Phase transitions in Kinetic Theory

Transcription

1 Phase transitions in Kinetic Theory Brown - FRG Meeting - May 2010 Raffaele Esposito esposito@univaq.it Dipartimento di Matematica pura ed applicata Università dell Aquila, Italy Phase transitions in Kinetic Theory p. 1/48

2 Outline The particle model. Kinetic description. Equilibrium, phase transition. Results: Stability for V-B and V-F-P in R; Instability for V-B in R. Joint works with Yan Guo and Rossana Marra: Arch.Rat.Mech.Anal.195, (2010), Commun.Math.Phys.296,1-33 (2010). Work in progress in a finite interval. Phase transitions in Kinetic Theory p. 2/48

3 The model Model introduced by Bastea and Lebowitz (1997). System ofn 1 red and N 2 blue hard spheres of radius a = 1 with unit mass in Λ R 3. Interactions: elastic collisions (color blind); long range Kac-type interaction between red and blue particles with a two body potential: ( ) x φ l (x) = A l U, U(r) = 0 for r > 1. l U is assumed smooth, non negative, decreasing and normalized ( dxu( x ) = 1) and A l 0 as l suitably. Phase transitions in Kinetic Theory p. 3/48

4 Boltzmann-Grad+Vlasov limit Λ = O(l 3 ) Mean free path λ = γ = λ l fixed. Λ (N 1 +N 2 )a 2. A l = γ 3 a 2 λ 2 = O(N 1 +N 2 ). Note that N 1 +N 2 = O(l 2 ), low density. By scaling space and time variables (with λ 1 ) and taking the formal limit in the BBGKY hierarchy one gets the kinetic equations below. Warning : The analog of Lanford theorem is not been proved. Phase transitions in Kinetic Theory p. 4/48

5 Boltzmann-Vlasov equation B-V t f 1 +v x f 1 +F 1 [f 2 ] v f 1 = B(f 1,f 1 )+B(f 1,f 2 ), t f 2 +v x f 2 +F 2 [f 1 ] v f 2 = B(f 2,f 2 )+B(f 2,f 1 ), f 1 = f red and f 2 = f blue probability distribution functions on the phase space Ω R 3, Ω = λ 1 Λ; Self-consistent forces F 1 [f 2 ], F 2 [f 1 ]: F i [f j ](x,t) = x dx γ 3 U(γ x x ) dvf j (x,v,t), i j. R 3 Ω Collisions: B(f,g) = dv R 3 ω =1 dω (v v ) ω [f(v )g(v ) f(v)g(v )], Phase transitions in Kinetic Theory p. 5/48

6 Remark Why not just one species? The interesting case for phase transitions, with one species, is the attractive one: U(r) 0. In order to have thermodynamic stability one needs to keep the hard core size finite on the kinetic scale. Not compatible with the Grad-Boltzmann limit. Phase transitions in Kinetic Theory p. 6/48

7 Vlasov-Fokker-Planck eq.n V-F-P No elastic collisions; system in contact with a thermal bath at inverse temperature β > 0, modeled by white noise force added to the deterministic evolution with long range interaction. Standard mean field arguments: limiting system t f 1 +v x f 1 +F 1 [f 2 ] v f 1 = Lf 1, t f 2 +v x f 2 +F 2 [f 1 ] v f 2 = Lf 2, Lf(v) = v (µ β v ( fµβ )), µ β = exp[ βv2 ] (2πβ 1 ) 3/2. Phase transitions in Kinetic Theory p. 7/48

8 Segregation Phase transitions in Kinetic Theory p. 8/48

9 Free energy functional F kin Ω (f1,f 2 ) = E Ω (f 1,f 2 )+β 1 H Ω (f 1,f 2 ), E Ω (f 1,f 2 ) = dx dv 1 [ ] Ω R 2 v2 f 1 (x,v)+f 2 (x,v) 3 + dx dx U( x x )ρ 1 (x)ρ 2 (x ), Ω Ω ρ i (x) = H Ω (f 1,f 2 ) = i=1,2 Ω dvf i (x,v), R 3 dx dvf i (x,v)lnf i (x,v). R 3 Phase transitions in Kinetic Theory p. 9/48

