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

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Twenty five years after KLS a celebration of
non-equilibrium statistical mechanics
SMM100, Rutgers, December 2008
R. K. P. Zia Physics Department, Virginia Tech,
Blacksburg, Virginia, USA
Many here at SMM100
supported in part by
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Whats KLS ? and 25 years after?
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Outline

• Overview/Review (devoted to students and
  newcomers)
• Whats 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

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Outline

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

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Whats the context of KLS? Why study these
systems?
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
?t P(C , t) S R(C ? ? C) P(C ?, t) ? R(C ?
C ?) P(C , t) C ?
.

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cartoon of equilibrium vs. non-quilibrium
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Whats the context of KLS? Why study these
systems?
Over/Review

• Non-equilibrium Statistical Mechanics
• Fundamental issue
• Systems in non-equilibrium steady states
  cannot be understood in the Boltzmann-Gibbs
  framework.
• Whats the new game in town?

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Whats the context of KLS? Why study these
systems?
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
• ?

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Whats the context of KLS? Why study these
systems?
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!
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Whats the original KLS?
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 ?

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Ising Lattice Gas
Over/Review

• Take a well-known equilibrium system

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Ising Lattice Gas
Over/Review

• Take a well-known equilibrium system,
• evolving with a simple dynamics

going from C to C ? with rates R(C ? C ?) that
obey detailed balance
R(C ? C ?) / R(C ? ? C) expH(C ?) ?
H(C)/kT
so that, in long times, the system is described
by the Boltzmann distribution P(C) ? exp ?
H(C) / kT
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Ising Lattice Gas
Over/Review

• Take a well-known equilibrium system,
• evolving with a simple dynamics

R(C ? C ?) / R(C ? ? C) expH(C ?) ?
H(C)/kT
one favorite R is Metropolis, e.g.,
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Driven Ising Lattice Gas
Over/Review

• Take a well-known equilibrium system
• 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 J0 case

• Cant have PBC !!
• Get to equilibrium with
  extra potential term NOTHING new!

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Driven Ising Lattice Gas
Over/Review

• Take a well-known equilibrium system
• 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 gt 2J
E tends to break bonds T tends to satisfy bonds
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Driven Ising Lattice Gas
Over/Review

• How does this differ from the equilibrium case?

• Dynamics violates detailed balance.
• System goes into non-equilibrium steady state
• non-trivial particle current and
• energy through-flux.

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Driven Ising Lattice Gas
Over/Review

• 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.

A simple, exactly solvable, example half
filled, 2?4 lattice
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Over/Review
Largest P normalized to unity
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Driven Ising Lattice Gas
Over/Review

• 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.

• Even simpler example 2?3 (E?)
• specific heat ???U? has a peak at ?n3 /4J
• energy fluctuations ??U2? monotonic in ?

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Driven Ising Lattice Gas
Over/Review

• 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!!

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Driven Ising Lattice Gas
Over/Review

• The surprises they bring!!
• breakdown of well founded intuition
• for example, consider phase
  diagram

KLS
Lenz-Ising, Onsager
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Whats your bet?
Over/Review
Tc goes up!!
My first guess just go into co-moving frame!
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Typical configurations
Over/Review
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Worse details depend on microscopics
Over/Review
E along one axis
Yet qualitative behaviour is the same for DC
drive, AC, or random drives !!
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Driven Ising Lattice Gas
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 Physics 70, 384
(2002)
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Driven Ising Lattice Gas
Over/Review

• The surprises they bring!!
• breakdown of well founded intuition
• negative responses
• generic long range correlations r d (all T gt
  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

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Driven Ising Lattice Gas
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
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Driven Ising Lattice Gas
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/(qc) 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!
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Driven Ising Lattice Gas
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
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Over/Review
DILG with Shifted PBC J.L Valles, K.-t. Leung,
RKPZ (1989)
100x100 T 0.8 E 8
similar to equilibrium Ising
SINGLE strip, multiple winding
meaning/existence of surface tension unclear!
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Over/Review
DILG with Shifted PBC T0.7 72x36 shift
6 M.J. Anderson, PhD thesis Virginia Tech (1998)
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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
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Driven Ising Lattice Gas
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

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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 5K MCS
t 10K MCS
T 0.6 E 0.7J
can modify rules of DILG to get icicles cannot
modify Cahn-Hilliard to get toll plazas
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Driven Ising Lattice Gas
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 study many other, similar
systems
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Driven Ising Lattice Gas
Over/Review

• The surprises continue
• E 0 J ? 0 d 1,2 (Lenz-Ising, Onsager,
  Lee-Yang, )
• E gt 0 J gt 0 d 2 KLS
• E gt 0 J gt 0 d 1
• lose anisotropy (no SPBC)
• stationary distribution still unknown
• no ordered state at low T for PBC
• non-trivial states for OBC

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Driven Ising Lattice Gas
Over/Review

• The surprises continue
• E 0 J ? 0 d 1,2 (Lenz-Ising, Onsager,
  Lee-Yang, )
• E gt 0 J gt 0 d 2 KLS
• E gt 0 J 0 d 1 Asymmetric Simple
  Exclusion Process
• E8 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)
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Driven Ising Lattice Gas
Over/Review

• The surprises continue
• E 0 J ? 0 d 1,2 (Lenz-Ising, Onsager,
  Lee-Yang, )
• E gt 0 J gt 0 d 2 KLS
• E gt 0 J 0 d 1 Asymmetric Simple
  Exclusion Process
• E8 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,)

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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!

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Other Driven Systems
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)
• ?

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Other Driven Systems
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
• ?

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Other Driven Systems
Outlook

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

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Other Driven Systems
Outlook

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

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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 EMgravity
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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
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Conclusions
Let's celebrate Non-equilibrium Stat Mech
come, join the party!
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Thank you... Joel for the last 100 SMM's
Looking forward to the 150th!