Density large-deviations of nonconserving driven models

1. Density large-deviations of nonconserving driven models Or Cohen and David Mukamel 5th KIAS Conference on Statistical Physics, Seoul, Korea, July 2012

2. Ensemble theory out of equilibrium ? Equilibrium T , µ

3. Ensemble theory out of equilibrium ? Driven diffusive systems Equilibrium conserving steady state T , µ q p w- w+

4. Ensemble theory out of equilibrium ? Driven diffusive systems Equilibrium T , µ q p w- w+ conserving steady state Can we infer about the nonconserving system from the steady state properties of the conserving system ?

5. Ensemble theory out of equilibrium ? Driven diffusive systems Equilibrium T , µ q p w- w+

6. Generic driven diffusive model wLC wRC L sites w-NC w+NC conserving (sum over η’ with same N) nonconserving (sum over η’ with N’≠N)

7. Generic driven diffusive model wLC wRC L sites w-NC w+NC conserving (sum over η’ with same N) nonconserving (sum over η’ with N’≠N) Guess a steady state of the form :

8. Generic driven diffusive model wLC wRC L sites w-NC w+NC conserving (sum over η’ with same N) nonconserving (sum over η’ with N’≠N) Guess a steady state of the form : In many cases : It is consistent if :

9. Slow nonconserving dynamics To leading order in ε we obtain

10. Slow nonconserving dynamics To leading order in ε we obtain = 1D - Random walk in a potential

11. Slow nonconserving dynamics To leading order in ε we obtain = 1D - Random walk in a potential Steady state solution :

12. Outline Limit of slow nonconserving Example of the ABC model Nonequilibrium chemical potential (dynamics dependent !) Conclusions

13. ABC model B C A Ring of size L Dynamics : q AB BA 1 q BC CB 1 q CA AC 1 Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998

14. ABC model B C A Ring of size L Dynamics : q AB BA 1 q BC CB 1 q CA AC 1 ABBCACCBACABACB q=1 q<1 AAAAABBBBBCCCCC Evans, Kafri , Koduvely & Mukamel - Phys. Rev. Lett. 1998

15. ABC model B C A x t

16. Nonconserving ABC model 1 2 q 1 ABBA 0X X0 X=A,B,C 1 1 q BCCB 1 fixed q CAAC 1 Conserving model (canonical ensemble) + 1 2 A B C 0 Lederhendler & Mukamel - Phys. Rev. Lett. 2010

17. Conserving model Weakly asymmetric thermodynamic limit Clincy, Derrida & Evans - Phys. Rev. E 2003

18. Conserving model Weakly asymmetric thermodynamic limit Density profile For low β’s known 2nd order Clincy, Derrida & Evans - Phys. Rev. E 2003

19. Conserving model Weakly asymmetric thermodynamic limit Density profile For low β’s known 2nd order Clincy, Derrida & Evans - Phys. Rev. E 2003

20. Nonconserving ABC model 1 2 q 1 ABBA 0X X0 X=A,B,C 1 1 q BCCB 3 pe-3βμ 1 ABC 000 q CAAC p 1 Conserving model (canonical ensemble) + 1 2 A B Nonconserving model (grand canonical ensemble) + + 1 2 3 C 0 Lederhendler & Mukamel - Phys. Rev. Lett. 2010

21. Slow nonconserving model Slow nonconserving limit pe-3βμ ABC 000 p

22. Slow nonconserving model Slow nonconserving limit pe-3βμ ABC 000 p saddle point approx.

23. Slow nonconserving model pe-3βμ ABC 000 p

24. Slow nonconserving model pe-3βμ ABC 000 p This is similar to equilibrium :

25. Large deviation function of r High µ Low µ First order phase transition (only in the nonconserving model)

26. Inequivalence of ensembles For NA=NB≠NC : Conserving = Canonical Nonconserving = Grand canonical disordered disordered ordered ordered 1st order transition 2nd order transition tricritical point

27. Why is µS(N) the chemical potential ? N1 N2 SLOW

28. Why is µS(N) the chemical potential ? N1 N2 SLOW SLOW Gauge measures

29. Conclusions Nonequlibrium ‘grand canonical ensemble’ - Slow nonconserving dynamics Example to ABC model 1st order phase transition for nonmonotoneousµs(r) and inequivalence of ensembles. Nonequilibrium chemical potential ( dynamics dependent ! ) Thank you ! Any questions ? Cohen & Mukamel- PRL 108, 060602 (2012)