SOLVABLE VARIATIONAL PROBLEMS IN N STATISTICAL MECHANICS

Transcription

1 SOLVABLE VARIATIONAL PROBLEMS IN NON EQUILIBRIUM STATISTICAL MECHANICS University of L Aquila October 2013 Tullio Levi Civita Lecture 2013

2 Coauthors Lorenzo Bertini Alberto De Sole Alessandra Faggionato Giovanni Jona Lasinio Claudio Landim

3 The general Framework Non equilibrium models of statistical mechanics Presence of fluxes (matter, energy,...) in the stationary state Non reversible stochastic models Large deviations analysis = Variational problems

4 Large deviations X N sequence of random variables taking values on a metric space M satisfies a Large deviations Principle (LDP) with speed α(n) and rate I : M R + {+ } if lim sup N + 1 α(n) log P (X N C) inf x C I(x), 1 lim inf N + α(n) log P (X N O) inf I(x), x O C closed O open ( ) P X N x e α(n)i(x)

5 Examples X N R d Gaussian 1 Z N e N 2 (x,c 1 x) dx satisfies a LDP with speed N and rate I(x) = 1 2 ( x, C 1 x ) A gas of independent coins ( ) η = η(1),..., η(n) {0, 1} N, i.i.d. Bernoulli

6 Examples EMPIRICAL MEASURE η = π N (η) := 1 N N i=1 η(i)δ i N

7 LD for the empirical measure π N (η) M + ([0, 1]), random positive measure, satisfies LDP P ( π N (η) ρ(x)dx ) e NI(ρ) the rate function is I(ρ) = 1 0 h m( ρ(x) ) d x where h m (ρ) = ρ log ρ m + (1 ρ) log 1 ρ 1 m

8 Contraction P Average number of coins 1 η(i) = dπ N (η) N ( 1 N i [0.1] satisfies LDP ) η(i) y e NJ(y) i BY CONTRACTION J(y) = inf {ρ : 1 0 ρ(x) d x=y} I(ρ)

9 A simple non equilibrium model CONTINUOUS TIME MARKOV CHAINS WITH EXPONENTIALLY SMALL RATES OF TRANSITION V finite set x, y, z V ; TRANSITION RATES r N (y, z) = e NR(y,z) INVARIANT MEASURE N 1 T, T =temperature µ N (x) = G Gx (y,z) G r N (y,z) Z N

10 The graphs in G x

11 A simple non equilibrium model NW (x) µ N (x) e W (x) = min G Gx { (y,z) G R(y, z) } + constant The reversible case (Example) H(z) H(y) R(y, z) = 2 Combinatorial optimization problem is solved (tricky) W = H + constant

12 From dynamic to static LD Stochastic differential equation in R d dx ɛ (t) = b (X ɛ (t)) dt + ɛdw (t) Dynamic sample path LD ( ) P X ɛ (t) x(t), t [T 1, T 2 ] e ɛ 1 T2 2 T ẋ b(x) 2 dt 1 { b = globally attractive vector field x such that b ( x) = 0 unique equilibrium point Invariant measure µ ɛ satisfies LD with rate the QUASIPOTENTIAL V (x) = inf T >0 inf {x(t) : x( T )= x,x(0)=x} I [ T,0] (x(t))

13 The quasipotential

14 The quasipotential b = S reversible = V = 2S In general V not differentiable (phase transitions, WASEP)

15 The general case If b has several stable attractors the quasipotential becomes V (x) = inf i W i + V i (x)

16 A solvable case A. Faggionato, D.G. (2012) Stochastic differential equation on S 1 (unit circle) b is periodic of period one dx ɛ (t) = b (X ɛ (t)) dt + ɛdw (t) S(x) := 2 x 0 b(y)dy S periodic 1 0 b(y)dy = 0 reversible = V = S Add an external field b = b + E

17 Sunshine transformation

18 Boundary driven 1-d simple exclusion

19 Scaling limit Diffusive rescaling L N = N 2 L N Hydrodynamic scaling limit π N (η t ) N + ρ t = ρ ρ(0, t) = ρ r0 ρ(1, t) = ρ r1 ρ(x, t)dx

20 Large deviations and quasipotential BDGJL (2002) DYNAMIC LARGE DEVIATIONS { P ( π N (η t ) ρ(x, t)dx, t [T 1, T 2 ] ) e NI [T 1,T 2 ](ρ) T2 I [T1,T 2 ](ρ) = 1 4 T 1 dt 1 0 dx ρ(1 ρ) ( H)2 t ρ = ρ (ρ(1 ρ) H), H(0, t) = H(1, t) = 0 THE QUASIPOTENTIAL r 0 = r 1 = reversible, gas of independent coins r 0 r 1 = not reversible, long range correlations { 1 [ V (ρ) = 0 hf (ρ) + log f ] ρ d x f f(1 f) + f = ρ, f(0) = ρ ( f) 2 r0, f(1) = ρ r1

21 The minimization path The minimizer for V (ρ 0 ) is the time reversal of the following coupled differential problem ( ) t ρ = ρ ρ(1 ρ) f(1 f) f ρ(u, 0) = ρ 0 (u) f(1 f) f + f = ρ ( f) 2 and b.c. A computation (magic transformation!) shows it is equivalent to t f = f f(1 f) f ( f) 2 + f = ρ ρ(u, 0) = ρ 0 (u) and b.c.

22 2-class TASEP

23 The invariant measure

24 Collapsing particles ( η 1, η T ) : x η 1 (x) x η T (x) = (η 1, η T ) = C [ ( η 1, η T )) ] Flux across bond (x, x + 1) [ ] J(x) = sup η 1 (z) η T (z) y z [y,x] +

25 Collapsing measures D.G. (08) ( ρ 1, ρ T )) : where d ρ 1 d ρ T = (ρ 1, ρ T ) = C [ ( ρ 1, ρ T )) ] S 1 S 1 Definition (a,b] dρ 1 = (a,b] d ρ 1 + J(a) J(b) J(x) := sup y [ d ρ 1 (y,x] (y,x] d ρ 2 ] +

26 Collapsing measures

27 Large deviations LD for the ( η 1, η T ) variables Ṽ ( ρ 1, ρ T ) = [h m1 ( ρ 1 ) + h m2 ( ρ T ))] d x S 1 LD for the SNS (not convex!) V (ρ 1, ρ T ) = inf Ṽ ( ρ 1, ρ T ) {( ρ 1, ρ T ) : C[( ρ 1, ρ T )]=(ρ 1,ρ T )} = [h m1 (ˆρ 1 ) + h m2 (ρ T ))] d x S 1 On any (a, b) where ρ 1 = ρ T x [ x ˆρ 1 (y)dy = CE a a ] ρ 1 (y)dy