F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
Spectral gaps. Since L N is not self adjoint in L 2 ( R N ) it is convenient to write F ( V ) = Γ( V ) H ( V ) where now H satisfy the new evolution ˙ H = − L N H with 2 λ � (˜ � L N = µ T i − I ) + ( R i , j − I ) N − 1 i i < j with � ˜ 1 , V ( 1 ) ) d θ dw . γ β ( w ) F ( v ∗ T 1 F ( V ) = It is now easy to see that L N is self adjoint on L 2 ( R N , Γ( V )) . F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
The convergence to the steady state is dominated by the thermostat. The spectral gap of L N is given by = − µ Λ ( 1 ) N 2 with eigenfunction i − 1 H ( 1 ) ( V ) = � v 2 � β = h 2 ( v i ) i i where h 2 is the Hermite polynomial of degree 2. To see the effect of the particle-particle collision we compute the second eigenvalue of L N and find, when N → ∞ , ∞ = − λ 2 − 5 Λ ( 2 ) 8 µ with eigenfunction � H ( 2 ) ( V ) = h 4 ( v i ) . i F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
Convergence in entropy One can also study the convergence in entropy. In this case one finds, thanks to the thermostat, that µ 2 S ( F ( 0 ) | Γ) S ( F ( t ) | Γ) ≤ e To obtain this one can reduce the problem to a one particle system using a Loomis-Whitney style inequality and then map the evolution of the one particle system into a Ornstein-Uhlembeck process. F. Bonetto, School of Mathematics, GeorgiaTech The Kac model with a thermostat.
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