From Hamiltonian particle systems to Kinetic equations

From Hamiltonian particle systems to Kinetic equations Università di Roma, La Sapienza WIAS, Berlin, February 2012

Plan of the lectures 1 Particle systems and BBKGY hierarchy (the paradigm of the Kinetic Theory)

Plan of the lectures 1 Particle systems and BBKGY hierarchy (the paradigm of the Kinetic Theory) 2 The Boltzmann equation (for hard spheres)

Plan of the lectures 1 Particle systems and BBKGY hierarchy (the paradigm of the Kinetic Theory) 2 The Boltzmann equation (for hard spheres) 3 The hard-spheres dynamics

Plan of the lectures 1 Particle systems and BBKGY hierarchy (the paradigm of the Kinetic Theory) 2 The Boltzmann equation (for hard spheres) 3 The hard-spheres dynamics 4 H-S hierarchy

Plan of the lectures 1 Particle systems and BBKGY hierarchy (the paradigm of the Kinetic Theory) 2 The Boltzmann equation (for hard spheres) 3 The hard-spheres dynamics 4 H-S hierarchy 5 Rigorous derivation of the Boltzmann equation for short times 6 Comments and remarks

Particle systems and BBKGY hierarchy N identical particles, unitary mass, in all the space R 3. Configurations in the phase space z 1... z N = Z N = (X N, V N ) = (x 1... x N, v 1... v N ) z i = (x i, v i ) denotes position and velocity of the i th particle. Two-body interaction ϕ : R 3 R, ϕ is spherically symmetric. ẋ i = v i, v i = ϕ(x i x j ) j=1...n j i

Particle systems and BBKGY hierarchy Probability measure W N (Z N )dz N on R 3N R 3N Symmetry: W N (z 1... z i... z j... z N ) = W N (z 1... z j... z i... z N )

Particle systems and BBKGY hierarchy Probability measure W N (Z N )dz N on R 3N R 3N Symmetry: W N (z 1... z i... z j... z N ) = W N (z 1... z j... z i... z N ) The time evolved measure is defined by W N (Z N ; t) = W N (Φ t (Z N )) Φ t (Z N ) the flow with initial datum Z N. An evolution equation for W N (Z N ; t).

Particle systems and BBKGY hierarchy t W N (t) = L N W N (t)

Particle systems and BBKGY hierarchy t W N (t) = L N W N (t) L N = N [v i xi + F i vi ] i=1 N (v i xi ) i=1 N F (x i x j ) ( vi vj ) i,j=1 i<j F i = j:j i ϕ(x i x j )

Particle systems and BBKGY hierarchy We are interested in the limit N. N particle description vs one-particle description. Define the j-particle marginals fj N (Z j ; t) = dz j+1... dz N W N (Z j, z j+1... z N ; t) j = 1... N. Evolution equation for f j (t). dz j+1... dz N ( t + N v i xi )W N (Z j, z j+1... z N ) = ( t + i=1 j i=1 v i xi )f N j free term

Particle systems and BBKGY hierarchy N N i=1 k=1,k i dz j+1... dz N N i=1 F i vi W N (Z j, z j+1... z N ) = dz j+1... dz N xi ϕ(x i x k ) vi W N (Z j, z j+1... z N ),

Particle systems and BBKGY hierarchy N N i=1 k=1,k i N i=j+1 dz j+1... dz N N i=1 F i vi W N (Z j, z j+1... z N ) = dz j+1... dz N xi ϕ(x i x k ) vi W N (Z j, z j+1... z N ), N k=1,k i... =0.

Particle systems and BBKGY hierarchy N N i=1 k=1,k i dz j+1... dz N N i=1 F i vi W N (Z j, z j+1... z N ) = dz j+1... dz N xi ϕ(x i x k ) vi W N (Z j, z j+1... z N ), N i=j+1 N k=1,k i... =0. j i=1 j k=1,k i... + the free term yields the j-particle Liouville operator j N i=1 k=j+1 (N j) dz j+1... dz N xi ϕ(x i x k ) vi W N (Z j, z j+1... z N ) = j i=1 dz j+1 xi ϕ(x i x j+1 ) vi f N j+1(z j, z j+1 ).

