Théorie des grandes déviations: Des mathématiques à la physique Hugo Touchette National Institute for Theoretical Physics (NITheP) Stellenbosch, Afrique du Sud CERMICS, École des Ponts Paris, France 30 novembre 2015 Hugo Touchette (NITheP) Grandes déviations Novembre 2015 1 / 22 Plan Themes Typical states Fluctuations around typicality Many components Outline A bit of history Basics of large deviations Equilibrium systems Nonequilibrium systems. Lewis (80s) Ellis (1984) Graham (80s) Lanford (1973) Onsager (1953) Einstein (1910) Boltzmann (1877). Gärtner (1977) Freidlin-Wentzell (70s) Donsker-Varadhan (70s) Sanov (1957) Cramér (1938) Hugo Touchette (NITheP) Grandes déviations Novembre 2015 2 / 22
Boltzmann (1877) Energy levels j. 4 3 2 1 1 2 3 4 N Particles i Energy distribution: w j = # particles in level j Multinomial distribution: ln N! j w j! N j w j ln w j = Ns(w) P(w) e Ns(w) Hugo Touchette (NITheP) Grandes déviations Novembre 2015 3 / 22 Einstein (1910) Generalize Boltzmann Macrostate: M N Density of states (complexion): W (m) = # microstates with M N = m Einstein s postulate W (m) = e Ns(m) Probability: P(m) = e N[s(m) s(m )] Equilibrium: s(m ) is max Hugo Touchette (NITheP) Grandes déviations Novembre 2015 4 / 22
Cramér (1938) Sample mean: S n = 1 n X i, n Cumulant: i=1 λ(k) = ln E[e kx ] = X i p(x) IID p(x) e kx dx R Harald Cramér (1893-1985) Probability density: ( ni (s) 1 P(S n = s) = e b 0 + b 1 n n + Rate function: I (s) = max{ks λ(k)} k R Hugo Touchette (NITheP) Grandes déviations Novembre 2015 5 / 22 ) Sanov (1957) Sequence of IID RVs: X 1, X 2,..., X n X i p(x) Empirical distribution: L n (x) = 1 n n i=1 δ Xi,x P(L n = ρ) e nd(ρ p) Relative entropy: D(ρ p) = dx ρ(x) ln ρ(x) p(x) Ivan Nikolaevich Sanov (1919-1968) (1919-1968) Law of Large Numbers: L n ρ Hugo Touchette (NITheP) Grandes déviations Novembre 2015 6 / 22
Large deviation theory Random variable: A n Probability density: P(A n = a) Large deviation principle (LDP) ni (a) P(A n = a) e Meaning of : lim n 1 n ln P(a) = ni (a) + o(n) ln P(a) = I (a) Rate function: I (a) 0 Goals of large deviation theory 1 Prove that a large deviation principle exists 2 Calculate the rate function Hugo Touchette (NITheP) Grandes déviations Novembre 2015 7 / 22 Varadhan s Theorem LDP: ni (a) P(A n = a) e Exponential expectation: E[e nf (An) ] = e nf (a) P(A n = a) da Limit functional: λ(f ) = lim n Theorem: Varadhan (1966) 1 n ln E[enf (A n) ] S. R. Srinivasa Varadhan Abel Prize 2007 Special case: f (a) = ka λ(f ) = max{f (a) I (a)} a λ(k) = max{ka I (a)} a Hugo Touchette (NITheP) Grandes déviations Novembre 2015 8 / 22
Gärtner-Ellis Theorem Scaled cumulant generating function (SCGF) λ(k) = lim n 1 n ln E[enkA n ], k R Theorem: Gärtner (1977), Ellis (1984) If λ(k) is differentiable, then 1 LDP: 2 Rate function: ni (a) P(A n = a) e Richard S. Ellis I (a) = max{ka λ(k)} k I (a) is the Legendre transform of λ(k) J. Gärtner Hugo Touchette (NITheP) Grandes déviations Novembre 2015 9 / 22 Cramer s Theorem Sample mean: SCGF: Gaussian S n = 1 n n X i, X i p(x), IID i=1 λ(k) = lim n λ(k) = µk + σ2 2 k2, k R I (s) = 1 (s µ) 2, 2σ 2 s R 1 n ln E[enkS n ] = ln E[e kx ] Exponential λ(k) = ln(1 µk), k < 1 µ I (s) = s µ 1 ln s µ, s > 0 p(s n=s) n=500 p(s =s) n n=500 n=100 I(s) n=10 n=100 n=10 I(s) µ s µ s Hugo Touchette (NITheP) Grandes déviations Novembre 2015 10 / 22
