Effects of aerogel-like disorder on the critical behavior of O(m)-vector models. Recent simulations and experimental evidence
We review recent results on the effect of a specific type of quenched disorder on well known O(m)-vector models in three dimensions: the XY model (3DXY, m = 2) and the Ising model (3DIS, m = 1). Evidence of changes of criticality in both systems, when confined in aerogel pores, is briefly refere...
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irk-123456789-1213252017-06-15T03:04:37Z Effects of aerogel-like disorder on the critical behavior of O(m)-vector models. Recent simulations and experimental evidence Vasquez, C. Paredes, R. We review recent results on the effect of a specific type of quenched disorder on well known O(m)-vector models in three dimensions: the XY model (3DXY, m = 2) and the Ising model (3DIS, m = 1). Evidence of changes of criticality in both systems, when confined in aerogel pores, is briefly referenced. The 3DXY model represents the universality class to which the λ-transition of bulk superfluid 4He belongs. Experiments report interesting changes of critical exponents for this transition, when superfluid 4He is confined in aerogels. Numerical evidence has also been presented that the 3DXY model, confined in aerogel-like structures, exhibits critical exponents different from those of bulk, in agreement with experiments. Both results seem to contradict Harris criterion: being the specific heat exponent negative for the pure system (α3DXY ' −0.011 < 0), changes should be explained in terms of the extended criterion due to Weinrib and Halperin, which requires disorder to be long-range correlated (LRC) at all scales. In numerical works, aerogels are simulated by the diffusion limited cluster-cluster aggregation (DLCA) algorithm, known to mimic the geometric features of aerogels. These objects, real or simulated, are fractal through some decades only, and present crossovers to homogeneous regimes at finite scales, so the violation to Harris criterion persists. The apparent violation has been explained in terms of hidden LRC subsets within aerogels [Phys. Rev. Lett., 2003, 90, 170602]. On the other hand, experiments on the liquid-vapor (LV) transition of ⁴He and N₂ confined in aerogels, also showed changes in critical-point exponents. Being the LV critical-point in the O(1) universality class, criticality may be affected by both short-range correlated (SRC) and LRC subsets of disorder. Simulations of the 3DIS in DLCA aerogels can corroborate experimental results. Both experiments and simulations suggest a shift in critical exponents to values closer to the SRC instead of those of the LRC fixed point. 2006 Article Effects of aerogel-like disorder on the critical behavior of O(m)-vector models. Recent simulations and experimental evidence / C. Vasquez, R. Paredes // Condensed Matter Physics. — 2006. — Т. 9, № 2(46). — С. 305–317. — Бібліогр.: 43 назв. — англ. 1607-324X PACS: 64.60.Cn, 64.60.Fr, 64.70.-p DOI:10.5488/CMP.9.2.305 http://dspace.nbuv.gov.ua/handle/123456789/121325 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
We review recent results on the effect of a specific type of quenched disorder on well known O(m)-vector
models in three dimensions: the XY model (3DXY, m = 2) and the Ising model (3DIS, m = 1). Evidence of
changes of criticality in both systems, when confined in aerogel pores, is briefly referenced. The 3DXY model
represents the universality class to which the λ-transition of bulk superfluid 4He belongs. Experiments report
interesting changes of critical exponents for this transition, when superfluid 4He is confined in aerogels. Numerical
evidence has also been presented that the 3DXY model, confined in aerogel-like structures, exhibits
critical exponents different from those of bulk, in agreement with experiments. Both results seem to contradict
Harris criterion: being the specific heat exponent negative for the pure system (α3DXY ' −0.011 < 0), changes
should be explained in terms of the extended criterion due to Weinrib and Halperin, which requires disorder to
be long-range correlated (LRC) at all scales. In numerical works, aerogels are simulated by the diffusion limited
cluster-cluster aggregation (DLCA) algorithm, known to mimic the geometric features of aerogels. These
objects, real or simulated, are fractal through some decades only, and present crossovers to homogeneous
regimes at finite scales, so the violation to Harris criterion persists. The apparent violation has been explained
in terms of hidden LRC subsets within aerogels [Phys. Rev. Lett., 2003, 90, 170602]. On the other hand,