10 The functional FΩ kin is non increasing for any β > 0 under the V-B dynamics and for β given by the thermal bath inverse temperature for the V-F-P evolution. Entropy-energy competition. β small: entropy dominates. Disordered state. β large: energy dominates. Segregated states. Phase transitions in Kinetic Theory p. 10/48

11 Equilibrium Equilibrium solutions (both V-B and V-F-P): f 1 (x,v) = ρ 1 (x)µ β (v), β 1 lnρ 1 (x)+ β 1 lnρ 2 (x)+ Ω Ω f 2 (x,v) = ρ 2 (x)µ β (v), dx U( x x )ρ 2 (x ) = C 1, dx U( x x )ρ 1 (x ) = C 2. Euler-Lagrange equations for the spatial free energy functional F Ω [ρ 1,ρ 2 ] = β 1 dx[ρ 1 (x)lnρ 1 (x)+ρ 2 (x)lnρ 2 (x)] Ω + dx dx U( x x )ρ 1 (x)ρ 2 (x ). Ω Ω Phase transitions in Kinetic Theory p. 11/48

12 Local free energy Homogeneous minimizers: minimizers of the local free energy ϕ(ρ 1,ρ 2 ) = β 1 [ρ 1 lnρ 1 +ρ 2 lnρ 2 ]+ρ 1 ρ 2 Indeed, Ω dx F Ω [ρ 1,ρ 2 ] = dxϕ(ρ 1 (x),ρ 2 (x)) Ω dx U( x x )(ρ 1 (x) ρ 1 (x ))(ρ 2 (x ) ρ 2 (x)). Ω By rearrangement arguments (U decreasing) the non local term is non negative on minimizers, so minimizers are spatially homogeneous as much as they can. Phase transitions in Kinetic Theory p. 12/48

13 Homogeneous equilibrium states Set ρ = ρ 1 +ρ 2 ; m = ρ 1 ρ 2 ρ. ϕ(ρ 1,ρ 2 ) = β 1 ρlnρ+ 1 4 ρ2 +ρ 2 f ρ (m) f ρ (m) = m2 4 +(ρβ) 1[ 1 m 2 ln 1 m m 2 ln 1+m ] 2 Graph ofm f ρ (m) forρβ > 2. Phase transitions in Kinetic Theory p. 13/48

14 Homogeneous equilibrium states The critical points for f have to solve with σ = 1 2 ρβ. m = tanh(σm), If σ 1: m = 0 is the only solution. If σ > 1: again m = 0 solution; Moreover m σ > 0 such that m = ±m σ is solution. Set ρ ± := 1 2 ρ(1±m1 2 ρβ). Phase transitions in Kinetic Theory p. 14/48

15 Homogeneous equilibrium states Therefore: ρβ 2 : m = 0: Miminizer of ϕ: ρ 1 = ρ 2 (mixed phase). ρβ > 2 : Minimizer: ρ 1 = ρ +,ρ 2 = ρ,(red rich phase); Minimizer: ρ 1 = ρ,ρ 2 = ρ +,(blue rich phase); Maximizer (local) : ρ 1 = ρ 2. Note: the minimizing total density is always unique. Phase transitions in Kinetic Theory p. 15/48

16 Non homogeneous equilibrium states Conservation of masses = Minimizers with the constraints 1 dxρ i (x) = n i, i = 1,2 Ω Ω n i given by the initial conditions. If < ρ n i < ρ +, i = 1,2 non homogeneous minimizers, regions blue rich and red rich separated by interfaces. Assume Ω a torus of size L. From rearrangement arguments (using U monotone), for L sufficiently large, the minimizers are symmetric monotone with values close to ρ + and ρ in regions separated by a small interface [Carlen, Carvalho, R.E., Lebowitz, Marra, Nonlinearity 16, (2003)]. Phase transitions in Kinetic Theory p. 16/48

17 Equilibrium states in R Ω = R. Fix ρ = 2 = σ = β; Critical point β c = 1. For β 1: the only possible equilibrium is the homogeneous mixed phase, with ρ 1 = ρ 2 = 1. For β > 1: three homogeneous states: the mixed one, ρ 1 = ρ 2, the red rich phase ρ 1 = 1+m β, ρ 2 = 1 m β and the blue rich phase ρ 1 = 1 m β, ρ 2 = 1+m β. For β > 1, non spatially homogeneous solutions (phase coexistence) by forcing the asymptotic values to ρ ± at : Two half-lines of red rich and blue rich phases separated by an interface of finite size. Phase transitions in Kinetic Theory p. 17/48

