From Newton s law to the linear Boltzmann equation without cut-off

From Newton s law to the linear Boltzmann equation without cut-off Nathalie Ayi 1,2 1 Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis 2 Project COFFEE, INRIA Sophia Antipolis Méditerranée 13 Mai 216 Nathalie AYI MATKIT Cambridge 13 Mai 216 1 / 23

Context Kinetic theory of gases = Describe a gas as a physical system constituted of a large number of small particles. Nathalie AYI MATKIT Cambridge 13 Mai 216 2 / 23

Context Kinetic theory of gases = Describe a gas as a physical system constituted of a large number of small particles. Statistical point of view : we are interested in the evolution of the density of particles f (t, x, v) where Nathalie AYI MATKIT Cambridge 13 Mai 216 2 / 23

Historical results : A fundamental example, the Boltzmann equation (1872) = the evolution equation for the density of particles of a sufficiently rarefied gas. t f + v. x f = }{{} Q(f, f ) }{{} free transport localized binary collisions In the sixth problem of Hilbert (19), idea = Boltzmann equation as an intermediate step in the transition between atomistic and contiuous model for gas dynamics. Nathalie AYI MATKIT Cambridge 13 Mai 216 3 / 23

Lanford proved the derivation of the Boltzmann equation from systems of particles in the context of hard-spheres (1975). Nathalie AYI MATKIT Cambridge 13 Mai 216 4 / 23

Lanford proved the derivation of the Boltzmann equation from systems of particles in the context of hard-spheres (1975). Particles bounces off according to the laws of elastic reflection. Nathalie AYI MATKIT Cambridge 13 Mai 216 4 / 23

Recently rigorously proved by Gallagher, Saint-Raymond and Texier, Pulvirenti, Saffirio and Simonella in the context of hard-spheres and short range potentials. (Figure : The Boltzmann equation and its application, Cercignani.) Nathalie AYI MATKIT Cambridge 13 Mai 216 5 / 23

The Boltzmann equation without cut-off t f + v. x f = Q(f, f ) where Q(f, f ) = [f f 1 ff 1 ]B(v v 1, ω)dv 1 dω, with f = f (v), f = f (v ), f 1 = f (v 1 ), f 1 = f (v 1 ). B = cross-section, B( v v 1, cosθ) with θ = deviation angle. Example Inverse-power law potentials : The cross-section satisfies Φ(r) = 1, s > 2. rs 1 B( v v 1, cos θ) = b(cosθ) v v 1 γ, sin d 2 θb(cos θ) Cθ 1 α, α > Nathalie AYI MATKIT Cambridge 13 Mai 216 6 / 23

Grazing collisions = Singularity. Nathalie AYI MATKIT Cambridge 13 Mai 216 7 / 23

Grazing collisions = Singularity. Cauchy theory : Alexandre, Villani (22), Gressman et al (21), Alexandre et al (211)... Nathalie AYI MATKIT Cambridge 13 Mai 216 7 / 23

Grazing collisions = Singularity. Cauchy theory : Alexandre, Villani (22), Gressman et al (21), Alexandre et al (211)... f (v)f (v 1 ) f (v )f (v 1) Nathalie AYI MATKIT Cambridge 13 Mai 216 7 / 23

Result Microscopic Model : for i {1,..., N}, x i T d, v i R d, dx i = v i, dt dv i = 1 Φ( x i x j ). dt ε ε j i Nathalie AYI MATKIT Cambridge 13 Mai 216 8 / 23

Result Microscopic Model : for i {1,..., N}, x i T d, v i R d, dx i = v i, dt dv i = 1 Φ( x i x j ). dt ε ε Small perturbation around the equilibrium : j i f N(Z N ) := M N,β (Z N )ρ (x 1 ) (1) ρ density of probability on T d, M N,β Gibbs measure : for β > with H N (Z N ) := M N,β (Z N ) := 1 ( ) dn/2 β exp( βh N (Z N )) (2) Z N 2π 1 i N 1 2 v i 2 + 1 i<j N Φ( (x i x j ). ε Nathalie AYI MATKIT Cambridge 13 Mai 216 8 / 23

Theorem (A., 216) Consider the initial distribution fn defined above, then the distribution f (1) N (t, x, v) of the tagged particle converges in D (T d R d ) when N goes to under the Boltzmann-Grad scaling Nε d 1 = 1 to M β (v)h(t, x, v) where h(t, x, v) is the solution of the linear Boltzmann equation without cut-off t h + v. x h = [h(v) h(v 1 )]M β (v 1 )b(v v 1 )dv 1 dν (3) with initial data ρ (x 1 ) and where M β (v) := ( ) d/2 ) β 2π exp ( β2 v 2, β >. Nathalie AYI MATKIT Cambridge 13 Mai 216 9 / 23

