Thermodynamics and dynamics of systems

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Thermodynamics and dynamics of systems with 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
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Systems with long range interactions
two-body interaction
v(r) a 1/rs at large r with sltd, d
dimensions
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Thermodynamics
since
the entropy may be neglected in
the thermodynamic limit.
The equilibrium state is just the ground
state. Nevertheless, entropy can not always be
neglected.
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In finite systems, although EgtgtS, if T is high
enough may be comparable to S, and the
full free energy need to be considered.
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Self gravitating systems
e.g. in globular clusters clusters of the order
105 stars within a distance of 102 light years.
May be considered as a gas of massive objects
Thus although
since T is large
becomes comparable to S
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Ferromagnetic dipolar systems
(for ellipsoidal samples)
D is the shape dependent demagnetization factor
Models of this type, although they look
extensive, are non-additive.
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For example, consider the Ising model
Although the canonical thermodynamic functions
(free energy, entropy etc) are extensive, the
system is non-additive
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Features which result from non-additivity
Thermodynamics
Negative specific heat in microcanonical ensemble
Inequivalence of microcanonical (MCE)
and canonical (CE) ensembles
Temperature discontinuity in MCE
Dynamics
Breaking of ergodicity in microcanonical ensemble
Slow dynamics, diverging relaxation time
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Ising model with long and short range
interactions.
d1 dimensional geometry, ferromagnetic long
range interaction Jgt0
The model has been analyzed within the canonical
ensemble Nagel (1970), Kardar (1983)
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T/J
2nd order
1st order
K/J
0
-1/2
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Canonical (T,K) phase diagram
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Microcanonical analysis
Mukamel, Ruffo, Schreiber (2005) Barre, Mukamel,
Ruffo (2001)
U number of broken bonds in a configuration
Number of microstates
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sS/N , E/N , mM/N , uU/N
but
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continuous transition
discontinuous transition
In a 1st order transition there is a
discontinuity in T, and thus there is a T region
which is not accessible.
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m0
discontinuity in T
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Microcanonical phase diagram
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canonical
microcanonical
The two phase diagrams differ in the 1st order
region of the canonical diagram
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In general it is expected that whenever the
canonical transition is first order the
microcanonical and canonical ensembles differ
from each other.
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Dynamics
Microcanonical Ising dynamics
Problem by making single spin flips it is
basically impossible to keep the energy fixed
for arbitrary K and J.
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Microcanonical Ising dynamics Creutz (1983)
This is implemented by adding an auxiliary
variable, called a demon such that
systems energy demons energy
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Creutz algorithm

• Start with
• 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.

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Yields the caloric curve T(E).
N400, K-0.35 E/N-0.2416
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To second order in ED the demon distribution is
And it looks as if it is unstable for CV lt
0 (particularly near the microcanonical
tricritical point where CV vanishes). However
the distribution is stable as long as the
entropy increases with E (namely Tgt0) since the
next to leading term is of order 1/N.
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Breaking of Ergodicity in Microcanonical
dynamics. Borgonovi, Celardo, Maianti, Pedersoli
(2004) Mukamel, Ruffo, Schreiber (2005).
Systems with short range interactions are defined
on a convex region of their extensive parameter
space.
If there are two microstates with magnetizations
M1 and M2 Then there are microstates
corresponding to any magnetization M1 lt M lt M2
.
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This is not correct for systems with long range
interactions where the domain over which the
model is defined need not be convex.
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Ising model with long and short range interactions
mM/N (N - N-)/N u U/N number of broken
bonds per site in a configuration
corresponding to isolated down spins -
- - -
Hence
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K-0.4
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Local dynamics cannot make the system cross from
one segment to another. Ergodicity is thus
broken even for a finite system.
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Breaking of Ergodicity at finite systems has to
do with the fact that the available range of m
dcreases as the energy is lowered.
Had it been
then the model could have moved between the right
and the left segments by lowering its energy with
a probability which is exponentially small in the
system size.
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Time scales
Relaxation time of a state at a local maximum of
the entropy (metastable state)
s
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critical radius above which droplet grows.
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For systems with long range interactions the
relaxation time 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
a potential V(m)-s(m)
Griffiths et al (1966) (CE Ising) Antoni et al
(2004) (XY model) Chavanis et al (2003)
(Gravitational systems)
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M0 is a local maximum of the entropy K-0.4
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Relaxation of a state with a local minimum of the
entropy (thermodynamically unstable)
One would expect the relaxation time of the
m0 state to remain finite for large systems (as
is the case of systems with short range
interactions..
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M0 is a minimum of the entropy K-0.25
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One may understand this result by considering the
following Langevin equation for m
With D1/N
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Fokker-Planck Equation
This is the dynamics of a particle moving in a
double well potential V(m)-s(m), with TD1/N
starting at m0.
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Taking for simplicity s(m)am2, agt0, the
problem becomes that of a particle moving in a
potential V(m) -am2 at temperature TD1/N
This equation yields at large t
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Diverging time scales have been observed in a
number of systems with long range interactions.
The Hamiltonian Mean Field Model (HMF) (an XY
model with mean field ferromagnetic interactions)
There exists a class of quasi-stationary m0
states with relaxation time
Yamaguchi, Barre, Bouchet, Dauxois, Ruffo (2004)
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Analysis of the relaxation times (K. Jain, F.
Bouchet, D. Mukamel 2007)
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Distribution function
Vlasov equation
Linear stability analysis
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Linearly stable, power law relaxation time
Linearly unstable, logarithmic relaxation time
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Anisotropic XY model
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Summary
Some general thermodynamic and dynamical
properties of system with long range interactions
have been considered. Canonical and
microcanonical ensembles need not be
equivalent whenever the canonical transition is
first order (yielding negative Specific heat,
temperature discontinuity etc.) Breaking of
ergodicity in microcanonical dynamics due
to non-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
interactions but they are expected to be valid
for algebraically decaying potentials.