Density large-deviations of nonconserving driven models

1. Density large-deviations of nonconserving driven models Or Cohen and David Mukamel STATPHYS 25 Conference, SNU, Seoul, Korea, July 2013

2. Grand canonical ensemble out of equilibrium ? Equilibrium T , µ

3. Grand canonical ensemble out of equilibrium ? Equilibrium T , µ conserving steady state Helmholtzfree energy

4. Grand canonical ensemble out of equilibrium ? Equilibrium T , µ conserving steady state Helmholtzfree energy q p Driven system conserving steady state

5. Grand canonical ensemble out of equilibrium ? Equilibrium T , µ conserving steady state Helmholtzfree energy q p e-βμ 1 Driven system conserving steady state

6. Grand canonical ensemble out of equilibrium ? Equilibrium T , µ conserving steady state Helmholtzfree energy q p e-βμ 1 Driven system conserving steady state Dynamics-dependentchemical potential of conserving system

7. General particle-nonconserving driven model wLC wRC L sites w-NC w+NC

8. General particle-nonconserving driven model wLC wRC L sites w-NC w+NC conserving (sum over η’ with same N) nonconserving (sum over η’ with N’≠N)

9. General particle-nonconserving driven 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 :

10. General particle-nonconserving driven 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 : For diffusive systems It is consistent if :

11. Slow nonconserving dynamics To leading order in L we obtain

12. Slow nonconserving dynamics = 1D - Random walk in a potential

13. Slow nonconserving dynamics = 1D - Random walk in a potential

14. Slow nonconserving dynamics w-NC w+NC

15. Outline Limit of slow nonconserving Example of the ABC model Corrections to the rate function using MFT Conclusions

16. ABC model B C A Ring of size L Dynamics : q AB BA 1 q BC CB 1 q CA AC 1 M R Evans, Y Kafri , H M Koduvely, D Mukamel - Phys. Rev. Lett. 80 425 (1998 )

17. 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 M R Evans, Y Kafri , H M Koduvely, D Mukamel - Phys. Rev. Lett. 80 425 (1998 )

18. ABC model B C A site index time

19. Conserving 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 A Lederhendler, D Mukamel - Phys. Rev. Lett. 105 105602 (2010) A Lederhendler, OC, D Mukamel- J. Stat. Mech. 11 11016 (2010)

20. Conserving ABC model Density profile Weakly asymmetric thermodynamic limit1 M Clincy, B Derrida, M R Evans - Phys. Rev. E 58 2764 (2003)

21. Conserving ABC model Density profile Weakly asymmetric thermodynamic limit1 known2 For low β’s 2nd order M Clincy, B Derrida, M R Evans - Phys. Rev. E 58 2764 (2003) OC, D Mukamel - J. Phys. A 44 415004 (2011)

22. Conserving ABC model Density profile Weakly asymmetric thermodynamic limit1 known2 For low β’s 2nd order Stationarymeasure3: M Clincy, B Derrida, M R Evans - Phys. Rev. E 58 2764 (2003) OC, D Mukamel - J. Phys. A 44 415004 (2011) T Bodineau, B Derrida - ComptesRendusPhysique 8 540 (2007)

23. 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 A Lederhendler, D Mukamel - Phys. Rev. Lett. 105 105602 (2010) A Lederhendler, OC, D Mukamel- J. Stat. Mech. 11 11016 (2010)

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

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

26. Slow nonconservingABC model pe-3βμ ABC 000 p

27. Slow nonconservingABC model pe-3βμ ABC 000 p This is similar to equilibrium : f = Helmholtz free energy density

28. Rate function of r, G(r) High µ Low µ First order phase transition (only in the nonconserving model) OC, D Mukamel - Phys. Rev. Let. 108, 060602 (2012)

29. Inequivalence of ensembles For NA=NB≠NC : Nonconserving (Grand canonical) Conserving (Canonical) disordered disordered ordered ordered 1st order transition 2nd order transition tricritical point Stability line OC, D Mukamel - Phys. Rev. Let. 108, 060602 (2012) Different nonconserving ABC model: J Barton, J L Lebowitz, E R Speer - J. Phys. A 44 065005 (2011) Discussion about ensemble inequivalence: OC, D Mukamel - J. Stat. Mech. 12 12017 (2012)

30. Corrections to G(r) using MFT pe-3βμ ABC 000 p - conserving current - conserving action1 - nonconserving current - nonconserving action2,3 T Bodineau, B Derrida - ComptesRendus Physique 8 540 (2007) G Jona-Lasinio, C Landimand M E Vares - Probability theory and related fields 97 339 (1993) T Bodineau, M Lagouge - J. Stat. Phys. 139 201 (2010)

31. Corrections to G(r) using MFT pe-3βμ ABC 000 p r rμ τ 0 T Instanton path:

32. Conclusions Nonequlibrium ‘grand canonical ensemble’ - Slow nonconserving dynamics Example to ABC model 1st order phase transition for nonmonotoneous µs(r)and inequivalence of ensembles.( µs(r) is dynamics dependent ! ) Corrections to rate function of r using MFT Thank you for listening ! Any questions ?