Particle systems and BBKGY hierarchy Conclusion: t f N j (t) = L j fj N (t) + (N j)c j+1 fj+1, N j = 1... N

Particle systems and BBKGY hierarchy Conclusion: t f N j (t) = L j fj N (t) + (N j)c j+1 fj+1, N j = 1... N C j+1 f N j+1(z j ) = j i=1 dz j+1 xi ϕ(x i x j+1 ) vi f N j+1(z j, z j+1 ) j < N and C N+1 = 0. For j = N Liouville eq.n f N N = W N. BBKGY hierarchy.

Particle systems and BBKGY hierarchy However if f N 2 (x 1, v 1, x 2, v 2 ) = f N 1 (x 1, v 1 )f N 1 (x 2, v 2 ) we get a single nonlinear eq.n. Not true.

Particle systems and BBKGY hierarchy However if f N 2 (x 1, v 1, x 2, v 2 ) = f N 1 (x 1, v 1 )f N 1 (x 2, v 2 ) we get a single nonlinear eq.n. Not true. But, if for some reason f N 2 (x 1, v 1, x 2, v 2 ) f N 1 (x 1, v 1 )f N 1 (x 2, v 2 ), in some limiting situations...

Boltzmann equation Boltzmann (1872) looked for an equation for µ(dz, t) = 1 N N δ(z z i (t))dz. i=1 the empirical distribution

Boltzmann equation Boltzmann (1872) looked for an equation for µ(dz, t) = 1 N N δ(z z i (t))dz. i=1 the empirical distribution or for f (x, v; t), the one-particle distribution.

Boltzmann equation Boltzmann (1872) looked for an equation for µ(dz, t) = 1 N N δ(z z i (t))dz. i=1 the empirical distribution or for f (x, v; t), the one-particle distribution. The dynamics is that of hard spheres. v =v n[n (v v 1 )] v 1 =v 1 + n[n (v v 1 )].

Boltzmann equation v 1 v n v 1 v

Boltzmann equation The Boltzmann equation Q is the collision operator Q(f, f )(x, v) = dv 1 R 3 ( t + v x )f = Q(f, f ) n (the impact parameter) is a unitary vector and S 2 + = {n n (v v 1 ) 0} S 2 + dn (v v 1 ) n [f (x, v )f (x, v 1) f (x, v)f (x, v 1 )]

Boltzmann equation Formal conservation in time of following five quantities dx dvf (x, v; t)v α with α = 0, 1, 2, (conservation of the probability [mass], momentum and energy).

Boltzmann equation Formal conservation in time of following five quantities dx dvf (x, v; t)v α with α = 0, 1, 2, (conservation of the probability [mass], momentum and energy). The entropy defined by H(t) = dx dv f log f (x, v, t) (1) is decreasing along the solutions. (The famous H-theorem) Equilibrium.

Boltzmann equation ϕ = ϕ(v) a test function, consider the scalar product in L 2 (v): (ϕ, Q(f, f )) = dv dv 1 dn B(v v 1 ; n) ϕ[f f 1 ff 1 ]. Here we use the standard notation f = f (v), f = f (v ), f 1 = f (v 1 ), f 1 = f (v 1 ). S 2 +

Boltzmann equation ϕ = ϕ(v) a test function, consider the scalar product in L 2 (v): (ϕ, Q(f, f )) = dv dv 1 dn B(v v 1 ; n) ϕ[f f 1 ff 1 ]. S 2 + Here we use the standard notation f = f (v), f = f (v ), f 1 = f (v 1 ), f 1 = f (v 1 ). Using the symmetry v v 1 and the fact that the Jacobian of is unitary, (ϕ, Q(f, f )) = 1 2 dv v, v 1 v, v 1 dv 1 S 2 + dn B(v v 1 ; n) ff 1 [ϕ +ϕ 1 ϕ ϕ 1 ] The conservation of mass, momentum and energy follows for ϕ = 1, v, v 2. (ϕ, Q(f, f )) = 0

Boltzmann equation H-theorem is consequence of the identity ϕ = log f, Ḣ(t) = (log f, Q(f, f ))

Boltzmann equation H-theorem is consequence of the identity ϕ = log f, we arrive to (log f, Q(f, f )) = 1 4 Entropy production. Boundary conditions. Paradoxes. Ḣ(t) = (log f, Q(f, f )) dx dv [ff 1 f f 1] log f f 1 ff 1 dv 1 S 2 + dn B(v v 1 ; n)