18 Equilibrium states in R Define ˆρ 1 (x) = { ρ, x < 0 ρ +, x > 0, ˆρ 2(x) = { ρ +, x < 0 ρ, x > 0. Excess free energy: [ ] F[ρ 1,ρ 2 ] = lim F ( l,l) [ρ 1,ρ 2 ] F ( l,l) [ˆρ 1, ˆρ 2 ]. l Remark: F[ρ 1,ρ 2 ] is not finite if lim ρ 1 ρ ± or x ± lim ρ 2 ρ. x ± Phase transitions in Kinetic Theory p. 18/48

19 Front solution Theorem[Carlen, Carvalho, R.E., Lebowitz, Marra; ARMA 194, (2009)] Letβ > 1. There is a unique (up to translations) minimizer to the excess free energy F. Let ρ = ( ρ1, ρ 2 ) be the one such that ρ 1 (0) = ρ 2 (0). ρ is smooth;ρ < ρ i (x) < ρ +, i=1,2; ρ 1 is increasing and ρ 2 is decreasing; β 1 ln ρ 1 +U ρ 2 = C = β 1 ln ρ 2 +U ρ 1 ; Phase transitions in Kinetic Theory p. 19/48

20 Front solution Moreover β 1 ρ 1 + ρ 1U ρ 2 = 0 = β 1 ρ 2 + ρ 1U ρ 1 ; ρ 1 (x) = ρ 2 ( x), ρ 1 (x) = ρ 2 ( x); α > 0: ρ 1 (x) ρ ± e α x 0, x ± ; ρ 2 (x) ρ e α x 0, x ±. Phase transitions in Kinetic Theory p. 20/48

21 Front solution Phase transitions in Kinetic Theory p. 21/48

22 Stability Question: Are the equilibrium states stable w.r.t. perturbations of the initial conditions in the time evolutions V-B and V-F-P? Expected answer: Minimizers are stable, maximizers are unstable. Results: Case Ω = R; V-F-P evolution: asymptotic stability of the minimizers. V-B evolution: stability of the minimizers, instability of the maximizer. Open problems: Instability of the maximizer for V-F-P and asymptotic stability of the minimizers for V-B. Ω = [ L,L], periodic b.c.,llarge enough. Phase transitions in Kinetic Theory p. 22/48

23 Stability for the V-F-P Main problem: the force cannot be small: small force = no phase transition. For small force, convergence to equilibrium and rate by Villani. This would cover the high temperature regime. For low temperatures, multiple equilibrium states. Difficult case: front. Homogeneous minimizers are simpler. Assume f i 0 (x,v) = ρ i(x)µ β (v)+h i (x,v,0), x R, v R 3. Define h i (x,v,t) := f i (x,v,t) ρ i (x)µ β (v). Phase transitions in Kinetic Theory p. 23/48

24 Stability for the V-F-P Equations for h i : t h i +G i h i = Lh i F i [h] vx h i, G i h i = v x x h i (U ρ j) vx h i ) + (U x dvh j (,v,t) βv x M ρ i R 3 Norms: denotes the L 2 (R R 3 ) with weight ( ρ i µ β ) 1 : f 2 = i=1,2 R R3 dxdv f(x,v)2 ρ(x)µ β (v) Phase transitions in Kinetic Theory p. 24/48

25 Stability for the V-F-P Null space of L: kern(l) = {(u 1,u 2 ) = (a 1 µ β,a 2 µ β )}. P projector on kern(l),p = 1 P ; = ( x, t ). Dissipation norm: f 2 D = P f 2 + v P f 2. Weighted norms: z γ = (1+ x 2 ) γ, f γ = z γ f ; f D,γ = z γ f D Symmetry: Assume h 1 (x,v,0) = h 2 ( x,rv,0) with Rv = ( v x,v y,v z ). Note that this is the same symmetry of the front. The symmetry is preserved in time. Phase transitions in Kinetic Theory p. 25/48

