- 46 Views
- Uploaded on
- Presentation posted in: General

Twenty five years after KLS a celebration of non-equilibrium statistical mechanics

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

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

Journal of Statistical Physics, 34, 497 (1984)

What’s KLS?and 25 years after?

- 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

- Over/Review – what did we learn?
- Outlook – what else can we look forward to?

Over/Review

- 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 .

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

T

T

E

P*=?

Over/Review

- 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?

Over/Review

- 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

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!

Over/Review

- 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 ?

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

- Take a well-known equilibriumsystem…

Over/Review

- 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 ]

Go with rate e2J/kT

Just go!

Over/Review

- 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.,

Go with rate emga/kT

g

Just go!

Over/Review

- 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!

Go with rate e(E-2J)/kT

E

Just go!

T

E

Over/Review

- 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

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

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.

Over/Review

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, 24 lattice

Over/Review

Largest P normalized to unity

Over/Review

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: 23 (E=)
- “specific heat” –U has a peak at n3 /4J
- energy fluctuations U2 monotonic in

Over/Review

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!!

T

disordered

ordered

E

Over/Review

The surprises they bring!!

- breakdown of well founded intuition
for example, consider phase diagram:

KLS

Lenz-Ising,

Onsager

Guesses based on energy-entropy intuition.

Over/Review

Tc goes up!!

My first guess…

… just go into co-moving frame!

2.2 Tc

1.1 Tc

Drive induces ORDERin the system!

1.1 Tc

Over/Review

These possible if E has components along all axes

Over/Review

E along one axis

Yet…

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

Over/Review

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)

Over/Review

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

Over/Review

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”

Over/Review

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!

Over/Review

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

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!

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)

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

Over/Review

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

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

Over/Review

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

Over/Review

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

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)

Over/Review

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,…)

- 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!

Outlook

What can we look forward to?

- 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

- 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

- Two species (e.g., for ionic conductors, bio-motors,…)

Outlook

- Various drives
- Multi-species
- Anisotropic interactions and jump rates
- Layered compounds
- Lamella amphiphilic structures.

Outlook

- Various drives
- Multi-species
- Anisotropic interactions and jump rates
- Quenched impurities

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

Let's celebrate

Non-equilibrium Stat Mech

…come, join the party!

Thank you... Joel

for the last 100 SMM's

Looking forward to the 150th!