Twenty five years after KLS
a celebration of
non

equilibrium statistical mechanics
R. K. P. Zia
Physics Department, Virginia Tech,
Blacksburg, Virginia, USA
SMM100, Rutgers,
December 2008
B. Schmittmann
supported in part by
Many here at
SMM100
What’s
KLS
?
and 25 years after?
Journal of Statistical Physics,
34
, 497 (1984)
Outline
•
Overview/Review
(devoted to students and newcomers)
What’s the context of KLS?
………….…….
………
Why study these systems?
Driven Ising Lattice Gas
(the “standard” model

KLS)
………….
and Variations
Novel properties: many surprises…
……
some understood, much
yet to be
understood
Outline
•
Over/Review
–
what did we learn?
•
Outlook
–
what else can we look forward to?
What’s the context of KLS?
Why study these systems?
•
Non

equilibrium Statistical Mechanics
○
d
etailed
b
alance
respecting
/
violating
dynamics
○
t

dependent phenomena
vs.
“being stuck”
○
stationary states with
d.b.
v.
dynamics
○
non

trivial probability currents and through

flux
…….
of energy, matter (particles), etc.
○
ps: Master equation approach, detailed balance, & Kolmogorov criterion
Over/Review
∂
t
P
(
C
,
t
)
=
Σ
{
R(
C
C
)
P
(
C
Ⱐ
t
)
刨
C
C
⤠
P
(
C
,
t
)
}
C
.
P
*,
P
*
cartoon of
equilibrium
vs
.
non

quilibrium
P
*
(
C
)
[
E

H
(
C
)
]
P
*
(
C
)
exp
[

H
]
P
*
=
?
What’s the context of KLS?
Why study these systems?
•
Non

equilibrium Statistical Mechanics
•
Fundamental issue:
Systems in
non

equilibrium
steady states
cannot
be understood in the Boltzmann

Gibbs framework.
What’s the
new game
in town?
Over/Review
What’s the context of KLS?
Why study these systems?
•
Non

equilibrium Statistical Mechanics
•
Physics of many systems
“all around us”
○
fast ionic conductors (KLS)
○
micro/macro biological systems
○
vehicular/pedestrian traffic, granular flow
○
social/economic networks
Over/Review
What’s the context of KLS?
Why study these systems?
•
Non

equilibrium Statistical Mechanics
•
Physics of many systems
“all around us”
Over/Review
What’s the original KLS?
•
Take a simple interacting many

particle
system…
(Ising model
–
lattice gas version, for the ions)
•
Drive it far from thermal equilibrium…
(by an external DC “electric” field)
•
Does anything “new” show up
?
Over/Review
Ising Lattice Gas
•
Take a well

known
equilibrium
system…
C
:
{
n
(
x,y
) } with
n =
0,1
e.g., Ising lattice gas
(2

d, Onsager)
H
(
C
) =
J
x,a
n
(
x
)
n
(
x
+
a
)
+ periodic boundary condtions (PBC)
Over/Review
Ising Lattice Gas
•
Take a well

known
equilibrium
system,
•
evolving with a simple dynamics…
Over/Review
…
going from
C
to
C
with rate
s
R(
C
C
⤠
that obey
detailed balance:
R(
C
C
⤠
/
R(
C
C
)
=
exp
[
{
H
(
C
⤠
H
(
C
)}
/
kT
]
…so that, in long times, the system is described by
the Boltzmann distribution:
P
*
(
C
)
exp
[
H
(
C
)
/
kT
]
Ising Lattice Gas
•
Take a well

known
equilibrium
system,
•
evolving with a simple dynamics…
Over/Review
…one favorite
R
is Metropolis, e.g.,
R(
C
C
⤠
/
R(
C
C
)
=
exp
[
{
H
(
C
⤠
H
(
C
)}
/
kT
]
Just go!
Go with rate
e
2
J/kT
Driven
Ising Lattice Gas
•
Take a well