experiments on the liquid-vapor (LV) transition of ⁴He and N₂ confined in aerogels, also showed changes in
critical-point exponents. Being the LV critical-point in the O(1) universality class, criticality may be affected by
both short-range correlated (SRC) and LRC subsets of disorder. Simulations of the 3DIS in DLCA aerogels
can corroborate experimental results. Both experiments and simulations suggest a shift in critical exponents
to values closer to the SRC instead of those of the LRC fixed point. |
| format |
Article |
| author |
Vasquez, C. Paredes, R. |
| spellingShingle |
Vasquez, C. Paredes, R. Effects of aerogel-like disorder on the critical behavior of O(m)-vector models. Recent simulations and experimental evidence Condensed Matter Physics |
| author_facet |
Vasquez, C. Paredes, R. |
| author_sort |
Vasquez, C. |
| title |
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models. Recent simulations and experimental evidence |
| title_short |
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models. Recent simulations and experimental evidence |
| title_full |
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models. Recent simulations and experimental evidence |
| title_fullStr |
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models. Recent simulations and experimental evidence |
| title_full_unstemmed |
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models. Recent simulations and experimental evidence |
| title_sort |
effects of aerogel-like disorder on the critical behavior of o(m)-vector models. recent simulations and experimental evidence |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121325 |
| citation_txt |
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models. Recent simulations and experimental evidence / C. Vasquez, R. Paredes // Condensed Matter Physics. — 2006. — Т. 9, № 2(46). — С. 305–317. — Бібліогр.: 43 назв. — англ. |
| series |
Condensed Matter Physics |
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| first_indexed |
2025-07-08T19:39:28Z |
| last_indexed |
2025-07-08T19:39:28Z |
| _version_ |
1837108901949997056 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 2(46), pp. 305–317
Effects of aerogel-like disorder on the critical behavior
of O(m)-vector models. Recent simulations and
experimental evidence ∗
C.Vásquez1, R.Paredes2,3
1 Departamento de Fı́sica, Universidad Simón Bolı́var,
Apartado 89000, Caracas 1080A, Venezuela
2 Centro de Fı́sica, Instituto Venezolano de Investigaciones Cientı́ficas,
Apartado 21827, Caracas 1020A, Venezuela
3 Particle Technology Group, Delft University of Technology,
Jualianalaan 136, 2628 BL Delft, The Netherlands
Received April 10, 2006, in final form May 12, 2006
We review recent results on the effect of a specific type of quenched disorder on well known O(m)-vector
models in three dimensions: the XY model (3DXY, m = 2) and the Ising model (3DIS, m = 1). Evidence of
changes of criticality in both systems, when confined in aerogel pores, is briefly referenced. The 3DXY model
represents the universality class to which the λ-transition of bulk superfluid 4He belongs. Experiments report
interesting changes of critical exponents for this transition, when superfluid 4He is confined in aerogels. Nu-
merical evidence has also been presented that the 3DXY model, confined in aerogel-like structures, exhibits
critical exponents different from those of bulk, in agreement with experiments. Both results seem to contradict
Harris criterion: being the specific heat exponent negative for the pure system (α3DXY ' −0.011 < 0), changes
should be explained in terms of the extended criterion due to Weinrib and Halperin, which requires disorder to
be long-range correlated (LRC) at all scales. In numerical works, aerogels are simulated by the diffusion lim-
ited cluster-cluster aggregation (DLCA) algorithm, known to mimic the geometric features of aerogels. These
objects, real or simulated, are fractal through some decades only, and present crossovers to homogeneous
regimes at finite scales, so the violation to Harris criterion persists. The apparent violation has been explained
in terms of hidden LRC subsets within aerogels [Phys. Rev. Lett., 2003, 90, 170602]. On the other hand,
experiments on the liquid-vapor (LV) transition of 4He and N2 confined in aerogels, also showed changes in
critical-point exponents. Being the LV critical-point in the O(1) universality class, criticality may be affected by
both short-range correlated (SRC) and LRC subsets of disorder. Simulations of the 3DIS in DLCA aerogels
can corroborate experimental results. Both experiments and simulations suggest a shift in critical exponents
to values closer to the SRC instead of those of the LRC fixed point.
