The document describes a model of zero-temperature Glauber dynamics for a one-dimensional Ising spin chain undergoing partially synchronous updating. Monte Carlo simulations are used to study the phase transitions of the system as the number of spins updated per time step (cL) varies according to different probability distributions. The model considers both cluster and random sequential updating schemes, where c is assigned at each time step from a distribution before flipping spins based on their energy differences. The goal is to examine how varying the updating parameter c according to different distributions affects the critical behavior of the system.

1. Phase diagram for a zero-temperature Glauber dynamics under partially synchronous
updates
Daniel Kosalla
Institute of Theoretical Physics, University of Wroclaw, pl. Maxa Borna 9, 50-204 Wroclaw, Poland
(Dated: June 12, 2013)
The model of zero-temperature Glauber dynamics one-dimensional system undergoing partially
synchronous, distribution dependent updating mode is being considered. Monte Carlo simulations
are being used to study phase transitions.
I. INTRODUCTION
In the presence of recent developments of SCM (Single
Chain Magnets) [1–4] the issue of criticality in 1D Ising-
like magnet chains has turned out to be an promising
field of study [5–8]. Some practical applications has been
already suggested [2]. However, the details of general
mechanism driving this changes in real world is yet to be
discovered.
II. HYPOTHESIS
Even though the idea of partially synchronous updat-
ing scheme has been suggested [5–7]. This mode was pre-
viously determined by fixed parameter of spins being up-
dated in one step-time. However, one can imagine, that
the number of updated spins/molecules (often referred to
as cL, where: L denotes size of the chain and c ∈ (0, 1]) is
changing as the simulation progresses. If so then it either
is linked to some characteristics of the system or may be
expressed with some probability distribution (described
in Section IV). This approach of changing c parameter
can be applied while choosing spins randomly as well as
in cluster (Section V) and will be considered in this pa-
per.
III. MODEL
In the proposed model cL sequential updating is
used with c due to provided distribution. Monte Carlo
simulations are being used. The considered environment
consist of one dimensional array of L spins si = ±1.
Index of each spin is denoted by i = 1, 2, . . . , L. Periodic
boundary conditions are assumed, i.e. sL+1 = s1.
It has been shown in [8] that the system under
synchronous Glauber dynamics reaches one of two
absorbing states - ferromagnetic or antiferromagnetic.
Therefore, we introduce the density of bonds (ρ) as an
order parameter:
ρ =
L
i=1
(1 − sisi+1)
2L
(1)
FIG. 1: The average density of active bonds in the stationary
state ρst as a function of W0 for c = 0.9 and several lattice
sizes L. [7] B. Skorupa, K. Sznajd-Weron, and R. Topolnicki.
Phase diagram for a zero-temperature Glauber dynamics un-
der partially synchronous updates, Phys. Rev. E 86, 051113
(2012)
As stated in [8] phase transitions in synchronous up-
dating modes and c-sequential [7] ought to be rather
continuous (in cases different then c = 1 for the later).
Smooth phase transition can be observed in the Figure
1.
We consider the systm in low temperatures (T) and
therefore assume T = 0. We consider Metropolis algo-
rithm as a special case of zero-temperature Glauber dy-
namics for 1/2 spins. Each spin is flipped (si = −si)
with rate W(δE) per unit time. While T = 0:
W(δE) =



1 if δE < 0,
W0 if δE = 0,
0 if δE > 0
(3)
In the case of T = 0 The ordering parameter
W0 = [0; 1] (e.g. Glauber rate - W0 = 1/2 or Metropolis
rate W0 = 1) is assumed to be constant. One can
imagine that W0 parameter can in fact be changed
during simulation process but that’s out of scope of
proposed model.
System starts in the fully ferromagnetic state
(ρ = ρf = 0). After each time-step changes are applied
to the system and the next time-step is being evaluated.
After predetermined number of time steps state of
the system is investigated. If the chain has obtained
antiferromagnetic state (ρ = ρaf = 1) or sufficiently
large number of time-steps has been inconclusive then

