Large deviations, metastability, and
effective interactions
Robert Jack (University of Bath)
Warwick workshop : metastability / KCMs, Jan 2016
Thanks to:
Peter Sollich (Kings College London)
Juan Garrahan (Nottingham) and David Chandler (Berkeley)
Fred van Wijland, Vivien Lecomte (Paris-Diderot)
Ian Thompson (Bath) and Yael Elmatad (Tapad)
General idea
Consider a stochastic system evolving in time
Focus on large deviations of time-averaged quantities.
General idea
Consider a stochastic system evolving in time
Focus on large deviations of time-averaged quantities.
What kinds of structure appear in the trajectories
when we condition on (rare) deviations from the
typical steady state behaviour?
General idea
Consider a stochastic system evolving in time
Focus on large deviations of time-averaged quantities.
What kinds of structure appear in the trajectories
when we condition on (rare) deviations from the
typical steady state behaviour?
Approach today: what can we say about the
“effective interactions” that stabilise this rare structure? (cf “driven processes” a la Touchette-Chetrite)
General idea
Consider a stochastic system evolving in time
Focus on large deviations of time-averaged quantities.
What kinds of structure appear in the trajectories
when we condition on (rare) deviations from the
typical steady state behaviour?
(Glass transitions?)
Approach today: what can we say about the
“effective interactions” that stabilise this rare structure? (cf “driven processes” a la Touchette-Chetrite)
Setup
Markov jump process with finite set of configurations C1 , C2 , . . .
(Examples: symmetric simple exclusion process, 1d Ising model)
“Activity” K: number of configuration changes in time
interval [0, tobs ]
Setup
Markov jump process with finite set of configurations C1 , C2 , . . .
(Examples: symmetric simple exclusion process, 1d Ising model)
“Activity” K: number of configuration changes in time
interval [0, tobs ]
Take large tobs and consider trajectories conditioned to
non-typical K
Setup
Markov jump process with finite set of configurations C1 , C2 , . . .
(Examples: symmetric simple exclusion process, 1d Ising model)
“Activity” K: number of configuration changes in time
interval [0, tobs ]
Take large tobs and consider trajectories conditioned to
non-typical K
Hop rate k = K/tobs obeys a large deviation principle,
p(k) ⇠ e t (k)
Setup
Markov jump process with finite set of configurations C1 , C2 , . . .
(Examples: symmetric simple exclusion process, 1d Ising model)
“Activity” K: number of configuration changes in time
interval [0, tobs ]
Take large tobs and consider trajectories conditioned to
non-typical K
Hop rate k = K/tobs obeys a large deviation principle,
p(k) ⇠ e t (k)
Conditioned set of trajectories can be described⇤ by
“auxiliary” (driven) process with “effective interactions”
(⇤ Terms and conditions apply)
[ RML Evans 2003, Maes & Netocny 2008, RLJ & Sollich 2010, Touchette & Chetrite 2015 ]
Master equations…
p(C, t): probability that the system is in C at time t
Master equations…
p(C, t): probability that the system is in C at time t
Master equation. . . transition rates W :
X
0
0
0
@t p(C, t) =
[p(C , t)W (C ! C) p(C, t)W (C ! C )]
C 0 6=C
Master equations…
p(C, t): probability that the system is in C at time t
Master equation. . . transition rates W :
X
0
0
0
@t p(C, t) =
[p(C , t)W (C ! C) p(C, t)W (C ! C )]
C 0 6=C
Write
@t pC (t) = W0 pC (t)
(W0 is the (forward) generator)
(pC is a discrete probability distribution over C)
Large deviations…
[ Lebowitz-Spohn, Bodineau-Derrida, Lecomte-van Wijland, etc ]
p(C, K, t): probability that the system is in C at time t,
having accumulated K hops so far.
Large deviations…
[ Lebowitz-Spohn, Bodineau-Derrida, Lecomte-van Wijland, etc ]
p(C, K, t): probability that the system is in C at time t,
having accumulated K hops so far.
@t P (C, K, t) =
X
C 0 (6=C)
0
0
0
[P (C , K 1, t)W (C ! C) P (C, K, t)W (C ! C )]
Large deviations…
[ Lebowitz-Spohn, Bodineau-Derrida, Lecomte-van Wijland, etc ]
p(C, K, t): probability that the system is in C at time t,
having accumulated K hops so far.
Large deviations…
[ Lebowitz-Spohn, Bodineau-Derrida, Lecomte-van Wijland, etc ]
p(C, K, t): probability that the system is in C at time t,
having accumulated K hops so far.
@t pC,K (t) = W1 pC,K (t)
Large deviations…
[ Lebowitz-Spohn, Bodineau-Derrida, Lecomte-van Wijland, etc ]
p(C, K, t): probability that the system is in C at time t,
having accumulated K hops so far.
