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Gibbs Point Processes in finite Volume

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Gibbs Point Processes in finite Volume GIBBSIAN POINT PROCESSES SABINE JANSEN Contents 1. The ideal gas 4 1.1. Uniform distribution on energy shells. Boltzmann entropy 4 1.2. Large deviations for Poisson and normal laws 8 1.3. Statistical ensembles and thermodynamic potentials 11 1.4. Exercises 13 2. Point processes 15 2.1. Configuration space 15 2.2. Probability measures 18 2.3. Observables 19 2.4. Poisson point process 21 2.5. Janossy densities 24 2.6. Intensity measure, one-particle density 24 2.7. Correlation functions 26 2.8. Generating functionals and Ruelle bound 29 2.9. Moment problem. Inversion formulas 33 2.10. Local convergence 39 2.11. Summary 42 2.12. Exercises 43 3. Gibbs measures in finite volume 45 3.1. Energy functions and interaction potentials 45 3.2. Boundary conditions 47 3.3. Grand-canonical Gibbs measure 48 3.4. The pressure and its derivatives 49 3.5. Correlation functions 51 3.6. Summary 53 3.7. Exercises 54 4. The infinite-volume limit of the pressure 55 4.1. An intermezzo on real-valued random variables 55 4.2. Existence of the limit of the pressure 58 4.3. A first look at cluster expansions 63 4.4. Summary 68 4.5. Exercises 68 5. Gibbs measures in infinite volume 71 5.1. A structural property of finite-volume Gibbs measures 71 5.2. DLR equations 72 5.3. Existence 74 Date: March 31, 2018. 1 2 SABINE JANSEN 5.4. GNZ equation 75 5.5. Correlation functions and Mayer-Montroll equation 79 5.6. Kirkwood-Salsburg equation 80 5.7. Proof of Theorem 5.8 83 5.8. Uniqueness for small z 84 5.9. Summary 86 5.10. Exercises 87 6. Phase transition for the Widom-Rowlinson model 89 6.1. The two-color Widom-Rowlinson model. Color symmetry 89 6.2. Contours. Peierls argument 91 6.3. Phase transition for the two-color Widom-Rowlinson model 93 6.4. Phase transition for the one-color Widom-Rowlinson model 93 6.5. Summary 93 7. Cluster expansions 95 7.1. Tree-graph inequality for non-negative interactions 95 7.2. Tree-graph inequality for stable interactions 97 7.3. Activity expansion of the pressure 99 7.4. Summary 101 Appendix A. Monotone class theorems 102 References 104 GIBBSIAN POINT PROCESSES 3 4 SABINE JANSEN 1. The ideal gas 1.1. Uniform distribution on energy shells. Boltzmann entropy. For m > 0 and n 2 N, consider the function n 3 n 3 n X 1 2 1 2 Hn :(R ) × (R ) ! R;Hn(x; p) = 2m jpij = 2 jpj : (1.1) i=1 3 Hn is the Hamiltonian for an ideal gas of n particles of mass m in R . Particles have positions x1; : : : ; xn and momenta p1; : : : ; pn. Hamilton functions are important in classical mechanics because they encode dynamics via an associated ordinary differential equation, given by x_ (t) = rpHn(x(t); p(t)); p_ (t) = −∇xHn(x(t); p(t)); (1.2) 1 which in our case become x_ (t) = m p(t), p_ (t) = 0, hence pj(t) = mx_ j(t); mx¨j(t) = 0 (j = 1; : : : ; n): (1.3) The momentum is the mass times the velocity, and for the Hamiltonian defined above, the ODE consists of a system of n independent ODEs (no interaction between particles). From now on we choose m = 1. Fix L; E; " > 0, set L L 3 Λ = [− 2 ; 2 ] (1.4) and consider the energy shell n" n 3 n ΩE;Λ;n := f(x; p) 2 Λ × (R ) j E − n" ≤ Hn(x; p) ≤ Eg: (1.5) n" n" n" Let PE;Λ;n be the uniform distribution on ΩE;Λ;n. We would like to know if PE;Λ;n has a limit, in some sense, when E = En, L = Ln and n all go to infinity in such a way that En n 3 lim = u; lim = ρ, jΛnj = Ln (1.6) n!1 jΛnj n!1 jΛnj at fixed particle density ρ > 0 and energy density u > 0. We investigate first the n" asymptotics of the normalization constant jΩE;Λ;nj, i.e., the Lebesgue volume of the energy shell. This is not strictly