Key words: phase transitions, vector models, correlated disorder, aerogels
PACS: 64.60.Cn, 64.60.Fr, 64.70.-p
1. Harris criterion in brief: original and extended
Whether the presence of disordered impurities affects the critical behavior of an ideal system,
or not, has been the task of numerous works through years. Since Harris’ seminal work [1], a robust
theoretical background has emerged to establish conditions for the relevance of disorder to phase
transitions, which concern the criticality of the original pure system as well as the geometrical
features of the disordered distribution of defects [2–5]. On the other hand, many experimental con-
tributions in the last two decades posed interesting questions about criteria of relevance, supplying
results that challenge predictions made by previous theoretical works. Amongst all, those on the
superfluid transition of 4He in light aerogels are rather intriguing [6–11].
Changes of critical exponents for the so called λ-transition, when superfluid 4He is confined
in aerogel pores, were reported repeatedly from late 80’s through late 90’s, with almost the same
∗Contribution to the Mochima theoretical physics spring school. Joint CEA–IVIC–SFP workshop on Foundations
of Statistical and Mesoscopic Physics. 2005 – World Year of Physics.
c© C.Vásquez, R.Paredes 305
C.Vásquez, R.Paredes
question left open: do these results violate Harris and/or other relevance criteria? The question
arises because aerogels are homogeneous (non-correlated) beyond a finite scale [12,13], and given
that the specific heat exponent (α) is negative for this transition, exponents should not change
after Harris criterion. Many numerical and theoretical works have since emerged to explain these
contradictory results [4,5,14–17].
According to Harris criterion [1], short-range correlated (SRC) disorder is irrelevant for the crit-
ical behaviour of any d-dimensional pure system which undergoes a second order phase transition
with a correlation length exponent νpure > 2/d. The criterion was shown valid if disorder presents a
correlation function δ(~r) and, after Josephson hyperscaling [18], αpure < 0. This original work was
improved several years later by Weinrib and Halperin (WH) [2], who established a more general
criterion of relevance: even if α < 0 for the pure system, it will change critical exponents if disorder
is “correlated enough”. A proper definition of this long-range correlated (LRC) disorder makes use
of the impurity-impurity correlation function g(r) = 〈n(r)n(0)〉 and its long-range scaling exponent
(g(r) ∼ r−a, r →∞). Depending on how fast the tail of g(r) decays, the extended criterion reads as
follows:
Figure 1. Correlation length exponent for O(m)-
vector models (m = 1, 2, 3) with LRC defects,
depicted as functions of df = d − a/2. WH
m-independent result plotted for comparison
(dashed line). Data from table V in [5].
2− dνpure > 0 for a > d, (1)
2− aνpure > 0 for a < d. (2)
In both cases disorder is relevant. Case (1) ex-
tends the particular definition of short-range
correlated (SRC) disorder, made by Harris, to
disorder distributions with a “fast-decaying”
tail (a > d). The second case (2) is valid even
if νpure > 2/d. Based on one-loop expansions
in ε = 4 − d ¿ 1 and δ = 4 − a ¿ 1, WH
also argued that the correlation length expo-
nent would be νLRC = 2/a, independent of the
internal dimension (m) of the order parameter.
Prudnikov et al. [5] performed more accurate
estimates to deduce m-dependent corrections,
i.e., under the same kind of LRC distribution of
defects (same a), their two-loop field theoreti-
cal expansions give different critical exponents
for different O(m) systems. Figure 1 graphically resumes what they obtain for the correlation
length exponent νLRC, at disorder correlation exponents 2 < a < 3 and m = 1, 2, 3, quite different
from the m-independent plot predicted by WH [2].
2. Phase transitions in aerogels: experiments
In this section, we review some important experimental results about critical systems confined
in aerogels. In 1988, Moses Chan et al. [6] reported on the effect of quenched disorder on the
superfluid (SF) transition of confined 4He, using three types of porous glass: Vycor, aerogels and
xerogels. They measured the temperature dependence of the relative superfluid density, which
scales as ρs/ρ ∼ tζ near the critical point, being t = |T − Tc|/Tc the reduced temperature, and
the exponent ζ ≡ ν due to hyperuniversality [9,11]. In the first case (4He-Vycor), they observed
no change in the critical exponent from ζ ' 0.6705 of bulk SF 4He, a fact clearly explained in
terms of Harris criterion, provided that the pure critical system exhibits a negative specific heat
exponent (αpure ' −0.0105) [19]. Internal microchannel structures of Vycor, randomly oriented
and randomly distributed, present scattering intensities that exhibit a peak near rmax ∼ 20 nm,
and fall off exponentially above this peak, clearly faster than r−d. In terms of the WH model, this
4He-Vycor system falls into the SRC regime (1), so theoretical predictions agree with experiments.
