1. Phase diagram and density large deviation of a nonconserving ABC model Or Cohen and David Mukamel International Workshop on Applied Probability, Jerusalem, 2012

2. T2T2 Driven diffusive systems T1T1 Boundary driven Bulk driven Studied via simplified

3. Motivation What is the effect of bulk nonconserving dynamics on bulk driven system ? pq w-w- w+w+ Can it be inferred from the conserving steady state properties ?

4. Outline 1.ABC model 2.Phase diagram under conserving dynamics 3.Slow nonconserving dynamic 4.Phase diagram and inequivalence of ensembles 5.Conclusions

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

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

7. ABC model Evans, Kafri, Koduvely & Mukamel - Phys. Rev. Lett. 1998 t x A BC

8. Equal densities For equal densities N A =N B =N C AAAAABBAABBBCBBCCCCCC BB BBB BB Potential induced by other species

9. Weak asymmetry Clincy, Derrida & Evans - Phys. Rev. E 2003 Coarse graining

10. Weak asymmetry Clincy, Derrida & Evans - Phys. Rev. E 2003 Weakly asymmetric thermodynamic limit Coarse graining

11. Phase transition Clincy, Derrida & Evans - Phys. Rev. E 2003 For low β is minimum of F[ρ α ]

12. Phase transition 2 nd order phase transition at Clincy, Derrida & Evans - Phys. Rev. E 2003 For low β is minimum of F[ρ α ]

13. Nonequal densities ? AAAAABBAABBBCBBCCC No detailed balance (Kolmogorov criterion violated) Steady state current Stationary measure unknown

14. Nonequal densities ? Hydrodynamics equations : Drift Diffusion AAAAABBAABBBCBBCCC No detailed balance (Kolmogorov criterion violated) Steady state current Stationary measure unknown

15. Nonequal densities ? Hydrodynamics equations : Drift Diffusion Full steady-state solution or Expansion around homogenous AAAAABBAABBBCBBCCC No detailed balance (Kolmogorov criterion violated) Steady state current Stationary measure unknown

16. Nonconserving ABC model 0X X0 X=A,B,C 1 1 AB C0 Lederhendler & Mukamel - Phys. Rev. Lett. 2010 AB BA BC CB CA AC q 1 q 1 q 1 1 2 12 Conserving model (canonical ensemble) +

17. Nonconserving ABC model 0X X0 X=A,B,C 1 1 AB C0 Lederhendler & Mukamel - Phys. Rev. Lett. 2010 AB BA BC CB CA AC q 1 q 1 q 1 ABC 000 pe -3βμ p 1 2 3 12 123 Conserving model (canonical ensemble) Nonconserving model (grand canonical ensemble) + ++

18. Nonequal densities Hydrodynamics equations : Drift Diffusion Deposition Evaporation

19. Nonequal densities ABC 000 pe -3βμ p 0X X0 1 1 AB BA e -β/L 1 BC CB e -β/L 1 CA AC e -β/L 1 X= A,B,C Hydrodynamics equations : Drift Diffusion Deposition Evaporation

20. Conserving steady-state Conserving model Steady-state profile Drift Diffusion Nonequal densities : Cohen & Mukamel - Phys. Rev. Lett. 2012 Equal densities : Ayyer et al. - J. Stat. Phys. 2009 Nonequal densities : Cohen & Mukamel - Phys. Rev. Lett. 2012 Equal densities : Ayyer et al. - J. Stat. Phys. 2009

21. Nonconserving steady-state Drift Diffusion Deposition Evaporation

22. Nonconserving steady-state Nonconserving model with slow nonconserving dynamics Drift + Diffusion Deposition + Evaporation

23. Dynamics of particle density

24. After time τ 1 :

25. Dynamics of particle density After time τ 2 :

26. Dynamics of particle density After time τ 1 :

27. Dynamics of particle density After time τ 2 :

28. Dynamics of particle density After time τ 1 :

29. Large deviation function of r After time τ 1 :

30. Large deviation function of r = 1D - Random walk in a potential

31. Large deviation function of r = 1D - Random walk in a potential ABC 000 pe -3βμ p Large deviation function

32. Large deviation function of r High µ

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

34. Inequivalence of ensembles Conserving = Canonical Nonconserving = Grand canonical 2 nd order transition ordered 1 st order transitiontricritical point disordered ordered disordered For N A =N B ≠N C :

35. Conclusions 1.ABC model 2.Slow nonconserving dynamics 3.Inequivalence of ensemble, and links to long range interacting systems. 4.Relevance to other driven diffusive systems.