@t pC,K (t) = W1 pC,K (t) . . . but W1 is not a nice object, so. . .
Large deviations…
[ Lebowitz-Spohn, Bodineau-Derrida, Lecomte-van Wijland, etc ]
p(C, K, t): probability that the system is in C at time t,
having accumulated K hops so far.
@t pC,K (t) = W1 pC,K (t) . . . but W1 is not a nice object, so. . .
Transform P
p(C, s, t) = K p(C, K, t)e
sK
Large deviations…
[ Lebowitz-Spohn, Bodineau-Derrida, Lecomte-van Wijland, etc ]
p(C, K, t): probability that the system is in C at time t,
having accumulated K hops so far.
@t pC,K (t) = W1 pC,K (t) . . . but W1 is not a nice object, so. . .
Transform P
p(C, s, t) = K p(C, K, t)e
sK
Finally end up with
C
C
@t p (s, t) = W(s)p (s, t)
. . . and W(s) typically has a simple representation
Auxiliary process
[ following RLJ-Sollich 2010 ]
@t pC (s, t) = W(s)pC (s, t)
Auxiliary process
[ following RLJ-Sollich 2010 ]
@t pC (s, t) = W(s)pC (s, t)
W(s)
P is not a generator because
@t C p(C, s, t) 6= 0 for s 6= 0
Auxiliary process
[ following RLJ-Sollich 2010 ]
@t pC (s, t) = W(s)pC (s, t)
W(s)
P is not a generator because
@t C p(C, s, t) 6= 0 for s 6= 0
Define Waux = u 1 W(s)u
(s)
with
(s): largest eigenvalue of W(s)
u: a diagonal operator (matrix) whose elements are the
elements of the dominant left eigenvector of W(s)
Auxiliary process
[ following RLJ-Sollich 2010 ]
@t pC (s, t) = W(s)pC (s, t)
W(s)
P is not a generator because
@t C p(C, s, t) 6= 0 for s 6= 0
Define Waux = u 1 W(s)u
(s)
with
(s): largest eigenvalue of W(s)
u: a diagonal operator (matrix) whose elements are the
elements of the dominant left eigenvector of W(s)
Then Waux is the (transposed) generator for the auxiliary
0
process [for conditioning K/tobs ⇡
(s)]
Auxiliary process
[ following RLJ-Sollich 2010 ]
This is an explicit construction of the auxiliary process that
mimics the conditioning
Auxiliary process
[ following RLJ-Sollich 2010 ]
This is an explicit construction of the auxiliary process that
mimics the conditioning
The transition rates for the auxiliary process are
W aux (C ! C 0 ) = u(C) 1 e s · W (C ! C 0 )u(C 0 )
Auxiliary process
[ following RLJ-Sollich 2010 ]
This is an explicit construction of the auxiliary process that
mimics the conditioning
The transition rates for the auxiliary process are
W aux (C ! C 0 ) = u(C) 1 e s · W (C ! C 0 )u(C 0 )
If the original process has detailed balance wrt
p0 (C) = e E(C)/T then. . .
auxiliary process has detailed balance with energy
function E aux (C) = E(C) T ln u(C)
Auxiliary process
[ following RLJ-Sollich 2010 ]
This is an explicit construction of the auxiliary process that
mimics the conditioning
The transition rates for the auxiliary process are
W aux (C ! C 0 ) = u(C) 1 e s · W (C ! C 0 )u(C 0 )
If the original process has detailed balance wrt
p0 (C) = e E(C)/T then. . .
auxiliary process has detailed balance with energy
function E aux (C) = E(C) T ln u(C)
Effective interaction
V (C)/T =
2 ln u(C)
Auxiliary process
[ following RLJ-Sollich 2010 ]
This is an explicit construction of the auxiliary process that
mimics the conditioning
The transition rates for the auxiliary process are
W aux (C ! C 0 ) = u(C) 1 e s · W (C ! C 0 )u(C 0 )
If the original process has detailed balance wrt
p0 (C) = e E(C)/T then. . .
auxiliary process has detailed balance with energy
function E aux (C) = E(C) T ln u(C)
Effective interaction
V (C)/T =
2 ln u(C)
Generalisation to diffusions and to conditioning on other
observables is straightforward
Auxiliary process
[ following RLJ-Sollich 2010 ]
This is an explicit construction of the auxiliary process that
mimics the conditioning
The transition rates for the auxiliary process are
W aux (C ! C 0 ) = u(C) 1 e s · W (C ! C 0 )u(C 0 )
If the original process has detailed balance wrt
p0 (C) = e E(C)/T then. . .
auxiliary process has detailed balance with energy
function E aux (C) = E(C) T ln u(C)
Effective interaction
V (C)/T =
2 ln u(C)
Generalisation to diffusions and to conditioning on other
observables is straightforward
[ ?? can we do this on infinite lattices ?? see later ]
Effective interactions
What can we say about the effective interactions?
