Twenty five years after kls a celebration of non equilibrium statistical mechanics
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SMM100, Rutgers, December 2008. B. Schmittmann. Twenty five years after KLS a celebration of non-equilibrium statistical mechanics . R. K. P. Zia Physics Department, Virginia Tech, Blacksburg, Virginia, USA. Many here at SMM100. supported in part by.

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Twenty five years after kls a celebration of non equilibrium statistical mechanics

SMM100, Rutgers, December 2008

B. Schmittmann

Twenty five years after KLSa celebration of non-equilibrium statistical mechanics

R. K. P. Zia

Physics Department, Virginia Tech,

Blacksburg, Virginia, USA

Many here at SMM100

supported in part by


Twenty five years after kls a celebration of non equilibrium statistical mechanics

Journal of Statistical Physics, 34, 497 (1984)

What’s KLS?and 25 years after?


Outline

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


Outline1

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

Over/Review

What’s the context of KLS? Why study these systems?

  • Non-equilibrium Statistical Mechanics

    • detailed balance 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

P*,P*

∂tP(C , t)= Σ { R(C C) P(C , t)  R(C  C ) P(C , t) }

C  .


Cartoon of equilibrium vs non quilibrium

P*(C) [E-H(C)] P*(C) exp[-H]

T

T

E

P*=?

cartoon of equilibriumvs.non-quilibrium


What s the context of kls why study these systems1

Over/Review

What’s the context of KLS? Why study these systems?

  • Non-equilibrium Statistical Mechanics

  • Fundamental issue:

    Systems innon-equilibrium steady states cannot be understood in the Boltzmann-Gibbs framework.

    What’s the new game in town?


What s the context of kls why study these systems2

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


What s the context of kls why study these systems3

Over/Review

What’s the context of KLS?Why study these systems?

Perhaps we can gain some insight through

SIMPLE systems, like the Ising model

  • Non-equilibrium Statistical Mechanics

  • Physics of many systems “all around us”

But, real life is VERY COMPLEX!


What s the original kls

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 ?


Ising lattice gas

e.g., Ising lattice gas (2-d, Onsager)

C : { n(x,y) } with n = 0,1

H(C) =  J x,an(x) n(x+a)

+ periodic boundary condtions (PBC)

Over/Review

Ising Lattice Gas

  • Take a well-known equilibriumsystem…


Ising lattice gas1

Over/Review

Ising Lattice Gas

  • Take a well-known equilibriumsystem,

  • evolving with a simple dynamics…

…going fromC toC  with ratesR(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 gas2

Go with rate e2J/kT

Just go!

Over/Review

Ising Lattice Gas

  • Take a well-known equilibriumsystem,

  • evolving with a simple dynamics…

R(C C ) / R(C  C) =exp[{H(C )  H(C)}/kT ]

…one favorite R is Metropolis, e.g.,


Driven ising lattice gas

Go with rate emga/kT

g

Just go!

Over/Review

Driven Ising Lattice Gas

  • Take a well-known equilibriumsystem

  • Drive it far from thermal equilibrium…..... (by some additional external force, so particles suffer biased diffusion.)

e.g., effects of gravity (uniform field)

a - lattice spacing J=0 case

  • Can’t have PBC !!

  • Get to equilibrium with ……………extra potential term… NOTHING new!


Driven ising lattice gas1

Go with rate e(E-2J)/kT

E

Just go!

T

E

Over/Review

Driven Ising Lattice Gas

  • Take a well-known equilibriumsystem

  • Drive it far from thermal equilibrium…..... (by some additional external force, so particles suffer biased diffusion.)

  • PBC possible with “electric” field, E(non-potential, rely on tB)

LOTS of

surprises!

unit “charge” and a with E > 2J

E tends to break bonds

T tends to satisfy bonds


Driven ising lattice gas2

In most cases, this is not easy to see!

In this case, it has to do with the PBC.

Irreversible K loops are global!

Over/Review

Driven Ising Lattice Gas

How does this differ from the equilibriumcase?

  • Dynamics violates detailed balance.

  • System goes into non-equilibrium steady state:

  • non-trivial particle current and

  • energy through-flux.


Driven ising lattice gas3

Over/Review

Driven Ising Lattice Gas

How does this differ from the equilibriumcase?

  • 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


Twenty five years after kls a celebration of non equilibrium statistical mechanics

Over/Review

Largest P normalized to unity


Driven ising lattice gas4

Over/Review

Driven Ising Lattice Gas

How does this differ from the equilibriumcase?

  • 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 U2 monotonic in 


Driven ising lattice gas5

Over/Review

Driven Ising Lattice Gas

How does this differ from the equilibriumcase?

  • 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 manysurprisesthey bring!!


Driven ising lattice gas6

T

disordered

ordered

E

Over/Review

Driven Ising Lattice Gas

The surprises they bring!!

  • breakdown of well founded intuition

    for example, consider phase diagram:

KLS

Lenz-Ising,

Onsager


What s your bet

Guesses based on energy-entropy intuition.

