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