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Effective long-range interactions in driven systems David Mukamel.

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1 Effective long-range interactions in driven systems David Mukamel

2 Systems with long range interactions in d dimensions two-body interaction for σ<0 the energy is not extensive -strong long-range interactions

3 self gravitating systems (1/r) σ=-2 ferromagnets σ=0 2d vortices log(r) σ=-2

4 These systems are non-additive AB As a result, many of the common properties of typical systems with short range interactions are not shared by these systems.

5 Driven systems T1T1 T2T2 T 1 >T 2 E heat current charge current Local and stochastic dynamics No detailed balance (non-vanishing current) What is the nature of the steady state?

6 drive in conserving systems result in many cases in long range correlations What can be learned from systems with long-range interactions on steady state properties of driven systems?

7 Free Energy: since the entropy may be neglected in the thermodynamic limit. In finite systems, although E>>S, if T is high enough E may be comparable to TS, and the full free energy need to be considered. (Self gravitating systems, e.g. globular clusters) Systems with long range interactions

8 Globular clusters are gravitationally bound concentrations of approximately ten thousand to one million stars, spread over a volume of several tens to about 200 light years in diameter.

9 Thus although and E may be comparable to TS For the M2 cluster N=150,000 stars R= 175 light years Kg

10 One may implement the large T limit by rescaling the Hamiltonian

11 Although the canonical thermodynamic functions (free energy, entropy etc) are extensive, the system is non-additive + _ For example, consider the Ising model:

12 Features which result from non-additivity Negative specific heat in microcanonical ensemble Inequivalence of microcanonical (MCE) and canonical (CE) ensembles Breaking of ergodicity in microcanonical ensemble Slow dynamics, diverging relaxation time Thermodynamics Dynamics Temperature discontinuity in MCE

13 Some general considerations Negative specific heat in microcanonical ensemble of non-additive systems. Antonov (1962); Lynden-Bell & Wood (1968); Thirring (1970), Thirring & Posch coexistence region in systems with short range interactions E 0 = xE 1 +(1-x)E 2 S 0 = xS 1 +(1-x)S 2 hence S is concave and the microcanonical specific heat is non-negative S

14 On the other hand in systems with long range interactions (non-additive), in the region E 1 <E<E 2 S E 0 = xE 1 +(1-x)E 2 S 0 xS 1 +(1-x)S 2 The entropy may thus follow the homogeneous system curve, the entropy is not concave. and the microcanonical specific heat becomes negative. compared with canonical ensemble where

15 Ising model with long and short range interactions. d=1 dimensional geometry, ferromagnetic long range interaction J>0 The model has been analyzed within the canonical ensemble Nagel (1970), Kardar (1983)

16 Canonical (T,K) phase diagram

17 Microcanonical analysis U = number of broken bonds in a configuration Number of microstates: U/2 (+) segments U/2 (-) segments Mukamel, Ruffo, Schreiber (2005); Barre, Mukamel, Ruffo (2001)

18 s=S/N, =E/N, m=M/N, u=U/N but Maximize to get

19 canonicalmicrocanonical The two phase diagrams differ in the 1st order region of the canonical diagram Ruffo, Schreiber, Mukamel (2005)

20 s m discontinuous transition: In a 1st order transition there is a discontinuity in T, and thus there is a T region which is not accessible.

21 S E

22 In general it is expected that whenever the canonical transition is first order the microcanonical and canonical ensembles differ from each other. S

23 Dynamics Systems with long range interactions exhibit slow relaxation processes. This may result in quasi-stationary states (long lived non-equilibrium states whose relaxation time to the equilibrium state diverges with the system size). Non-additivity may facilitate breaking of ergodicity which could lead to trapping of systems in non- Equilibrium states.

24 Long range correlations in driven systems Conserved variables tend to produce long range correlations. Thermal equilibrium states are independent of the dynamics (e.g. Glauber and Kawasaki dynamics result in the same Boltzmann distribution) Non-equilibrium steady states depend on the dynamics (e.g. conserving or non-conserving) Conserving dynamics in driven, non-equilibrium systems may result in steady states with long range correlations even when the dynamics is local Can these correlations be viewed as resulting from effective long-range interactions? Driven systems

25 ABC model One dimensional driven model with stochastic local dynamics which results in phase separation (long range order) where the steady state can be expressed as a Boltzmann distribution of an effective energy with long-range interactions.

