Intro Response Lin resp Variational Summary Large deviations of the dynamical activity in the East model: analysing structure in biased trajectories Peter Sollich, Robert L Jack Disordered Systems Group, King’s College London; Physics, Bath J. Phys. A, 47:015003, 2014 P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Motivation Large deviations in time-integrated quantities Can be thought of as steady states generated by effective interaction Two-body? Many-body? Range? Dependence on strength of bias? Relevant e.g. for stable glassy states from activity bias, systems under steady shear Work so far mainly on one-body problems (Evans, Baule, Simha, Chetrite, Touchette) or extreme bias (Popkov, Schütz, Simon) Study effective interactions for East model . . . with bias towards large activity . . . across range of biases Hierarchy of responses, mirrors aging dynamics P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Outline 1 Timescales, activity vs escape rate bias, observables 2 Response to bias: overview 3 Linear response theory 4 Variational approaches 5 Summary & outlook P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Outline 1 Timescales, activity vs escape rate bias, observables 2 Response to bias: overview 3 Linear response theory 4 Variational approaches 5 Summary & outlook P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary East model & timescale hierarchy Chain of N spins, ni = 1 up, ni = 0 down Facilitated spins: up-spin to left Facilitated spins flip up with rate c, down with rate 1 − c Up-spin concentration c = 1/(1 + eβ ) Hierarchy of timescales: to relax up-spin at distance `, energy barrier α` = dlog2 `e, timescale τ` ∼ eβα` ∼ c−α` Path entropy matters once ` ∼ 1/c, timescale 2 τ1/c = τ0 ∼ eβ /(2 ln 2) For larger `, τ` ∼ c`τ0 from ≈ independent events on length scale 1/c P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary East model with activity bias Activity K = nr. of spin flips over time tobs Biased ensemble of trajectories [C(t)] Prob[C(t), s] ∝ Prob[C(t), 0]e−sK /he−sK i0 s < 0 favours large activity s > 0 gives inactive state Dynamical free energy e−N tobs ψK (s) = he−sK i0 ψK (s) has kink at s = 0 Dynamical phase transition there, bimodal distribution of K P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Analogy with equilibrium statistical mechanics Equilibrium: Bias configurations by factor ehM Gibbs free energy Space-time: Bias trajectories by factor e−sK Dynamical free energy P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Activity vs escape rate bias Dynamical free energy is largest eigenvalue of deformed master operator WK (s) (Spohn Lebowitz) WK (s) = W but with all off-diagonal elements mult. by e−s So es WK (s) = W with all diagonal elements mult. by es Diagonal elements are (negative) escape rates −r(C) X r(C) = ri , ri = (1 − c)ni−1 ni + cni−1 (1 − ni ) i So es WK (s) = W − r(C)(es − 1) on diagonals Deformed master operator for ensemble biased w.r.t. Rt integrated escape rate R[C(t)] = 0 obs dt r(C(t)) Trajectory weights Prob[C(t), ν] ∝ Prob[C(t), 0]eνR Here ν = 1 − es , free energies related by ψR (ν) = es ψK (s) P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Auxiliary master operator & effective interaction Biased trajectories reach a steady state away from transients near t = 0 and t = tobs Steady state dynamics is governed by auxiliary master operator (e.g. Simon, Jack PS) Obtained from deformed (biased) operator by multiplying 0 transition rates à la Metropolis, by e[∆V (C)−∆V (C )]/2 (Evans) Steady state Ps (C) ∝ e−β P i ni −∆V (C) ∆V (C) is effective interaction Can in principle be got from uC = e−∆V (C)/2 = leading left eigenvector of deformed master operator P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Observables To understand the effects of bias ν will use: 0 (ν) mean escape rate r(ν) = hRiν /(N tobs ) = −ψR 00 (ν), susceptibility χR (ν) = r0 (ν) = −ψR also gives variance of R spatial correlations C(x) = hδni δni+x iν , at equilibrium C(x) = c(1 − c)δx,0 domain size distribution p(d), for domains defined as 10 . . . 