Cluster Expansion in the Canonical Ensemble

Transcription

1 Università di Roma Tre September 2, 2011 (joint work with Dimitrios Tsagkarogiannis)

2 Background Motivation: phase transitions for a system of particles in the continuum interacting via a Kac potential, as in the LMP model (Lebowitz, Mazel, Presutti Liquid-vapor phase transitions for systems with finite-range interactions., J. Stat. Phys. 94, (1999)), with extra short range interaction. General technique: eliminate some degrees of freedom of the system by partitioning the space into boxes and defining a coarse-grained functional for the order parameter (multi-canonical set-up). Prototype example: calculation of the free energy functional with respect to the density in a single box. Method: cluster expansion of the canonical partition function. (new, more direct proof of Mayer s virial expansion)

3 The Model Configuration q {q 1,..., q N } of N particles in a box Λ R d which interact with a pair potential V : R d R, such that: (stable) 1 i<j N V (q i q j ) BN, N, q 1,..., q N (tempered) C(β) := e βv (q) 1 dq <, R d β > 0

4 The Model Configuration q {q 1,..., q N } of N particles in a box Λ R d which interact with a pair potential V : R d R, such that: (stable) 1 i<j N V (q i q j ) BN, N, q 1,..., q N (tempered) C(β) := e βv (q) 1 dq <, R d β > 0 The hamiltonian is: H Λ (q) := V (q i, q j ) 1 i<j N The canonical partition function is given by: Z β,λ,n := 1 dq 1... dq N e βh Λ(q) N! Λ N

5 The Result Theorem (with D. Tsagkarogiannis): If ρ C(β) is small enough then: 1 Λ log Z β,λ,n = log Λ N N! + N Λ F N,Λ (n) n 1 F N,Λ (n) Ce cn, n 1; F N,Λ (n) = 1 n+1 P N, Λ (n)b β, Λ (n), for all n 1 P N, Λ (n) = (N 1)... (N n) Λ n ρ n and lim Λ B β,λ(n) = β n βf β (ρ) = ρ(log ρ 1) n 1 1 n + 1 β nρ n+1 (β n : Mayer s coefficients)

6 Mayer s idea Grand canonical partition function: Ξ β,λ (z) := N 0 z N Z β,λ,n z : activity βp β (z) := ρ β (z) := lim Λ lim Λ 1 Λ log Ξ β,λ(z) = b n z n n 1 (1) 1 Λ z z log Ξ β,λ(z) = nb n z n n 1 (2)

7 Mayer s idea Grand canonical partition function: Ξ β,λ (z) := N 0 z N Z β,λ,n z : activity βp β (z) := ρ β (z) := lim Λ lim Λ 1 Λ log Ξ β,λ(z) = b n z n n 1 (1) 1 Λ z z log Ξ β,λ(z) = nb n z n n 1 (2) Inverting (2) and replacing z(ρ) in (1): βp β (ρ) = ρ 1 m m + 1 β mρ m m 1 virial expansion

8 Cluster Expansion for the Canonical Partition Function 1) Normalize the measure with the volume: Z ideal Λ,N Λ N := N! Z β,λ,n = Z ideal Λ,N Z int β,λ,n, and Zβ,Λ,N int := dq 1 Λ N Λ... dq N Λ e βh Λ(q) 2) Use the ±1 trick (Mayer s idea) for e βhλ(q) : e β i<j V (q i q j ) = (e βv (q i q j ) 1 + 1) = 1 i<j N g G N {i,j} E(g) G N : set of graphs on up to N vertices (labels of particles) f i,j := e βv (q i q j ) 1 f i,j

9 Cluster Expansion for the Canonical Partition Function 3) Compatibility: g g supp g supp g =, (supp g: vertices set) where: = g {g 1,..., g k }, g G N Z int β,λ,n = ζ Λ (g) := {g 1,...,g k } i=1 Λ g i supp g dq i Λ k ζ Λ (g i ) {i,j} E(g) C(N): set of connected graphs on up to N vertices g i C(N): polymers of the expansion ( g : cardinality of supp g) f i,j

