Transcript Slide 1

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
Systems with long range interactions
two-body interaction
v(r) a 1/rs
ER
at large r
with s<d, d dimensions
2d s
1
V
s
  1  0
d
Thermodynamics:
since
S  V
F  E  S
E V
1
, S V
the entropy may be neglected in the
thermodynamic limit.
The equilibrium state is just the ground state.
Nevertheless, entropy can not always be neglected.
F  E  S
In finite systems, although E>>S, if T is high enough
E may be comparable to S, and the full free energy
need to be considered.
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
1
2
k BT  Mv v  30 km/sec
3
Thus although
E
E V
5/3
T  1062 K
since T is large
becomes comparable to S
F  E  S
Ferromagnetic dipolar systems
v ~ 1/r3
D 2
H H M
V
 0
(for ellipsoidal samples)
D is the shape dependent demagnetization factor
J

H  H    Si 
N  i 1 
N
2
Models of this type, although they look extensive,
are non-additive.
For example, consider the Ising model:
J N
H 
( S i ) 2
2 N i 1
Si  1
Although the canonical thermodynamic functions (free energy,
entropy etc) are extensive, the system is non-additive
+
E 0
_
E  E   JN / 4
E  E1  E2
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
Ising model with long and short range interactions.
J N
K
2
H 
( S i ) 
2 N i 1
2
N
 (S S
i 1
i
i 1
 1)
d=1 dimensional geometry, ferromagnetic long range
interaction J>0
The model has been analyzed within the canonical
ensemble Nagel (1970), Kardar (1983)
Ground state
+-+-+-+-
+++++
-1/2
K/J
T/J
2nd order
1st order
-1/2
0
K/J
Canonical (T,K) phase diagram
M 0
M 0
Microcanonical analysis
Mukamel, Ruffo, Schreiber (2005); Barre, Mukamel, Ruffo (2001)
J N
K
2
H 
( S i ) 
2 N i 1
2
N
 (S S
i 1
i
i 1
 1)
J
E
M 2  KU
2N
M  N  N
N  N  N
U = number of broken bonds in a configuration
Number of microstates:
U/2 (+) segments
 N  N 
( N  , N  ,U )      
U / 2 U / 2 
U/2 (-) segments
s=S/N ,  =E/N , m=M/N , u=U/N
1
s (u, m)  ln  
N
1
1
(1  m) ln( 1  m)  (1  m) ln( 1  m)  u ln u
2
2
1
1
 (1  m  u ) ln( 1  m  u )  (1  m  u ) ln( 1  m  u )
2
2
J 2
   m  Ku
2
but
Maximize
s( , m)
to get
s( )  s( , m( ))
s(u, m)  s( , m)
m( )
1/ T  s / 
s( , m)  s0 ( )  am  bm  ...
2
4

continuous transition:
s
m
discontinuous transition:


In a 1st order transition there is a discontinuity in T, and thus there
is a T region which is not accessible.
m=0
m0


discontinuity in T

Microcanonical phase diagram
canonical
microcanonical
ln 3
 0.317
2 3
TMTP / J  0.359
TCTP / J  
The two phase diagrams differ in the 1st order region of the canonical diagram
In general it is expected that whenever the canonical transition
is first order the microcanonical and canonical ensembles
differ from each other.
S
E1
E2
E
Dynamics
Microcanonical Ising dynamics:
J N
K
2
H 
( S i ) 
2 N i 1
2
N
 (S S
i 1
i
i 1
 1)
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 the
system with energy  E
This is implemented by adding an auxiliary variable,
called a demon such that
ES  E D  E
system’s energy
ES
demon’s energy
ED  0
Creutz algorithm:
1. Start with
ES  E
ED  0
2. Attempt to flip a spin:
accept the move if energy decreases
and give the excess energy to the demon.
ES  ES  E
ED  ED  E , E  0
if energy increases, take the needed energy from the
demon. Reject the move if the demon does not have
the needed energy.
ES  ES  E ,
ED  ED  E  0
P( ED )  e
 E D
  1 / k BT
Yields the caloric curve T(E).
N=400, K=-0.35
E/N=-0.2416
To second order in ED the demon distribution is
P( ED )  e
 ED  ED2 / 2CV T 2
S
1 2S 2
S ( Es )  S ( E ) 
ED 
ED
2
E
2 E
S 1
2S
1
 ,
 2
2
E T
E
T CV
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 entropy
increases with E (namely T>0) since the next to leading term is
of 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 convex
region of their extensive parameter space.
M1
E
M2
M
If there are two microstates with magnetizations M1 and M2
Then there are microstates corresponding to any magnetization
M1 < M < M2
.
This is not correct for systems with long range interactions
where the domain over which the model is defined need not
be convex.
E
M
Ising model with long and short range interactions
J 2
   m  Ku
2
m=M/N= (N+ - N-)/N
u =U/N = number of broken bonds per site in a configuration
for N  N one has U  2 N  N  M
corresponding to isolated down spins
+++-++++-++-++++-++
Hence:
0  u  1 m
The available
( , m) is not convex.
J 2
0 /K 
m  1 m
2K
K=-0.4
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 fact
that 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 left
segments by lowering its energy with a probability which is
exponentially small in the system size.
Time scales
Relaxation time of a state at a local maximum of the entropy
(metastable state)
0  ms
s
0
m1 ms
For a system with short range interactions
 (0  ms ) is finite. It does not diverge with N.
m=0
ms
R
0
m1ms
dR