The hard-sphere dynamics N identical hard spheres of diameter ε of unitary mass. Z N = (z 1,..., z N ) = (X N, V N ) = (x 1,..., x N ; v 1,..., v N ) z i = (x i, v i ) R 6. The phase space Γ ε N (R6 ) N = {Z N x i x j ε for i j} Dynamics is the free flow Z N (t) = (x 1 + v 1 t,..., x N + v N t; v 1,..., v N ) up to the first impact time when x i x j = ε. Then an instantaneous collision takes place, according to the collision law and the flow goes on up to the next collision instant.

The hard-sphere dynamics Symmetric probability measure with density W N 0 on Γε N

The hard-sphere dynamics Symmetric probability measure with density W N 0 on Γε N W N (Z N ; t) = W N 0 (Φ t Z N ). The j-particle distributions fj N of a measure W N (or marginals) are fj N (z 1,... z j ; t) = dz j+1... dz N W N (z 1,... z N ; t) Looking for an evolution equation for f j (t).

H-S hierarchy For Z N Γ ε N t W N (Z N, t) = N v j xj W N (Z N, t). j=1

H-S hierarchy For Z N Γ ε N t W N (Z N, t) = N v j xj W N (Z N, t). j=1 Integrating on the last N 1 variables: t f 1(x, v; t) = where N j=1 Γ(x 1) v j xj W N (x 1, v 1... x N, v N )dx 2, dv 2... dx N, dv N, Γ(x 1 ) = {x 2, v 2... x N, v N x 1 x s > ε, s = 2... N}.

H-S hierarchy We deal first with the term j = 1 v 1 x1 f 1 (x 1, v 1 ) =v 1 x1 W N (x 1, v 1... x N, v N )dx 2, dv 2... dx N, dv N = N Γ(x 1) Γ(x 1) s=2 Γ s (x 1) v 1 x1 W N dx 2, dv 2... dx N, dv N + v 1 n s W N dx 2, dv 2... dσ s, dv s... dx N, dv N Γ s (x 1 ) = {x 2, v 2... x N, v N x 1 x r > ε, r = 2... N, r s, x 1 x s = ε}.

H-S hierarchy By the div lemma: N j=2 Γ(x 1 ) v j xj W N (x 1, v 1... x N, v N )dx 2, dv 2... dx N, dv N N j=2 Γ j (x 1 ) v j n j W N dx 2, dv 2... dσ j dv j... dx N, dv N = (N 1) dv 2 dσ 2 v 2 n 2 f 2 (x 1, v 1, x 2, v 2 ) = (N 1)ε 2 dv 2 dnv 2 nf 2 (x 1, v 1, x 1 + εn, v 2 ).

H-S hierarchy Using the symmetry of W N, ( t + v 1 x1 )f 1 (x 1, v 1 ; t) =(N 1)ε 2 dv 2 (v 2 v 1 ) n dn f 2 (x 1, v 1, x 1 + nε, v 2 )

H-S hierarchy Using the symmetry of W N, ( t + v 1 x1 )f 1 (x 1, v 1 ; t) =(N 1)ε 2 dv 2 (v 2 v 1 ) n dn f 2 (x 1, v 1, x 1 + nε, v 2 ) again not very useful.

H-S hierarchy Using the symmetry of W N, ( t + v 1 x1 )f 1 (x 1, v 1 ; t) =(N 1)ε 2 dv 2 (v 2 v 1 ) n dn f 2 (x 1, v 1, x 1 + nε, v 2 ) again not very useful. The size of the r.h.s. Nε 3 10 4 cm 3, Nε 2 = O(1)

H-S hierarchy Using the symmetry of W N, ( t + v 1 x1 )f 1 (x 1, v 1 ; t) =(N 1)ε 2 dv 2 (v 2 v 1 ) n dn f 2 (x 1, v 1, x 1 + nε, v 2 ) again not very useful. The size of the r.h.s. Nε 3 10 4 cm 3, Nε 2 = O(1) the probability that two tagged particles collide is O(ε 2 ). The probability that a given particle performs a collision is O(Nε 2 ) = O(1). f 2 (x, v, x 2, v 2 ) = f 1 (x, v)f 1 (x 2, v 2 ) statistical independence may be ok only before the collision.