known
equilibrium
system
•
Drive
it far from thermal equilibrium
….....
(by some additional external force, so particles suffer
biased
diffusion.)
Over/Review
e.g., effects of gravity
(uniform field)
g
Just go!
Go with rate
e
mga/kT
•
Can’t have PBC !!
•
Get to equilibrium
with
……………
extra potential term… NOTHING new!
a

lattice spacing
J
=0 case
Driven Ising Lattice Gas
•
Take a well

known
equilibrium
system
•
Drive
it far from thermal equilibrium
….....
(by some additional external force, so particles suffer
biased
diffusion.)
Over/Review
PBC possible with “electric” field,
E
(non

potential, rely on
t
B
)
E
Just go!
Go with rate
e
(
E

2J
)
/kT
unit “charge” and
a
with
E > 2J
LOTS of
surprises!
E
tends to break bonds
T
tends to satisfy bonds
Driven Ising Lattice Gas
How does this differ from the
equilibrium
case?
Over/Review
Dynamics violates detailed balance.
System goes into
non

equilibrium
steady state:
non

trivial
particle current
and
energy through

flux.
In most cases, this is not easy to see!
In this case, it has to do with the PBC.
Irreversible K loops are
global
!
Driven Ising Lattice Gas
How does this differ from the
equilibrium
case?
Over/Review
Dynamics violates detailed balance.
System goes into
non

equilibrium
steady state
Stationary distribution,
P
*
(
C
)
, exists…
...but very different from Boltzmann.
A simple, exactly solvable, example:
half filled, 2
4 lattice
Over/Review
Largest P
normalized
to unity
Driven Ising Lattice Gas
How does this differ from the
equilibrium
case?
Over/Review
Dynamics violates detailed balance.
System goes into
non

equilibrium
steady state
Stationary distribution,
P
*
(
C
)
, exists…
…………….
...but very different from Boltzmann.
Usual fluctuation

dissipation theorem violated.
Even simpler example: 2
3 (
E=
)
•
“specific heat”
–
U
has a
peak
at
n3
/
4J
•
energy fluctuations
U
2
monotonic
in
Driven Ising Lattice Gas
How does this differ from the
equilibrium
case?
Dynamics violates detailed balance.
System goes into
non

equilibrium
steady state
Stationary distribution,
P
*
(
C
)
, exists…
…………….
...but very different from Boltzmann.
Usual fluctuation

dissipation theorem violated.
The many
surprises
they bring!!
Over/Review
Driven Ising Lattice Gas
The surprises they bring!!
•
breakdown of well founded intuition
for example, consider phase diagram:
T
E
ordered
disordered
Lenz

