VALIDITY OF THE BOLTZMANN EQUATION

Transcription

1 VALIDITY OF THE BOLTZMANN EQUATION BEYOND HARD SPHERES based on joint work with M. Pulvirenti and C. Saffirio Sergio Simonella Technische Universität München Sergio Simonella - TU München Academia Sinica October 3, / 26

2 OUTLINE 1. INTRODUCTION 2. HIERARCHIES 3. SMOOTH POTENTIALS 4. RESULT Sergio Simonella - TU München Academia Sinica October 3, / 26

3 1. INTRODUCTION. THE VALIDITY PROBLEM Many-body classical system: i = 1,, N, z i = (x i, v i ) R 3 R 3, ϕ : R 3 R N = 1 2 ẋ i = v i v i = ϕ(x i x j ) j :j i z i () = z i, (z i, ) N i=1 random variable on ( (R 3 R 3 ) N,µ N ) = KINETIC THEORY: f P ( R 3 R 3) ( ) Chaos: f N t f + v x f = Q(f, f ). 1 (t), f 2 N (t) first and second marginal of ( ): t f N 1 + v x f N 1 = QN (f N 2 ) f N 2 f N 1 f N 1? Mean-field (Vlasov) Collisions (Boltzmann) Sergio Simonella - TU München Academia Sinica October 3, / 26

4 1. FROM NEWTON... Low density gas N = number of particles ; ε = collision length rate of coll. Nε 2 = 1 ; volume density Nε 3 = ε ε : low density limit (Boltzmann-Grad limit) i = 1,, N, (x i, v i ) R 3 R 3, ϕ : R 3 R ẋ i = v i v i = 1 ( ) xi x j ϕ ε j :j i ε (x i, v i )() = z i, Sergio Simonella - TU München Academia Sinica October 3, / 26

5 1....TO BOLTZMANN Grad s conjecture : the first marginal f1 N (t) f (t) as ε ( t + v x )f (x, v, t) = dv 1 dω B(ω,V ) R 3 S { 2 } f (x, v 1, t)f (x, v, t) f (x, v 1, t)f (x, v, t) { v = v ω[ω (v v 1 )] v 1 = v 1 + ω[ω (v v 1 )] V V = v 1 v, V = v 1 v B(ω,V )/ V = differential cross section 1 f (x, v, t)dxdv = probability of finding a particle in position x with velocity v, at time t. Sergio Simonella - TU München Academia Sinica October 3, / 26

6 1. VALIDITY THEOREM (1) Setting M N canonical phase space of N hard spheres in R 3 (diameter ε) { } M N = z N = (z 1,, z N ), z i = (x i, v i ) R 3 R 3, x i x j > ε for i j z N > z N (t) = hard-sphere flow (a.e.-defined) Initial distribution µ N : N symmetric density f s.t.: N particles almost i.i.d. prob. density f P (R 3 R 3 ) Nε 2 = 1 uniform bounds Example: Minimally correlated state f N := 1 Z f N N, f e µ+β(v 2 /2) L < + µ,β >, Z N = dz N f N Sergio Simonella - TU München Academia Sinica October 3, / 26

7 1. VALIDITY THEOREM (2) Observables g 1, g 2, test functions over R 3 R 3, F i (t)(z 1,, z N ) := ε 2 N j =1 g i (z j (t)) F 1,F 2, observables over M N (e.g. number of particles in a cell) LLN: F i (t) g i f (t) in the BG limit (local Poisson) Theorem. There exists t > such that, if t [, t ), j = 1,2, E [ j i=1 ( F i (t) g i f (t)) ] as ε. OP: 1 t 2t 2 ϕ(x) x k, k > Sergio Simonella - TU München Academia Sinica October 3, / 26

8 1. SOME REFERENCES SHORT TIME, HARD SPHERES [Lanford ( 75) See also: King, Spohn, Ilner, Pulvirenti, Uchiyama, Ukai...] PERTURBATION OF VACUUM. [Illner, Pulvirenti ( 86)] PERTURBATION OF EQUILIBRIUM. [van Beijeren, Lanford, Lebowitz, Spohn ( 8), Bodineau, Gallagher, Saint-Raymond ( 16, 17)] M N,β = Gibbs state (N hard spheres, inv.temp. β) (i) M N,β (z 1,, z N )h (z 1 ) linear BE ( Brownian motion) (ii) M N,β (z 1,, z N ) ) N i=1 (1 + N 1 h (z i ) linearized BE QUANTITATIVE CHAOS FAR FROM EQUILIBRIUM. [Pulvirenti, S. ( 17)] Sergio Simonella - TU München Academia Sinica October 3, / 26