The hard-sphere case The BBGKY hierarchy : for s < N, ( t + s i=1 v i. xi )f (s) N (t, Z s) = (C s,s+1 f (s+1) N )(t, Z s ) with C s,s+1 = the collision term, Z s = (z 1,..., z s ), z i = (x i, v i ). Nathalie AYI MATKIT Cambridge 13 Mai 216 1 / 23

The hard-sphere case The BBGKY hierarchy : for s < N, ( t + s i=1 v i. xi )f (s) N (t, Z s) = (C s,s+1 f (s+1) N )(t, Z s ) with C s,s+1 = the collision term, Z s = (z 1,..., z s ), z i = (x i, v i ). Duhamel s formula : f (s) N (t) = T s(t)f (s) N () + t T s (t t 1 )C s,s+1 f (s+1) N (t 1 )dt 1. Nathalie AYI MATKIT Cambridge 13 Mai 216 1 / 23

Iterated Duhamel s formula. The BBGKY series : f (s) N N s (t) = n= t t1 tn 1... T s (t t 1 )C s,s+1 T s+1 (t 1 t 2 )C s+1,s+2... T s+n (t n ) f (s+n) N ()dt n... dt 1. Nathalie AYI MATKIT Cambridge 13 Mai 216 11 / 23

Iterated Duhamel s formula. The BBGKY series : f (s) N N s (t) = n= t t1 tn 1... T s (t t 1 )C s,s+1 T s+1 (t 1 t 2 )C s+1,s+2... T s+n (t n ) f (s+n) N ()dt n... dt 1. The Boltzmann series : g (s) (t) = n t t1 tn 1... Ts (t t 1 )Cs,s+1T s+1(t 1 t 2 )Cs+1,s+2... T s+n (t n ) g (s+n) ()dt n... dt 1 Nathalie AYI MATKIT Cambridge 13 Mai 216 11 / 23

t 2 t 1 t Figure : Representation of a collision tree associated to the term t t1 1 5 2 4 3 T 3 (t t 1 )C j1,2 3,4 T 4 (t 1 t 2 )C j2,1 4,5 T 5(t 2 )f (5) N ()dt 2dt 1. 5 4 1 2 3 Nathalie AYI MATKIT Cambridge 13 Mai 216 12 / 23

t 2 t 1 t Figure : Representation of a collision tree associated to the term t t1 1 5 2 4 3 T 3 (t t 1 )C j1,2 3,4 T 4 (t 1 t 2 )C j2,1 4,5 T 5(t 2 )f (5) N ()dt 2dt 1. Strategy : Coupling of the pseudo-trajectories. 5 4 1 2 3 Nathalie AYI MATKIT Cambridge 13 Mai 216 12 / 23

Recollision v 1 v 2 v 3 t t 2 Figure : An example of a recollision between particles 1 and 2 at time t. t 1 t Nathalie AYI MATKIT Cambridge 13 Mai 216 13 / 23

Recollision v 1 v 2 v 3 t t 2 Figure : An example of a recollision between particles 1 and 2 at time t. Strategy : Geometrical control of the recollisions t 1 t Nathalie AYI MATKIT Cambridge 13 Mai 216 13 / 23

The linear Boltzmann equation case Bodineau, Gallagher and Saint-Raymond (214) : overcome the difficulty of the short time validity in the case of a fluctuation around the equilibrium. Nathalie AYI MATKIT Cambridge 13 Mai 216 14 / 23

The linear Boltzmann equation without cut-off Truncated marginals : (s) f N,R (t, Z s) := f N (t, Z s, z s+1,..., z N ) T d(n s) R d(n s) 1 i s s+1 j N 1 { xi x j >Rε}dz s+1... dz N Nathalie AYI MATKIT Cambridge 13 Mai 216 15 / 23