001 . . . P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Range of ∆V ? p(d) useful probe of interaction range Exponential in equilibrium: p(d) = c(1 − c)d−1 Can show: if ∆V has finite range, p(d) remains exponential for d > interaction range Will find that at any ν > 0, p(d) decays faster than exponential So ∆V must have infinite range May be related to question of whether effective potentials are ”Gibbsian” P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Outline 1 Timescales, activity vs escape rate bias, observables 2 Response to bias: overview 3 Linear response theory 4 Variational approaches 5 Summary & outlook P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Numerical methods To sample biased ensemble numerically, can use transition path sampling (N = 32 . . . 64) Alternatively diagonalize deformed master operator exactly to find ψR (ν) and ∆V (N = 14) Checks for finite size effects convergence to large tobs results in TPS comparison of N = 12 vs N = 14 for exact method check that typical domain sizes < N P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Mean escape rate, density, susceptibility c=0.1 (b) (a) 0.6 0.5 0.4 r(ν) Exact diagonalisation TPS simulation QE prediction 0.04 0.3 r(ν) 0.02 0.2 TPS simulation QE prediction linear response 0.1 00 (c) 0.2 0.4 ν 0.6 0.8 00 1 1 (d) 100 0.8 80 ρ(ν) 0.6 χR(ν) 0.4 0.2 00 0.005 ν 0.01 0.0005 0.001 60 40 20 0.2 0.4 ν 0.6 0.8 1 P Sollich & R Jack 00 ν East model with biased activity Intro Response Lin resp Variational Summary Susceptibility and linear response regime Susceptibility χR (ν) grows for ν → 0 Have (proportionality factor is 1/(N tobs )) X Z tobs 2 2 χR (ν) ∝ hR iν − hRiν = dt dt0 hδri (t) δrj (t0 )iν ij 0 For ν → 0, correlation hδri (t)δrj (t0 )i0 decays on timescale of order τ0 This gives dominant c-dependence of χR (ν) Linear response up to ν ' r(0)/χR (0) ∼ τ0−1 Smallest of a hierarchy of scales for ν P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Quasiequilibrium Some degrees of freedom can remain quasiequilibrated Formally, some probability ratios Ps (C)/Ps (C 0 ) stay as for ν = 0, and ∆V (C) − ∆V (C 0 ) remains small E.g. if spin i is facilitated, typical lifetime of facilitating spin i − 1 is ∼ τ0 1/c ⇒ spin i can flip many times Effect of ν on these local flips small if ν 1 So probability ratio of configurations C = . . . 0 . . . 010 . . . C 0 = . . . 0 . . . 011 . . . is as in equilibrium For correlations get hni−1 ni iν ≈ hni−1 iν c . . . and for mean escape rate r(ν) = 2c(1 − c)ρ(ν) where ρ = up-spin density Each up-spin contributes ≈ 2c to escape rate P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Spatial correlations c=0.1 (a) 0.04 ν = 0 -4 ν = 10-3 ν = 10-2 ν = 10 0.02 C(x) 0 -0.02 0 5 x 10 15 Up spins repel each other, corresponding nearest-neighbour peak But relatively weak effects P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Domain size distribution c=0.1 p(d) 0.01 (c) 0.001 0 1 -5 0 0.1 ν=0 -5 ν = 10 -4 ν = 10 -3 ν = 10 ν = 0.01 ν = 0.04 p(d) / p (d) 1 (b) ν = 10 LR 0.9 0.8 10 20 30 40 50 60 70 d 0 10 20 30 d Large domains suppressed as ν grows Eventually get peak at emergent lengthscale d∗ Quasiequilibrium: p(1) ≈ c, ok P Sollich & R Jack East model with biased activity 40 50 60 Intro Response Lin resp Variational Summary Scaling for c → 0 ν = c2 0.3 2 c = 0.1 c = 0.067 c = 0.05 [ν=c ] 0.2 p(d) 0.1 00 2 4 6 d 8 10 12 14 Consider small c limit with ν = cb (here b = 2) Peak becomes sharper: p(d) for d < d∗ shrinks Consider linear response next to understand this P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Outline 1 Timescales, activity vs escape rate bias, observables 2 Response to bias: overview 3 Linear response theory 4 Variational approaches 5 Summary & outlook P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Effective interaction By linear response theory can relate effects of small ν to equilibrium correlations For effective interaction find ∆VC = −2νRC + O(ν 2 ) Here the propensity is RC ≡ XZ j 0 ∞ dt hδrj (t)iC Mean change in escape rate when starting in configuration C Still need to understand how RC depends on C (2N numbers) But useful for intuition P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Propensity differences d=3 C C′ × ℓ=5 m=8 If αd < αm−d , αl then C relaxes to C 0 on timescale τd Main contribution to propensity difference from site × Facilitating up-spin has lifetime τd , so RC − RC 0 ≈ 2cτd Hence ∆VC 0 − ∆VC ≈ 4νcτd Depends strongly on d (for d = 1 get O(1) difference), e.g. ∼ νc−b for d = 1 + 2b Configurations are favoured for having more spins, but far apart (large d) P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Domain size distribution With same arguments can estimate linear response of p(d): p(d) p0 (d) p(d) p0 (d) p(d) p0 (d) ' 1 + A1 ν + O(ν 2 ), d=1 ' 1 + Ad ν/cαd −1 + O(ν 2 ), 2 ≤ d . 