Twenty five years after KLS

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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


t
)



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!