Alternatively, Zassenhaus and Reppy [20], reported calorimetric studies for this 4He-Vycor system,
306
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models
showing that the singular part of specific heat curves fit well a λ-like curve with the same exponent
α as bulk SF 4He. Further experiments, using porous gold to confine SF 4He, instead of Vycor,
confirm these results [21]. Microchannels within these structures are similar to those of Vycor, but
porosities are greater.
The challenge to theory comes in the case of 4He confined in silica aerogels (AE) and xerogels
(XE), where authors observed ζAE ' 0.81 and ζXE ' 0.89, respectively, larger than the bulk
exponent. After the WH model, these results suggest that LRC distributions of defects should be
present within AE and XE. In practice, correlation functions are measured by means of (neutron/X-
ray) scattered intensities [12], thus algebraic (power-law) regimes on these plots correspond to
algebraic regimes on g(r). At these regimes, objects present self-similar (fractal) structure. Thus,
provided that the condition (2) is fulfilled by the exponent (a), critical exponents of the system
will be modified by disorder.
Nevertheless, authors argue that silica AE certainly exhibit fractal regimes for several length
scales, but they point out that beyond a finite cutoff ΛAE curves enter homogeneous SRC regimes.
Thus, close enough to the critical point, where the correlation length of SF 4He diverges (ξ ∼ t−ν),
disorder may appear homogeneous at the typical length scale of the system, and exponents should
cross over to bulk values. For instance, in the case of 95% porous aerogels ΛAE ' 150 nm, and
authors estimate ξ(t) ' 480 nm already at t ' 10−4, but although they run experiments up
to t ' 10−5, the crossover never appears [11]. They actually rule out an apparent violation to
Harris criterion, but leave the open question about the LRC character of disorder, and propose the
existence of LRC within aerogels, that cannot be observed through conventional techniques [7].
An attempt to explain these changes on the critical behaviour of SF 4He will be discussed below,
in section 3, showing that well defined LRC structures actually exist within aerogels. In the XE
case, instead, a clear explanation to changes in critical exponents does not exist so far. As aerogels,
these structures are created through silica sol-gel aggregation, but microstructures (strands) are
broken and reorganized at the drying stage of the process, resulting in more compact (less porous)
materials. It has been suggested [22] that, close to the flocculation-percolation transition in the
aggregation process, strands present the same fractal dimension as the LRC structure, but still
comparative studies do not reveal self-similarity in xerogels [23].
On the other hand, the liquid-vapor (LV) critical point, known to belong to the 3DIS universality
class [24], gives another scope into the effect of aerogel-like disorder on phase transitions. In this
case, any type of disorder, correlated or not, is relevant for criticality of the pure system. After
Harris criterion [1], a positive specific heat exponent for the pure system (α3DIS ' 0.109) makes
random SRC disorder already relevant. In addition, theory predicts [2,5] that the O(1) universality
class is also subject to changes in criticality under a LRC distribution of defects.
As we quote [13,17] and show below, both SRC and LRC structures are present within aerogels.
Thus, the 3DIS confined in aerogel-like distributions of defects (AEIS) may be subject to these
two competing effects. Results presented by Wong et al. [25], show that the LV critical point of
fluid 4He in aerogels exhibits specific heat curves with a finite peak at Tc, which suggests that
αAEIS < 0, clearly different from the pure system exponent. In these and further experiments, using
N2 instead [26], authors also report on the order parameter exponent concluding that, within error
bars, it is indistinguishable from the pure system exponent. Theoretical predictions on the 3DIS
with defects, give second order corrections to magnetic exponents with respect to those of the pure
system. In addition, the coexistence curve is about 12 times narrower than that of the pure system
[25], so it is not surprising that changes were so difficult to detect.
