INTRODUCTION TO STATISTICAL MECHANICS Exercises

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1 École d hiver de Probabilités Semaine 1 : Mécanique statistique de l équilibre CIRM - Marseille 4-8 Février 2012 INTRODUCTION TO STATISTICAL MECHANICS Exercises Course : Yvan Velenik, Exercises : Loren Coquille The Ising model in dimension 1 Consider the graph V N := {1, 2..., N} Z, on which live the random variables σ i { 1, +1}, i V N denoted spins. We write σ := (σ i ) i VN. Let β R + be the inverse temperature, and h R be the magnetic field. Hamiltonian ) of the system is: The energy (called H N,β,h (σ) := β i,j V N i j σ i σ j h We introduce the following probability measure on Ω N := { 1, +1} V N : N σ i P N,β,h (σ) := 1 Z N,β,h exp ( H N,β,h (σ)) where Z N,β,h = σ Ω N exp( H N,β,h (σ)) is the normalizing constant called partition function. The following limit, if it exists, is called the free energy : f β (h) = lim f 1 N,β(h) := lim N N N log Z N,β,h QUESTION 1. Show that for all β > 0 and all h R, 1. the free energy exists and equals: Hint : f β (h) = log ( e β cosh(h) + ) e 2β cosh 2 (h) 2 sinh(2β). Calculate the partition function Z N,β,h and then the free energy f β (h) for the graph V N where we add an edge between 1 and N (periodic boundary condition): show that Z N,β,h can be rewritten as ( ) Z N,β,h = T r(t N e β+h e ) where T = β+h is called transfert matrix. e β h e β h

2 Show that f β (h) = f β (h) by upper-bounding H N,β,h H N,β,h. 2. f β (h) is a convex function of h. Hint : Show that log Z N,β,h is convex in h for all N and use the following theorem: Let (g n ) n 1 be a sequence of convex functions from an open interval I R into R converging pointwise towards a function g, then g is also convex. QUESTION 2. Show that, for h = 0, the average magnetization of a system of size N concentrates exponentially (in N) on 0, i.e. show that for all β (0, ) and for all ε > 0, there exists c(β, ε) > 0 such that for all N : [ ] 1 N P N,β,0 σ i ( ε, ε) 1 2e c(β,ε)n N in particular 1 N N σ i 0 in probability. The first bound implies also the almost sure convergence. This hence prove a strong law of large numbers for the (non independant) random variables σ i. Hint : [ ] [ 1 N Use the Chebychev inequality to show that P N,β,0 N σ i ε e hεn E N,β,0 e h P N Show that sup h 0 {hε Λ β (h)} > 0 by writing Λ β (h) in terms of f β (h) and using Question 1. σi ], h 0. QUESTION 3. On the graph V N magnetization, i.e. with periodic boundary condition, calculate the finite volume m(β, h) := lim Ẽ N,β,h [σ 1 ]. N Draw the graph of m(β, h) as a function of h for two values of β. Hint : Show that ẼN,β,h[σ 1 ] = ẼN,β,h[ 1 N N σ i] = h f N,β(h), and use the following theorem: Let (g n ) n 1 be a sequence of convex functions from an open interval I R into R converging pointwise towards a function g. If g is differentiable at x, then lim n + g n (x) = lim n g n (x) = g (x) QUESTION 4. Show that for k N, Cov N,β,0 [σ 1 σ k ] = E N,β,0 [σ 1 σ k ] = th(β) k 1 i.e., when h = 0, the covariance of two spins decays exponentially in the distance between the spins. Hint : Rewrite the Hamiltonien H N,β,0 (σ) := β N 1 J iσ i σ i+1 with J i = 1 for all i, and show that 1 k 1 Z N,β,0 E N,β,0 [σ 1 σ k ] = β k 1. Z N,β,0 J 1... J k 1 J i Calculate Z N,β,0 in terms of Z N 1,β,0 to show that Z N,β,0 = N 2ch(βJ i) and conclude.

3 The Mean Field Ising model Consider the complete graph with N vertices, denoted K N. Let σ i { 1, +1}, i = 1... N be the spin variables, with the notation : σ := (σ i ) N. For β R +, and h R, the Hamiltonian is: H N,β,h (σ) := β 2N N N σ i σ j h i,j=1 We consider the following probability measure on Ω N := { 1, +1} K N : P N,β,h (σ) := 1 Z N,β,h exp ( H N,β,h (σ)) where Z N,β,h = σ i σ Ω N exp( H N,β,h (σ)). QUESTION 5. Calculate the distribution of the magnetization m N := 1 N N σ i. Hint : Express the Hamiltonien in terms of m N : observe that H N,β,h (σ) = H N,β,h (m N (σ)). QUESTION Let x k := 1 + 2k N with k = 0, 1,..., N. Using the Stirling formula 1, show that there exists c 1, c 2 (0, ) uniform in x k such that ( ) N c 1 N 1/2 e Ns(xk) c 2 N 1/2 e Ns(x k) k with 2. Conclude that with s(x) := 1 + x 2 c 1 N 1/2 e Nf(x k) ( 1 + x log 2 ) 1 x log 2 ( ) 1 x σ:m N (σ)=x k e HN,β,h(mN ) c 2 N 1/2 e Nf(xk) f(x) = s(x) + βx2 2 + hx. 2 (1) 1 N! = N N e N 2πN(1 + O(1/N))

