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

1. 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

2. Journal of Statistical Physics, 34, 497 (1984) What’s KLS?and 25 years after?

3. 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

4. Outline • Over/Review – what did we learn? • Outlook – what else can we look forward to?

5. 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  .

6. P*(C) [E-H(C)] P*(C) exp[-H] T T E P*=? cartoon of equilibriumvs.non-quilibrium

7. 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?

8. 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 

9. 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!

10. 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 ?

11. 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…

12. 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 ]

13. 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.,

14. 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!

15. 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

16. 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.

17. 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

18. Over/Review Largest P normalized to unity

19. 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 

20. 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!!

21. 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

22. Guesses based on energy-entropy intuition. Over/Review What’s your bet? Tc goes up!! My first guess… … just go into co-moving frame!

23. 2.2 Tc 1.1 Tc Drive induces ORDERin the system! 1.1 Tc Over/Review Typical configurations

24. 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 !!

25. 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)

26. 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

27. 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”

28. 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!

29. 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

30. 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!

31. Over/Review DILG with Shifted PBC T=0.7 72x36 shift = 6 M.J. Anderson, PhD thesis Virginia Tech (1998)

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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)

38. 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,…)

39. 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!

40. 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) 

41. 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 

42. Outlook Other Driven Systems • Various drives • Multi-species • Anisotropic interactions and jump rates • Layered compounds • Lamella amphiphilic structures. 

43. Outlook Other Driven Systems • Various drives • Multi-species • Anisotropic interactions and jump rates • Quenched impurities 

44. 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

45. 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…

46. Conclusions Let's celebrate Non-equilibrium Stat Mech …come, join the party!

47. Thank you... Joel for the last 100 SMM's Looking forward to the 150th!