Phase transitions and coarse graining for a system of particles in the continuum

Transcription

1 Phase transitions and coarse graining for a system of particles in the continuum Elena Pulvirenti (joint work with E. Presutti and D. Tsagkarogiannis) Leiden University April 7, 2014 Symposium on Statistical Mechanics: Combinatorics and Statistical Mechanics University of Warwick Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

2 Introduction Problem: derivation of phase diagrams of fluids. P S G L T Phase transition (1st order, order parameterm ρ): there is a forbidden interval [ρ, ρ ] of density values, namely: if we put a mass ρ Λ, ρ (ρ T, ρ ), T small enough, then we see T c ρ in a set Λ Λ and ρ in Λ \ Λ We want to study the liquid-vapour coeistence line Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

3 Introduction Model: identical point particles which interact pairwise via a potential repulsive at the origin and with an attractive tail at large distances (the prototype is the Lennard-Jones potential) Conjecture: fluids interacting via a Lennard-Jones potential behave according to the phase diagram.... but no rigorous proof of this conjecture! Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

4 Introduction State of the art: In the discrete: first proof of liquid-vapor type phase transitions was given for lattice systems as the Ising model (argument of Peierls). In the continuum: Widom-Rowlinson model of two component fluids (Ruelle, 71) 1d with long range interaction (Johansson, 95) Lebowitz, Mazel and Presutti (LMP) in 99 prove liquid-vapor phase transition for a model of particles in the continuum d 2 interacting via a 2-body attractive plus a 4-body repulsive interaction (both long range). Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

5 Microscopic model Local mean field: local energy density e( ) is a function of local particle density ρ γ (r; q) := J γ (r, q i ) q i q local particle density at r R d Kac potentials J() J γ (r, r ) = γ d J(γr, γr ) H γ (q) = γ > 0 : Kac scaling parameter (γ 0 mean field limit) drj(r, r ) = 1 e(ρ γ (r; q)) dr R d Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

6 LMP model LMP model (Lebowitz, Mazel, Presutti): Take the energy density e(ρ) = ρ2 2 ρ4 4! Hγ LMP (q) = e(ρ γ (r; q)) dr, R d Hγ LMP (q) = 1 J γ (2) (q i, q j ) 1 J γ (4) (q i1,..., q i4 ) 2! 4! i j i 1... i 4 }{{}}{{} 2-body 4-body J γ (2) (q i, q j ) := J γ J γ (q i, q j ) = J γ (r, q i )J γ (r, q j ) dr Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

7 Towards our model Theorem (LMP): There are β c > 0, β > β c such that for β (β c, β ) for any γ small enough there is a forbidden interval (ρ β,γ, ρ β,γ ). Drawbacks: scale attraction scale repulsion towards crystal structures and solid phase...need repulsion on scale 1 Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

8 Towards our model Theorem (LMP): There are β c > 0, β > β c such that for β (β c, β ) for any γ small enough there is a forbidden interval (ρ β,γ, ρ β,γ ). Drawbacks: scale attraction scale repulsion towards crystal structures and solid phase...need repulsion on scale 1 LMP h.c. model: add the interaction given by H hc (q) := i<j V hc (q i, q j ) where V hc : R d R V hc (q i, q j ) = { if q i q j R 0 if q i q j > R with R the radius of the hard spheres and ɛ = B 0 (R) their volume. Our goal: to etend this result to: LMP hard core (perturbation theory for small values of the radius R) (...work in progress) Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

9 Statistical mechanics formulation Grand canonical measure in the region Λ R d and b. c. q in Λ c is: µ Λ γ,β,r,λ (dq q) = Z 1 γ,β,r,λ (Λ q)e βh γ,r,λ(q q) ν Λ (dq) First order phase transition if β is large enough, the limiting (Λ R d ) Gibbs state is not unique, i.e. we have instability with respect to boundary conditions Diluted Gibbs measure: µ ± Λ,β,γ,R = µλ γ,β,r 1 {on the boundary of Λ we fi a frame with the phase ±} Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

10 Result R: radius of the hard spheres, γ: Kac scaling parameter Theorem (Liquid-vapor phase transition for LMPhc) Consider the LMPhc model in dimensions d 2. For: 0 < R R 0, β (β c,r, β 0,R ), γ γ β,r, there is λ β,γ,r such that: There are two distinct infinite volume measures µ ± β,γ,r with chemical potential λ β,γ,r and inverse temperature β and two different densities: 0 < ρ β,γ,r, < ρ β,γ,r,. Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

11 Remark 1. lim ρ β,γ,r,± = ρ β,r,±, γ 0 lim γ 0 λ β,γ,r = λ β,r with are respectively densities and chemical potential for which there is a phase transition in the LMP h.c. mean field model. ρ β,r, < ρ β,r,. Remark 2. β c,r = β LMP c β LMP c = 3 2 ɛ (β LMP c ) 2/3 O(ɛ 2 ) 3 2, ɛ = B0 (R) Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