needed for the limiting behavior of the probability distributions but is of interest in its own. d Proposition 1.1. Fix u; ρ > 0. Let (En)n2N, (Ln)n2N, and Λn = [−Ln=2;Ln=2] be sequences in R+ that satisfy (1.6). Then for every " > 0, 1 1 n" lim log jΩE ;Λ ;nj = s(u; ρ) (1.7) n!1 jΛnj n! n n where 3 3 4πu s(u; ρ) = −ρ(log ρ − 1) + ρ 2 + 2 log 3ρ (1.8) The function s(u; ρ) = is the Boltzmann entropy per unit volume of the ideal gas. Eq. (1.8) can be written more compactly as 3=2 5 1 4πu s(u; ρ) = ρ 2 + log ρ 3ρ (1.9) which is a version of the Sackur-Tetrode equation. GIBBSIAN POINT PROCESSES 5 Proof of Proposition 1.1. Clearly p p log 1 jΩn" j = log 1 L3n + log B ( 2E ) n B ( 2E − n") n! En;Ln;n n! n 3n n 3n n p 3n n where B3n(r) is the ball of radius r in R . Stirling's formula n! ∼ 2πn(n=e) yields n 3 1 1 3n 1 n Ln n log n! Ln = n log n! + log n ! 1 − log ρ. (1.10) R 1 x−1 −t Volumes of balls are given in terms of the Gamma function Γ(x) = 0 t e dt as 3n=2 π 3n jB3n(r)j = 3n r : (1.11) Γ( 2 + 1) The Gamma function satisfies Γ(n + 1) = n! for n 2pN0 and Stirling's formula applies to non-integer input as well, i.e., Γ(x + 1) ∼ 2πx(x=e)x as x ! 1, see Exercise 1.3. It follows that 1 p p n log B3n( 2En) n B3n( 2En − n") 3n=2 3n 1 π 1 p p 3n = n log 3n + n log 2En − 2(En − n") Γ( 2 + 1) p 3n ! p 3 1 (3n=2)3n=2 1 2E = 3 log π + log + log n 3n p 3n 2 3n=2 Γ( 2 + 1) n 3n=2 1 q 3n + log 1 − 1 − n" n En 3 3 4πu ! 3 log π + 1 + log 4 u = + log 2 3 ρ 2 2 3ρ and the proof is easily completed. Next we address the behavior of the probability measures. To that aim it is convenient to single out test functions F that are local, i.e., they depend only on the particles in some bounded Borel set ∆ ⊂ R3. We further assume that the test function does not depend on the labelling of the particle. Such a function F can be specified in the following form: let n X N∆(x) := #fj 2 f1; : : : ; ng j xj 2 ∆g = δxj (∆): (1.12) j=1 be the number of particles in ∆. Suppose we are given a scalar f0 2 R and 3 k a family (fk)k2N0 of functions fk : (∆ × R ) ! R that are symmetric, i.e., 3 k fk(yσ(1); : : : ; yσ(k)) = fk(y1; : : : ; yk) for all σ 2 Sk and (y1; : : : ; yk) 2 (∆ × R ) . We define ( f0; if N∆(x) = 0; F (x; p) = (1.13) fk ((xj; pj))j2[n]:xj 2∆ ; if N∆(x) = k 2 N: Define @s 1 @s β = (u; ρ); µ = − (u; ρ); z = exp(βµ): (1.14) @u β @ρ Equivalently, 3 ρ ρ 1 β = ; z = 3 ; µ = log z: (1.15) 2 u p2π/β β 6 SABINE JANSEN Proposition 1.2. Fix u; ρ > 0 and let β; z; µ be as in (1.15). Fix a bounded Borel 3 set ∆ ⊂ R . Let (En) and (Ln) be as in Proposition 1.1 and " > 0. Then for 3 n families of bounded symmetric functions (fn)n2N0 , fn : (∆ × R ) ! R and F as in (1.13), such that F is bounded, we have Z lim F dPn" En;Ln;n n!1 Ωn" En;Ln;n 1 k Z X z k 1 2 −ρj∆j − 2 βjpj = e fk (xj; pj)j=1 e dxdp k! k 3 k k=0 ∆ ×(R ) 1 1 Z −ρj∆j X k −β[Hk(x;p)−µk] = e fk (xj; pj)j=1 e dxdp: k! k 3 k k=0 ∆ ×(R ) In this sense the sequence of equidistributions on the energy shell Ωn" does En;Λn;n indeed admit a limit. Moreover the limit does not depend on the shell's thickness ". Lemma 1.3. Under the assumptions of Proposition 1.2, we have k n" (ρj∆j) −ρj∆j lim PE ;Λ ;n(N∆ = k) = e (1.16) n!1 n n k! for all k 2 N0. In particular, in the limit n ! 1 the number N∆ of particles in ∆ converges to a Poisson random variable with parameter ρj∆j. Proof. To lighten notation, we abbreviate Pn" = P . Let us first look at the En;Λn;n n distribution of the particle number N∆. Fix k 2 f0; 1 : : : ; ng. Then X 1 Z Y Y c Pn(N∆ = k) = n 1l∆(xj) 1l∆ (xj)dx jΛnj n J⊂[n] Λn j2J j2[n]nJ #J=k n k n−k j∆j = qn(1 − qn) qn := k jΛnj "k−1 # 1 Y ` (ρj∆j)k = 1 − (nq )k(1 − q )n−k ! e−ρj∆j: (1.17) k! n n n k! `=0 Lemma 1.4.
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