Then, the only trace left of an effect of aerogel-like disorder on the LV critical point is that
concerning the finite peak of the specific heat. In a recent paper [27], Paredes and Vásquez show
that this finite peak is consitent with a shift in the correlation length exponent to a value close to
2/3, which is closer to the corresponding exponent for the randomly diluted Ising system (RDIS)
[28–30], which is SRC at concentrations of disorder c below the percolation critical value pc. How-
ever, it is hard to reach definitive asymptotic values for the exponents, due to an apparent osci-
llating approach to the fixed point. Thus the same question emerges from both experiments and
simulations: why criticality would shift to the SRC and not to the LRC fixed point?
307
C.Vásquez, R.Paredes
3. Correlations within aerogel-like distributions of disorder: DLCA
In the sol-gel process of aerogel construction, silica dust is suspended in a solvent (sol phase),
which allows diffusion and cluster-cluster aggregation to take place. Once the process has ended
and the gel has been dried, detailed analysis of structure reveals that these objects are neither
fractal nor LRC in the WH sense (2): only finite regions of power-law scaling are observed in
scattering intensities from these objects [12]. However, at a given time tg of the process, one of the
clusters spans all the space of the flask, the gelling cluster (GC). Although it has not been tested
experimentally, evidence exists that these GC are fractal at all scales [13] and LRC in the WH
sense [17]. A numerical tool, the diffusion limited cluster-cluster aggregation algorithm (DLCA),
developed in early 80’s separately by Meakin [31] and Kolb, Botet, and Jullien [32], has proved to
well reproduce the geometrical features of real aerogels [13].
As silica dust in suspension, in DLCA, particles that initially occupy random positions in a
3D box, are allowed to perform independent Brownian motions, and then keep together whenever
two of them enter in contact. Undergoing this process, monomers and aggregates continue to move
randomly, and attach themselves to each other until a unique cluster has been finally formed. In
the on-lattice version of the algorithm, under periodic boundary conditions, cL3 particles occupy
random sites in a simple cubic L3 box, being c the concentration and ϕ = 1 − c the porosity. A
mass dependent diffusive constant is taken into account through a probability p ∼ m−ς for an
aggregate of mass m to move, where ς is a positive mobility exponent.
As in experiments, the GC is defined as the first cluster to reach opposites sides of the box in
any direction. Right after the GC is built, many other smaller clusters (islands) continue to diffuse,
and finally attach themselves at random sites to the GC. In the fabrication of an aerogel, the
resulting structure is preserved by hypercritical evaporation of the solvent [12], so the GC matrix
is kept almost intact, with randomly distributed strands attached to it. Islands represent the SRC
subset within the whole cluster.
Figure 2. Average density n(r) versus box size
r < L/2 for different subsets of c = 0.05 DLCA
clusters (L = 128). The dashed line represents the
slope a/2 = 1.5.
Figure 2 shows the geometrical features
of different subsets of DLCA clusters, built
in L = 128 boxes at concentration c =
0.05. Following a box-counting procedure [33],
the scaling of corresponding average densities
n(r) = m(r)/rd have been depicted as function
of the box size r. The average over 32 realiza-
tions is taken. Plots marked as BBDLCA and
BBGC correspond to backbones for the com-
plete DLCA structure and the GC, respecti-
vely. Plots corresponding to complete DLCA
clusters (circles) fast enter the homogeneous
regime, due to islands attached to random sites
at gelling clusters (squares): the slow power-
law decay of GC alone suggest that these
objects are LRC, but when islands have al-
ready reached the GC to finally form the whole
DLCA structure, the LRC regime disappears
screened (hidden) by the homogeneous distri-
bution of islands. Backbones BBDLCA and BBGC also exhibit power-law long regimes, but as we
argue just below, exponents larger than 3/2 give a > d = 3, which makes these subsets enter the
SRC class, after WH criterion (1).
The average mass m(r) scales as rdf , being df the fractal dimension of the object. Recall that
in the box-counting procedure one only takes non-empty boxes to estimate the average. This is
equivalent to considering n(r) as a one-point correlation function: considering only occupied sites as
centers for d-dimensional cubes, the algorithm effectively measures the probability for surrounding
sites to be occupied, normalized to rd. This method rends a decaying exponent (d − df) for n(r).
We have argued [17] that the decaying exponent for this one-point correlation function is related
308
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models
to that of the impurity-impurity correlation function g(r) ∼ r−a as a = 2(d− df).