4 QUESTION Study the function f in terms of β and h. 2. Show that N max f(x) max f(x k) const. x [ 1,1] 0 k N 3. Using (1), show that there exists c 3 (0, ) such that c 3 N 1/2 e N max x [ 1,1] f(x) Z N,β,h (N + 1)c 2 N 1/2 e N max x [ 1,1] f(x) QUESTION 8. Let 1 a < b 1. Show that 1 ( N log (P(m N [a, b])) max f(x) x [a,b] ) ( ) log N max f(y) = O y [ 1,1] N QUESTION Let M(β, h) be the set of global maxima of f. Let M ɛ (β, h) := {x [ 1, 1] : min y M(β,h) x y ɛ}. Show that for all ɛ > 0, lim N 1 N log P N,β,h(m N M ɛ (β, h)) < 0 2. Conclude that the law of large numbers for the magnetization is verified when h 0 as well as when h = 0 and β 1, but it is violated when h = 0 and β > Draw the graph of the magnetization as a function of h when β 1 and when β > 1.

5 Canonical ensemble for the ideal lattice gas A lattice gas in a box Λ Z d is a random configuration in {0, 1} Λ. Every site i Λ carries a state σ i {0, 1}. We say that the gas is ideal if the random variables σ i are independent. We define the average energy per particle in the box by: H Λ (σ) = 1 N N σ i, where N = Λ The microcanonical ensemble describes many-particle closed systems at constant energy. define: P (micro) Λ,E = Uniform {σ : HΛ (σ)=e/n} i.e. P (micro) Λ,E (σ) = ( N E) 1 1 {HΛ (σ)=e/n} We QUESTION 10. Canonical ensemble Let Λ such that = n. We denote by ω the configuration σ restricted to. Let ε R + and e {0,..., n}. Show that lim E,N E/N ε P (micro) Λ,E (H (ω) = e/n) = ω : H (ω)=e/n n e β(ε)ω i for some β(ε) R +, where Z n,ε = (1 ε) n. This result naturally leads to the definition: P (can),ε (ω) = n e β(ε)ω i Z n,ε = e β(ε)h (ω) Z n,ε Verify that H = ε under P (can),ε. This marginal measure on as Λ Z d describes an open system exchanging energy with a heat bath at constant temperature 1/β(ε), we are in the canonical ensemble. Z n,ε QUESTION 11. P (can),ε minimizes the informational entropy under the constraint H = ε Let P be a probability measure on a finite space {x 1,..., x n }. We write p i = P(x i ). The informational entropy of P is defined by: s(p) = n p i log p i. 1. Let P, Q be two probability measures on {x 1,..., x n } which have finite entropy and such that q i > 0 for all i. Show that n p i log q i = s(q) s(p) s(q) with equality if and only if p i = q i for all i. 2. Let Ω be a finite state space and H : Ω R (the energy function). Show that for all ε between min Ω H and max Ω H, the unique probability distribution P on Ω that satisfies the condition H P = ε and has maximum entropy is: for a unique β = β(ε) R. P(ω) = e βh(ω) ω Ω e βh(ω)