12 Pirogov-Sinai Theory Contour method approach and Pirogov-Sinai theory Idea (for Ising): to look at configurations at small T as of perturbations of the two ground states ( and ) Instead: perturb mean field (γ 0), where two g.s. are ρ, ρ Analogy: T small γ small spin-flip symmetry no symmetry... towards an abstract contour model! Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

13 Abstract contour model Scaling parameters: l = γ (1 α), l = γ (1α), ζ = γ a, 1 α a > 0 Two partitions with cubes C (l ), C (l) η (ζ,l ) (q; r) = θ (ζ,l,l) (q; r) = { ±1 if ρ (l ) (q; r) ρ β,r,± ζ 0 otherwise { ±1 if η (ζ,l ) (q; r ) = ±1 r C r (l) 0 otherwise Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

14 Abstract contour model Scaling parameters: l = γ (1 α), l = γ (1α), ζ = γ a, 1 α a > 0 Two partitions with cubes C (l ), C (l) η (ζ,l ) (q; r) = { ±1 if ρ (l ) (q; r) ρ β,r,± ζ 0 otherwise { θ (ζ,l,l) ±1 if η (ζ,l ) (q; r ) = ±1 r C r (l) (q; r) = 0 otherwise { Θ (ζ,l,l) ±1 if η (ζ,l ) (q; r ) = ±1 r C r (l) δ l (q; r) = out[c r (l) ] 0 otherwise which are the phase indicators! Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

15 Abstract contour model o o o oo o o o o o o o o o o o oo o o o o o oooooooo o ooo o oo o Figure: Values of θ (l,l,ζ) o _ o o Figure: Discontinuity lines of Θ (l,l,ζ) and regions where it is, and 0. Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

16 Abstract contour model o o o oo o o o o o o o o o o o oo o o o o o oooooooo o ooo o oo o Figure: Values of θ (l,l,ζ) o _ o o Figure: Discontinuity lines of Θ (l,l,ζ) and regions where it is, and 0. Definition (Contour) A contour is a pair Γ = ( sp(γ), η Γ ), where sp(γ) is a maimal connected component of the incorrect set {r R d : Θ (ζ,l,l ) (q; r) = 0} and η Γ is the restriction to sp(γ) of η (ζ,l ) (q; ). Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

17 Abstract contour model Figure: The set c(γ) Figure: A i (Γ) are the regions made of the marked squares contained in int i (Γ) Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

18 Abstract contour model Figure: The set c(γ) Figure: A i (Γ) are the regions made of the marked squares contained in int i (Γ) Γ = ( sp(γ), η Γ ) is called (or ) contour if Θ(q; r) = 1 (or 1) on the frame of width l around the set c(γ). Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

19 Abstract contour model Figure: The set c(γ) Figure: A i (Γ) are the regions made of the marked squares contained in int i (Γ) The weight W γ,r,λ (Γ; q) of a contour Γ is Prob. of having a contour Prob. of having the phase Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

20 Abstract contour model Figure: The set c(γ) Figure: A i (Γ) are the regions made of the marked squares contained in int i (Γ) The weight W γ,r,λ (Γ; q) of a contour Γ is ) µ (η(q c(γ) q c(γ) ; r) = η Γ (r), r sp(γ); Θ(q c(γ) ; r) = ±1, r A ± (Γ) ) µ (η(q c(γ) q c(γ) ; r) = 1, r sp(γ); Θ(q c(γ) ; r) = 1, r A ± (Γ) Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

21 Peierls bounds The main technical point is to prove that contours are improbable, i.e. Peierls estimates: Theorem For: 0 < R R 0, β (β c,r, β 0,R ), γ γ β,r, there is λ β,γ,r so that: for any ± contour Γ and any ± boundary condition q ± outside Γ, Corollary { } W ± γ,r,λ (Γ; q± ) ep βc (ζ 2 l d ) N Γ, N Γ = sp(γ) l d...then for any bounded, simply connected, D (l ) -measurable region Λ, any ± boundary condition q ± and any r Λ, { µ Λ,± ({Θ(q; r) = ±1}) 1 ep β c } q ± 2 (ζ2 l d ). Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

22 Problems: both numerator and denominator in the weight defined in terms of epressions which involve not only the support of Γ but also its whole interior (bulk quantities!). the desired bound involves only the volume of the support of Γ which is a surface quantity! Cancellation of bulk terms difficult when there is no simmetry between phases (as for Ising). Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

23 Three steps Step one: Equality of pressures For any λ [λ(β, R) 1, λ(β, R) 1] the following holds lim n 1 β Λ n log Z± λ (Λ n q ± n ) = P ± λ moreover, for all γ small enough there is λ := λ β,γ,r such that P λ = P λ, λ λ(β, R) cγ 1/2 (coarse graining argument a la Lebowitz-Penrose, closeness of the pressures to their mean field values p ±;mf R,λ ) Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

