Driven ABC model under particle-nonconserving dynamics. Or Cohen and David Mukamel

Transcription

1 Diven BC model unde paticle-nonconseving dynamics O Cohen and David Mukamel Intenational Semina on Lage Fluctuations in Non-Equilibium Systems, Desden, July 0

2 Motivation Equilibium systems with Long-ange inteactions System diven out of equilibium v( ) GMm T T Much is known Exhibit long-ange coelations Exhibit unique phenomena : inequivalence of ensembles, negative specific heat in MC ensemble, slow elaxation, quasi-stationay states Much emains unknown Exhibit long-ange coelations Exhibit simila phenomena?

3 Outline Equilibium Long-ange. Long-ange inteactions. Inequivalence of ensembles Diven Models 3. BC model 4. Inequivalence of ensembles 5. Conclusions

4 Enegy scaling Long-ange inteactions Shot ange E V v( ) d Long ange σ>0 dditive YES NO E V / d S S Micocanonical C V <0 0 C V <0 E E E E E E Canonical C V 0 C V 0

5 Inequivalence of ensembles T Micocanonical disodeed T Canonical disodeed odeed C V <0 K odeed K T K = inteaction stength st ode tansition disodeed nd ode tansition odeed inequivalence K

6 Outline Equilibium Long-ange. Long-ange inteactions. Inequivalence of ensembles Diven Models 3. BC model 4. Inequivalence of ensembles 5. Conclusions

7 BC model Ring of size L B Dynamics : q B BC q CB B C C q C L q= q< BBCCCBCBCB BBBBBCCCCC Evans, Kafi, Koduvely & Mukamel - Phys. Rev. Lett. 998

8 Cuents & Detailed Balance. Equal densities N =N B =N C lthough q detailed balance obeyed with espect with to H ({ X i }) L i L k k L B i ik B C i ik C i ik ~ L P H ({ X i }) ({ X i }) q. Nonequal densities, e.g. N B N C No effective Hamiltonian

9 Weak asymmety E S ~ L ~ L q exp( L ) / ( ) P( E) ( E) q E e S E L e L f Clincy, Deida & Evans - Phys. Rev. E 003

10 Weak asymmety E S ~ L ~ L q exp( L ) / ( ) P( E) ( E) q E e S E L e L f nd ode phase tansiton at the citical temp. c Clincy, Deida & Evans - Phys. Rev. E 003

11 Outline Equilibium Long-ange. Long-ange inteactions. Inequivalence of ensembles Diven Models 3. BC model 4. Inequivalence of ensembles 5. Conclusions

12 BC model + vacancies Canonical ensemble : 0X X0 X=,B,C B C 0 Ledehendle & Mukamel - Phys. Rev. Lett. 00

13 Nonconseving dynamics Gand Canonical ensemble : B C 0 0X pe -3βμ BC 000 p X0 X=,B,C Fluctuating paamete : Conjugate field : N N N B C L Ledehendle & Mukamel - Phys. Rev. Lett. 00

14 Inequivalence of ensembles Fo N =N B =N C : Conseving = Canonical Nonconseving = Gand canonical disodeed disodeed T= odeed T= odeed st ode tansition nd ode tansition ticitical point Ledehendle, Cohen & Mukamel - J. Stat. Mech: Theoy Exp. 00

15 Nonequal densities Hydodynamics equations : d dt i ibi i B ( ) i L d B C 0 L dx Dift Diffusion Deposition Evapoation d 3 p e 3 dx i B C

16 Nonequal densities Hydodynamics equations : d dt i ibi i B ( ) i L d B C 0 L dx Dift Diffusion Deposition Evapoation d 3 p e 3 dx i B C B e -β/l B BC e -β/l CB 0X X0 BC pe -3βμ 000 p C e -β/l C X=,B,C

17 Conseving steady-state Conseving model 0 p Steady-state pofile L N N N C B d x c sn b a d x c sn x,, ), ( * C B C B e p dx d dx d L dt d Dift Diffusion Nonequal densities : Cohen & Mukamel - Pepint Equal densities : yye et al. - J. Stat. Phys. 009

18 Nonconseving steady-state C B C B e p dx d dx d L dt d Dift Diffusion Deposition Evapoation

19 Nonconseving steady-state d dt d B C 0 L dx d 3 p e 3 dx B C Dift + Diffusion Deposition + Evapoation Nonconseving model with slow nonconseving dynamics p ~ L, Steady-state pofile * ( x, ) Steady-state density?

20 Dynamics of paticle density ~ L ~ L (x) B(x) C (x)

21 Dynamics of paticle density fte time τ : ~ L ~ L * ( x, )

22 Dynamics of paticle density fte time τ : ~ L ~ L

23 Dynamics of paticle density fte time τ : ~ L ~ L * ( x, )

24 Dynamics of paticle density fte time τ : ~ L ~ L 3

25 Dynamics of paticle density fte time τ : ~ L ~ L 3 * ( x, 3 )

26 Lage deviation function of fte time τ : ~ L ~ L 3 * ( x, 3 ) * ( x, 3 ) 3 R( 4 3 ), R( 4 3 L 3 ) L

27 Lage deviation function of V () R () R () min = D - Random walk in a potential max

28 Lage deviation function of V () R () R () min = D - Random walk in a potential P( ) ' 0 R R ( ') ( ') exp[ LF ( )] max Lage deviation function 3 * 3 e dx( 0 ) pe -3βμ 0 F ( ) d'log BC 000 * * * p 0 dx BC 0

29 Inequivalence of ensembles Fo N =N B N C : Conseving = Canonical B, C Nonconseving = Gand canonical disodeed disodeed odeed odeed st ode tansition nd ode tansition ticitical point

30 Locating st ode tansition F Lage deviation function ( ) 0 d'( ( ')) Chemical potential in conseving system ( ') log 3 0 dx( * * B * C * ( ) 0 3 ) Conseving Nonconseving μ nd ode tans. Citical point Odeed phase μ st ode tans. Maxwell s constuction F ( ) F ( ) Homogenous phase

31 Fast evapoation & deposition p ~ L Conseving Nonconseving Flat vacancies pofile ( 0 x) Oscillatoy vacancies pofile 0 ( x) const No moving solutions d d ( x) 0 Moving solutions ( x, ) ( x v ) c 3 36 c ( k ) 3 36 ( k ) NESS is sensitive to the dynamics

32 Results & Conclusions. Inequivalence of ensembles in the BC model Open questions : Othe similaities to system with LRI? (dynamical featues etc.) In othe diven models?. Dynamical definition of ensembles in diven models? Conseving BC model + slow nonconseving dynamics Obtain LDF of paticle density pplies to othe diven models