Effective long-range interactions in driven systems David Mukamel.

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Effective long-range interactions in driven systems David Mukamel

Transcript of Effective long-range interactions in driven systems David Mukamel.

Page 1: Effective long-range interactions in driven systems David Mukamel.

Effective long-range interactions in driven systems

David Mukamel

Page 2: Effective long-range interactions in driven systems David Mukamel.

Systems with long range interactions

in d dimensions

two-body interaction

dVVRE /1

dr

rv1

)(

for σ<0 the energy is not extensive

-strong long-range interactions

Page 3: Effective long-range interactions in driven systems David Mukamel.

self gravitating systems (1/r) σ=-2

ferromagnets σ=0

2d vortices log(r) σ=-2

)/1( 3r

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These systems are non-additive BA FFF

A B

As a result, many of the common properties of typicalsystems with short range interactions are not sharedby these systems.

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Driven systems

T1 T2

T1>T2

E

heat current

charge current

Local and stochastic dynamics

No detailed balance (non-vanishing current)

What is the nature of the steady state?

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drive in conserving systems result in many cases in long range correlations

What can be learned from systems with long-range interactions

on steady state properties of driven systems?

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TSEF Free Energy:

VSVE d , /1 since

S E the entropy may be neglected in thethermodynamic limit.

In finite systems, although E>>S, if T is high enoughE may be comparable to TS, and the full free energyneed to be considered. (Self gravitating systems, e.g.globular clusters)

Systems with long range interactions

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Globular clusters are gravitationally bound concentrationsof approximately ten thousand to one million stars, spreadover a volume of several tens to about 200 light years in diameter.

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KT 62102

3

1MvTkB km/sec 30v

Thus although andE may be comparable to TS

3/5VE VS

TSEF

For the M2 clusterN=150,000 starsR= 175 light years

3010 2M Kg

1~1

~

~ ~22

)O(R

GNM

TTS

E

NMSR

MGNE

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One may implement the large T limit by rescalingthe Hamiltonian

VEHVH d /

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1S )(2 i

2

1

N

iiSN

JH

Although the canonical thermodynamic functions (free energy,entropy etc) are extensive, the system is non-additive

+ _

4/JNEE 0E

21 EEE

For example, consider the Ising model:

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Features which result from non-additivity

Negative specific heat in microcanonical ensemble

Inequivalence of microcanonical (MCE) andcanonical (CE) ensembles

Breaking of ergodicity in microcanonical ensemble

Slow dynamics, diverging relaxation time

Thermodynamics

Dynamics

Temperature discontinuity in MCE

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Some general considerations

Negative specific heat in microcanonical ensembleof non-additive systems.Antonov (1962); Lynden-Bell & Wood (1968); Thirring (1970), Thirring & Posch

coexistence region in systems with short range interactions

E0 = xE1 +(1-x)E2 S0 = xS1 +(1-x)S2

hence S is concave and the microcanonicalspecific heat is non-negative

S

1E 2E E0E

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On the other hand in systems with long range interactions(non-additive), in the region E1<E<E2

S

1E 2E E

E0 = xE1 +(1-x)E2

S0 xS1 +(1-x)S2

The entropy may thus follow the homogeneoussystem curve, the entropy is not concave. andthe microcanonical specific heat becomesnegative.

0 222 EECT V

compared with canonical ensemble where

0E

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Ising model with long and short range interactions.

)1(2

)(2 1

1

2

1

i

N

ii

N

ii SS

KS

N

JH

d=1 dimensional geometry, ferromagnetic long range interaction J>0

The model has been analyzed within the canonical ensemble Nagel (1970), Kardar (1983)

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Canonical (T,K) phase diagram

0M

0M

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Microcanonical analysis

)1(2

)(2 1

1

2

1

i

N

ii

N

ii SS

KS

N

JH

KUMN

JE 2

2

NNN

NNMU = number of broken bonds in a configuration

Number of microstates:

2/2/),,(

U

N

U

NUNN

U/2 (+) segments U/2 (-) segments

Mukamel, Ruffo, Schreiber (2005); Barre, Mukamel, Ruffo (2001)

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s=S/N , =E/N , m=M/N , u=U/N

)1ln()1(2

1)1ln()1(

2

1

ln)1ln()1(2

1)1ln()1(

2

1

ln1

),(

umumumum

uummmm

Nmus

but

Maximize to get ),( ms )(m

))(,()( mss

KumJ

2

2

//1 sT

),(),( msmus

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canonical microcanonical

359.0/

317.032

3ln/

JT

JT

MTP

CTP

The two phase diagrams differ in the 1st order region of the canonical diagram

Ruffo, Schreiber, Mukamel (2005)

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s

m

discontinuous transition:

In a 1st order transition there is a discontinuity in T, and thus thereis a T region which is not accessible.

//1 sT

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S

E

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In general it is expected that whenever the canonical transitionis first order the microcanonical and canonical ensemblesdiffer from each other.

S

1E 2E E

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Dynamics

Systems with long range interactions exhibit slow relaxation processes.

This may result in quasi-stationary states (long livednon-equilibrium states whose relaxation time to the equilibrium state diverges with the system size).

Non-additivity may facilitate breaking of ergodicity which could lead to trapping of systems in non-Equilibrium states.

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Long range correlations in driven systems Conserved variables tend to produce long range correlations.

Thermal equilibrium states are independent of the dynamics(e.g. Glauber and Kawasaki dynamics result in the same Boltzmann distribution)

Non-equilibrium steady states depend on the dynamics(e.g. conserving or non-conserving)

Conserving dynamics in driven, non-equilibrium systemsmay result in steady states with long range correlationseven when the dynamics is local

Can these correlations be viewed as resulting from effectivelong-range interactions?