Effective interactions
What can we say about the effective interactions?
Some interesting cases:
1. Exclusion processes (SSEP and ASEP)
2. East model
3. 1d Ising model
4. Model sheared systems
Exclusion processes
Particles on periodic 1d lattice, at most one per site,
attempt to hop right (or left) with rate 1 + a0 (or 1 a0 ).
Joint conditioning on activity (bias s) and current (bias h)
Exclusion processes
Particles on periodic 1d lattice, at most one per site,
attempt to hop right (or left) with rate 1 + a0 (or 1 a0 ).
Joint conditioning on activity (bias s) and current (bias h)
W(s, h) =
X
e
s+h
(1+a0 )
i
+
s h
(1
i+1 +e
a0 )
+
i i+1
i
ni =
+
i i
2ni (1
ni )
ð5Þ
dt
−ψ
L
KJ
obs hOehJ−sK i
e
is hOis;h ¼
0 [22]. Within fluctuating
hydrodynamics (assuming a ¼ 0 as above), the response to
the bias depends only on the quantity B ¼ sκ 000 − 12 h2 σ 000
[22]. For on
B¼
0, then1dSðqÞ
hasata most
finiteone
(nonzero)
Particles
periodic
lattice,
per site,limit as
q → 0; for
B >right
0, the
hyperuniform,
attempt
to hop
(or system
left) withisrate
1 + a0 (or 1 while
a0 ). for
B < 0, one has phase separation. The resulting dynamical
Joint
activity (bias
s) and
(bias h)
phaseconditioning
diagram ison shown
in Fig.
2: current
the fluctuating
Exclusion processes
ðsÞ
this
ical
ore
[ RLJ, Thompson, Sollich, PRL 114, 060601 (2015) ]
the
ular
(c)
(a)
(b)
mic
PS
HU
vior
HU PS
PS
HU
g. 1,
obs )
HU
PS
5) is
Þ as
SSEP
ASEP
tobs
HU: hyperuniform state
her
FIG. 2 (color online).
Proposed dynamic(inhomogeneous)
phase diagrams for
PS: “phase-separated”
state
mo-
Hyperuniformity
[ Torquato and Stillinger, 2003- ]
In HU states, the variance of the number of points
d
in a region of volume R scales as
h n(Rd )2 i ⇠ Rd 1
In ‘normal’ equilibrium states
d 2
d
h n(R ) i ⇠ R
where  is a compressibility.
Hyperuniformity
[ Torquato and Stillinger, 2003- ]
In HU states, the variance of the number of points
d
in a region of volume R scales as
h n(Rd )2 i ⇠ Rd 1
or maybe h n(Rd )2 i ⇠ Rd ↵ , ↵ > 0
In ‘normal’ equilibrium states
d 2
d
h n(R ) i ⇠ R
where  is a compressibility.
Hyperuniformity
[ Torquato and Stillinger, 2003- ]
In HU states, the variance of the number of points
d
in a region of volume R scales as
h n(Rd )2 i ⇠ Rd 1
or maybe h n(Rd )2 i ⇠ Rd ↵ , ↵ > 0
In ‘normal’ equilibrium states
d 2
d
h n(R ) i ⇠ R
where  is a compressibility.
HU states have strong suppression of large-scale
density fluctuations…
… jammed particle packings, biological systems,
novel photonic materials, galaxies…
Hyperuniformity
[Gabrielli,
Jancovici,. .Joyce,
Lebowitz, Pietronero and
Labini, 2002]
GENERATION OF PRIMORDIAL
COSMOLOGICAL
.
PHYSICAL
REVIEW
FIG. 2. Fragmentation step for the ‘‘pi
plane.
a SSP, is not statistically isotropic bec
lattice structure is not completely eras
#20$.
To construct a statistically isotropic
particle distribution with such a behavi
trivial. A particular example is the so-ca
ing of the plane #21,22$. The generatio
defined by taking a right angled triangle
tive lengths one and two %and hypoten
Hyperuniformity
[Gabrielli,
Jancovici,. .Joyce,
Lebowitz, Pietronero and
Labini, 2002]
GENERATION OF PRIMORDIAL
COSMOLOGICAL
.
PHYSICAL
REVIEW
FIG. 2. Fragmentation step for the ‘‘pi
plane.
Ideal gas
a SSP, is not statistically isotropic bec
lattice structure is not completely eras
#20$.
To construct a statistically isotropic
particle distribution with such a behavi
trivial. A particular example is the so-ca
ing of the plane #21,22$. The generatio
defined by taking a right angled triangle
tive lengths one and two %and hypoten
Hyperuniformity
[Gabrielli,
Jancovici,. .Joyce,
Lebowitz, Pietronero and
Labini, 2002]
GENERATION OF PRIMORDIAL
COSMOLOGICAL
.