6 Extremal Gibbs measures are weak limits of finite volume measures Let Σ be a finite set called spin space, and denote S the set of all parts of Σ. We consider the σ-algebra on Ω = Σ Zd given by F = S Zd. It is generated by the cylinders {C Λ,ω } Λ Z d,ω Ω where C Λ,ω = {ω Ω : ω Λ = ω}, where ω Λ denotes the restriction of ω to Λ Z d. We write F Λ = S Λ, and F = Λ Z df Λ c. The later is called asymptotic σ-algebra and contains the events whose realization does not depend on the spins states inside any finite region. An infinite volume Gibbs measure is by definition a probability measure µ on (Ω, F) such that for all Λ Z d, µπ Λ = µ where π Λ is the Gibbsian specification given by : π Λ (ω) = µ( F Λ c)(ω) = µ ω Λ( ) with µ ω Λ a finite volume Gibbs measure with boundary condition ω in the volume Λ, corresponding to your favourite Hamiltonian (sum of local potentials). In other words, µ satisfies: σ Ω, µ(σ σ Λ c = ω) = µ ω Λ(σ) for µ-almost every ω Ω and Λ Z d. These are called the DLR (for Dobrushin, Landford, Ruelle) equations, and we write G(π) = {µ : µ satisfies DLR equations for the specification π}. It is easy to check that the accumulation points of sequences (µ ωn Λ n ) n of finite volume Gibbs measures (w.r.t. the weak topology) are in G(π), our goal is to show that G(π) do not contain other physically relevant measures. It is possible to show that G(π) is a convex set. Therefore, the relevant measures are the extremal ones. Any extremal measure µ satisfies by definition the following: if µ = αµ 1 + (1 α)µ 2 with α (0, 1) and µ 1, µ 2 G(π) then µ 1 = µ 2. Moreover, G(π) is a simplex, i.e. every µ G(π) can be written in a unique way as an integral over the set of extremal measures. We will not prove it here. QUESTION 12. For f : Ω R such that µ(f) = 1, we define fµ(a) := A µ(dω)f(ω) for all A F. Show that if µ G(π) and g is F -measurable, g 0 and µ(g) = 1, then gµ G(π). QUESTION 13. Show that if µ G(π) is extremal then µ is trivial on F. (i.e. µ(a) {0, 1} for all A F.) QUESTION 14. Show that for all integrable f, the sequence of random variables X n = µ(f F Λ c n ) is a reversed martingale with respect to the filtration (F Λ c n ) n. (i.e. for all n we have, F n+1 F n, µ( X n ) < and µ(x n F n+1 ) = X n+1.) QUESTION 15. Show that if µ G(π) is extremal, then it is a limit of a sequence of finite volume Gibbs measures. This shows that G(π) and the set of accumulation points of (µ ω Λ n ) n coincide on the physically relevant measures (the extremal ones). Hint : Show that lim Λ Z d µ ω Λ µ for µ almost every ω Ω.

7 Some consequences of the GKS inequalities Consider the Ising model in a box Λ Z d, with the following Hamiltonian: HΛ,J,h ω := J ij σ i σ j h i σ i i Λ We write: µ ω Λ,J,h (σ) = e Hω Λ,J,h (σ) Z ω Λ,J,h {i,j} Λ i j et f ω Λ,J,h = Recall the GKS inequalities : For all A, B Λ, and h 0 we have σ A + Λ,J,h 0 σ Ω Λ f(σ)µ ω Λ,J,h (σ) σ A σ B + Λ,J,h σ A + Λ,J,h σ B + Λ,J,h were we write σ A := i A σ i. These inequlities are also true for free boundary condition (written ). QUESTION 16. Show that 1. σ A + Λ,J,h is an non decreasing function of J and of h. 2. For all J 0, h 0 and A Λ 1 Λ 2 Z d we have σ A Λ 1,J,h σ A Λ 2,J,h QUESTION 17. Show that 1. σ A + β,h 2. σ A β,h 3. σ A + β,h is an increasing function of the dimension d of the lattice when h 0. is left-continuous as a function of β when h 0. is right-continuous as a function of β when h 0. Hint : Use the preceding question, and for the two last questions, use the following lemma : Let a m,n be a non decreasing double sequence bounded above. Then, lim lim a m,n = lim lim a m,n = sup{a m,n } m n n m The same result holds if the double sequence is nonincreasing and bounded below. m,n QUESTION 18. Show that the critical inverse temperature β c (d) is a non increasing function of d.

8 High temperature expansion and exponential decay of correlation QUESTION 19. High temperature expansion. We saw in the course that the high temperature expansion lead to a representation of Z + Λ,β,0 and σ 0 + Λ,β,0. Derive similar relations for σ iσ j + Λ,β,0 and σ iσ j Λ,β,0. QUESTION 20. Exponential decay of correlations. Prove that, for all β sufficiently small (for which we have in particular uniqueness of the infinite volume Gibbs measure µ β,0 ), there exists c = c(β) > 0 such that Hint : Use the high temperature expansion. σ i σ j β,0 1 c e c i j 2 About the Mermin-Wagner theorem QUESTION 21. Consider Λ Z d, σ = (σ i ) i Λ (S 1 ) Λ, and the Hamiltonian H σ Λ,β (σ) = β {i,j} E σ Λ V (σ j σ i ) where σ (S 1 ) Λ is the boundary condition, and V : S 1 R is a twice differentiable even function such that V (0) = 0. As usual we define µ Λ,β (dσ) = 1 ZΛ,β σ exp( H Λ,β (σ))d Λ σ We recall that in the following notations σ S 1 is parametrized by its angle, so we write σ [0, 2π), and we identify 0 and 2π. Note that the Hamiltonian is invariant under the action of the group of rotations of the circle SO(2). 1. For d = 2, adapt the computation done in the course to show that, in the box Λ N = { N,... N} 2, the energy required to set to π all the spins in the box { l,... l} 2 goes to 0 when N. 2. For d 3, use the Cauchy-Schwarz inequality to show that for any function f : N R such that f(r) = π for r [0, l] and f(r) = 0 for r > N, we have {i,j} E 0 Λ N (f( i ) f( j )) 2 2d ( N+1 r=l r 2 ) 1 Conclude that the minimal energy required to set to π all spins in the box { l, l} d is not bounded uniformly in l.