The linear Boltzmann equation without cut-off Truncated marginals : (s) f N,R (t, Z s) := f N (t, Z s, z s+1,..., z N ) T d(n s) R d(n s) The BBGKY hierachy : for s < N, t f (s) N,R + s i=1 = C s,s+1 f (s+1) N,R v i. xi f (s) N,R 1 ε +C s,s+1 f (s+1) N,R + (N s) ε i=1 s i,j=1 i j + 1 ε 1 i s s+1 j N Φ < ( x i x j (s) ). vi f N,R ε s i,j=1 i j Φ > ( x i x j (s) ). vi f N,R ε 1 { xi x j >Rε}dz s+1... dz N s Φ( x i x s+1 ). vi f N (t, Z N ) T d(n s) R ε d(n s) 1 l s s+1 k N 1 { xl x k >Rε}dZ (s+1,n) Nathalie AYI MATKIT Cambridge 13 Mai 216 15 / 23

Duhamel s formula : f (s) N,R (t, Z s) = S s (t) f (s) N,R (, Z s) + + t t + 1 ε + S s (t t 1 )C s,s+1 f (s+1) N,R (t 1, Z s )dt 1 S s (t t 1 )C s,s+1 f (s+1) N,R (t 1, Z s )dt 1 s t [ S s (t t 1 ) i,j=1 i j (N s) ε s i=1 t Φ > ( x i x j (s) ). vi f N,R ε ] (t 1, Z s )dt 1 [ S s (t t 1 ) Φ( x i x s+1 ). vi f N T d(n s) R ε d(n s) 1 l s s+1 k N 1 { xl x k >Rε}dZ (s+1,n) (t 1, Z s )dt 1 Nathalie AYI MATKIT Cambridge 13 Mai 216 16 / 23

Obstacles to the convergence Four possible obstacles to the convergence : - the very long-range interactions, - multiple simultaneous interactions, - the presence of recollisions, - super-exponential collision process. Nathalie AYI MATKIT Cambridge 13 Mai 216 17 / 23

Strategy Iteration on the term : t S s (t t 1 )C s,s+1 f (s+1) N,R (t 1, Z s )dt 1. Nathalie AYI MATKIT Cambridge 13 Mai 216 18 / 23

Definition of the operators : Q s,s (t) := S s (t) Q s,s+n (t) := t δ t1 δ... tn 1 δ S s (t t 1 )C s,s+1 χ Hs+1 ( 1 χgeom(s+1) ) χηs+1...... S s+n 1 (t n 1 t n )C s+n 1,s+n χ Hs+n ( 1 χgeom(s+n) ) χηs+n S s+n (t n )dt n... dt 1. Remainders associated to the long-range part : r Pot,a s,m+1 (, t, Z s) := m n= t nδ Q s,s+n (t t n+1 ) 1 ε s+n [ i,j=1 i j Φ > ( x i x j (s+n) ). vi f N,R ε ] (t n+1, Z s )dt n+1 Nathalie AYI MATKIT Cambridge 13 Mai 216 19 / 23

r Pot,b s,m+1 (, t, Z s) := m n= (N (s + n)) ε t nδ Q s,s+n (t t n+1 ) s+n [ Φ( x i x s+n+1 ). vi f N i=1 T d(n (s+n)) R ε d(n (s+n)) 1 l s+n s+n+1 k N 1 { xl x k >Rε}dZ (s+n,n) (t n+1, Z s )dt n+1. Nathalie AYI MATKIT Cambridge 13 Mai 216 2 / 23

Advantage of the iteration method no pathological situations easy to pass from Z m (t m ) to z (state of particle 1 at time t) via changes of variables such as T d R d [, t δ] S d 1 R d [, t m 1 δ] S d 1 R d T (m+1)d R (m+1)d (z, t 1, ν 2, v 2,..., t m, ν m+1, v m+1 ) Z m+1 = Z m+1 (t m). Key point : z Lipschitz function of ( x 1, ṽ 1,..., x m+1, ṽ m+1 ) Tools to prove it : study of the reduced dynamics - bound on the microscopic time of interaction thanks to the lower bound on relative velocites, - use of the Cauchy-Lipschitz theorem, Φ being Lipschitz, - Lipschitz character of the collision times. Nathalie AYI MATKIT Cambridge 13 Mai 216 21 / 23

Lipschitz control associated to the pseudo-trajectories = bound of the remainder associated to the long-range part controlled by ( ) 2 K+1 e C R2 η Φ >. Nathalie AYI MATKIT Cambridge 13 Mai 216 22 / 23

Lipschitz control associated to the pseudo-trajectories = bound of the remainder associated to the long-range part controlled by ( ) 2 K+1 e C R2 η Φ >. Limit of the approach : Very decreasing potentials. Nathalie AYI MATKIT Cambridge 13 Mai 216 22 / 23

Thank you for your attention. Nathalie AYI MATKIT Cambridge 13 Mai 216 23 / 23