Over/Review

What’s your bet?

Tc goes up!!

My first guess…

… just go into co-moving frame!


Typical configurations

2.2 Tc

1.1 Tc

Drive induces ORDERin the system!

1.1 Tc

Over/Review

Typical configurations


Worse details depend on microscopics

These possible if E has components along all axes

Over/Review

Worse … details depend on microscopics:

E along one axis

Yet…

qualitative behaviour is the same for DC drive, AC, or random drives !!


Driven ising lattice gas7

Over/Review

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 Physics70, 384 (2002)


Driven ising lattice gas8

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 > Tc )

    • 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


Driven ising lattice gas9

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 Tc )

  • anisotropic scaling & new universality classes, e.g.,

    dc = 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)

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 gas10

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 Tc )

  • 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 !

1/q2

K.t. Leung and RKPZ (1993)

meaning/existence of surface tension unclear!


Driven ising lattice gas11

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 Tc )

  • new universality classes

  • anomalous interfacial properties

  • new ordered states if PBC  SPBC, OBC

reminder: Interesting, new, but understandable, phenomena


Twenty five years after kls a celebration of non equilibrium statistical mechanics

5

shift = 5

shift = 20

20

Over/Review

DILG with Shifted PBC

J.L Valles, K.-t. Leung, RKPZ (1989)

100x100 T = 0.8 E = ∞

“similar” to equilibrium Ising

SINGLE strip, multiple winding

meaning/existence of surface tension unclear!


Twenty five years after kls a celebration of non equilibrium statistical mechanics

Over/Review

DILG with Shifted PBC

T=0.7 72x36 shift = 6

M.J. Anderson, PhD thesis Virginia Tech (1998)


Twenty five years after kls a celebration of non equilibrium statistical mechanics

Over/Review

DILG with Open BC

D. Boal, B. Schmittmann, RKPZ (1991)

100x200

100x100 T = 0.7 E = 2J

Fill first row

“ICICLES”

instead of strips

How many icicles if system is really long and thin?

Empty last row


Driven ising lattice gas12

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 Tc )

  • new universality classes

  • anomalous interfacial properties

  • new ordered states if PBC  SPBC, OBC

  • complex phase separation dynamics


Twenty five years after kls a celebration of non equilibrium statistical mechanics

Over/Review

Coarsening in DILG

F.J. Alexander. C.A. Laberge, J.L. Lebowitz, RKPZ (1996)

“Inverted” icicles, or “Toll plaza effect”…

… but, modified Cahn-Hilliard eqn. leads to “icicles”!

  • no simple dynamic scaling

  • transverse and longitudinal exponents differ

128x256

in

512x1024

t = 1K MCS

t = 10K MCS

t = 5K MCS

T = 0.6 E = 0.7J

can modify rules of DILG to get icicles

cannot modify Cahn-Hilliard to get toll plazas


Driven ising lattice gas13

Over/Review

Driven Ising Lattice Gas

The surprises they bring!!

  • breakdown of well founded intuition

  • …need new intuition/paradigm…

How about if we look at

even simpler versions of KLS?

How about if we follow Ising? and consider d = 1 systems?

One way forward is

to studymany other, similar systems


Driven ising lattice gas14

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

    • lose anisotropy (no SPBC)

    • stationary distribution still unknown

    • no ordered state at low T for PBC

    • non-trivial states for OBC


Driven ising lattice gas15

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 Asymmetric Simple Exclusion Process

  • E=∞ J =0 d = 1 Totally ASEP (Spitzer 1970)

    • for PBC, P* trivial, but dynamics non-trivial (Spohn,…)

    • for OBC, P* non-trivial (Derrida, Mukamel, Schütz,…)

    • …boundary induced phases (Krug,…)

(G. Schütz,…, H. Widom)


Driven ising lattice gas16

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 Asymmetric Simple Exclusion Process

  • E=∞ J =0 d = 1 Totally ASEP (Spitzer 1970)

    • for PBC, P* trivial, but dynamics non-trivial (Spohn,…)

    • for OBC, P* non-trivial (1992: Derrida, Mukamel, Schütz,…)

    • …boundary induced phases (1991: Krug,…)


D 1 dilg

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

Outlook

What can we look forward to?

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)


Other driven systems1

Outlook

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


Other driven systems2

Outlook

Other Driven Systems

  • Various drives

  • Multi-species

  • Anisotropic interactions and jump rates

    • Layered compounds

    • Lamella amphiphilic structures.


Other driven systems3

Outlook

Other Driven Systems

  • Various drives

  • Multi-species

  • Anisotropic interactions and jump rates

  • Quenched impurities


Take home message

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


Twenty five years after kls a celebration of non equilibrium statistical mechanics

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…


Twenty five years after kls a celebration of non equilibrium statistical mechanics

Conclusions

Let's celebrate

Non-equilibrium Stat Mech

…come, join the party!


Twenty five years after kls a celebration of non equilibrium statistical mechanics

Thank you... Joel

for the last 100 SMM's

Looking forward to the 150th!


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