Thermodynamics and dynamics of systemswith long range interactions

David Mukamel

Systems with long range interactions

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

two-body interaction

examples: dipolar interactions, self gravitating systems,coulomb interactions, vortices in two dimensions,electrons interacting with laser field etc.

12 VRE sd 01 d

s

SEF Thermodynamics:

VSVE , 1 since

VS the entropy may be neglected in theThermodynamic limit

the equilibrium state is just the ground state

In order to make the energy and the entropy scale in the same way one may rescale the Hamiltonian(Kac prescription)

HV

H

1

This is equivalent to taking the thermodynamic limittogether with the limit asT

VT since SEF

self gravitating systems ferromagnetic dipolar systems

2RT

In gravitational systems this would mean

3/2since

Is this limit realized in self gravitating systems?

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.

SEF

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

KR

GNMT 62

2

10

2

3

1MvTkB km/sec 30v

Ferromagnetic dipolar systems

v ~ 1/r3 0

2MV

DHH

D is the shape dependent demagnetization factor

2

1

N

iiSN

JHH

Discuss features which result from the non-additivity

Inequivalence of microcanonical (MCE) andcanonical (CE) ensembles

Breaking of ergodicity in microcanonical ensemble

Slow dynamics, diverging relaxation time

Negative specific heat in microcanonical ensemble

Thermodynamics

Dynamics

Some general considerations

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

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

To study this question of ensembles’ inequivalence in more detailit is instructive to consider mean-field type interactions, e.g. inIsing models

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

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)

K/J0

T/J

-1/2

1st order

2nd order

K/J-1/2

+ - + - + - + - + + + + +

Ground state

Canonical (T,K) phase diagram

0M

0M

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)

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

)1ln()1(2

1)1ln()1(

2

1

ln)1ln()1(2

1)1ln()1(

2

1

ln1

),(

umumumum

uummmm

Nms

KumJ

2

2where

Maximize to get ),( ms )(m

))(,()( mss //1 sT

s

m

continuous transition:

discontinuous transition:

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

...)(),( 420 bmamsms

Microcanonical phase diagram

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

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

Microcanonical Ising dynamics:

)1(2

)(2 1

1

2

1

i

N

ii

N

ii SS

KS

N

JH

Problem:by making single spin flips it is basically impossible to keep the energy fixed for arbitrary K and J.

Microcanonical Ising dynamics:Creutz (1983)

In this algorithm one probes the microstates of thesystem with energy E

This is implemented by adding an auxiliary variable,called a demon such that

EEE DS

system’s energy demon’s energySE 0DE

Creutz algorithm:

1. Start with

2. Attempt to flip a spin: accept the move if energy decreases and give the excess energy to the demon.

if energy increases, take the needed energy from the demon. Reject the move if the demon does not have the needed energy.

0 DS EEE

0 , EEEEEEE DDSS

0 , EEEEEE DDSS

TkeEP BE

DD /1 )(

Yields the caloric curve T(E).

N=400, K=-0.35E/N=-0.2416

22 2/)( TCEED

VDDeEP

To second order in ED the demon distribution is

And it looks as if it is unstable for CV < 0(particularly near the microcanonical tricritical point where CV vanishes).

However the distribution is stable as long as the entropyincreases with E (namely T>0) since the next to leading term isof order 1/N.

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

1M2M

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

Ising model with long and short range interactions

m=M/N= (N+ - N-)/Nu =U/N = number of broken bonds per site in a configuration

MNNUNN 2 has one for

corresponding to isolated down spins

+ + + - + + + + - + + - + + + + - + +

Hence:

KumJ

2

2

mu 10

The available is not convex.),( m

K=-0.4

012

/0 2 mmK

JK

m

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

Ergodicity is thus broken even for a finite system.

Breaking of Ergodicity at finite systems has to do with the factthat the available range of m dcreases as the energy is lowered.

m

Had it been

then the model could have moved between the right and the leftsegments by lowering its energy with a probability which isexponentially small in the system size.

Time scales

0 m1 ms

s

Relaxation time of a state at a local maximum of the entropy(metastable state)

sm0

For a system with short range interactions is finite. It does not diverge with N. )0( sm

ms

m=0

R

energy free surface

differenceenergy free ,0)()0(

smfffR

fdt

dR

0 m1ms

fRc / critical radius above which droplet grows.

)()0( )0( 1msssem sNs

For systems with long range interactions the relaxationtime grows exponentially with the system size.

In this case there is no geometry.The dynamics depends only on m.The dynamics is that of a single particle moving in apotential V(m)=-s(m)

0 m1ms

Griffiths et al (1966) (CE Ising); Antoni et al (2004) (XY model); Chavanis et al (2003) (Gravitational systems)

M=0 is a local maximum of the entropyK=-0.4 0.318

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

Diverging time scales have been observed in a numberof systems with log range interactions.

The Hamiltonian Mean Field Model (HMF)(an XY model with mean field ferromagnetic interactions)

N

jiji

N

ii NpH

1,

2

1

2 ))cos(1(2

1

2

1

There exists a class of quasistationary m=0 stateswith relaxation time

7.1 with N

Yamaguchi, Barre, Bouchet, Dauxois, Ruffo (2004)

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.

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