10 Cluster Expansion for the Canonical Partition Function 3) Compatibility: g g supp g supp g =, (supp g: vertices set) V := supp g, g G N where: Z int β,λ,n = {V 1,...,V k } i=1 ζ Λ (V ) := g C V ζ Λ (g) k ζ Λ (V i ) V(N) := {V : V {1,..., N}} V i V(N): polymers of the expansion C V : the set of all connected graphs with support V

11 Cluster Expansion for the Canonical Partition Function 4) Exponentiation of the partition function. We work in the space of vertices sets: V(N) := {V : V {1,..., N}} Z int β,λ,n = {V 1,...,V k } i=1 { } k ζ Λ (V i ) c.e. = exp c I ζλ I where: I : V(N) {0, 1,...} with supp I := {V V(N) : I (V ) > 0} is a multi-index, ζ I Λ = V ζ Λ(V ) I (V ), and c I = 1 V I (V ) log Z int β,λ,n I! I (V1) ζ Λ (V 1 ) I (Vn) ζ Λ (V n ) I ζλ (V )=0 c I 0 supp I is an incompatible collection of V i s

12 Convergence of the Cluster Expansion Kotecký - Preiss convergence condition: There exist two positive functions a, c : V(N) R such that for any V V(N): ζ Λ (V ) e a(v ) δ holds for some δ > 0 small. Moreover, for any V V(N) V : V V ζ Λ (V ) e a(v )+c(v ) a(v ). Theorem (Cluster expansion) If the K-P condition holds, convergence is absolute.

13 We have to prove: 1 c I ζλ I Λ = N F N,Λ (n) Λ where: 1 F N,Λ (n) Ce cn, n 1 2 F N,Λ (n) = 1 n+1 P N, Λ (n)b β, Λ (n), β n := lim N, Λ, N/ Λ =ρ lim Λ I n 1 P N, Λ (n) = ρ n and lim Λ B β, Λ (n) = β n, 1 Λ n! g B n+1 w Λ (g) Mayer s coefficients, B n+1 : set 2-connected graphs on (n + 1) vertices n+1 w Λ (g) := f i,j dq i, f i,j := e βv (q i q j ) 1 Λ n+1 {i,j} E(g) i=1

14 Proof of (2) Let s work in the space C(n + 1), with: Remark: I c I ζλ I = Ĩ c Ĩ ζĩλ. Ĩ : C(n + 1) {0, 1,...}. F N,Λ (n) = 1 ( N 1 n + 1 n ) Ĩ : supp Ĩ {1,...,n+1} cĩ ζĩ Λ = 1 n + 1 P N, Λ (n)b β,λ (n) where: P N, Λ (n) := (N 1)... (N n) Λ n ρ n

15 Proof of (2) Let s work in the space C(n + 1), with: Remark: I c I ζλ I = Ĩ c Ĩ ζĩλ. Ĩ : C(n + 1) {0, 1,...}. F N,Λ (n) = 1 ( N 1 n + 1 n ) Ĩ : supp Ĩ {1,...,n+1} cĩ ζĩ Λ = 1 n + 1 P N, Λ (n)b β,λ (n) where: P N, Λ (n) := (N 1)... (N n) Λ n ρ n B β,λ (n) = Λ n n! g C n+1 Ĩ : g g supp Ĩ =g cĩ ζĩ Λ?

16 Proof of (2) Infinite volume cancellations: Since: ζ Λ (g) 1 Λ g 1, then: B β,λ (n) = Λ n n! g C n+1 Ĩ : g supp Ĩ g =g ( 1 ) cĩ ζĩλ + O Λ where the terms in the satisfy the factorization property: ζ Λ ( g Ag ) = g A ζ Λ (g ), A supp Ĩ and, moreover: Ĩ (g ) = 1, g supp Ĩ

17 Proof of (2) Finite volume cancellations: Only 2-connected graphs give non-zero contribution in the! Lemma Given g C n+1 \ B n+1, then: cĩ ζĩλ = 0 Ĩ : g supp Ĩ g =g We remain with those terms such that: g B n+1 and Ĩ (g) = 1 B β,λ (n) = Λ n n! g B n+1 ζ Λ (g) β n

18 Next steps 1 Finite volume corrections to the free energy; 2 Optimal radius of convergence of the expansion in powers of the density (also for hard spheres); 3 Applications to the LMP model (coarse-grained Hamiltonians)...