9 2. HIERARCHIES. THE GENERAL STRATEGY Newton equation Liouville equation kinetic equation kinetic hierarchy { ( t + v x )f = Q(f, f ) f () = f P (R 3 R 3 ) f f (t) = T t f π P ( P (R 3 R 3 ) ) π(t, f ) := π (T t f ) Statistical solution Moments: f j (t) := P (R 6 ) f j dπ(t, f ) = P (R 6 ) ( Tt f ) j dπ (f ), j = 1,2, ( t + j v i xi )f j = C j +1 (f j +1 ) i=1 C j +1 (f j +1 ) = j i=1 Q i,j +1(f j +1 ) Boltzmann hierarchy Sergio Simonella - TU München Academia Sinica October 3, / 26

10 2. KINETIC HIERARCHY ( t + j v i xi )f j = C j +1 (f j +1 ) i=1 Boltzmann hierarchy C j +1 (f j +1 ) = j i=1 Q i,j +1(f j +1 ) = j ] k=1 dωdv j +1 B(ω, v k v j +1 )[ f j +1 ( x k, v k,, x k, v j +1 ) f j +1( x k, v k,, x k, v j +1 ) Uniqueness: Spohn ( 84) Remark. For any (f j ) j 1, f j = f j (z 1,, z j ) P (R 6j ) (i) symmetric and (ii) compatible ( d f j +1 (z j +1 ) = f j ),! Borel π on P (R 6 ) s.t. f j = P (R 6 ) f j dπ(f ) Propagation of chaos: π = δ f π(t) = δ f (t) Sergio Simonella - TU München Academia Sinica October 3, / 26

11 2. PARTICLE HIERARCHY N particles, Hamiltonian H N Prob. density f N P ((R 3 R 3 ) N ) t f N = {H N, f N } f N () = f N Marginals: f N j := R 6(N j ) f N dz j +1 dz N, j = 1,2, ( j t + v i xi 1 i=1 ε C N j +1 (f N j +1 ) = N j ε j i,k=1 ( xi x k ϕ ε ) ) vi f N j = C N j +1 (f N j +1 ) j i=1 R 6 ϕ( x i x k ) ε vi f N j +1 dx j +1dv j +1 BBGKY Uniform bounds for (f N j ) 1 j N : Lanford, King ( 75) Sergio Simonella - TU München Academia Sinica October 3, / 26

12 2. CONVERGENCE Particle chaos: f N j f j as N ( f N,j f j f N j (t) f (t) j ) Sergio Simonella - TU München Academia Sinica October 3, / 26

13 2. CONVERGENCE Particle chaos: f N j f j as N ( f N,j f j f N j (t) f (t) j ) Remark: Formal comparison BBGKY j t + v i xi 1 i=1 ε = N j ε j i,k=1 i k i=1 ( xi x ) k ϕ vi f N j ε j ( xi x j +1 dx j +1 dv j +1 ϕ ε vs. Boltzmann hierarchy? not so enlightening... ) vi f N j +1. Sergio Simonella - TU München Academia Sinica October 3, / 26

14 2. CONVERGENCE... unless for hard spheres (Cercignani 72): ( t + v x )f1 R N (x, v, t) = (N 1)ε2 dv 1 dω B(ω,V ) 3 S { 2 } f2 N (x εω, v 1, x, v, t) f2 N (x + εω, v 1, x, v, t) vs. ( t + v x )f (x, v, t) = dv 1 dω B(ω,V ) R 3 S { 2 } f (x, v 1, t)f (x, v, t) f (x, v 1, t)f (x, v, t) B(ω,V ) = ω V 1 {ω V } Sergio Simonella - TU München Academia Sinica October 3, / 26

15 3. SMOOTH POTENTIALS. STATE OF THE ART V V 1 1 =? Infinite range: poorly understood [Ayi ( 17)] Finite range [Gallagher, Saint-Raymond, Texier ( 14), Pulvirenti, Saffirio, S. ( 14)] Sergio Simonella - TU München Academia Sinica October 3, / 26