24 Three steps Step two: Energy estimates Given γ small and R < R 0 : N λ (Γ, q ) D λ (Γ, q ) e β(cζ2 c γ 1/2 2αd )l d N Γ eβi (int (Γ)) Z λ (int (Γ) χ ) e βi (int (Γ)) Z λ (int (Γ) χ ) where: I ± (int (Γ)) is a surface term χ ± (r) = ρ β,±1 r, χ ± = χ ± R { } d ep βc ζ 2 l d N Γ is Peierls gain term } ep {βcγ 1/2 2αd l d N Γ small error Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

25 Three steps Step three: Surface correction to the pressure For any γ small enough and all plus contours Γ: { e βi±(int (Γ)) log Z ± λ (int (Γ) χ ± )} c γ 1/2 l d N Γ e β int (Γ) P ± λ Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

26 Three steps Let i be the centers of the cubes C (l ) D (l ). f 1, 2 = 1 2! q i1 C (l ) 1,q i2 C (l ) 2 Theorem (Eponential decay of correlations) (2) Jγ (q i1, q i2 ) There are positive constants δ, c and c so that for all f 1,.., n ) ( ) E µ 1( f1,.., n Eµ 2 f1,.., n c e c[γ δ l 1 dist(c(l ) 1,Λ c )] where µ 1 is the Gibbs measure in the volume Λ with b.c. q and µ 2 is the Gibbs measure on a torus T much larger than Λ, both referring to the hamiltonian H γ,r. Eventually take the limit T R d... Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

27 Eponential decay of correlations To prove eponential decay of correlations we need two ingredients: 1 Cluster epansion 2 Dobrushin condition Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

28 Restricted ensembles A central point in P-S theory is a change of measure. The diluted partition function in a region Λ can be written as a partition function in Q Λ = {q Q Λ : η(q, r) = 1, r Λ} (restricted ensemble). Let us consider Λ bounded and D (l ) measurable and q b.c. Z γ,r,λ (Λ q) = ν Λ (dq)e βh γ,r(q q) W γ,r,λ (Γ, q) Γ Γ Γ:sp(Γ) Λ Q Λ Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

29 Coarse graining Coarse graining: fi the number of particles in each cube C (l ) density configuration ρ = {ρ } (l X ). Hence: Λ Z γ,r,λ (Λ q) = ρ e h(ρ q) where h is finite volume effective hamiltonian (function only of the cell variables).... cluster epansion to find h, multi-canonical constraint... Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

30 Cluster epansion Why it works: Kac: Because we can confuse H γ with its coarse grained version H (l ) γ making a small error H γ, e β Hγ is ok for cl. ep.! Hard core:... Cluster epansion in the canonical ensemble (E. P., D. Tsagkarogiannis) Commun. Math. Phys. (2012) Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

31 Cluster epansion in the canonical ensemble Model with stable and tempered pair potential: Theorem c 0, indep. of N and Λ, s.t. if ρc(β) < c 0 : 1 Λ log Z β,λ,n = 1 Λ N log Λ N! with N = ρ Λ and F β,λ,n (n) Ce cn. Furthermore in the thermodynamic limit: N Λ F β,λ,n (n) n 1 lim F β,n,λ(n) = 1 N, Λ, N= ρ Λ n 1 β nρ n1, n 1 where β n are the Mayer s coefficients. Remark: C(β) is equal to the volume of hard spheres ɛ. Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

32 Dobrushin condition Weak dependence on the boundaries decay of correlations Space for spin block model: { } X Λ = n = (n ) XΛ N X Λ : l d n ρ β,r, ζ, for all X Λ p 1, p 2 conditional Gibbs measures on X : p i (n ) := p(n ρ i, q i ) = Vaserstein distance is ( ) R p 1 ( ), p 2 ( ) 1 Z (ρ i, q i ) ep { h(ρ ρ i, q i ) } i = 1, 2 := inf Q n 1,n 2 where Q are joint representations of p 1 and p 2 Q(n 1, n 2 )d(n 1, n 2 ) Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

33 Dobrushin condition Theorem There are u, c 1, c 2 > 0 s.t. Λ ( ) R p 1 ( ), p 2 ( ) r γ,r (, z)d(n 1 z, n 2 z) r γ,r (, z)d z ( q 1, q 2 ) z X Λ,z z X Λ c with r γ,r (, z) u < 1 z r γ,r (, z) c 1 e c2γ z, z l Then: local bounds can be made global if µ 1, µ 2 Gibbs measures on X Λ, there eists a joint repr. P(n 1, n 2 q 1, q 2 ) s.t. E [ (d(n 1, n 2 ) ] := d(n 1, n 2 )P(n 1, n 2 ) n 1,n 2 { } c ep c γ δ l 1 dist(, Λ c ) Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30

34 Conclusions and open problems Phase transitions in the canonical ensemble Cluster epansion in the canonical ensemble compute correlations (work in progress...) get a better radius of convergence finite volume corrections to the free energy (work in progress...) Quantum models Cluster epansion (Ginibre) Phase transitions for systems of bosons (Kuna-Merola-Presutti, for quantum gas with Boltzmann statistics) Elena Pulvirenti (Leiden University) Phase transitions April 7, / 30