PHYSICAL
REVIEW
FIG. 2. Fragmentation step for the ‘‘pi
plane.
Ideal gas
a SSP, is not statistically isotropic bec
lattice structure is not completely eras
#20$.
To construct a statistically isotropic
FIG. 1. A super-homogeneous distribution %bottom& and a
particle distribution with such a behavi
son distribution %top& with approximately the same numb
trivial. A particular
example
the so-ca
points. Both are projections
of thin slices
of threeisdimensiona
ing of the plane one
#21,22$.
The generatio
tributions. The super-homogeneous
is a ‘‘pre-initial’’
con
defined
a right representing
angled triangle
ration used in setting
up by
the taking
configuration
the i
lengths one
andsimulations
two %and of
hypoten
conditions for thetive
cosmological
N-body
#18$. A
Hyperuniformity
[Gabrielli,
Jancovici,. .Joyce,
Lebowitz, Pietronero and
Labini, 2002]
GENERATION OF PRIMORDIAL
COSMOLOGICAL
.
PHYSICAL
REVIEW
FIG. 2. Fragmentation step for the ‘‘pi
plane.
Ideal gas
a SSP, is not statistically isotropic bec
lattice structure is not completely eras
#20$.
To construct a statistically isotropic
FIG. 1. A super-homogeneous distribution %bottom& and a
particle
distribution with such a behavi
son distribution %top&Hyperuniform
with approximately the same numb
trivial. A particular
example
the so-ca
points. Both are projections
of thin slices
of threeisdimensiona
ing of the plane one
#21,22$.
The generatio
tributions. The super-homogeneous
is a ‘‘pre-initial’’
con
defined
a right representing
angled triangle
ration used in setting
up by
the taking
configuration
the i
lengths one
andsimulations
two %and of
hypoten
conditions for thetive
cosmological
N-body
#18$. A
Hyperuniformity and effective interactions
General picture for SEPs at weak bias can be obtained by
macroscopic fluctuation theory
[ Bertini, de Sole, D Gabrielli, Jona-Lasinio, Landim, 2002- ]
Hyperuniformity and effective interactions
General picture for SEPs at weak bias can be obtained by
macroscopic fluctuation theory
[ Bertini, de Sole, D Gabrielli, Jona-Lasinio, Landim, 2002- ]
For bias to high activity or high current in exclusion
processes…
Hyperuniformity and effective interactions
General picture for SEPs at weak bias can be obtained by
macroscopic fluctuation theory
[ Bertini, de Sole, D Gabrielli, Jona-Lasinio, Landim, 2002- ]
For bias to high activity or high current in exclusion
processes…
… hyperuniformity comes from very long ranged
repulsions in the “auxiliary dynamics”…
Hyperuniformity and effective interactions
General picture for SEPs at weak bias can be obtained by
macroscopic fluctuation theory
[ Bertini, de Sole, D Gabrielli, Jona-Lasinio, Landim, 2002- ]
For bias to high activity or high current in exclusion
processes…
… hyperuniformity comes from very long ranged
repulsions in the “auxiliary dynamics”…
For extreme bias, can get exact results (Bethe ansatz):
[ Schuetz, Simon, Popkov, Lazarescu, 2009- ]
⇡(i j)
2L
Repulsive potential V (i j) ⇠ log sin
That is, particles on a circle interact by (2d) Coulomb
repulsion
FIG. 5:
FIG. 3: (Color online) Scaling of the transition with system
constant-vo
size. (a) The activity k(s) collapses onto a single curve as
with N = 1
a function of sN . (b) Similarly, the peak in the dynamic
surements
⇤
susceptibility occurs at s ⇠ N . Inset: The maximal suscepphase-sepa
⇤
tibility
increases
increasing
size, showing
equilibrium
Hard
particles
in 1d with
evolving
withsystem
Brownian
motion that
(Langevin),
the magnitude of the activity fluctuations is increasing.
“Phase separation”
conditioned on low activity
(a) Ideal gas
(b) Low activity
uncorrelat
ticles. Fo
density ⇢
tor
Space
Time
Void
then we fi
Now defi
its right n
Can think of attractive interactions, or Langevin noises with FIG. 4: (Color online) Trajectories of a constant-density sysnon-zero
at =
the
of the void
tem at Nmean,
= 60, for
= particles
0.88 and tobs
20⌧edge
B . Particles are shown
FIG. 5:
FIG. 3: (Color online) Scaling of the transition with system
constant-vo
size. (a) The activity k(s) collapses onto a single curve as
with N = 1
a function of sN . (b) Similarly, the peak in the dynamic
surements
⇤ [ RLJ, Thompson, Sollich, PRL 114, 060601 (2015) ]
susceptibility occurs at s ⇠ N . Inset: The maximal suscepphase-sepa
⇤
tibility
increases
increasing
size, showing
equilibrium
Hard
particles
in 1d with
evolving
withsystem
Brownian
motion that
(Langevin),
the magnitude of the activity fluctuations is increasing.