Driven systems

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ABC model

One dimensional driven model with stochastic local dynamicswhich results in phase separation (long range order) where thesteady state can be expressed as a Boltzmann distribution of aneffective energy with long-range interactions.

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A B C

ABC Model

AB BA1

q

BC CB1

q

CA AC1

q

dynamics

Evans,Kafri, Koduvely, Mukamel PRL 80, 425 (1998)A model with similar features was discussed by Lahiri, Ramaswamy PRL 79, 1150 (1997)

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Simple argument:

AB BA1

q

BC CB1

q

CA AC1

q

ACCCC CCCCA

CBBBB BBBBC

BAAAA AAAAB

…AACBBBCCAAACBBBCCC…

…AABBBCCCAAABBBCCCC…

…AAAAABBBBBCCCCCCAA…

fast rearrangement

slow coarsening

The model reaches a phase separated steady state

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logarithmically slow coarsening

…AAAAABBBBBCCCCCCAA…

tlqt l ln

needs n>2 species to have phase separation

strong phase separation: no fluctuation in the bulk;only at the boundaries.

…AAAAAAAAAABBBBBBBBBBBBCCCCCCCCCCC…

Phase separation takes place for any q (except q=1)

Phase separation takes place for any density N , N , N A B C

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N N NA B C Special case

The argument presented before is general, independent of densities.

For the equal densities case the model has detailed balance for arbitrary q.

We will demonstrate that for any microscopic configuration {X}

One can define “energy” E({X}) such that the steady state

Distribution is

})({})({ XEqXP

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AAAAAABBBBBBCCCCC E=0

……AB….. ……BA….. E E+1

……BC….. ……CB….. E E+1

……CA….. ……AC….. E E+1

...)(...)(...)(...)( BAPABBAWABPBAABW

With this weight one has:

=q =1

qABPBAP ...)(.../...)(...

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AAAAABBBBBCCCCC AAAABBBBBCCCCCA

E E+NB-NC

NB = NC

Thus such “energy” can be defined only for NA=NB=NC

This definition of “energy” is possible only for N N NA B C

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AABBBBCCCAAAAABBBCCCC

The rates with which an A particle makes a full circle clockwiseAnd counterclockwise are equal

CB NN qq

Hence no currents for any N.

For the current of A particles satisfies CB NN CB NNA qqJ

The current is non-vanishing for finite N. It vanishes only in thelimit . Thus no detailed balance in this case.N

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…AAAAAAAABBBABBBBBBCCCCCCCCCAA…

The model exhibits strong phase separation

The probability of a particle to be at a distance on the wrong side of the boundary is lql

The width of the boundary layer is -1/lnq

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N N NA B C The “energy” E may be written as

summation over modulo )( ki N• long range

Local dynamics

xEqxP

1

2

1 1

( / 3)N N i

i i k i i k i i ki k

E x C B AC B A N

N

i

N

kkiikiikii ABCABC

N

kxE

1

1

1

1

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Partition sum

nqnpqZ )()(1

Excitations near a single interface: AAAAAAABBBBBB

P(n)= degeneracy of the excitation with energy n

P(0)=1P(1)=1P(2)=2 (2, 1+1)P(3)=3 (3, 2+1, 1+1+1)P(4)=5 (4, 3+1, 2+2, 2+1+1, 1+1+1+1)

P(n)= no. of partitions of an integer n

nn

nnp

34

)3/2exp(~)(

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Weakly asymmetric ABC modelNeq /

The model exhibits a phase transition atfor the case of equal densities

32 c

c

c

homogeneous

inhomogeneous

9/1 1

1 1

NABCABCN

xEN

i

N

kkiikiikii

effective rescaled “energy”

Clincy, Derrida, Evans, PRE 67, 066115 (2003)

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Generalized ABC model

add vacancies: LNNNNN CBA , 3/

A , B, C, 0

AB BA1

q

BC CB1

q

CA AC1

q

0X 0X1

1X=A,B,C

Dynamics

A. Lederhendler, D. Mukamel

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grand-canonical dynamics

AB BA1

q

BC CB1

q

CA AC1

q

0X 0X1

1

ABC 0001

q

(6 1) 3N L

X=A,B,C

…A00ACBABCCA00AACBBB00000CCC…

ABC000

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1

2

1 1

L L i

i i k i i k i i ki k

E x C B AC B A N NL

3)16()},({)3},({ NNXENXE

The dynamics is local for 6/1

For NA=NB=NC: there is detailed balance with respect to

})({})({ XEqXP

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/1 , , )),(( / TkeqTSETxF BN

1 , 0CBA

continuum version of the model

21

0

1

0

1

0

)()()()()()()(

xdxzxxzxxzxxdzdxE BCCAAB

)(ln)()(ln)()(ln)()(ln)( 00

1

0

xxxxxxxxdxS CCBBAA

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canonical

grand canonical

)),(( TxF

)),((),( TxFTG

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6

1 ,T=0.02, Grand canonical

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6

1 ,T=0.02, Canonical – 2nd order transition at 2177.0c

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6

1 ,T=0.02, Grand canonical – 1st order transition

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049.0Correlations for both solutions with

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Summary

Local stochastic dynamics may result in effective long-range interactions in driven systems.

This is manifested in the existence of phase transitionsin one dimensional driven models.

Existence of effective long range interactions can be explicitlydemonstrated in the ABC model.

The model exhibits phase separation for any drive

Phase separation is a result of effective long-rangeinteractions generated by the local dynamics.

Inequivalence of ensembles in the driven model.

1q