H-S hierarchy Applying factorization (propagation of chaos) only for the incoming velocities if (v 1 v 2 ) n 0 f 2 (x 1, v 1, x 1 + nε, v 2 ) = f 1 (x 1, v 1 )f 1 (x 1 + nε, v 2 )

H-S hierarchy Applying factorization (propagation of chaos) only for the incoming velocities f 2 (x 1, v 1, x 1 + nε, v 2 ) = f 1 (x 1, v 1 )f 1 (x 1 + nε, v 2 ) if (v 1 v 2 ) n 0 and f 2 (x 1, v 1, x 1 + nε, v 2 ) = f 2 (x 1, v 1, x 1 + nε, v 2) if (v 1 v 2 ) n < 0 (here v 1, v 2 v 1, v 2 are outgoing. are incoming). Changing n n,

H-S hierarchy Applying factorization (propagation of chaos) only for the incoming velocities f 2 (x 1, v 1, x 1 + nε, v 2 ) = f 1 (x 1, v 1 )f 1 (x 1 + nε, v 2 ) if (v 1 v 2 ) n 0 and f 2 (x 1, v 1, x 1 + nε, v 2 ) = f 2 (x 1, v 1, x 1 + nε, v 2) if (v 1 v 2 ) n < 0 (here v 1, v 2 are incoming). Changing n n, v 1, v 2 are outgoing. We have the Boltzmann equation.

H-S hierarchy Applying factorization (propagation of chaos) only for the incoming velocities f 2 (x 1, v 1, x 1 + nε, v 2 ) = f 1 (x 1, v 1 )f 1 (x 1 + nε, v 2 ) if (v 1 v 2 ) n 0 and f 2 (x 1, v 1, x 1 + nε, v 2 ) = f 2 (x 1, v 1, x 1 + nε, v 2) if (v 1 v 2 ) n < 0 (here v 1, v 2 are incoming). Changing n n, v 1, v 2 are outgoing. We have the Boltzmann equation. However this can be true only in the limit called Boltzmann-Grad limit. ε 0, N, nε 2 const.

H-S hierarchy In general we have ( t + L ε j )f N j = (N j)ε 2 C ε j+1f N j+1, j = 1... N where L ε j is the generator of the dynamics of j hard-spheres of diameter ε. C ε j+1f N j+1(x 1, v 1,... x j, v j ) = j k=1 dn f N j+1(x 1, v 1,... x k, v k,..., x k + εn, v j+1 ) dv j+1 n (v k v j+1 ) where n is the unit vector. fj N = 0 if j > N and hence, for j = N, we have nothing else than the Liouville equation.

H-S hierarchy By the previous manipulations: C ε j+1f N j+1(x 1, v 1,... x j, v j ) = j k=1 dv j+1 S + dnn (v k v j+1 ) f N j+1(x 1, v 1,... x k, v k,..., x k εn, v j+1) f N j+1(x 1, v 1,... x k, v k,..., x k + εn, v j+1 )

H-S hierarchy and C ε j+1 = j k=1 C ε k,j+1 C ε k,j+1 = C ε+ k,j+1 C ε k,j+1 C ε+ k,j+1 g j+1(x 1,..., x j ; v 1,..., v j ) = dv j+1 S 2 + dωω (v k v j+1 ) [g j+1 (x 1,..., x j, x k εω; v 1,..., v k..., v j+1), C ε k,j+1 g j+1(x 1,..., x j ; v 1,..., v j ) = dv j+1 S 2 + dωω (v k v j+1 ) g j+1 (x 1,..., x j, x k + εω; v 1,..., v k,..., v j+1 )].

H-S hierarchy Therefore, by the Dyson (Duhamel) expansion: N j fj ε (t) = n=0 σ 1,...,σ n σ i =±1 k 1,...,k n t t1 ( 1) σ n dt 1 0 0 tn 1 dt 2... dt n 0 α ε n(j)u ε (t t 1 )C ε,σ 1 k 1,j+1... Uε (t n 1 t n )C ε,σn k n,j+n Uε (t n )f ε 0,n+j. α ε n(j) = ε 2m (N j)(n j 1)... (N j n + 1).