Density large-deviations of nonconserving driven models

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1 Density large-deviations of nonconserving driven models Or Cohen and David Mukamel eizmann Institute of Science Statistical Mechanical Day, eizmann Institute, June 202

2 Ensemble theory out of euilibrium? Euilibrium H ( ) T, µ P G. Can. (, ) P ( ) P ( ) Can. G. Can.

3 Ensemble theory out of euilibrium? Euilibrium T, µ H ( ) Driven diffusive systems p C P ( ) w - w + P G. Can. (, ) P ( ) P ( ) Can. G. Can. C C w P (, ) F P ( ),??? w Can we infer about the nonconserving system from the steady state properties of the conserving system?

4 Ensemble theory out of euilibrium? Euilibrium T, µ H ( ) Driven diffusive systems p C P ( ) w - w + P G. Can. (, ) P ( ) P ( ) Can. G. Can. C C w P (, ) F P ( ),??? w w, w, p C C C P (, ) P ( ) P ( )

5 Generic driven diffusive model w L C w R C w - C w + C P ( ) w P ( ') w P ( ) w P ( ') w P ( ) t C C C C C C C C C ' ' ' ' ' ' ' conserving (sum over η with same ) ' nonconserving (sum over η with )

6 Generic driven diffusive model w L C w R C w - C w + C P ( ) w P ( ') w P ( ) w P ( ') w P ( ) t C C C C C C C C C ' ' ' ' ' ' ' conserving (sum over η with same ) ' nonconserving (sum over η with ) Guess a steady state of the form : C C higher order P (, ) P ( ) f ( ) O in L

7 Generic driven diffusive model w L C w R C w - C w + C P ( ) w P ( ') w P ( ) w P ( ') w P ( ) t C C C C C C C C C ' ' ' ' ' ' ' conserving (sum over η with same ) ' nonconserving (sum over η with ) Guess a steady state of the form : It is consistent if : C C higher order P (, ) P ( ) f ( ) O in L w In many cases : C ' L 2

8 Slow nonconserving dynamics To leading order in L we obtain 0 f ( ')[ w P ( ')] f ( )[ w P ( ')] C C C C ' ' ' ' ' ' ' 0 f ( ') f ( ) ' ',, '

9 Slow nonconserving dynamics To leading order in L we obtain 0 f ( ')[ w P ( ')] f ( )[ w P ( ')] C C C C ' ' ' ' ' ' ' 0 f ( ') f ( ) ' ',, ' V log[ f ( )],, = D - Random walk in a potential * min max

10 Slow nonconserving dynamics To leading order in L we obtain 0 f ( ')[ w P ( ')] f ( )[ w P ( ')] C C C C ' ' ' ' ' ' ' 0 f ( ') f ( ) ' ',, ' V log[ f ( )],, = D - Random walk in a potential * min max Steady state solution : C f ( ) P ( ) 0 ', ' ' ', '

11 Outline. Limit of slow nonconserving 2. Example of the ABC model w C C C C P (, ) P ( ) P ( ) C C C P ( ) F[ P ( ), w ] L 3. oneuilibrium chemical potential ( ) (dynamics dependent!) 4. Conclusions

12 ABC model Ring of size L AB Dynamics : BA BC CB A B C CA AC Evans, Kafri, Koduvely & Mukamel - Phys. Rev. Lett. 998

13 ABC model Ring of size L AB Dynamics : BA BC CB A B C CA AC L = < ABBCACCBACABACB AAAAABBBBBCCCCC Evans, Kafri, Koduvely & Mukamel - Phys. Rev. Lett. 998

14 ABC model x t A B C Evans, Kafri, Koduvely & Mukamel - Phys. Rev. Lett. 998

15 Phase transition eakly asymmetric thermodynamic limit exp( L ) LF[ ] P[ ] e A, B, C Clincy, Derrida & Evans - Phys. Rev. E 2003

16 Phase transition eakly asymmetric thermodynamic limit exp( L ) LF[ ] P[ ] e A, B, C For low β s * ( x) r / L 2 nd order phase transition at Clincy, Derrida & Evans - Phys. Rev. E 2003 c 3 r 2 3 / ( ) 2

17 onconserving ABC model A 2 AB BC CA B BA CB AC 0X + 2 X0 X=A,B,C Conserving model (canonical ensemble) C 0 Lederhendler & Mukamel - Phys. Rev. Lett. 200

18 onconserving ABC model A 2 AB BC CA B BA CB AC 3 0X ABC pe -3βμ p X0 X=A,B,C Conserving model (canonical ensemble) C onconserving model (grand canonical ensemble) Lederhendler & Mukamel - Phys. Rev. Lett. 200

19 Conserving steady-state Hydrodynamic euations : i A( ) L A i A B Drift Diffusion da d da 2 A B C dt L dx dx i A B i i i oneual densities : Cohen & Mukamel - Phys. Rev. Lett. 202 Eual densities : Ayyer et al. - J. Stat. Phys. 2009

20 Conserving steady-state Hydrodynamic euations : i A( ) L Steady state can be solved analytically (, ) xr A i Drift A B Diffusion da d da 2 A B C dt L dx dx i A B i i i oneual densities : Cohen & Mukamel - Phys. Rev. Lett. 202 Eual densities : Ayyer et al. - J. Stat. Phys. 2009

21 Slow nonconserving model Slow nonconserving limit p ~ L, 2 V( ), 3, 3 ABC pe -3βμ 000 p * min max

22 Slow nonconserving model Slow nonconserving limit p ~ L, 2 V( ), 3, 3 ABC pe -3βμ 000 p * min max C C * 3, 3 ' ( ') ( 0 (, )) w P dx x L ' 0 saddle point approx.

23 Slow nonconserving model V( ), 3, 3 * min r S max * 3 dx( 0 ( x, r)) 0 S ( r) log[ ] 3 * * * dxa( x, r) B( x, r) C ( x, r) 0 LG C P ( r) exp[ L ( dr ' ( r ') r)] e ABC pe -3βμ 000 p ( r) r L

24 Slow nonconserving model V( ), 3, 3 * min r S max * 3 dx( 0 ( x, r)) 0 S ( r) log[ ] 3 * * * dxa( x, r) B( x, r) C ( x, r) 0 LG C P ( r) exp[ L ( dr ' ( r ') r)] e ABC pe -3βμ 000 p ( r) r L This is similar to euilibrium : G. Can P r L F r r ( ) exp[ ( (, ) )]

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

26 Ineuivalence of ensembles For A = B C : Conserving = Canonical r r r rb, rc A 0. 0 onconserving = Grand canonical disordered disordered ordered ordered st order transition 2 nd order transition tricritical point

27 hy is µ S () the chemical potential? 2 SLO ( / L ) ( / L ) S S 2 2 2

28 hy is µ S () the chemical potential? 2 SLO ( / L ) ( / L ) S S SLO Gauge measures ( S / L )

29 Conclusions. Slow nonconserving dynamics 2. Example to ABC model 3. st order phase transition for nonmonotoneous µ(r) and ineuivalence of ensembles. 4. oneuilibrium chemical potential ( dynamics dependent! )