“Phase separation”
conditioned on low activity
(a) Ideal gas
(b) Low activity
uncorrelat
ticles. Fo
density ⇢
tor
Space
Time
Void
then we fi
Now defi
its right n
Can think of attractive interactions, or Langevin noises with FIG. 4: (Color online) Trajectories of a constant-density sysnon-zero
at =
the
of the void
tem at Nmean,
= 60, for
= particles
0.88 and tobs
20⌧edge
B . Particles are shown
Linear response
[ RLJ, Thompson, Sollich, PRL 114, 060601 (2015) ]
General result: can obtain effective potential values from
“propensities” for activity
u(C, s) / he sK iC(0)=C
(average over long trajectories starting in C, need to take
care with normalisation)
Simple hydrodynamic argument shows that u-values for
phase-separated / hyperuniform configurations have
divergent du(C, s)/ds as system size L ! 1.
This comes from a diverging time scale: ⌧L ⇠ L2 is the
time required for these trajectories to relax to the steady
state. . .
Diverging interaction range linked to diverging length and
time scales. . .
ð5Þ
is hOis;h ¼ e
hOe
i0 [22]. Within fluctuating
hydrodynamics (assuming a ¼ 0 as above), the response to
1 2 00
00
the bias depends only on the quantity B ¼ sκ 0 − 2 h σ 0
[22]. For B ¼ 0, then SðqÞ has a finite (nonzero) limit as
[ RLJ, Thompson, Sollich, PRL 114, 060601 (2015) ]
q → 0; for B > 0, the system is hyperuniform, while for
B < 0, one has phase separation. The resulting dynamical
Long-ranged effective interactions (repulsive or attractive)
phase
diagram
is
shown
in
Fig.
2:
the
fluctuating
are generic in conditioned exclusion processes
Exclusion
processes
summary
ðsÞ
this
ical
ore
the
ular
mic
vior
g. 1,
obs )
5) is
Þ as
tobs
her
mod in
(a)
(c)
(b)
PS
HU
HU
PS
HU
SSEP
HU
PS
PS
ASEP
FIG. physical
2 (color origin
online).of the
Proposed
dynamic
phase diagrams
The
weak-bias
instabilities
is the for
biased exclusion
processes.time
HU,scale
hyperuniform
PS, phase
diverging
hydrodynamic
⌧R ⇠ R2 states;
.
East model
Site i has occupancy ni 2 {0, 1}
East model
Site i has occupancy ni 2 {0, 1}
For each site with ni = 1, with rate 1, refresh site i + 1 as
ni+1 = 1,
prob c
ni+1 = 0,
prob 1
c
East model
Site i has occupancy ni 2 {0, 1}
For each site with ni = 1, with rate 1, refresh site i + 1 as
ni+1 = 1,
prob c
ni+1 = 0,
prob 1
c
Simple model for glassy dynamics, stationary state has trivial
product structure
[ Jackle & Kronig 1991, Garrahan & Chandler, 2003- ]
East model
Site i has occupancy ni 2 {0, 1}
For each site with ni = 1, with rate 1, refresh site i + 1 as
ni+1 = 1,
prob c
ni+1 = 0,
prob 1
c
Simple model for glassy dynamics, stationary state has trivial
product structure
[ Jackle & Kronig 1991, Garrahan & Chandler, 2003- ]
Relaxation time diverges for small c as
log ⌧ ⇠ a(log c)
2
Hierarchical relaxation mechanism…
[ Sollich & Evans 1999, Aldous & Diaconis 2002, Toninelli, Martinelli & others 2007- ]
for our stochastic systems by
X
^
!sK"history#
ZK "s; t# $
Prob"history#e
;
(1)
East model : phase transition
histories
0.08
active
active
inactive
inactive
0.06
0.4
0.04
0.2
0.02
0
0
-0.2
-0.1
0
0.1
s
0.4
0.3
0.2 -0.2
-0.1
0
s
0.1
-0.02
0.2
K(s)/t =
model with
kinetic constraints
ρK(s)
As system size N 0.2
! 1, there is a jump in the first
derivative of the “free
energy”
(s)/N
.
kinetic constraints
0.1
removed
0
ψK(s) / N
0.6
K(s) / Nt
as
stnd
yis
ls.
me,
,a
ely
ly
n,
ce
els
in
on
ebe
First order dynamical phase
transition,
accompanied
by
-1
0
1
2
s
phase separation in space-time
0
(s)
for our stochastic systems by
X
^
!sK"history#
ZK "s; t# $
Prob"history#e
;
(1)
East model : phase transition
histories
12A531-4
Y. S. Elmatad and R. L. Jack
0.08
active
active
inactive
inactive
0.06
0.4
0.04
0.2
0.02
0
0
-0.2
-0.1
0
0.1
s
0.4
0.3
0.2 -0.2
-0.1
0
s
0.1
ψK(s) / N
0.6
K(s) / Nt
-0.02
0.2
K(s)/t =
model with
kinetic constraints
0
(s)
As system size N 0.2
! 1, there is a jump in the first
FIG. 2.