Sanov s Theorem n IID random variables: ω = ω 1, ω 2,..., ω n, P(ω i = j) = p j Empirical frequencies: L n,j = 1 n δ ωi,j = # (ω i = j), L n = (L n,1, L n,2,...) n n Gärtner-Ellis SCGF: Rate function: i=1 λ(k) = lim n 1 n ln E[enk L n ] = ln q D(µ) = inf{k µ λ(k)} = k q p j e k j j=1 j=1 µ j ln µ j p j Hugo Touchette (NITheP) Grandes déviations Novembre 2015 11 / 22 Beyond IID Markov processes {X t } T t=0 A T = 1 T T 0 f (X t) dt P(A T = a) e TI (a) Long time limit Donsker & Varadhan (1975) SDEs ẋ(t) = f (x(t)) + ɛ ξ(t) P[x] e I [x]/ɛ Low noise limit Freidlin & Wentzell (1970s) Onsager & Machlup (1953) Applications Noisy dynamical systems Interacting SDEs Stochastic PDEs Interacting particle systems RWs random environments Queueing theory Statistics, estimation Information theory Hugo Touchette (NITheP) Grandes déviations Novembre 2015 12 / 22
Summary ni (a) P(A n = a) e p (a) n I(a) p (a) n I(a) Law of Large Numbers Typical value = zeros of I (a) Central Limit Theorem Quadratic minima = Gaussian fluctuations Small deviations Large deviations Fluctuations away from typical value a a General theory of typical states and fluctuations Hugo Touchette (NITheP) Grandes déviations Novembre 2015 13 / 22 Equilibrium systems N particles u Microstate: ω = ω 1, ω 2,..., ω N Statistical ensemble: P(ω) Macrostate: M N (ω) Macrostate distribution: P(M N = m) = ω:m N (ω)=m P(ω) Problems Calculate P(M N = m) Find most probable values of M N (= equilibrium states) Study fluctuations around most probable values Thermodynamic limit N Hugo Touchette (NITheP) Grandes déviations Novembre 2015 14 / 22
Equilibrium large deviations Microcanonical Einstein (1910) u Canonical Landau (1937) T P u (M N = m) = e S(u,m)/k B Extensivity: S N LDP: P u (M N = m) e NI u(m) F (β,m) P β (M N = m) = e Extensivity: F N LDP: P β (M N = m) e NI β(m) Exponential concentration of probability Equilibrium states = minima and zeros of I Hugo Touchette (NITheP) Grandes déviations Novembre 2015 15 / 22 Maxwell distribution ρ(v) Velocity distribution: Sanov s Theorem L N (v) = # particles with v i [v, v + v] N v P u (L N = ρ) e NI u(ρ) v Equilibrium distribution: ρ (v) = c v 2 e mv 2 2k B T Hugo Touchette (NITheP) Grandes déviations Novembre 2015 16 / 22
Entropy and free energy Density of states: u Ω(u) = # ω with U/N = u Large deviation form: Ω(u) e Ns(u) Gärtner-Ellis Theorem Free energy: ϕ(β) = lim N 1 N s(u) = min{βu ϕ(β)} β ln Z(β), Z(β) = dω e βu(ω) Z(β) = partition function = generating function ϕ(β) = free energy = SCGF Basis of Legendre transform in thermodynamics Hugo Touchette (NITheP) Grandes déviations Novembre 2015 17 / 22 Sources and applications Finite-range systems Lanford (1973) Spin systems Ellis (1980s) Bose condensation Lewis (1980s) 2D turbulence Long-range systems Quantum systems Lenci, Lebowitz (2000) Spin glasses Large deviation structure Typical states and fluctuations Oscar Lanford III (1940-2013) John T. Lewis (1932-2004) Hugo Touchette (NITheP) Grandes déviations Novembre 2015 18 / 22
Nonequilibrium systems Process: X t One or many particles Markov process External forces Boundary reservoirs T a J T b Trajectory: {x t } T t=0 Path distribution: P[x] Observable: A N,T [x] Problems Calculate P(A N,T = a) Find most probable values of A N,T (= stationary states) Study fluctuations around typical values Scaling limits: N T noise 0 Hugo Touchette (NITheP) Grandes déviations Novembre 2015 19 / 22 Example: Pulled Brownian particle Glass bead in water Laser tweezers Langevin dynamics: mẍ(t) = αẋ }{{} k[x(t) vt] }{{} drag spring force + ξ(t) }{{} noise Fluctuating work: Q vt W T }{{} work = }{{} U + Q }{{} T potential heat U T LDP TI (w) P(W T = w) e Fluctuation relation P(W T = w) P(W T = w) = etcw Hugo Touchette (NITheP) Grandes déviations Novembre 2015 20 / 22