16 3. BASIC TOOL FOR SHORT RANGE ϕ Reduced marginals : f N = j S(x j ) N j dz j +1 dz N f N S(x j ) = { x x k > ε for all k = 1,, j } ( j t + v i xi 1 i=1 ε j i,k=1 ) ( xi x ) k ϕ vi ε f N j = C ε j +1 f N j +1 + E ε j C ε j +1 f N j +1 (z j, t) = ε2 (N j ) Error E ε (finite range): j Couplings to all f j +2, f j +3, E ε j = O(ε) j k=1 Grad hierarchy S 2 R 3 dνdv j +1 1 {minl k x k +νε x l >ε} (v j +1 v k ) νf N j +1 (z j, x k + νε, v j +1, t). does not worsen the uniform bounds on (f N j ) 1 j N [King 75 (ϕ )] Sergio Simonella - TU München Academia Sinica October 3, / 26

17 3. DIFFICULTY Is the Boltzmann equation valid for any compactly supported ϕ? Sergio Simonella - TU München Academia Sinica October 3, / 26

18 3. DIFFICULTY Is the Boltzmann equation valid for any compactly supported ϕ? t Caution: geometrical estimates on recollision sets Explicit solution of the hierarchy: weighted average over backward clusters. E.g. for two hard spheres with configuration (z 1, z 2 ), z i = (x i, v i ) at time t:, but also (Breakdown of chaos: f N j (t) (f N 1 ) j (t)) Standard kinetics may fail (Uchiyama model [ 88], external magnetic field [Bobylev et al 95]) Sergio Simonella - TU München Academia Sinica October 3, / 26

19 3. COLLISION HISTORIES (1) Iterated Duhamel: f j (t) = t t1 tn 1 dt 1 dt 2 dt n n S j (t t 1 )C j +1 S j +1 (t 1 t 2 ) C j +n S j +n (t n )f j +n () f N j (t) = N j n= t t1 dt 1 dt 2 tn 1 dt n S ε j (t t 1)C ε j +1 S ε j +1 (t 1 t 2 ) C ε j +n S ε j +n (t n) S ε (t) = flow operator for the j body dynamics j S j (t) = free flow operator O(ε) = multiple collisions Two steps: A. Absolute convergence of the series (uniform in ε) ; B. Term by term convergence f N j +n () +O(ε) Sergio Simonella - TU München Academia Sinica October 3, / 26

20 3. COLLISION HISTORIES (2) ζ ε (s) = (ζ ε 1 (s),ζε 2 (s), ) = trajectory of backward clusters f 2 N (z 1, z 2, t) f 1 N (z 1, t) f 1 N (z 2, t) = n 2 [ ] dλ ε n (ζε ) f n (ζ ε ()) χ rec 1,2 χov 1,2 +O(ε) dλ ε n = (signed) measure over trajectories of n particle backward clusters χ rec 1,2 = 1 there exists a recollision among the clusters 1 and 2. = 1 there exists an overlap among the clusters 1 and 2. χ ov 1,2 v 3 z z 1 2 recollision v 3 z z 1 2 overlap v 4 v 4 3 () 1 () () () () () 4 () 2 1 () Sergio Simonella - TU München Academia Sinica October 3, / 26

21 3. COLLISION HISTORIES (3) ζ ε (s) z j n t 1,, t n ν 1,,ν n v j +1,, v j +n (starting (time t) j particle configuration) (number of added particles) (times of creation of added particles) (impact vector of the added particles) (velocities of added particles). z z 1 2 recollision z z 1 2 overlap v 3 v 3 v 4 v 4 3 () 1 () () () () () 4 () 2 1 () Sergio Simonella - TU München Academia Sinica October 3, / 26

22 3. COLLISION HISTORIES (3) Smooth ϕ pathologies 1. interaction time BBGKY Boltzmann high-energy grazing collisions trapping orbits cross-section R > energy cutoff rec (V, ω) S 2 d ˆV R dv V 2 dωb(ω, V )χ rec = O(R 4 σ ϕ ε 2 ) σ ϕ = differential cross-section. Typically σ ϕ = rec Sergio Simonella - TU München Academia Sinica October 3, / 26

