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○ 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

Over/Review

∂t P(C , t) = Σ { R(C C) P(C , t) R(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 rates 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 e2J/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)

gJust go!

Go with rate emga/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 tB)

EJust 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, 24 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: 23 (E=)

• “specific heat” –U has a peak at n3 /4J

• energy fluctuations U2 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 GasThe 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!

Tc 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 Tc

1.1 Tc

2.2 Tc

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 GasThe 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 GasThe 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

Over/Review

Driven Ising Lattice GasThe 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)

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 GasThe 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 !

K.t. Leung and RKPZ (1993)

Over/Review

meaning/existence of surface tension unclear!

1/q2

Driven Ising Lattice GasThe 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

Over/Review

reminder: Interesting, new, but understandable, phenomena

Over/Review

DILG with Shifted PBCJ.L Valles, K.-t. Leung, RKPZ (1989)

shift = 55

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

100x100 T = 0.7 E = 2J

Fill first row

Empty last row

“ICICLES”

instead of strips

100x200

How many icicles if system is really long and thin?

Driven Ising Lattice GasThe 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

Over/Review

Over/Review

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

T = 0.6 E = 0.7J

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 iciclescannot modify Cahn-Hilliard to get toll plazas

Driven Ising Lattice GasThe 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 GasThe 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 GasThe 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,…)

Over/Review

(G. Schütz,…, H. Widom)

Driven Ising Lattice GasThe 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,…)

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!