1/c ' 1 − Ad ντ0 c2 d(cd − 1) + O(ν 2 ), d & 1/c Suppression of very large domains: reduction in propensity from down-spins in interior of domain significant P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Hierarchy of responses Generalization of quasiequilibrium argument for ν 1 Consider ν cb−1 Domains of sizes d ≤ 2b weakly affected by ν (relative linear response correction is 1) So p(d) = O(c) for d ≤ 2b Larger d: relative response large, expect p(d) = O(1) P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Revisit earlier results 2 ν=c 0.3 2 c = 0.1 c = 0.067 c = 0.05 [ν=c ] 0.2 p(d) 0.1 00 2 4 6 d 8 10 12 14 ν = c2 c = cb−1 with b = 2 So expect p(d) = O(c) for d ≤ 2b = 4 Larger domains have finite probability P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Plateau regions Consider ν between two scales, cb ν cb−1 p(d) = O(c) for d ≤ 2b as before Larger domains can have p(d) = O(1) System can maximize its escape rate by making most (all?) domains of size d = 2b + 1 So should get density ρ = 1/(2b + 1) E.g. ρ = 1/3 for c ν 1 Numerical results consistent with this (c) 1 0.8 ρ(ν) 0.6 0.4 0.2 00 0.2 P Sollich & R Jack 0.4 ν 0.6 0.8 1 East model with biased activity Intro Response Lin resp Variational Summary Summary of hierarchy ρ 1 1 3 1 5 1 9 1 17 c τ0−1 c3 c2 c 1 ν Close similarity to hierarchical density evolution during aging Entropy vanishing in plateaux so non-monotonic in ν? P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Outline 1 Timescales, activity vs escape rate bias, observables 2 Response to bias: overview 3 Linear response theory 4 Variational approaches 5 Summary & outlook P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Variational approach Escape rate bias acts only on diagonal elements of W so does not destroy detailed balance Can determine dynamical free energy ψR (ν) variationally Choose tractable classes of effective interactions Interactions of blocks of B spins: maximal range B − 1, up to B-body Bias on p(d): interaction of 1’s separated by string of 0’s Bias on p(f, e): same with domains defined as string of 1’s followed by string of 0’s d=3 (f, e) = (1, 2) d=3 (2, 2) P Sollich & R Jack d=5 (1, 4) East model with biased activity Intro Response Lin resp Variational Summary Free energy and mean escape rate c=0.1 (a) 10 -Fvar/ f0 1 0.0001 (b) pd model B=6 model pfe model 0.1 r(ν) 0.001 0.01 ν 0.1 1 0.01 0.0001 "exact" pd model B=6 model pfe model 0.001 0.01 ν 0.1 1 Free energy plotted as ratio with linear baseline −f0 = νr(0) All approximations decent for larger ν but become worse as ν decreases p(f, e) model best P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Domain size distributions c=0.1 (a) "exact" pd model B=6 model pfe model 0.4 0.3 p(d) (b) 0.3 p(d) 0.2 0.2 0.1 00 "exact" pd model B=6 model pfe model 0.4 0.1 [ν = 0.63] 2 4 6 d 8 (c) 10 12 "exact" pd model B=6 model pfe model 0.2 p(d) 0.1 [ν = 0.1] 00 14 (d) 2 4 0.1 6 00 2 4 10 12 14 0.1 0.01 p(d) 0.05 0.001 0 [ν = 0.01] d 8 10 20 30 40 50 −4 [ν = 10 ] 6 d 8 10 12 14 P Sollich & R Jack 00 10 20 d 30 East model with biased activity 40 50 Intro Response Lin resp Variational Summary Where do variational approaches fail? (a) (b) C1 C2 × × C1′ C2′ × × C1 has longer-lived ”superspin” than C2 so is favoured p(d) model cannot represent this; p(f, e) can, and captures quasiequilbrium by p(f + 1, e) ≈ cp(f, e + 1) But p(f, e) model fails similarly at next level up (all lengthscales doubled) Block model will fail once preferred domain size > B − 1 P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Outline 1 Timescales, activity vs escape rate bias, observables 2 Response to bias: overview 3 Linear response theory 4 Variational approaches 5 Summary & outlook P Sollich & R Jack East model with biased activity Intro Response Lin resp Variational Summary Summary & outlook In linear response, have found general link between effective interaction and propensity For East model specifically, hierarchical structure of response Similar to aging case Conjecture for simple ordered structure of system in plateau regions of ν, non-monotonic entropy Variational approaches can be useful but need physical insight Timescale separation leads to weak interactions on short lengthscales, may be helpful in other contexts P Sollich & R Jack East model with biased activity