2. 2
FIG. 2: The average density of active bonds in the stationary
state st as a function of W0 and c for lattice size L = 64.
Simulations were conducted for 5 105 MCS, and averaging
was done over 5 103 samples.
[7] B. Skorupa, K. Sznajd-Weron, and R. Topolnicki. Phase
diagram for a zero-temperature Glauber dynamics under par-
tially synchronous updates, Phys. Rev. E 86, 051113 (2012)
whole simulation is being shout down.
Presented algorithms (Sections V A and V B) are to
be investigated and compared to those with stationary
c values (Figure 2). In addition to that, influence of
changing c-value made on phase transition of the system
remains to be seen.
IV. DISTRIBUTIONS
During the simulation c will not be fixed in time but
rather change from [0; 1] according to some well known
probability distributions, such as:
1. Uniform
0.0 0.2 0.4 0.6 0.8 1.0
0.94
0.96
0.98
1.00
1.02
1.04
1.06
Meaning: c could be any value in the interval [0; 1] with
equal probability.
2. Triangular
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
Meaning: c could be any value in the interval [0; 1] but
is most likely to around value of c = 1/2. Other values
possible but their probabilities are inversely
proportional to their distance from c = 1/2.
3. Gaussian
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Meaning: c could be any value in the interval [0; 1] but
is most likely to be around value of c = 1/2.
4. Well
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Meaning: c could be any value in the interval [0; 1] but
it is least possible that c = 1/2. c is most likely to be at
around its extrema - either c = 0 (sequential) or c = 1
(synchronous).
During simulation c will vary accordingly to above-
mentioned methods. While studying different initial con-
ditions for simulations distributions are to be adjusted in
order to provide peak values in range {0, 1}. This is due
to the fact that the value of 0.5 (as presented on the plots)
would mean that in each time-step half of the spins gets
to be updated.
V. UPDATING
The following algorithms make use of selected proba-
bility density function to assign appropriate c value be-
fore each time step. After (on average) L updated spins
each Monte Carlo Step (MCS) can be distinguished.

3. 3
A. Cluster updating
Update cL consecutive spins starting from
randomly chosen one. Each change is saved to
the new array rather than the old one. After
each Stop updated spins are saved and new
time-step can be started.
1. Assign c value with given distribution
2. Choose random value of i ∈ [0, L]
3. max = i + cL
4. si is i-th spin
• if si = si+1 ∧ si = si−1 :
– si = si+1 = si−1
• otherwise:
– Flip si with probability W0
5. if i ≤ max
• i = i + 1
• Go to step 4
6. Stop
Pseudocode:
def update sequence cluster ( ) :
c = d i s t r i b u t i o n ()
f i r s t i = randrange (0 , L)
l a s t i = f i r s t i + ( c∗L)
S new = copy (S)
for i in xrange ( f i r s t i , l a s t i ) :
i f S [ i −1] == S [ i +1]:
S new [ i ] = S [ i −1]
e l i f w0 > randfloat ( 0 , 1 ) :
S new [ i ] = −S [ i ]
S = S new
B. Random sequence updating
Update cL random spins. Each change
is saved to the new array rather than
the old one. After each Stop, updated
spins are saved and new time-step can
be started.
1. Assign c value with given distribution
2. si is i-th spin
• if si = si+1 ∧ si = si−1 :
– si = si+1 = si−1
• otherwise:
– Flip si with probability W0
3. i = i + 1
4. If i = L: Go to step 3
5. Stop
Pseudocode:
def update sequence random ( ) :
c = d i s t r i b u t i o n ()
S new = copy (S)
for i in xrange (0 , L ) :
i f c < randfloat ( ) :
i f S [ i −1] == S [ i +1]:
S new [ i ] = S [ i −1]
e l i f w0 > randfloat ( 0 , 1 ) :
S new [ i ] = −S [ i ]
S = S new

4. 4
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