Ordering dynamics in zerotemperature randomfield Ising model

1
Ordering dynamics in zero-temperature
random-field Ising model

• Hiroki Ohta
• collaborated with Shin-ichi Sasa
• Department of Pure and Applied Science(Komaba),
• University of Tokyo

2
Outline of this presentation

• Definition of the model and problem related to a
  critical phenomanon
• Theoretical framework
• Derivation of the evolutional equation Structure
  of Bifurcation and critical exponents
• Direct numerical experiment of the model
• Concluding remarks
• (including a possible relation to jamming
  transition)

3
Definition of the model
zcoordition number, Bia set of connected sites
of site i
Fioperator which makes spin si flip
Transition rate(Glauber)
(ß-10)
Master equation
For example,(z2)
Distribution of random-field
ordering
h is decreasing
Setting R as that in Ferromagnetic phase
h
t0
jammed
at hc (R)
4
Definition of the problem in this model

• -gt Dynamical properties in ordering phase
• Specially, approaching hc, how characteristic
  length(size) and time grow.

no theory
(Related to Rosinbergs talk on yesterday)
m
jump
?0.7(3d at Rc)
h
F.J.Perez,Vives PRB,2004 Carpenter,Dahmen PRB,2003
h
jump disappears at Rc
We try to give a theoretical understanding to
this problem.
5
Derivation of the equation of effective variables
(1)
effective variables
Assumption1 Markov process
?1
?3
?0
?2
?
?-
?0
?
?2
?1
?1
?4
At the initial condition, densities of each
effective variables are determined, which
depend parameter space (R,h).
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Derivation of the equation of effective variables
(2)
Derivation of the evolutional equation
?1
?0
Assumption2 Occupied probabilities depend on
only those densities. (not exact)
?0
Generally, we can write
?3
?
?0
?0
?

?2
?
7
From the transition rate, we can also obtain
The behavior of this dynamical system depends on
the parameter space (R,h).
8
Bifurcation(RltRc)
This leads to near hc .(eh-hc)
f is a variable,which is linear coupling of
elements of ?
9
RRc (jump disappears)
With assuming diffusion coupling, We can predict
?2/3,?1/3,z?/?2(mean field system).
Naïve estimate leads to
?z?4/3,??G/d2/3(3-dimensional lattice system)
With using upper critical dimension 6 (Dahmen
Sethna,PRB,1996),
?G2(1/36).
10
Numerical experiment at RRc
in 3d(Ohta,Sasa,PRE), ?0.7(2/3),?1.3(4/3) This
result strongly suggests that we can understand
it as a deviation of the mean field system.
(naïve estimate)
11
Concluding remarks

• We theoretically clarify that in an ordering
  process of random-field ising model on a random
  graph, saddle-node bifurcation occurs at RltRc and
  the degenerate form appears at RRc.
• Our theory may be a departure point to understand
  the criticality at hhc(Rc) in 3-dimensional
  lattice system.

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A possible Relation to Jamming transition

• A typical Jamming transition

Contact number
Characteristic length
ß1/2
(Berhinger et al,PRL,2006)
Our model(on random graph)
Magnetization
m
ß1/2
Power-law distributed avalanches occur at
hc(R). the exponent is 3/2(rigorous). (Shukla,et
al,J.Stat.Phys,1999)
(Dahmen et al,Nature,2001)
h
13
phase diagram