Effect of Dilution on Spinodals and Pseudospinodals

0.19 MB·English

by  K. Liu

Effect of Dilution on Spinodals and Pseudospinodals
K. Liu,1,∗ W. Klein,1,2,† and C. A. Serino1,‡
1Department of Physics, Boston University, Boston, Massachusetts 02215
2Center for Computational Science, Boston University, Boston, Massachusetts 02215
(Dated: January 30, 2013)
We investigate the effect of quenched dilution on the critical and spinodal points in the infinite
range(mean-field)andlong-range(near-mean-field)Isingmodel. Wefindthatunliketheshort-range
3
Isingmodel,theeffectofthedilutionisnotsimplyrelatedtothedivergenceofthespecificheat,i.e.,
1
theHarris criterion. We also find the mean-field behavior differs from that of d=4 (uppercritical
0
dimension) nearest neighbor model at the critical point. These results are an important first step
2
forunderstandingtheeffectofthespinodalonthenucleationprocessaswellasthepropertiesofthe

n metastablestateinsystemsofconsiderableinterestinmaterialscience,geophysics,andeconophysics
a which in general havedefects.
J
9 PACSnumbers:
2

] Long range interactions are common as wellas impor- shortrangeIsing models in the presence of vacancies. In
h
tant in both physical and social systems. Metals [1, 2] this paper we will investigate the same question for mf
c
and earthquake faults [3], both with long-range elastic and nmf critical points, spinodals and pseudospinodals.
e
m interactions, as well as biological and polymer materi- As we will show, mf and nmf Ising systems have a more
als [4] all have processes strongly influenced by the long complicated response to defects than nearest neighbor
-
t range nature of their interactions. In addition, economic models. Moreover, the response is different than what
a
t systems as well as societal interactions that facilitate one finds in d = dc = 4, the Ising upper critical dimen-
s
. the transmission of diseases [5] have effective long range sion [14, 15].
t
a “interactions” due to globalization. In all of these sys- We beginby noting thatthe Ginzburg criterionfor mf
m tems metastablestatesarecommon[6,7]andnucleation to be a good approximation in spin systems is that the
- plays an important role. In addition, it has been estab- fluctuations in the magnetization are small compared to
d
lished that in long range systems the nucleation process its mean value. At the critical point [8]
n
is strongly affected by the pseudospinodal [8]. It is im-
o ξdχ
c portant to distinguish mean-field (mf) systems with in- G−1 = T =0 G=Rdǫ2−d/2 , (1)
[ finite range interactions [9, 10] from systems with long ξ2dφ2 → →∞