Ising,
Onsager
KLS
Over/Review
My first guess…
…
just go into co

moving frame!
T
c
goes up!!
0
1
2
E
T
What’s your bet?
Guesses based on energy

entropy intuition.
Over/Review
0
1
2
3
E
T
Typical configurations
1.1
T
c
1.1
T
c
2.2
T
c
Over/Review
Drive
induces
ORDER
in the system!
E
along one axis
0
1
2
E
T
Worse … details depend on microscopics:
These possible if
E
has
components along
all
axes
Over/Review
Yet…
qualitative behaviour is the
same for DC drive,
AC
, or
random drives
!!
Driven Ising Lattice Gas
The surprises they bring!!
•
breakdown of well founded intuition
•
negative responses
(
E
“adds” noise ~ higher
T ; but …
)
‘‘
Freezing by heating
’’
H. E. Stanley
,
Nature
404
,
718
(2000)
“
Getting more by pushing less
”
RKPZ, E.L. Praestgaard, and O.G. Mouritsen
American Journal of Physics
70,
384 (2002)
Over/Review
Driven Ising Lattice Gas
The surprises they bring!!
•
breakdown of well founded intuition
•
negative responses
•
generic long range correlations:
r
–
d
(
all T
>
T
c
)
–
related to
generic discontinuity singularity in
S
(
k
)
–
related to
number fluctuations in a window is
………………..
geometry/orientation dependent
–
traced to
generic violation of FDT
Over/Review
Driven Ising Lattice Gas
The surprises they bring!!
•
breakdown of well founded intuition
•
negative responses
•
generic long range correlations:
r
–
d
(
all T
not near
T
c
)
•
anisotropic scaling & new universality classes, e.g.,
d
c
= 5
[3]
for uniformly
[randomly]
driven case
K.t. Leung and
J.L. Cardy
(1986)
H.K. Janssen and
B. Schmittmann
(1986)
B. Schmittmann
and RKPZ (1991)
B. Schmittmann
(1993)
Over/Review
Fixed point
violates
detailed
balance: “truly NEq”
Mostly confirmed by
simulations, though a
controversy lingers!
J. Marro,
P. Garrido
, …
Fixed point
satisfies
detailed balance:
Equilibrium “restored under RG”
Driven Ising Lattice Gas
The surprises they bring!!
•
breakdown of well founded intuition
•
negative responses
•
generic long range correlations:
r
–
d
(
all T
not near
T
c
)
•
new universality classes
•
anomalous interfacial properties, e.g.,
G
(
q
) ~
q
–
0.67
[1/(
q
+
c
)]
for uniformly [randomly] driven case
interfacial widths do not diverge with
L
!
K.t. Leung and RKPZ (1993)
Over/Review
meaning/existence of surface tension unclear!
1/q
2
Driven Ising Lattice Gas
The surprises they bring!!
•
breakdown of well founded intuition
•
negative responses
•
generic long range correlations:
r
–
d
(
all T
not near
T
c
)
•
new universality classes
•
anomalous interfacial properties
•
new ordered states if PBC
SPBC, OBC
Over/Review
reminder:
Interesting, new, but
understandable, phenomena
Over/Review
DILG with
Shifted PBC
J.L Valles, K.

t. Leung, RKPZ (1989)
shift = 5
5
100x100
T
= 0.8
E
=
∞
20
shift = 20
“similar” to
equilibrium Ising
SINGLE strip,
multiple winding
meaning/existence of surface tension unclear!
Over/Review
DILG with
Shifted PBC
T=0.7 72x36 shift = 6
M.J. Anderson, PhD thesis
Virginia Tech (1998)
Over/Review
DILG with
Open BC
D. Boal,
B. Schmittmann
, RKPZ (1991)
100x100
T
= 0.7
E
= 2
J
Fill first row
Empty last row
“ICICLES”
instead of strips
100x
200
How many icicles if system
is
really
long and thin?
Driven Ising Lattice Gas
The surprises they bring!!
•
breakdown of well founded intuition
•
negative responses
•
generic long range correlations:
r
–
d
(
all T
not near
T
c
)
•
new universality classes
•
anomalous interfacial properties
•
new ordered states if PBC
SPBC, OBC
•
complex phase separation dynamics
Over/Review
Over/Review
Coarsening in DILG
F.J. Alexander. C.A. Laberge,
J.L. Lebowitz
, RKPZ (1996)
T
= 0.6
E
= 0.7
J
128x256
in
512x1024
t
= 1K MCS
t
= 5K MCS
t
= 10K MCS
“Inverted” icicles, or
“Toll plaza effect”…
… but, modified
Cahn

Hilliard eqn.
leads to “icicles”!
•
no simple dynamic scaling
•
transverse and longitudinal
exponents differ
can modify rules of DILG to get icicles
cannot modify Cahn

Hilliard to get toll plazas
Driven Ising Lattice Gas
The surprises they bring!!
•
breakdown of well founded intuition
•
…
need new intuition/paradigm
…
One way forward is
to study
many other, similar systems
Over/Review
How about if we look at
even simpler versions of KLS?
How about if we follow Ising?
and consider
d
= 1 systems?
Driven Ising Lattice Gas
The surprises continue…
•
E
= 0
J
≠
0
d
= 1,2
(Lenz