Applications Driven nonequilibrium systems Interacting particle models Current, density fluctuations Macroscopic, hydrodynamic limit Thermal activation Kramers escape problem Disorded systems Multifractals Chaotic systems Quantum systems T a T b 56 H. Touchette / Physics Reports 478 (20 a Fig. 20. (a) Exclusion process on the lattice Z n and (b) rescaled lattice Z n /n. A particle can (red arrow). The thin line at the bottom indicates the periodic boundary condition (0) = interest for these models comes from the F fact that their macroscopic o their microscopic dynamics, sometimes in an exact way. Moreover, be studied by deriving large deviation principles which characterize th the hydrodynamic evolution [164]. The interpretation of these large d x theory, in that a deterministic dynamical behavior here the hydrodyn zero of a given (functional) rate function. From this point of view, the hy motion describing the hydrodynamic behavior, can be characterized as t minimum dissipation principle of Onsager [214]. Two excellent review papers [113,215] have appeared recently on int so we will not review this subject in detail here. The next example il the results that are typically obtained when studying these models. T Varadhan [216], who were the first to apply large deviation theory for stud Hugo Touchette (NITheP) models. Grandes déviations Novembre 2015 21 / 22 Exponentially rare fluctuations Exponential concentration of typical states Same theory for equilibrium and nonequilibrium systems Summary Example 6.11 (Simple Symmetric Exclusion Process). Consider a system ranging from 0 to n, n > k; see Fig. 20(a). The rules that determine th following: A particle at site i waits for a random exponential time with mean 1, The particle at i jumps to j if j is unoccupied; if j is occupied, then the p before choosing another neighbor to jump to (exclusion principle). Random variables ensembles stochastic systems Most probable values equilibrium states typical states Fluctuations rare events Rate function = entropy Cumulant function = free energy (Lf )( ) = 1 X Scaling limit: N, T, ɛ 0 2 i j =1 Unified language for statistical mechanics We denote by t (i) the occupation of the site i 2 Z n at time t, and configuration or microstate of the system. Because of the exclusion princ conditions on the lattice by identifying the first and last site. The generator of the Markovian process defined by the rules above c a jump from i to j only if (i) = 1 and (j) = 0. Therefore, J (i)[1 (j)][f ( i,j ) f ( )], where f is any function of, and i,j is the configuration obtained afte exchanging the occupied state at i with the unoccupied state at j: ( (i) if k = j H. Touchette The large deviation approach to statistical mechanics i,j (k) = Physics Reports 478, 1-69, 2009 (j) (k) www.physics.sun.ac.za/~htouchette Prochain exposé if k = i otherwise. Markov processes conditioned on large deviations To obtain a hydrodynamic description of this dynamics, we rescale the la and take the limit n!1with r = k/n, the density of particles, fixed n 2 to overcome the fact that the diffusion dynamics of the particle syste proved that the empirical density of the rescaled dynamics, defined by n (x) = 1 X t n n 2 t(i) (x i2z n When a fluctuation happens, how does it happen? Hugo Touchette (NITheP) where Grandes x is déviations a point of the unit circle C, weakly Novembre converges 2015 22 in/ probability 22 t diffusion equation F(x) i/n), b