Thermodynamics and dynamics of systemswith long range interactions

David Mukamel

S. Ruffo, J. Barre, A. Campa, A. Giansanti, N. Schreiber,

P. de Buyl, R. Khomeriki, K. Jain, F. Bouchet, T. Dauxois

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

SEF Thermodynamics:

VSVE , 1 since

VS the entropy may be neglected in thethermodynamic limit.

The equilibrium state is just the ground state.Nevertheless, entropy can not always be neglected.

SEF

In finite systems, although E>>S, if T is high enough may be comparable to S, and the full free energyneed to be considered.E

SEF

Self gravitating systems

e.g. in globular clusters:clusters of the order 105 stars within a distanceof 102 light years. May be considered as a gasof massive objects

2

3

1MvTkB km/sec 30v KT 6210

Thus although 3/5VE since T is large

E becomes comparable to S

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

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 , 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

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

m=0

0m

discontinuity in T

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.

V

DDs

CTE

S

TE

S

EE

SE

E

SESES

22

2

22

2

1 ,

1

2

1)()(

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

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.

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~D~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 long range interactions.

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

N

jiji

N

ii N

pH1,

2

1

2 ))cos(1(2

1

2

1

There exists a class of quasi-stationary m=0 stateswith relaxation time

7.1 with N

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

4/3c

m=0m>0

N

jiji

N

ii N

pH1,

2

1

2 ))cos(1(2

1

2

1

iyixi

ii mm

dt

dpp

dt

d cossin

N

i iy

N

i ix Nm

Nm

11sin

1 cos

1

Analysis of the relaxation times:(K. Jain, F. Bouchet, D. Mukamel 2007)

4/3c

m=0m>0

12/7*

Nln N

),','())'cos(1(''),( tpfddptV

000

0 2

1)( ppp

ppf

),,()(2

1),,( 10 tpfpftpf

N

1

),,( tpf

0

p

fVfp

t

f

Distribution function

Vlasov equation

Linear stability analysis:

c *Linearly stable, power law relaxation time

* Linearly unstable, logarithmic relaxation time

)12/7(6 1 teN

m

12/7 4/3 * c

4/3c

m=0m>0

12/7*

Nln N

2

11,

2

1

2 cos2

))cos(1(2

1

2

1

N

ii

N

jiji

N

ii N

D

NpH

D

0

2

0.5 1

m=0

linearly stable

N

Linearly unstable

Nln

Anisotropic XY model

12/)7(

4/)3(* D

Dc

Summary

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

Canonical and microcanonical ensembles need not be equivalentwhenever the canonical transition is first order (yielding negativeSpecific heat, temperature discontinuity etc.)

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.

General framework to analyze time scales is still lacking.

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