Sample kinetic
trajectories
from the three state points id
derivative of the “free
energy”
(s)/N
.
constraints
[ Elmatad and RLJ, 2013 ]
0.1
ρK(s)
as
stnd
yis
ls.
me,
,a
ely
ly
n,
ce
els
in
on
ebe
0
ϵ ≤ ϵ c these trajectories
are rare, coming from the trough
removed
inactive sites are white (ni = 0). (a) Trajectory with ϵ < ϵ c . s
but the clusters no longer have a sharp interface. (c) Trajecto
First order dynamical phase
transition,
accompanied
by
-1
0
1
2
s
phase separation in space-time
As discussed in Sec. III, our main results are su
Fig. 1 where we show sample numerical results f
for our stochastic systems by
X
^
!sK"history#
ZK "s; t# $
Prob"history#e
;
(1)
East model : phase transition
histories
12A531-4
Y. S. Elmatad and R. L. Jack
0.08
active
active
inactive
inactive
0.06
0.4
0.04
0.2
0.02
0
0
-0.2
-0.1
0
0.1
s
0.4
What happens here?
0.3
0.2 -0.2
-0.1
0
s
0.1
ψK(s) / N
0.6
K(s) / Nt
-0.02
0.2
K(s)/t =
model with
kinetic constraints
0
(s)
As system size N 0.2
! 1, there is a jump in the first
FIG. 2.
Sample kinetic
trajectories
from the three state points id
derivative of the “free
energy”
(s)/N
.
constraints
[ Elmatad and RLJ, 2013 ]
0.1
ρK(s)
as
stnd
yis
ls.
me,
,a
ely
ly
n,
ce
els
in
on
ebe
0
ϵ ≤ ϵ c these trajectories
are rare, coming from the trough
removed
inactive sites are white (ni = 0). (a) Trajectory with ϵ < ϵ c . s
but the clusters no longer have a sharp interface. (c) Trajecto
First order dynamical phase
transition,
accompanied
by
-1
0
1
2
s
phase separation in space-time
As discussed in Sec. III, our main results are su
Fig. 1 where we show sample numerical results f
Variational principle
(s) is the largest eigenvalue of W(s)
Variational principle
(s) is the largest eigenvalue of W(s)
If the auxiliary process is time-reversal symmetric, W(s)
can be symmetrised by a similarity transformation
Wsym (s) = ⇡
1/2
W(s)⇡ 1/2
is symmetric, where ⇡ is a diagonal matrix whose
elements are the stationary probabilities.
Variational principle
(s) is the largest eigenvalue of W(s)
If the auxiliary process is time-reversal symmetric, W(s)
can be symmetrised by a similarity transformation
Wsym (s) = ⇡
1/2
W(s)⇡ 1/2
is symmetric, where ⇡ is a diagonal matrix whose
elements are the stationary probabilities.
Hence
hv|Wsym (s)|vi
(s) = maxv
hv|vi
Variational principle
(s) is the largest eigenvalue of W(s)
If the auxiliary process is time-reversal symmetric, W(s)
can be symmetrised by a similarity transformation
Wsym (s) = ⇡
1/2
W(s)⇡ 1/2
is symmetric, where ⇡ is a diagonal matrix whose
elements are the stationary probabilities.
Hence
hv|Wsym (s)|vi
(s) = maxv
hv|vi
Hence u(C) = hC|⇡
1/2
|v ⇤ i where |v ⇤ i is the maximiser
Variational principle
(s) is the largest eigenvalue of W(s)
If the auxiliary process is time-reversal symmetric, W(s)
can be symmetrised by a similarity transformation
Wsym (s) = ⇡
1/2
W(s)⇡ 1/2
is symmetric, where ⇡ is a diagonal matrix whose
elements are the stationary probabilities.
Hence
hv|Wsym (s)|vi
(s) = maxv
hv|vi
Hence u(C) = hC|⇡
1/2
|v ⇤ i where |v ⇤ i is the maximiser
(Variational problem with 2N d.o.f.)
