a r X i v :c o n d -m a t /0201472v 1 [c o n d -m a t .s t a t -m e c h ] 25 J a n 2002
Binder cumulants of an urn model and Ising model above critical dimension
Adam Lipowski 1),2)and Michel Droz 1)
1)
Department of Physics,University of Gen`e ve,CH 1211Gen`e ve 4,Switzerland 2)
Department of Physics,A.Mickiewicz University,61-614Poznan,Poland
(February 1,2008)
Solving numerically master equation for a recently introduced urn model,we show that the fourth-and sixth-order cumulants remain constant along an exactly located line of critical points.Obtained values are in very good agreement with values predicted by Br´e zin and Zinn-Justin for the Ising model above the critical dimension.At the tricritical point cumulants acquire values which also agree with a suitably extended Br´e zin and Zinn-Justin approach.
The concept of universality and scale invariance plays a fundamental role in the theory of critical phenomena [1].
It is well known that at criticality the system is characterized by critical exponents.Calculation of the exponents for dimension of the system d lower than the so-called critical dimension d c is a highly nontrivial task [2].On the other hand for d >d c the behaviour of a given system is much simpler and critical exponents take mean-field values which are usually simple fractional numbers.
However,not everything is clearly understood above the critical dimension.One of the examples is the Ising model (d c =4)where despite intensive rearch rious discrepancies between analytical [3]and numerical [4]calculations still persist.Of particular interest is the value of the Binder cumulant at the critical point.Several years ago Br´e zin and Zinn-Justin (BJ)calculated this quantity using field theory methods [5]and only recently numerical simulations for the d =5model are able to confirm it [6].Some other properties of the Ising model above critical dimension are still poorly explained by existing theories.For example,the theoretically predicted leading corrections to the susceptibility disagree even up the sign with numerical simulations [4].
In addition to direct simulations of the nearest-neighbour Ising model,there are also some other ways
february的音标to study the critical point of Ising model above critical dimension.For example,Luijten and Bl¨o te ud the model with d ≤3but with long-range interactions [7].Using such an approach they confirmed with good accuracy the BJ predictions for the Binder cumulant.
In the prent paper we propo yet another approach to the problem of cumulants above critical dimension.Namely,we calculate fourth-and sixth-order cumulants at the critical point of a recently introduced urn model [8].Albeit structureless,this model exhibits a mean-field Ising-type symmetry breaking.Along an exactly located critical line,the obtained values are in a very good agreement with values predicted by BJ.Let us notice that our calculations:(i)are not affected by the inaccuracy of the location of the critical point which is a rious problem in the ca of the Ising model (ii)are bad on the numerical solution of the master equation which offers a much better accuracy than Monte Carlo simulations.Moreover,we calculate the cumulants at the tricritical point and show that the obtained values are also in agreement with suitably extended calculations of BJ.That both the Ising model and the (structureless)urn model have the same cumulants is a manifestation of strong universality above the upper critical dimension:at the critical point not only the lattice structure but also the lattice itlf becomes irrelevant.What really matters is the type of symmetry which is broken and since in both cas it is the same Z 2symmetry,the equality of cumulants follows.
Our urn model was motivated by recent experiments on the spatial paration of shaken sand [9].In the prent paper we are not concerned with the relation with granular matter and a more detailed justification of rules of the urn model is omitted [8].The model is defined as follows:N particles are distributed between two urns A and B and the number of particles in each urn is denoted as M and N −M ,respectively.Particles in a given urn (say A)are subject to thermal fluctuations and the temperature T of the urn depends on the number of particles in it as:
T (x )=T 0+∆(1−x ),
(1)
where x is a fraction of a total number of particles in a given urn and T 0and ∆are positive constants.(For urn A and B,x =M/N and (N −M )/N ,respectively.)Next,we define dynamics of the model [8]:(i)One of the N particles is lected randomly.(ii)With probability exp[−1
ǫ=2M−Nstreet是什么意思
N
−
1
T(<M/N>)]=<N−M>exp[
−1
2+<ǫ>)exp[
−1
2
+<ǫ>)
]=(
1
noproblemT(1
∆/2−∆/2,0<∆<
2
3
,T0=
√
3
.Let us notice that a random lection of
particles implies basically the mean-field nature of this model.Conquently,at the critical pointβ=1/2andγ≈1 (measured from the divergence of the variance of the order parameter),which are ordinary mean-field exponents. However,the calculation of the dynamical exponent z gives z=0.50(1)[8]while the mean-field value is2.We do not have convincing arguments which would explain such a small value of z.Presumably,this fact might be related with a structureless nature of our model.