but finite range interactions. We will refer to the latter
1 where for interactions with range R, the correlation
v as near-mean-field (nmf). Mean-field systems have spin- length is ξ = Rǫ−1/2, the reduced temperature is ǫ =
1 odalswhilenmfsystemshavepseudospinodals[8,11,12].
(T T )/T ,andT is the criticaltemperature. The sus-
c c c
2 Pseudospinodals behave as spinodals unless they are ap- cep−tibility, χ ǫ−1, and the magnetization per spin,
68 proached too closely as we will discuss below. φ ǫ1/2. WT e∝can use G to distinguish between mf
∝
1. Although the spinodal and pseudospinodal have a (G→∞) and nmf (G≫1 but not infinite) for all d.
Near spinodals or pseudospinodals the same consider-
0 strong effect on the metastable state and nucleation in
3 defect free mf and nmf systems [4, 8], the systems men- ations lead to the criterion [8]
1
: tioned above generally have defects. In materials such Gs =Rd∆h3/2−d/4 1 (2)
v as metals, the defects can take the form of impurities ≫
i and/or vacancies [7]. In economic and other social sys- for there to be a well pronounced pseudospinodal; G
X s
→
tems agents that behave differently than the majority foratruespinodal[8,11]. Herethecorrelationlength
r ∞
a canbe thoughtofas defects. Examplesinclude the pres- is ξ = R∆h−1/4, the susceptibility is χ ∆h−1/2, and
T
∼
ence of immune individuals in models of disease trans- the difference between the magnetization and its value
mission [6] and agents that do not participate in wealth at the spinodal is ∆φ ∆h1/2 where ∆h = (h h)/h
s s
∼ −
exchange in economic models. This raises an important, whereh isthevalueofthemagneticfieldatthespinodal.
s
but as yet unanswered, question: How, if it all, does the Some of the simulations we discuss below involve fully
presence of defects affect the spinodal and pseudospin- connected models, i.e., every spin interacts with every
odal? This question is analogous to what was asked other spin. As in finite size scaling, we take the volume
about critical points in systems with defects and short of the system, N, to be either ξd = Rdǫ−d/2 (critical
range interactions several decades ago. In the 70’s, Har- point) or Rd∆h−d/4 (spinodal). Then G = Nǫ2(critical
ris [13] developed a criterion associated with the specific point) and G =N∆h3/2 (spinodal).
s
heatexponentinthedefectfree,orpure,systemthatpre- The Harris criterion [13] (HC) compares the width of
dictedwhetherthecriticalpointremaineswelldefinedin the distribution of vacant sites (p fraction of occupied
≡
2
sites) to the distance from the continuous transition. At
1.8
the critical point this leads to no change in exponents if 1D N=9000 R=450
1.6 α =1.51
[ξdp(1 p)]1/2 2D N=10000 R=5
− ǫ. (3) 1.4 α =1.00
ξd ≪
1.2
)
v
Using again the fact that ξ = Rǫ−1/2 Eq. (3) results in C( 1
G=Rdǫ2−d/2 1. og
≫ l0.8
How does the HC relate to the specific heat in mf and
nmf systems? We can obtain the specific heat by calcu- 0.6
latingtheenergyfluctuationsperunitvolume. Inmfand
0.4
nmf, the energy fluctuations are the square of the mag-
netization fluctuations. The magnetization fluctuations 0.−22 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6
log(ǫ)
per unit volume in mf and nmf do not scale as ǫβ (with
β =1/2 at mean-field critical points) but rather as [8]
FIG. 1: (color online) Plots of the fluctuations in the en-
ergy(specificheat)asafunctionofthedistancefromthecrit-
ǫ1/2
m . (4) ical point for thepuresystem in d=1 and 2 for T >Tc.
∼ (Rdǫ2−d/2)1/2
At critical points, the scaling in Eq. (4) can be seen

by looking at a Landau-Ginzburg-Wilson Hamiltonian C
150 V
(LGW) in zero field [15] We scale all lengths with ξ and
χ
m(~x) by ǫ1/2 and obtain
φ4(~y) 100
H(m)=Rdǫ2−d/2 d~y ( φ(~y))2+φ2(~y)+ ,
Z (cid:20) ∇ Rdǫ2−d/2 2(cid:21)
(cid:0) (cid:1)(5)
where ~y is the scaled length and φ(~y) is the scaled mag- 50
netization. As can be readily seen from using Eq. (5)
to calculate the partition function, φ(~y) is of order
(Rdǫ2−d/2)−1/2. This scaling reduces the LGW Hamil-
0
tonian to a Gaussian because we can ignore the scaled