Conditioned East model
[ RLJ and Sollich, J Phys A, 2014 ]
East model, biased to high activity
J. Phys. A: Math. Theor. 47 (2014) 015003
d=3
C
C
×
=5
m=8
Figure 4. Sketch illustrating the configurations C and
Conditioned East model
[ RLJ and Sollich, J Phys A, 2014 ]
East model, biased to high activity
Variational results using ansatze for effective interactions
1. local multi-spin interactions up to 6-body
2. interactions dependent on domain-size distribution
J. Phys. A: Math. Theor. 47 (2014) 015003
d=3
C
C
×
=5
m=8
Figure 4. Sketch illustrating the configurations C and
Conditioned East model
[ RLJ and Sollich, J Phys A, 2014 ]
East model, biased to high activity
Variational results using ansatze for effective interactions
1. local multi-spin interactions up to 6-body
2. interactions dependent on domain-size distribution
J. Phys. A: Math. Theor. 47 (2014) 015003
d=3
Also,
numerical results
(exact diagonalisation
and transition path sampling)
C
C
… and perturbative results for
small bias
×
=5
m=8
Figure 4. Sketch illustrating the configurations C and
Conditioned East model
[ RLJ and Sollich, J Phys A, 2014 ]
Main results:
J. Phys. A: Math. Theor. 47 (2014) 015003
d=3
C
C
×
=5
m=8
Figure 4. Sketch illustrating the configurations C and
Conditioned East model
[ RLJ and Sollich, J Phys A, 2014 ]
Main results:
Interactions are long-ranged, (no finite ranged interaction
can capture even the first-order response to the bias)
can attribute this long range to large time scales for relaxation on large length scale (cf SEP)
J. Phys. A: Math. Theor. 47 (2014) 015003
d=3
C
C
×
=5
m=8
Figure 4. Sketch illustrating the configurations C and
Conditioned East model
[ RLJ and Sollich, J Phys A, 2014 ]
Main results:
Interactions are long-ranged, (no finite ranged interaction
can capture even the first-order response to the bias)
can attribute this long range to large time scales for relaxation on large length scale (cf SEP)
J. Phys. A: Math. Theor. 47 (2014) 015003
Variational ansatze may be good for ψ but not capture structural C
features
C
d=3
×
=5
m=8
Figure 4. Sketch illustrating the configurations C and
Conditioned East model
[ RLJ and Sollich, J Phys A, 2014 ]
Main results:
Interactions are long-ranged, (no finite ranged interaction
can capture even the first-order response to the bias)
can attribute this long range to large time scales for relaxation on large length scale (cf SEP)
J. Phys. A: Math. Theor. 47 (2014) 015003
Variational ansatze may be good for ψ but not capture structural C
features
hierarchy of responses to bias for small c, mirrors metastable state structure,
aging behaviour of model…
C
d=3
×
=5
m=8
Figure 4. Sketch illustrating the configurations C and
Conditioned East model
J. Phys. A: Math. Theor. 47 (2014) 015003
Density of up spins ⇢
Main results:
ρ
R L Jack and P Sollich
[ RLJ and Sollich, J Phys A, 2014 ]
1
1
3
1
5
1
9
Interactions are long-ranged, (no finite ranged interaction
can capture even the first-order response to the bias)
1
can attribute this17long range to large time scales for relaxation on large length scale (cf SEP)
c
J. Phys. A: Math. Theor. 47 (2014) 015003
Variational ansatze may−1be good 3
2
d = c3
ν
τ0
1
c
c
for ψ but not capture structural C
⌫ on
⇡ν, fors
Figure 5. Sketch showing how the density of up-spins ρ = ⟨niBias
⟩ν depends
features
×
very small c. Both axes are logarithmic. The solid line shows the limiting behaviour as
c → 0, taken at fixed (ln ν )/(ln c), while the dashed line shows the expected behaviour
for a system with small positive c. The effect of the bias ν becomes significant when
unity, the
ν ≈ τ0−1 , the inverse bulk relaxation time. As=ν 5is increased further
m =towards
8
C of steps. We expect a weak dependence on ν
system eventually responds in a sequence
whenever cb ≪ ν ≪ cb−1 for integer b, leading to plateaus in ρ. Within each plateau
(b ! 1), our conjecture is that the spacing between up-spins converges to 1 + 2b as
1
−b−1
and
c → 0, so ρ → 1+2
b . For large b, the plateaux are then bounded by ρ = 2
ρ = 2b , corresponding to power-law behaviour ρ ∼ 2−(ln ν )/(ln c) ∼ ν T ln 2 (dashed lines).
