Statistical Mechanics of Systems withLong range interactions

David Mukamel

Systems with long range interactions

v(r) 1/rs at large r with s<d, d dimensions

two-body interaction

12 VRE sd 01 d

s

self gravitating systems s=1ferromagnets s=32d vortices s=0 (logarithmic)

SEF Free Energy:

VSVE , 1 since

VS the entropy may be neglected in thethermodynamic limit.

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

NEE /

Ferromagnetic dipolar systems

v ~ 1/r3 0

2MV

DHH

D is the shape dependent demagnetization factor

2

1

N

iiSN

JHH

Models of this type, although they look extensive,are non-additive.

(for ellipsoidal samples)

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:

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

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

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

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)

Canonical (T,K) phase diagram

0M

0M

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)

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

S

E

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

S

1E 2E E

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.

Slow Relaxation

In systems with short range interaction, typically the relaxation timefrom an unstable (or metastable) state to a stable one is finite

(independent of the system size).

ms

m=0

R

1 dd RfR

fRc /

free energy gain of a droplet

critical radius above which the droplet grows.

Since the critical radius is finite, the relaxation time scale insystems with short range interactions is finite.

This is not the case in systems with long range interactions.relaxation processes are typically slow, with relaxation time

which grows with the system size.

In the case of the Ising model, the relaxation time isfound to grow as logN.

In other cases it is found to grow with a power of N.

This results in non-equilibrium, quasi-stationary states.

Dynamics

Microcanonical Monte Carlo Ising dynamics:

)1(2

)(2 1

1

2

1

i

N

ii

N

ii SS

KS

N

JH

Relaxation of a state with a local minimum of the entropy(thermodynamically unstable)

0 ms

One would expect the relaxation time of the m=0state to remain finite for large systems (as is the caseof systems with short range interactions..

sm0

M=0 is a minimum of the entropyK=-0.25 0.2

Nln

One may understand this result by considering the followingLangevin equation for m:

)()()( )( '' ttDtttm

s

t

m

With D~1/N

0, )( 42 babmamms

Pm

s

mm

PD

t

P2

2

Fokker-Planck Equation:

)()0,( ),( mtmPtmP

This is the dynamics of a particle moving in a double wellpotential V(m)=-s(m), with T~1/N starting at m=0.

This equation yields at large t

D

eamtmP

at

2exp),(

22

mPm

am

PD

t

P

2

2

Taking for simplicity s(m)~am2, a>0, the problem becomes that of a particle moving in a potential V(m) ~ -am2 at temperature T~D~1/N

Since D~1/N the width of the distribution is Nem at /22

NNe a ln 1/2

i

N

i

N

jiji

N

ii W

NpH

1

2

1,

2

1

2 cos))cos(1(2

1

2

1

The anisotropic XY model

with Hamiltonian, deterministic dynamics

ii

ii

dt

dp

pdt

d

slow relaxation with algebraically increasing time scale

m=0m>0

N

7.1

Nln

Dynamical phase diagram of the anisotropic XY model

Relaxation of the thermodynamically unstable m=0 state

One would expect the relaxation time of the m=0state to remain finite for large systems (as is the caseof systems with short range interactions.

Yamaguchi, Barre, Bouchet, Dauxois, Ruffo (2004)Jain, Bouchet, Mukamel, J. Stat. Mech. (2007)

N=500

N=10000

Relaxation of the quasi-stationary m=0 state:

In fact depending on energy and initial distributionone seems to have for the XY model:

NNln

1 c

Breaking of Ergodicity in Microcanonical dynamics.

Borgonovi, Celardo, Maianti, Pedersoli (2004); Mukamel, Ruffo, Schreiber (2005).

Systems with short range interactions are defined on a convexregion of their extensive parameter space.

E

M

1M 2M

If there are two microstates with magnetizations M1 and M2 Then there are microstates corresponding to any magnetizationM1 < M < M2

.

This is not correct for systems with long range interactionswhere the domain over which the model is defined need notbe convex.

E

M

K=-0.4

mmK

JK 1

2/0 2

m

Local dynamics cannot make the system cross from one segment to another.

Ergodicity is thus broken even for a finite system.

Summary

Some general thermodynamic and dynamical propertiesof system with long range interactions have been considered.

Negative specific heat in microcanonical ensembles

Canonical and microcanonical ensembles need not be equivalentwhenever the canonical transition is first order.

Breaking of ergodicity in microcanonical dynamics due tonon-convexity of the domain over which the model exists.

Long time scales, diverging with the system size.

Quasi-staionary states.

The results were derived for mean field long range interactionsbut they are expected to be valid for algebraically decayingpotentials.

S. Ruffo, (Fitenze)J. Barre, (Nice)A. Campa, (Rome)A. Giansanti, (Rome)N. Schreiber, (Weizmann)P. de Buyl, (Firenze)R. Khomeriki, (Georgia)K. Jain, (Weizmann)F. Bouchet, (Nice)T. Dauxois (Lyon)