Ising, Onsager, Lee

Yang, …)
•
E
> 0
J
>
0
d
= 2
KLS
•
E
> 0
J
>
0
d
= 1
–
lose anisotropy
(no SPBC)
–
stationary distribution still unknown
–
no ordered state at low
T
for PBC
–
non

trivial states for OBC
Over/Review
Driven Ising Lattice Gas
The surprises continue…
•
E
= 0
J
≠
0
d
= 1,2
(Lenz

Ising, Onsager, Lee

Yang, …)
•
E
> 0
J
>
0
d
= 2
KLS
•
E
> 0
J
=
0
d
= 1 A
symmetric
S
imple
E
xclusion
P
rocess
•
E=
∞
J
=
0
d
= 1 T
otally
ASEP
(Spitzer 1970)
–
for PBC,
P
*
trivial, but dynamics non

trivial
(
Spohn
,…)
–
for OBC,
P
*
non

trivial
(
Derrida
, Mukamel, Sch
ü
tz,…)
–
…boundary induced phases
(
Krug
,…)
Over/Review
(G. Schütz,…,
H. Widom
)
Driven Ising Lattice Gas
The surprises continue…
•
E
= 0
J
≠
0
d
= 1,2
(Lenz

Ising, Onsager, Lee

Yang, …)
•
E
> 0
J
>
0
d
= 2
KLS
•
E
> 0
J
=
0
d
= 1 A
symmetric
S
imple
E
xclusion
P
rocess
•
E=
∞
J
=
0
d
= 1 T
otally
ASEP
(Spitzer 1970)
–
for PBC,
P
*
trivial, but dynamics non

trivial
(
Spohn
,…)
–
for OBC,
P
*
non

trivial
(1992:
Derrida
, Mukamel, Sch
ü
tz,…)
–
…boundary induced phases
(1991:
Krug
,…)
Over/Review
d
= 1 DILG
•
HUGE body of literature on ASEP and
TASEP!!
•
Many exact results; much better understood
•
Nevertheless, there are still many surprises
•
Topic for a whole conference … not just the
next 5 minutes!
Other Driven Systems
•
Various drives:
–
AC or random
E
field
(more accessible experimentally)
–
Two
(or more)
temperatures
(as in cooking)
–
Open boundaries
(as in real wires)
–
Mixture of Glauber/Kawasaki dynamics
(e.g., bio

motors)
Outlook
What can we look forward to?
Other Driven Systems
•
Various drives
•
Multi

species:
–
Two species (e.g., for ionic conductors, bio

motors,…)
Baseline Study: driven in opposite directions, with “no” interactions
“American football, Barber poles, and Clouds”
–
Pink model
(with 10 or more species)
for bio

membranes
Outlook
Other Driven Systems
•
Various drives
•
Multi

species
•
Anisotropic interactions and jump rates
–
Layered compounds
–
Lamella amphiphilic structures.
Outlook
Other Driven Systems
•
Various drives
•
Multi

species
•
Anisotropic interactions and jump rates
•
Quenched impurities
Outlook
Take

home message:
Many

body systems, with very simple
constituents and rules

of

evolution
(especially “non

equilibrium” rules),
often display a rich variety of complex and
amazing
behavior.
Atoms and
E&M+gravity
Conclusions
•
Lots of exciting things
yet
to be discovered and understood:
–
in driven lattice gases
(just tip of iceberg here)
–
in other non

equilibrium steady states
(e.g., reaction diffusion)
–
in full dynamics
•
Many possible applications
(
biology, chemistry, …, sociology, economics,…
)
•
A range of methods (from simple MC to rigorous proofs)
Come, join the party, and…
Conclusions
…come, join the party!
Enter the password to open this PDF file:
File name:

File size:

Title:

Author:

Subject:

Keywords:

Creation Date:

Modification Date:

Creator:

PDF Producer:

PDF Version:

Page Count:

Preparing document for printing…
0%
Comments 0
Log in to post a comment