26 Stability for the V-F-P Theorem[ARMA 195, (2010)] There is δ > 0 s.t. if h(,0) + h(,0) < δ, then there is a unique global solution to the equation forh. Moreover there isk > 0 such that d ( h(t) 2 + t h(t) 2 + x h(t) 2) dt +K ( h(t) 2 D + th(t) 2 D + xh(t) 2 D) 0. If, forγ > 0 sufficiently small, h(,0) γ + h(,0) γ+ 1 2 < δ then the same norms are bounded at any time by the initial data and we have the decay estimate h(t) 2 + h(t) 2 C [ 1+ t ] [ ] 2γ h(0) 2 γ + h(0) 2 1 2γ +γ. 2 Phase transitions in Kinetic Theory p. 26/48

27 Main tools Spectral gap for L: there is ν 0 > 0 s.t. (g,lg) ν 0 P g 2 D. This is used to control P h(t). Control P h(t) = (g 1 µ β,g 2 µ β ): (Ag) 1 = β 1g 1 ρ 1 +U g 2, (Ag) 2 = β 1g 2 ρ 2 +U g 1. Note that (g,ag) = d2 ds 2 ˆF( ρ+sg) s=0. Phase transitions in Kinetic Theory p. 27/48

28 Main tools The operator A has a null space kern(a) = {α( ρ 1, ρ 2 )}. Let P is the projector on kern(a) and P = 1 P. (Transl. invar.) Spectral gap for A: there is λ > 0 s.t. (g,ag) λ(p g,p g). The component of h(t) in kern(a) is not under control. The symmetry condition ensures that h(t) is orthogonal to kern(a). Lower bound for (Au) : There is C > 0 such that (Au) 2 C Qu 2. where Q is the orthogonal projection on the orthogonal complement of ρ. Phase transitions in Kinetic Theory p. 28/48

29 Remarks The same results hold for the mixed phase above the critical temperature. We do not have the instability of the mixed phase below the critical temperature. The construction of the growing mode presented below for V-B, fails for V-F-P because of the unboundedness of the F-P operator L. Phase transitions in Kinetic Theory p. 29/48

30 Results for V-B Theorem[CMP 296,1-33 (2010)]: Assume ρ = 2. β < 1: The unique equilibrium(f 1,f 2 ) = (µ β,µ β ) is stable. β > 1: the homogeneous equilibrium states (f 1,f 2 ) = (ρ + µ β,ρ µ β ) and(f 1,f 2 ) = (ρ µ β,ρ + µ β ) are stable; the equilibrium(f 1,f 2 ) = ( ρ 1 (x)µ β, ρ 2 (x)µ β ) is stable w.r.t. symmetric perturbations; the homogeneous equilibrium(f 1,f 2 ) = (µ β,µ β ) is unstable. Here stability and instability are in L (R R 3 ) and in H 1 (R R 3 ). Symmetric perturbation means again h 1 (x,v) = h 2 ( x,rv), where Rv = ( v x,v y,v z ). Phase transitions in Kinetic Theory p. 30/48

31 Remarks No convergence to the equilibrium is stated. This has to be compared with the Vlasov-Fokker-Plank case where there is an algebraic rate of convergence. No instability result for VFP. In order to have phase transitions: force not small. Treating the force terms as perturbations does not work. Strategy based on entropy-energy arguments: L 2 estimates promoted to L by analysis of the characteristics. Crucial step: spectral gap for the second derivative of the free energy. The instability is based on the construction of a growing mode for the linear collisionless case, perturbation arguments and persistence of the gorwing mode at non linear level. Phase transitions in Kinetic Theory p. 31/48

32 Free energy functional Given the equilibrium state (M 1,M 2 ) = (ρ 1 µ β,ρ 2 µ β ), let g = (g 1,g 2 ) with g i = f i M i Mi be the deviation from the equilibrium. Define: M i (g) = R dx dv M R 3 i g i (x,v), 2 ] H(g) = dx dv [f i logf i M i logm i, + R R i=1 E(g) = R 2 i=1 R dx dv v2 2 g i Mi ( ) dxdyu( x y ) ρ f1 (x)ρ f2 (y) ρ 1 (x)ρ 2 (y), ρ fi = dvf i (x,v). Phase transitions in Kinetic Theory p. 32/48