23 3. EXPLORING SINGULARITIES: σ ϕ = ρ 2 sin(2θ) dρ dθ V 1 Θ(ρ) = arcsinρ + ρ 1 1 r dr r 2 1 2ϕ(r ) V 2 ρ2 r 2 dθ dρ = ρ ϕ (1 ) V 2 ρ 2 2ϕ(r ) V 2 + ρ2 r 2 = sin2 α + [ π/2 arcsinρ dα sinα ρ ( ( ) 3 y ρ V 2 y 2 ϕ ( ρ y ) V 2 y 2 ϕ ρ y ) ( ) + 2 V 2 y ϕ ρ y + ρ ( V 4 y 4 ( )) ϕ ρ 2 ] y ρ Θ(ρ) monotonic if q ϕ ( q ) + 2ϕ ( q ) [Sone, Aoki] Sergio Simonella - TU München Academia Sinica October 3, / 26

24 3. EXPLORING SINGULARITIES: σ ϕ = Hidden : ρ 2 sin(2θ) dρ dθ Θ(ρ) E =.5 k=4 δ= ρ δ k+2 k q k + δ δ2 (1 + k 1 ) < q < δ ϕ(q) = δ(1 q ) δ < q < 1 q 1 Sergio Simonella - TU München Academia Sinica October 3, / 26

25 3. EXPLORING SINGULARITIES: σ ϕ = Hidden : ρ 2 sin(2θ) dρ dθ Θ(ρ) E =.5 k=4 δ= ρ δ k+2 k q k + δ δ2 (1 + k 1 ) < q < δ ϕ(q) = δ(1 q ) δ < q < 1 q 1 Is the Boltzmann equation valid for any compactly supported ϕ? Sergio Simonella - TU München Academia Sinica October 3, / 26

26 4. RESULT (1) Hypothesis 1. The two body potential ϕ = ϕ(q) is radial, with support q < 1, class C 2 (R 3 \ {}) and stable ( i<k ϕ(q i q k ) C N, C > ). Hypothesis 2. The initial datum for Boltzmann is f C (R 6 ), f (x, v) Ce (β/2)v 2, C,β >. Hypothesis 3. The initial datum for the N particle system is the symmetric probability density f N N, with marginals f,j eαj e (β/2) [ i v 2 i + i,k ϕ ( xi x k ε Hypothesis 4. f N,j f j uniformly on compact sets outside the diagonals (x i = x k ). )], α,β >. THEOREM (PULVIRENTI, SAFFIRIO, S.) { } Let Ω j = z j t.c. x i x k (v i v k )s > s. In the hypotheses 1 4, there exists t s.t. for any t < t, j >, uniformly on compact sets in Ω j. lim f N ε j (t) = f (t) j Nε 2 =1 Sergio Simonella - TU München Academia Sinica October 3, / 26

27 4. RESULT (2) Hypothesis 5. ϕ is non increasing. Moreover, for some C,L >, sup e β j 2 i=1 v 2 i f j f N,j (C ) j ε (convergence), x i x k >ε e β 2 v 2 f (x, v) f (x, v) L x x (regularity). THEOREM (PULVIRENTI, SAFFIRIO, S.) In the hypotheses 1 5, for any t < t, j >, z j Ω j and ε small enough, with suitable C,γ >. ( ) f N (z j, t) f (t) j (z j ) C j ε γ, j Remark. z j Ω j irreversibility. Improved: one-sided convergence [Bodineau, Gallagher, Saint-Raymond, S. 17] Sergio Simonella - TU München Academia Sinica October 3, / 26

28 4. PROOF Boltzmann equation emerging in the form ( t + v x )f (x, v, t) = dv R 3 1 dν (v v 1 ) ν S+ 2 { } f (x, v 1, t)f (x, v, t) f (x, v 1, t)f (x, v, t) Case A Case B t 1 t 2 t 3 t 4 t l t t * t Y Y 1 Y 2 3 Y Y 4 l Y W W 1 W 2 W 3 W 4 W l t 1 t 2 t 3 t 4 t l t t * t Y = Y 1 Y 2 Y 3 Y 4 l Y Y =, t = t < t f W W 1 W 2 W 3 W 4 W l Global approach: work on the Boltzmann flow ( Y l W l s < δ) integrate over times; exploit the global structure of ζ(s); keep using ν, not ω. Sergio Simonella - TU München Academia Sinica October 3, / 26

29 OPEN - unstable interactions - Vlasov-Boltzmann - long range Sergio Simonella - TU München Academia Sinica October 3, / 26

30 OPEN - unstable interactions - Vlasov-Boltzmann - long range THANKS! Sergio Simonella - TU München Academia Sinica October 3, / 26