 f 
dt
R
f  f (0)  f (ms )  0, free energy difference
 surface free energy
Rc   / f
critical radius above which droplet grows.
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)
0
 (0  ms )  e
Ns
m1ms
s  s(0)  s(m1 )
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 entropy
K=-0.4   0.318
Relaxation of a state with a local minimum of the entropy
(thermodynamically unstable)
0
ms
0  ms
One would expect the relaxation time of the m=0
state to remain finite for large systems (as is the case
of systems with short range interactions..
M=0 is a minimum of the entropy
K=-0.25   0.2
  ln N
One may understand this result by considering the following
Langevin equation for m:
m s

  (t )
t m
With
  (t ) (t ' )  D (t  t ' )
D~1/N
s(m)  am2  bm4
a, b  0
Fokker-Planck Equation:
P(m, t )
P(m, t  0)   (m)
P
 2 P   s 
D 2 
P

t
m m  m 
This is the dynamics of a particle moving in a double well
potential V(m)=-s(m), with T~D~1/N starting at m=0.
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
P
2P

mP
 D 2 a
t
m
m
This equation yields at large t
 am 2e 2 at 

P(m, t )  exp  
2D 

Since D~1/N the width of the distribution is
e 2 a / N  1
  ln N
 m  e
2
2 at
/N
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)
1 N 2 1 N
2
H   pi 
(
1

cos(



))

i
j
2 i 1
2 N i , j 1
m>0
m=0
c  3/ 4

There exists a class of quasi-stationary m=0 states
with relaxation time
  N
with
  1.7
Yamaguchi, Barre, Bouchet, Dauxois, Ruffo (2004)
Analysis of the relaxation times:
(K. Jain, F. Bouchet, D. Mukamel 2007)
m=0
m>0
  ln N
  N
c  3/ 4
 *  7 / 12

1 N 2 1 N
2
H   pi 
(
1

cos(



))

i
j
2 i 1
2 N i , j 1
d i
 pi
dt
dpi
 mx sin  i  m y cos  i
dt
1 N
mx  i 1 cos  i
N
1 N
m y  i 1 sin  i
N
Distribution function
f ( , p, t )
Vlasov equation
f
f V f
p

0
t
  p

V ( , t )   dp'  d '(1  cos(   ' )) f ( ' , p' , t )

f 0 ( p) 
1
2 p0
 p0  p  p0
Linear stability analysis:
f ( , p, t ) 
1
f 0 ( p)  f1 ( , p, t )
2

1
N
 *    c
  *
Linearly stable, power law relaxation time
Linearly unstable, logarithmic relaxation time
c  3/ 4
m
1 t
e
N
 * 7 / 12
  6(7 / 12   )
m=0
m>0
  ln N
 *  7 / 12
  N
c  3/ 4

Anisotropic XY model
1
1
D 

2
2
H   pi 
(1  cos(i   j )) 
  cos i 

2 i 1
2 N i , j 1
2 N  i 1

N
N
N
2
2
linearly stable
Linearly unstable
 c  (3  D) / 4
  N
  ln N
D
 *  (7  D) / 12
m=0
0
0.5

1
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.