From the quasi-equilibrium argument,
one expects
alsoillustrating
r(ν ) ≈ 2c⟨n
= 2cρ(ν ): whenC and
Figure
4. Sketch
the
i ⟩ν configurations
hierarchy of responses to bias for small c, mirrors metastable state structure,
aging behaviour of model…
Summary so far
In East model and exclusion processes,
several things go together
Summary so far
In East model and exclusion processes,
several things go together
Long time scales (and metastability)
Summary so far
In East model and exclusion processes,
several things go together
Long time scales (and metastability)
Long length scales
Summary so far
In East model and exclusion processes,
several things go together
Long time scales (and metastability)
Long length scales
Long-ranged effective interactions
(not just long-ranged correlations)
Summary so far
In East model and exclusion processes,
several things go together
Long time scales (and metastability)
Long length scales
Long-ranged effective interactions
(not just long-ranged correlations)
Also, sometimes, dynamical phase transitions
Summary so far
In East model and exclusion processes,
several things go together
Long time scales (and metastability)
Long length scales
Long-ranged effective interactions
(not just long-ranged correlations)
Also, sometimes, dynamical phase transitions
There are good reasons to expect this to be general:
(eg perturbative arguments, small spectral gaps…)
… also, plenty of other examples
1d Ising model
Consider 1d Glauber-Ising chain (periodic) conditioned on
time-integrated energy
1d Ising model
Consider 1d Glauber-Ising chain (periodic) conditioned on
time-integrated energy
Can solve this model exactly by free fermions
[ see eg RLJ and Sollich, 2010 ]
1d Ising model
Consider 1d Glauber-Ising chain (periodic) conditioned on
time-integrated energy
Can solve this model exactly by free fermions
[ see eg RLJ and Sollich, 2010 ]
Trajectories in (1 + 1)d space-time are configurations of a
2d Ising model.
1d Ising model
Consider 1d Glauber-Ising chain (periodic) conditioned on
time-integrated energy
Can solve this model exactly by free fermions
[ see eg RLJ and Sollich, 2010 ]
Trajectories in (1 + 1)d space-time are configurations of a
2d Ising model.
For appropriate bias, get ferromagnetic states, even if
unconditioned chain has T ! 1 (!)
1d Ising model
Consider 1d Glauber-Ising chain (periodic) conditioned on
time-integrated energy
Can solve this model exactly by free fermions
[ see eg RLJ and Sollich, 2010 ]
Trajectories in (1 + 1)d space-time are configurations of a
2d Ising model.
For appropriate bias, get ferromagnetic states, even if
unconditioned chain has T ! 1 (!)
Effective interactions clearly long-ranged, also
non-Gibbsian.
[ Maes, Redig, von Enter, 1999 ]
1d Ising model
Consider 1d Glauber-Ising chain (periodic) conditioned on
time-integrated energy
“The traditional solution is to define an infinite-volume
Can solve
thismeasure
model exactly
free fermions
Gibbs
[as] a by
measure
which is a limit. . . of
[ see eg RLJ and Sollich, 2010 ]
finite-volume Gibbs measures
with some chosen
boundary
conditions”
Trajectories
in (1 +
1)d space-time are configurations of a
2d Ising model.
von Enter, Fernandez, Sokal, J Stat Phys 1993
For appropriate bias, get ferromagnetic states, even if
unconditioned chain has T ! 1 (!)
Effective interactions clearly long-ranged, also
non-Gibbsian.
[ Maes, Redig, von Enter, 1999 ]
1d Ising model
Consider 1d Glauber-Ising chain (periodic) conditioned on
time-integrated energy
“The traditional solution is to define an infinite-volume
Can solve
thismeasure
model exactly
free fermions
Gibbs
[as] a by
measure
which is a limit. . . of
[ see eg RLJ and Sollich, 2010 ]
finite-volume Gibbs measures
with some chosen
boundary
conditions”
Trajectories
in (1 +
1)d space-time are configurations of a
2d Ising model.
von Enter, Fernandez, Sokal, J Stat Phys 1993
For appropriate bias, get ferromagnetic states, even if
unconditioned chain has T ! 1 (!)
Effective interactions clearly long-ranged, also
non-Gibbsian.
[ Maes, Redig, von Enter, 1999 ]
(how general is this?)
Final summary
Conditioning stochastic systems to non-typical values of
time-integrated observables leads to rich phenomenology
Hyperuniformity, phase separation in space and/or time,
dynamical phase transitions, hierarchical responses…
Final summary
Conditioning stochastic systems to non-typical values of
time-integrated observables leads to rich phenomenology
Hyperuniformity, phase separation in space and/or time,
dynamical phase transitions, hierarchical responses…
These phenomena can be characterised by effective
interactions, but these are often different from familiar
equilibrium interactions
Long-ranged, non-Gibbsian, absence of dissipation
Final summary
Conditioning stochastic systems to non-typical values of
time-integrated observables leads to rich phenomenology
Hyperuniformity, phase separation in space and/or time,
dynamical phase transitions, hierarchical responses…
These phenomena can be characterised by effective
interactions, but these are often different from familiar
equilibrium interactions
Long-ranged, non-Gibbsian, absence of dissipation
Slow degrees of freedom respond most strongly to bias (or
conditioning), this can be one origin of long-ranged
interactions
[ more info: RLJ & Sollich, EPJE 2015, Touchette and Chetrite arXiv 1506.05291 ]