Defining p(M,t)as the probability that in a given urn(say A)at the time t there are M particles,the evolution of the model is described by the following master equation
bathroom怎么读p(M,t+1)=N−M+1
N
p(M+1,t)ω(M+1)+ p(M,t){
M
N
[1−ω(N−M)]}for M=N−1
p(0,t+1)=
1
N
p(N−1,t)ω(1)+p(N,t)[1−ω(N)],(6) whereω(M)=exp[−1
<ǫ2>2,x6=
<ǫ6> N
−
1
8,1
2
imply翻译
and2
x 4=
1
toe
4
)]4≈,x 6=
3
4
)]4≈
(9)
The fact that one can restrict the expansion of the free energy to the lowest order term is by no means obvious [3].Such a restriction leads to the correct results but only above critical dimension where the model behaves according to the mean-field scenario with fluctuations playing negligible role.For d <d c additional terms in the expansion are also important and cumulants take different value.Numerical confirmation of the above results requires extensive Monte Carlo simulations,and a satisfactory confirmation was obtained only for x 4[6,10].
Omitting detailed field theory analysis,we can extend the BJ approach to the tricritical point.At such a point also the quartic term vanishes which makes the sixth-order term the leading one and the probability distribution gets theprinceton
form p (x )∼e −x 6
.Simple calculations for such a distribution yield
linqx 4=
Γ(5
6)2)2
=2,x 6=
Γ(1
6Γ(1
3
(tricritical point).
Arrows indicate the BJ results for the critical and the tricritical point.
3
4
5
6
7
00.0020.004
0.0060.0080.01
x 6(N )
1/N
FIG.2.The same as in Fig.1but for the sixth-order cumulant x 6(N ).
The BJ results (9)-(10)are indicated by small arrows in Figs.1-2.Even without any extrapolation one can e,especially for critical points,a good agreement with our results.Data in Figs.1-2shows strong finite-size corrections.To have a better estimations of asymptotic values in the limit N →∞we assume finite size corrections of the form
x 4,6(N )=x 4,6(∞)+AN −ω.
(11)
The least-square fitting of our finite-N data to eq.(11)gives x 4,6(∞)which agree with BJ values (9)-(10)within the accuracy better than 0.1%.A better estimation of the correction exponent ωis obtained assuming that x 4,6(∞)are given by the BJ values.The exponent ωequals then the slope of the date in the logarithmic scale as prented in Figs.3-4.Our data shows that for the critical(tricritical)point ω=13).
Let us notice that leading finite-size corrections to the Binder cumulant in the d =5Ising model at the critical point are also of the form N −0.5(with N being the linear system size)[7].Moreover,for the tricritical point but d <d c the probability distribution is known to exhibit a three-peak structure [11],which is different than the single-peak form
p (x )∼e −x 6
.
-2
-1.5-1-0.500.512 2.53
3.54
4.55
l o g 10[x 4,6(B J )-x 4,6(N )]
log 10(N)
FIG.3.Logarithmic plot of x 4(BJ )−x 4(N )(+)and x 6(BJ )−x 6(N )(×)as a function for N for ∆=0.5.Dotted straight lines have slope 0.5.
-1.4
-1.2-1-0.8-0.6-0.4-0.20
0.20.42
2.5
3
3.54
4.5
chestnut5
l o g 10[x 4,6(B J )-x 4,6(N )]
log 10(N)
FIG.4.Logarithmic plot of x 4(BJ )−x 4(N )(+)and x 6(BJ )−x 6(N )(×)as a function fo N for ∆=
2
3
.In summary,we calculated fourth-and sixth-order cumulants at the critical and tricritical points in an urn model
which undergoes a symmetry breaking transition.Our results confirm that,as predicted by Br´e zin and Zinn-Justin,appetite
the critical probability distributions of the rescaled order parameter has the form p (x )∼e −x 4
.Similarly,for the
tricritical point our results suggest that p (x )∼e −x 6
.
Although in our opinion convincing,the results are obtained using numerical methods.It would be desirable to have analytical arguments for the generation of such probability distributions.It ems that for the prented urn model this might be easier than for the Ising-type models.Let us notice that for the simplest urn model,which was introduced by Ehrenfest [12],the steady-state probability distribution can be calculated exactly in the continuum
limit of the master equation and the result has the form p (x )∼e −x 2
,where x is now proportional to the difference
of occupancy ǫ.In the Ehrenfest model there is no critical point and we expect that a distribution of the type e −x
2
might characterize our model but offthe critical line(in the symmetric pha).We hope that when suitably extended, an analytic approach to our model might extract critical and tricritical distributions as well.Such an approach is left as a future problem.
ACKNOWLEDGMENTS
This work was partially supported by the Swiss National Science Foundation and the project OFES00-0578”COSYC OF SENS”.