φ4 term. For T > T and h = 0, the specific heat at 3.6 3.65 3.7 3.75
c
T
constant h is
C ǫ1/2 4Rdǫ−d/2 = 1 . (6) FIG. 2: (color online) Susceptibility and specific heat as a
V ∼(cid:18)√Rdǫ2−d/2(cid:19) Rdǫ2−d/2 functionofǫforasystemwithaveragedilutionof10%. Sites
were designated as empty with a 0.1 probability. Twenty
Thisisthesquareoftheenergyfluctuationsperunitvol- systems were simulated.
ume and hence for T > T , the HC for the mf (or nmf)
c
critical point to be unaffected by the dilution becomes
C 1. In Fig. 1 we plot the energy fluctuations (spe- system crosses over to d = 3 short-range Ising behavior,
V
≪
cific heat) of the non-diluted model as a function of ǫ in the vacancies alter the critical behavior from what one
d=1,2 (blue and red, respectively). For T >T the di- obtains in the pure model.
c
vergenceisconsistentwithEq.(6),namelyǫ−3/2ind=1 Our resultis somewhatdifferent thanone obtainedby
and ǫ−1 in d=2. Ballesteros et al. [14] who looked at the nearest neigh-
There are several points to be made. The first is that bor dilute Ising model in d = dc = 4 where the specific
at the critical point the nmf criterion that G 1 is the heat exponent α = 0, so the HC is indeterminate [13].
≫
sameastheHCsothatifthesystemiswellapproximated Despite no power-law divergence, the pure system does
by mf theory, the critical point remains well defined and have a logarithmically diverging specific heat. It was
unchanged in the presence of vacancies. In addition, for found in Ref. [14] that the logarithmiccorrectionsto the
G (mf) the criticalpoint is unchangedby the dilu- exponents changed; however, the exponents themselves
→∞
tion. The second point is that for R 1 but finite, the did not. For the mf (G ) system we consider, the
≫ → ∞
affect of the dilution depends on the distance from the HC predicts that the nature ofthe criticalpointremains
critical point ǫ. Hence dilution in the d= 3 Ising model the same as the pure system.
has no affect on the apparent mean-field critical point In Fig. 2 we plot the susceptibility χ and specific heat
if G 1 and finite. However, as ǫ decreases and the C asafunctionofǫ>0forthesystemwithdilutionand
V
≫
3
G 1 but finite for the fully connected model. As we
app≫roach the critical point and G decreases, the spread 1.8
N=40000 R=10 q=0
inboth CV andχ increasesconsistentwith the HC.(See 1.7 α=0.5029
Fig. 5)
We next turn to the spinodal. The HC is 1.6
[ξdp(1 p)]1/2 constant )v1.5
− = ∆h. (7) C
ξd ξd/2 ≪ (1.4
g
o
With ξ =R∆h−1/4, the HC becomes l1.3
1.2
∆h−1/2
1. (8)
Rd∆h3/2−d/4 ≪ 1.1
The specific heat is again related to the fluctuations of 1
−1.8 −1.6 −1.4 −1.2 −1 −0.8
the energy. Near the spinodal the magnetization is non- log(∆h)
zero. Byusingessentiallythesameargumentasoutlined
in Eq. (5), the fluctuations in the magnetizationscale as FIG. 3: The specific heat for the pure system plotted as a
functionofhford=2. Thefitisconsistentwithadivergence
∆h1/2 with exponent1/2 consistent with the scaling analysis.
m , (9)
∼ √Rd∆h3/2−d/4
where we have expanded the LGW Hamiltonian around
the magnetization at the spinodal [8].

The specific heat (energy fluctuations)at the spinodal C
V
is 20 χ
∆h1/2 2 2
C +φ φ2 Rd∆h−d/4
SP
∝(cid:20)(cid:18)√Rd∆h3/2−d/4 (cid:19) − (cid:21) 15
∆h−1/2. (10)
∝
Note that near the spinodal the magnetization does not 10
go to zero as the spinodal is approached. Near the pseu-
dospinodal, the HC is not simply related to the specific
heat. What is required for the pseudospinodal to be un- 5
changedby the presence of dilution according to the HC
isthatthespecificheatdividedbytheGinzburgparame-
0
terbesmall. Thisresultimpliesthatinmean-fieldwhere