26 A BC ABC Model AB BA 1 q BC CB 1 q CA AC 1 q dynamics Evans,Kafri, Koduvely, Mukamel PRL 80, 425 (1998) A model with similar features was discussed by Lahiri, Ramaswamy PRL 79, 1150 (1997)

27 Simple argument: AB BA 1 q BC CB 1 q CA AC 1 q ACCCC CCCCA CBBBB BBBBC BAAAA AAAAB …AACBBBCCAAACBBBCCC… …AABBBCCCAAABBBCCCC… …AAAAABBBBBCCCCCCAA… fast rearrangement slow coarsening The model reaches a phase separated steady state

28 logarithmically slow coarsening …AAAAABBBBBCCCCCCAA… needs n>2 species to have phase separation strong phase separation: no fluctuation in the bulk; only at the boundaries. …AAAAAAAAAABBBBBBBBBBBBCCCCCCCCCCC… Phase separation takes place for any q (except q=1) Phase separation takes place for any density N, N, N ABC

29 ABC Special case The argument presented before is general, independent of densities. For the equal densities case the model has detailed balance for arbitrary q. We will demonstrate that for any microscopic configuration {X} One can define “energy” E({X}) such that the steady state Distribution is

30 AAAAAABBBBBBCCCCCE=0 ……AB….. ……BA….. E E+1 ……BC….. ……CB….. E E+1 ……CA….. ……AC….. E E+1 With this weight one has: =q=1

31 AAAAABBBBBCCCCC AAAABBBBBCCCCCA E E+N B -N C N B = N C Thus such “energy” can be defined only for N A =N B =N C This definition of “energy” is possible only for ABC

32 AABBBBCCCAAAAABBBCCCC The rates with which an A particle makes a full circle clockwise And counterclockwise are equal Hence no currents for any N. For the current of A particles satisfies The current is non-vanishing for finite N. It vanishes only in the limit. Thus no detailed balance in this case.

33 …AAAAAAAABBBABBBBBBCCCCCCCCCAA… The model exhibits strong phase separation The probability of a particle to be at a distance on the wrong side of the boundary is The width of the boundary layer is -1/lnq

34 ABC The “energy” E may be written as summation over modulo long range Local dynamics

35 Partition sum Excitations near a single interface: AAAAAAABBBBBB P(n)= degeneracy of the excitation with energy n P(0)=1 P(1)=1 P(2)=2 (2, 1+1) P(3)=3 (3, 2+1, 1+1+1) P(4)=5 (4, 3+1, 2+2, 2+1+1, 1+1+1+1) P(n)= no. of partitions of an integer n

36 Weakly asymmetric ABC model The model exhibits a phase transition at for the case of equal densities homogeneous inhomogeneous effective rescaled “energy” Clincy, Derrida, Evans, PRE 67, 066115 (2003)

37 Generalized ABC model add vacancies: A, B, C, 0 AB BA 1 q BC CB 1 q CA AC 1 q 0X 1 1 X=A,B,C Dynamics A. Lederhendler, D. Mukamel

38 grand-canonical dynamics AB BA 1 q BC CB 1 q CA AC 1 q 0X 1 1 ABC 000 1 qq X=A,B,C …A00ACBABCCA00AACBBB00000CCC… ABCABC 000

39 The dynamics is local for For N A =N B =N C : there is detailed balance with respect to

40 continuum version of the model

41 canonical grand canonical


43 , T=0.02, Grand canonical

44 , T=0.02, Canonical – 2 nd order transition at

45 , T=0.02, Grand canonical – 1 st order transition

46 Correlations for both solutions with

47 Summary Local stochastic dynamics may result in effective long- range interactions in driven systems. This is manifested in the existence of phase transitions in one dimensional driven models. Existence of effective long range interactions can be explicitly demonstrated in the ABC model. The model exhibits phase separation for any drive Phase separation is a result of effective long-range interactions generated by the local dynamics. Inequivalence of ensembles in the driven model.

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