33 Free energy functional The free energy functional is ( F(g) = H(g)+βE(g) C +1+log ( ) β 3/2 2 M i (g), 2π) The free energy functional does not increase: F(g(t)) F(g(0)) for any t > 0. Quadratic approximation. The coefficients have been chosen to cancel the linear part. For some f i = αf i +(1 α)m i, α (0,1): 2 R3 F(g) = dx dv (f i(t) M i ) 2 i=1 R 2 f i +β dx dyu( x y )(ρ f1 (t,x) ρ 1 (x))(ρ f2 (t,y) ρ 2 (y)). R R i=1 Phase transitions in Kinetic Theory p. 33/48

34 Free energy functional Lemma. Ifu i = ρ fi ρ i s.t. P(u 1,u 2 ) = 0, then there areα > 0 and κ sufficiently small so that α i=1,2 R { (f i (t) M i ) 2 dx dv 1 R M { fi (t) M i κm i } 3 i + f i (t) M i 1 { fi (t) M i κm i } F(g(0)). } Remark: It is crucial that the initial perturbation is orthogonal to the null space ofa. This is trivial for the spatially homogeneous equilibrium, while it is ensured by the symmetry condition for the phase coexisting equilibrium. Phase transitions in Kinetic Theory p. 34/48

35 Main proposition Theorem: Letw(v) = (Σ+ v 2 ) γ, withσsufficiently large and γ > 3 2. If wg(0) + F(g(0)) < δ forδ sufficiently small, then there ist 0 > such that wg(t 0 ) 1 2 wg(0) +C T0 F(g(0)). The stability follows by iteration on the time interval. Phase transitions in Kinetic Theory p. 35/48

36 Growing mode Idea: Without collisions there is a growing mode. Collisions do not destroy the growing mode. The linearization of the equation for the perturbation is: t g +Lg = 0, Notation : µ = µ β ;ξ first component of the velocity v = (ξ,ζ), (Lg) i = ξ x g i βf( µg i+1 )ξ µ αl i g, α = 1 and L i g = 1 (B( µg i,2µ)+b(µ, ) µ(g 1 +g 2 )). µ Seek for a growing mode of the form g 1 (t,x,ξ,ζ) = e λt e ikx q(ξ,ζ), g 2 (t,x,ξ,ζ) = e λt e ikx q( ξ,ζ). Phase transitions in Kinetic Theory p. 36/48

37 Eigenvalue problem {λ+iξk}q βkiû(k) { R 3 q µdv } ξ µ αlq = 0. Proposition 1: Letβ > 1. There exists sufficiently smallα > 0 such that there is an eigenfunctionq(v) and the eigenvalueλwithrλ > 0. Proof. First assume α = 0. λ is found by O. Penrose criterion, β ξ 2 Û(k)k 2 µ(v) R λ 2 +k 2 ξ 2 dv = 1. 3 Indeed, since βû(0) > 1, by continuity there is k 0 > 0 such that βû(k 0) > 1 and hence a λ > 0 so that this is satisfied. Use Kato perturbation theorem to extend to α > 0 small. Phase transitions in Kinetic Theory p. 37/48

38 Eigenvalue problem Proposition 2: Letα 0 be the supremum of theα s such that Proposition 1 is true. Thenα 0 = +. Proof. Indeed, if α 0 < then, λ 0 = lim α α0 λ α exists (up to subsequences) and is a purely imaginary eigenvalue. It can be shown that the corresponding eigenfunction must be in the null space of L and this implies λ = 0. Moreover, collisions disappear for such an eigenfunction and we can use again the Penrose criterion which implies βû(k 0) = 1. This is in contradiction with the definition of k 0. This provides a linear growing mode for any α > 0. Remark: It is crucial that L is a bounded perturbation. It does not work with the Fokker-Plank operator which is unbounded. Non linear analysis. Bootstrap argument. Very technical. Phase transitions in Kinetic Theory p. 38/48