the R before ∆h 0 [9, 10] the spinodal is unaf- 1 1.05 1.1 1.15
→ ∞ → h
fected by the dilution. In Fig.3 we plotthe specific heat
(energyfluctuations)inthe long-rangeIsingmodelwith-
FIG. 4: (color online) The susceptibility and specific heat as
out dilution as the spinodal is approached. The result is
a function of the distance from the spinodal for the dilute
consistent with C ∆h−1/2.
SP ∼ fully connected model. Twenty systems with a probability
In Fig. 4 we plot 20 realizations of the isothermal sus- of a site being diluted of 0.1 were simulated. The existence
ceptibility and specific heat as a function of h near the of the spread for small ∆h is consistent with the theoretical
pseudospinodalforthefullyconnectedmodelwith9 104 analysis.
×
spins. Theemptysitesaredistributedwithaprobability
of 0.10 as for the critical point. Note that the curves for
each simulation overlap until ∆h becomes too small at In Fig. 5 we plot the width or spread of the suscepti-
which point the pseudospinodal is smeared as evidenced bilitynearthe criticalpoint(normalizedbythe suscepti-
bythefactthatthecurvesshowadistinctspread. Thisis bility) for a 20%and40%dilution in the fully connected
consistent with our analysis. For both the critical point model for two system sizes. The width increases as ǫ de-
andthepseudospinodalthespreadcausedbythedilution creasesasexpected. Wedidnotdisplaydataforǫ<0.04
inthe fully connectedmodelnarrowsfora givenvalueof because the nmf picture breaks down for ǫ 0.05. Simi-
∼
thedistancefromthepseudospinodalorthecriticalpoint lar results are obtained for the pseudospinodal [17]. Be-
as the system size is increased and hence R increases as cause the interaction range is a function of the system
would be expected from our analysis. size,weexpectthespreadofthesusceptibilityassociated
4
on nucleation and pseudospinodals in more complicated
1.5 models associated with materials such as the long-range
6
q=0.20 N=10
Ising antiferromagnet [8]? What is the effect of other
q=0.20 N=9×106 types of damage such as fixed spins or dislocations?
q=0.40 N=106 What is the effect of variousforms ofdamage and impu-
1 q=0.40 N=9×106 rities in models in other fields such as econophysics and
thespreadofdisease? Theanswerstothesequestionsare
χ of considerable importance in a variety of fields of study
¯χ
δ ranging from materials and biophysics to economics and
social science.
0.5
ThisresearchwassupportedbytheDepartmentofEn-
ergy through grant DE-FG02-95ER14498. We acknowl-
edge useful conversations with H. Gould, J. Silva and J.
B. Rundle.
0
−0.1 −0.05 ǫ0 0.05 0.1
FIG. 5: (color online) Width of the susceptibility (variance
normalized by the mean) distribution near the critical point ∗ Electronic address: [email protected]
(see Fig. 2) for (a) q=0.20 and (b) q=0.40 with a cutoff of † Electronic address: [email protected]
ǫ=0.04. ‡ Electronic address: [email protected]
[1] W. Klein, T. Lookman, A. Saxena, and D. M. Hatch,
Phys. Rev.Lett. 88, 085701 (2002).
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forthesamevalueofǫ. Thisisconsistentwiththescaling
(1989).
analysis. Asimilarresultis foundnearthe spinodal[17]. [3] C. A. Serino, K. F. Tiampo, and W. Klein, Phys. Rev.
In conclusionwe have found that the effect of dilution Lett. 106, 108501 (2011).
in mf and nmf systems on critical points is more subtle [4] K. Binder, Phys. Rev.A 29, 341 (1984).
[5] P.S.DoddsandD.J.Watts,Phys.Rev.Lett.92,218701
than the effect in systems with short range interactions
(2004).
below the upper critical dimension, and it is different
[6] P. Ball, Complexus 1, 190 (2003).
from the behavior of short range systems in d = d di-
c [7] K. F. Kelton and A. L. Greer, Nucleation in Condensed
mensions. Wealsofoundthatinmfsystemsthespinodal Matter-Applications in Materials and Biology, Elsevier
isnotchangedbydilutionbutinmorephysicallyrealistic 2010.
nmf systems the spread and consequent rounding of the [8] W. Klein, H. Gould, N. Gulbahce, J. B. Rundle, and K.
pseudospinodal depends on the distance from the point F. Tiampo, Phys.Rev.E 75, 031114 (2007).
of the apparent divergence. At the spinodal the HC is [9] M. Kac, G. E. Uhlenbeck, and P. Hemmer, J. Math.
Phys. 4, 216 (1963).
not simply related to the mf specific heat exponent [16].
[10] J. L. Lebowitz and O. Penrose, J. Math. Phys. 7, 98
Near T for h = 0 the Harris criterion is simply related
c (1966).
to the specific heat for T > Tc. Near the critical point [11] N.Gulbahce,H.Gould,andW.Klein,Phys.Rev.E69,
but for T < Tc we found a result similar to what one 036119 (2004).
finds near the spinodal. These results will be published [12] D.W.Heermann,W.Klein,andD.Stauffer,Phys.Rev.
in Ref. [17]. Lett. 49, 1262 (1982).
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[14] H. G. Ballesteros, L. Fernandez, V. Martin-Mayor, A.
odal on nucleation in a wide variety of systems is not
Munoz Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo, Nu-
eliminated by the presence of dilution and that spinodal clear Physics B 512, 681 (1998).
nucleationremains animportantfeature ofsystems with [15] S.K.Ma, Modern Theory ofCritical Phenomena, West-
long range interactions. view Press (1976).
Our results raise many questions and suggest several [16] See<www.physik.uni-leipzig.de/~janke/ENRAGE-School08/talks/
new research directions. What is the effect of dilution [17] K. Liu, W. Klein, and C. A.Serino, in preparation.