Limitations of statistical mechanics: Hints from large deviation theory

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1 Limittions of sttisticl mechnics: Hints from lrge devition theory Hugo Touchette School of Mthemticl Sciences Queen Mry, University of London 3rd Interntionl Conference on Sttisticl Physics Lrnc, Cyprus, July 2011 Supported by the Europen Physicl Society Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17 Limits, limittions nd boundries Experimentl limits - incompleteness Reltivistic phenomen not described by Newtonin mechnics Photoelectric effect not explined by clssicl EM theory Theoreticl limittions QM does not describe nonliner evolutions (if ny) Clssicl EM theory does not explin prticle-like phenomen Conditions of vlidity / boundries Thermodynmics pply to lrge systems QM pplies when ction Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17

2 Questions nd pproch Questions 1 Are there ny phenomen not explined by sttisticl mechnics? (Boltzmnn-Gibbs equilibrium sttisticl mechnics = ESM) 2 Wht re the conditions of vlidity of ESM? 3 Wht re the boundries of ESM? Approch ESM = Lrge devition theory (LDT) Study known boundries of LDT Derive boundries of ESM Pln Recp on LDT / Limits of LDT ESM = LDT / Limits of ESM Conclusions Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17 Lrge devition theory Ellis (1985), Touchette Phys Rep 178 (2009) Rndom vrible: A n Probbility distribution: P(A n = ) Lrge devition principle (LDP) Mening of : P(A n = ) e ni (), n Rte function: I () 0 lim n 1 n ln P() = I () Gols of lrge devition theory 1 Prove tht lrge devition principle exists 2 Clculte the rte function Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17

3 Two importnt results Scled cumulnt generting function (SCGF): λ(k) = lim n 1 n ln enka n, k R Vrdhn (1966) If A n stisfies n LDP with rte function I (), then λ(k) = mx{k I ()} λ = I λ(k) lwys convex Gärtner (1977), Ellis (1984) If λ(k) is differentible, then 1 P(A n = ) e ni () 2 I () = mx{k λ(k)} I = λ k I () is convex in this cse Not pplicble when I is nonconvex Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17 Applictions Sum of rndom vribles Crmér 1938 Product of rndom vribles Mrkov processes Donsker & Vrdhn Stochstic differentil equtions Freidlin & Wentzell 1970s Stochstic field equtions... Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17

4 Exmple: Exponentil rndom vribles S n = 1 n n X i, p(x i = x) = 1 µ e x/µ, x > 0, IID i=1 SCGF: p(s =s) n n=500 λ(k) = ln(1 µk), k < 1 µ Rte function: I (s) = s µ 1 ln s µ, s > 0 µ n=100 n=10 I(s) s Concentrtion point: s = X = µ Gussin fluctutions round s Non-Gussin fluctutions wy from s Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17 Generl properties p () n I() p () n I() Lw of Lrge Numbers Typicl points = concentrtion points = zeros of I () Centrl Limit Theorem Qudrtic minim = Gussin fluctutions Smll devitions Lrge devitions Fluctutions wy from typicl points Generl theory of typicl sttes nd fluctutions Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17

5 Boundries of LDT LDP: ni () P(A n = ) e LDP SCGF: λ(k) = sup{k I ()} Rte function: λ GE I Boundry cses no LDP I () = sup{k λ(k)} k I = 0 or λ not differentible (smoothness problem) λ does not exist (existence problem) Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17 Smoothness problem: Nonconvex rte functions Dinwoodie (1993), Ellis (1995) λ(k) lwys convex I () not necessrily convex Convex λ = I nd I = λ Nonconvex I λ I λ = I but I λ λ differentible I = λ λ nondifferentible I is nonconvex or ffine k Legendre structure only if I is convex Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17

6 Existence problem: Non-exponentil LDs Existence of λ(k) Existence of LDP Sub-exponentil λ = if P(A n ) n α I = 0 Super-exponentil λ = 0 if P(A n ) e en I = Exmple: Cuchy smple men S n = 1 n X i, p(x i = x) = 1 n π SCGF: λ(k) = i=1 { 0 k = 0 k 0 1 x 2 + 1, x R No LDP LDT does not pply Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17 Applictions in sttisticl physics Oono Prog Theoret Phys Suppl (1989), HT Phys Rep (2009) Equilibrium sttisticl mechnics Lnford (1973) Ruelle (1960s) Ellis (1984) Noise-perturbed dynmicl systems, SDEs Freidlin & Wentzell (1970s) Onsger-Mchlup (1953) Grhm (1980s) Nonequilibrium systems Gllvotti & Cohen (1995) Derrid, Bodineu (1990s-2000s) Bertini, Gbrielli, Jon-Lsinio (2000s)... LDT is the mthemticl lnguge of sttisticl mechnics Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17

7 Entropy nd free energy Microstte: ω = ω 1, ω 2,..., ω N Energy: U N (ω) Density of sttes: Ω(U N = u) LDP: Ω(U N = u) e Ns(u) Gärtner-Ellis Theorem Free energy: s(u) = min{βu ϕ(β)} β ϕ(β) = lim N 1 N ln Z N(β), Z N (β) = e βu N(ω) dω Z N (β) = prtition function = generting function ϕ(β) = free energy = SCGF s(u) = entropy = rte function Bsis of Legendre trnsform in thermo Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17 Boundries of ESM LDT: P n ()? ni () e LDP Convex Nonconvex No LDP ESM: Ω N (u)? e Ns(u) Exponentil Concve Nonconcve Non-exponentil Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17

8 Nonconcve entropies Cmp, Duxois & Ruffo Phys Rep (2009); HT Phys Rep (2009) Concve entropy ϕ = s s = ϕ s slope = β ϕ slope = u Nonconcve entropy s u ϕ β ϕ = s s ϕ Long-rnge systems (men-field, grvittion, etc.) Generlized cnonicl ensemble recovers equivlence [HT PRE 2009] u β No Legendre trnsform for nonconcve entropy systems Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17 Non-exponentil density of sttes Accepted ide Free energy does not exist no ESM True for cnonicl ensemble Not true for microcnonicl ensemble Existence of ϕ(β) Ω N(u) exponentil Sub-exponentil ϕ = if Ω N (u) N α s = 0 Super-exponentil? Use probbilities Are there systems with non-exponentil density of sttes? Described by microcnonicl ensemble Possible generliztion of cnonicl ensemble? Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17

9 Conclusions ESM ESM bsed on LDP Sttisticl mechnics Lrge devition theory Ω N (u) nd Z N (β) exponentil in N = LDP Entropy s(u) = rte function Free energy ϕ(β) = SCGF Legendre trnsform Gärtner-Ellis Theorem Limittions 1 s(u) my be nonconcve 2 ϕ(β) my not exist Ω N (u) not exponentil Physiclly possible / observble? Systems with: long-rnge interction / correltion / order Hugo Touchette (QMUL) Limits of sttisticl mechnics July / 17