Effective long-range interactions in - PowerPoint Presentation

Slide1

Effective long-range interactions in driven systemsDavid MukamelSlide2

Systems with long range interactions

in d

dimensions

two-body interaction

for

σ

<0 the energy is not extensive

-strong long-range interactions Slide3

self gravitating systems (1/r) σ=-2

ferromagnets

σ

=02d vortices log(r)

σ=-2Slide4

These systems are non-additive

A

B

As a result, many of the common properties of typical

systems with short range interactions are not shared

by these systems.Slide5

Driven systems

T

1

T

2

T

1

>T

2

E

heat current

charge current

Local and stochastic dynamics

No detailed balance (non-vanishing current)

What is the nature of the steady state?Slide6

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?

Slide7

Free Energy:

since

the entropy may be neglected in the

thermodynamic limit.

In finite systems, although

E>>S

, if

T

is high enough

E

may be comparable to

TS

, and the full free energy

need to be considered. (Self gravitating systems, e.g.

globular clusters)

Systems with long range interactionsSlide8

Globular clusters

are gravitationally bound concentrations

of approximately ten thousand to one million stars, spread

over a volume of several tens to about 200 light years in

diameter. Slide9

Thus although and

E may be comparable to TS

For the M2 cluster

N=150,000 stars

R= 175 light years

Kg Slide10

One may implement the large T limit by rescalingthe HamiltonianSlide11

Although the canonical thermodynamic functions (free energy,

entropy etc) are extensive, the system is

non-additive

+

_

For example, consider the Ising model:Slide12

Features which result from non-additivity

Negative specific heat in microcanonical ensemble

Inequivalence of microcanonical (MCE) and

canonical (CE) ensembles

Breaking of ergodicity in microcanonical ensemble

Slow dynamics, diverging relaxation time

Thermodynamics

Dynamics

Temperature discontinuity in MCE Slide13

Some general considerations

Negative specific heat in microcanonical ensemble

of non-additive systems.

Antonov (1962); Lynden-Bell & Wood (1968); Thirring (1970), Thirring & Posch

coexistence region

in systems with short range interactions

E0 = xE

1 +(1-x)E2 S0 = xS

1

+(1-x)S

2

hence S is concave and the microcanonical

specific heat is non-negative

SSlide14

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

S

E

0

= xE

1

+(1-x)E

2

S

0

xS

1

+(1-x)S

2

The entropy may thus follow the homogeneous

system curve, the entropy is not concave. and

the microcanonical specific heat becomes

negative.

compared with canonical ensemble whereSlide15

Ising model with long and short range interactions.

d=1 dimensional geometry, ferromagnetic long range

interaction J>0

The model has been analyzed within the canonical

ensemble

Nagel (1970), Kardar (1983)Slide16

Canonical (T,K) phase diagramSlide17

Microcanonical analysis

U = number of broken bonds in a configuration

Number of microstates:

U/2

(+)

segments U/2

(-)

segments

Mukamel, Ruffo, Schreiber (2005); Barre, Mukamel, Ruffo (2001)Slide18

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

but

Maximize

to get Slide19

canonical

microcanonical

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

Ruffo, Schreiber, Mukamel (2005)Slide20

s

m

discontinuous transition:

In a 1st order transition there is a discontinuity in T, and thus there

is a T region which is not accessible.Slide21

S

ESlide22

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

SSlide23

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. Slide24

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 localCan these correlations be viewed as resulting from effective

long-range interactions?

Driven systemsSlide25

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. Slide26

A

B

C

ABC Model

AB BA

1

q

BC CB

1

q

CA AC

1

q

dynamics

Evans,Kafri

,

Koduvely

,

Mukamel

PRL 80, 425 (1998)

A model with similar features was discussed by

Lahiri

,

Ramaswamy

PRL 79, 1150 (1997) Slide27

Simple argument:

AB BA

1

q

BC CB

1

q

CA AC

1

q

A

CCCC

CCCC

A

C

BBBB

BBBB

C

B

AAAA

AAAA

B

…

AA

C

BBB

CC

AAA

C

BBB

CCC

…

…

AA

BBB

CCC

AAA

BBB

CCCC

…

…

AAAAA

BBBBB

CCCCCC

AA

…

fast rearrangement

slow coarsening

The model reaches a phase separated steady stateSlide28

logarithmically slow coarsening

…

AAAAA

BBBBB

CCCCCCAA…

needs n>2 species to have phase separation

strong phase separation: no fluctuation in the bulk;

only at the boundaries.

…

AAAAAAAAAA

BBBBBBBBBBBB

CCCCCCCCCCC

…

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

Phase separation takes place for any density N , N , N

A

B

CSlide29

A

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

Slide30

AAAAAABBBBBBCCCCC

E=0

……AB….. ……BA….. E E+1

……BC….. ……CB….. E E+1

……CA….. ……AC….. E E+1

With this weight one has:

=q

=1Slide31

AAAAABBBBBCCCCC AAAABBBBBCCCCCA

E E+N

B

-N

C

N

B

= NC

Thus such “energy” can be defined only for

N

A

=N

B

=N

C

This definition of “energy” is possible only for

A

B

CSlide32

AABBBBCCCAAAAABBB

CCCC

The rates with which an

A

particle makes a full circle clockwise

And counterclockwise are equal

Hence no currents for any N.

For the current of A particles satisfies

The current is

non-vanishing

for finite

N

. It vanishes only in the

limit . Thus no detailed balance in this case.Slide33

…AAAAAAAABBBABBBBBB

CCCCCCCCC

AA

…

The model exhibits strong phase separation

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

The width of the boundary layer is

-1/lnqSlide34

A

B

C

The “energy” E may be written as

summation over modulo

long range

Local dynamicsSlide35

Partition sum

Excitations near a single interface:

AAAAAAA

BBBBBB

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 nSlide36

Weakly asymmetric ABC model

The model exhibits a phase transition at

for the case of equal densities

homogeneous

inhomogeneous

effective rescaled “energy”

Clincy

, Derrida, Evans, PRE 67, 066115 (2003)Slide37

Generalized ABC model

add

vacancies:

A , B, C, 0

AB BA

1

q

BC CB

1

q

CA AC

1

q

0X 0X

1

1

X=A,B,C

Dynamics

A.

Lederhendler

, D.

MukamelSlide38

grand-canonical dynamics

AB BA

1

q

BC CB

1

q

CA AC

1

q

0X 0X

1

1

ABC 000

1

q



X=A,B,C

…

A

00

A

C

B

A

B

CC

A

00

AA

C

BBB

00000

CCC

…

A

B

C

000Slide39

The dynamics is

local

for

For N

A

=N

B

=N

C

: there is detailed balance with respect toSlide40

continuum version of the modelSlide41

canonical grand canonicalSlide42
Slide43

, T=0.02, Grand canonicalSlide44

, T=0.02, Canonical – 2

nd

order transition at Slide45

, T=0.02, Grand canonical – 1

st

order transitionSlide46

Correlations for

both solutions

withSlide47

Summary

Local stochastic dynamics may result in effective long-

range interactions in driven systems.

This is manifested in the existence of phase transitions

in 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.