39 Instability theorem Theorem. Assumeβ > 1. There exist constantsk 0 > 0,θ > 0, C > 0,c > 0 and a family of initial 2π k 0 periodic data f δ i (0) = µ+ µg δ i (0) 0, withgδ (0) satisfying x,ξ g δ (0) L 2 + wg δ (0) L Cδ, forδ sufficiently small, but the solutiong δ (t) satisfies sup wg δ (t) L c sup g δ (t) L 2 cθ > 0. 0 t T δ 0 t T δ Here the escape time ist δ = 1 Rλ ln θ δ, Note that the growing mode is symmetric. The instability does not depend on the absence of symmetry. Phase transitions in Kinetic Theory p. 39/48

40 Work in progress Finite volume, 1d torus (with Y. Guo and R. Marra) Stability of the non constant minimizer double front ) Operator A on L 2 (T L ),T L the 1-d torus of size L. Derivative of the front is in the null. Null space? Spectral gap? Phase transitions in Kinetic Theory p. 40/48

41 Spectral gap A has a negative eigenvalue. Indeed, let w = (w 1,w 2 ) be a front on the torus such that Aw = 0. Define w = ( w 1, w 2 ) ( w,a w) w = T L w 1 ( w 1 w 1 U w 2 )+ w 2 ( w 2 w 2 U w 1 ) = 2 L 0 w 1U ( w 2 +w 2) 2 We have used the E-L equations 0 L w 2U ( w 1 +w 1) < 0 w 1 w 1 = U w 2, x 0; w 2 w 2 = U w 1, x 0 Show that the mass constraint kills the negative eigenvalue. Phase transitions in Kinetic Theory p. 41/48

42 Neumann b.c. Spectral gap true for anti-symmetric (by reflection) functions. Problem on the torus for symmetric functions reduced to the case of Neumann boundary conditions on [0,L]: (Âg) 1 = β 1 g 1 w 1 +Û g 2, (Âg) 2 = β 1 g 2 w 2 +βû g 1 Û(z,z ) = U(z,z )+U(z,R 0 z )+U(z,R L z ) R 0 reflection around zero and R L reflection around L. λ L < 0 minimum eigenvalue and ê its eigenfunction. Phase transitions in Kinetic Theory p. 42/48

43 Spectral gap We need spectral gap for functions in the hyperplane H = {h : L 0 h = 0}. We have spectral gap for functions in the orthogonal to ê. (u,âu) δ(u,u), if (u,ê) = 0. If the angle α between ê and H is too small we are in trouble: Phase transitions in Kinetic Theory p. 43/48

44 Spectral gap Decompose h H as aê + bû with û orthogonal to ê (h,âh) = a2 λ L +b 2 (û,âû) a 2 = cos 2 α, b 2 = sin 2 α. with sin α = 1 L L 0 ê. If ê decays fast enough b 2 L 1. λ L is negative Competition (h,âh) λ L + c L δ If λ L decays faster than L 1 (h,âh) > d(h,h) G. Manzi (2007) we can prove spectral gap for L large Phase transitions in Kinetic Theory p. 44/48

45 Spectrum Analysis of the spectrum using Markov chains. Generalize method by De Masi, Olivieri, Presutti (1998), Ising case. bound on the minimum eigenvalue λ L c 1 e γl λ L c 2 e γl exponential bound on the minimum eigenfunction ê ce γ L x ê < 0 spectral gap in a suitable weighted L for functions u in the orthogonal to ê. Implies spectral gap in L 2. Phase transitions in Kinetic Theory p. 45/48

46 Spectral gap Operator S (Su) i = w i Û u j, i j,i = 1,2 (u,âu) = dxw i u i (Âu) i = (u,u)+(u,su) i (u,s 2 u) = i u i w i Û (w j Û u i ) = (u,tu) Negative eigenvalue for  means eigenvalue for S greater than 1. We study the operators T 1 h = w 1 Û (w 2 Û h), T 2 h = w 2 Û (w 1 Û h). Phase transitions in Kinetic Theory p. 46/48

47 Markov chain Ĵ(x,x ) = dzû(x z)w 2(z) w 2 (x) dyû(z x ) M(x,x ) = p(x)ĵ(x,x ); p(x) = w 1 (x)w 2 (x) For λ L > 0 and ê(x) positive, define the Markov kernel K(x,y) = M(x,y)ê(y) λ 0 ê(x). Phase transitions in Kinetic Theory p. 47/48

48 The End Thanks! Phase transitions in Kinetic Theory p. 48/48