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)