An example for this assertion comes from the percolation model, whose density-density cor-
relation function scales as g(r) ∼ r−(d−2+η) = r−a at the critical occupation fraction p = pc.
Using Rushbrooke hyperscaling relation one obtains d − 2 + η = 2β/ν, being β the critical expo-
nent for the order parameter P∞. Remind that the order parameter in the percolation problem is
P∞ = m∞/N , where m∞ is the mass (size) of the percolating cluster at pc. This quantity scales as
P∞ ∼ L−(d−df ), and being this the order parameter for the percolation transition [33], its exponent
is defined as d − df = β/ν. In this case, it follows that the relation stated above, a = 2(d − df)
seems to hold [34].
For the complete DLCA cluster, the crossover to a homogeneous regime occurs at r ¿ L/2,
while the fractal regime for backbones seems to span all scales. In the WH scheme, however,
backbones present the fast decaying SRC regime (1), comparing the corresponding n(r) with the
slope 1.5 (dashed line). The long-range decay for the GC seems slower than r−d, which enters the
LRC regime (2). This case is more complex as it becomes homogeneous at r ≈ L/2, probably
because some strands are already randomly attached to it at the gelation time tg. To show that
GC are in effect fractal at the thermodynamical limit L →∞, while complete DLCA are not, plots
of n(r) for different lattice sizes L = 32 − 128 are shown in figure 3. Arrows indicate crossover
Figure 3. Average density n(r) versus box size r for (a) DLCA complete clusters and (b) cor-
responding GCs. Crossover to homogeneous regimes indicated by arrows. The dashed curve in
(b) is the stretched exponential fit for the GC at L = 128 (see text). Plots have been vertically
displaced for better comprehension.
lengths which, for DLCA clusters (a), reach no more than a few lattice constants, independent of
L. On the other hand, arrows in figure 3 (b) indicate that crossover lengths of GCs scale with the
lattice size, a trace of fractality of these objects at L →∞. The complete DLCA cluster contains
this LRC structure within, with numerous islands randomly attached to it, which hide correlations
in the box-counting numerical method and make the whole structure appear homogeneous. This
screening effect is unavoidable in experimental measurements of correlations (small-angle scattered
intensities), resulting in finite fractal regimes for silica aerogels, which tend to disappear as density
increases [12]. For simulated aerogels, finite size modifies the long-range behaviour of the correlation
function. This effect is taken into account through the stretched exponential expression proposed
by authors in [22,35] n(r) = cg + Aor
−(d−df ) exp[−(r/ξ)δ], where cg is the actual density of the
GC (which tends to zero as L → ∞) and ξ is the characteristic radius of gyration (cuttoff). The
parameter δ > 1 describes the faster crossover to the homogeneous regime characteristic of finite
lattices. The dashed curve in figure 3 (b) represents the fit to this expression for L = 128, giving
the estimate df ≈ 1.7 (thus a ≈ 2.6) for the fractal dimension of the GC, with a cutoff at ξ ≈ 35.
309
C.Vásquez, R.Paredes
4. O(m)-vector models with aerogel-like distribution of defects
Monte Carlo simulations of O(m)-vector models consider the nearest neighbor interactions
between spins, placed on a 3D simple cubic lattice of size L (1 < i < L3), with periodical boundary
conditions. When defects are present, these interactions are described by the Hamiltonian:
H
kT
= −K
∑
〈ij〉
εiεj
~φi · ~φj , (3)
where K = J/kT is the coupling (inverse temperature, with k = J = 1), and εi = 1 if a spin
occupies the site, or εi = 0 otherwise. Depending on the internal dimension (m) of the order
parameter, spins are taken as ~φi = ±1 for the 3DIS (m = 1), and ~φi = (cos θi, sin θi) for the
3DXY (m = 2) model. For each independent disorder realization, an aerogel is first built by the
DLCA on-lattice algorithm described above [32], taking each site from the complete final cluster
as a defect. Aerogel pores, i. e., N = (1 − c)L3 empty sites left in the lattice, are then filled
with spins, being c is the concentration of defects. Monte Carlo sweeps (MCS) are then taken
as follows, using Hamiltonian (3): Wolff cluster update algorithm [36] is implemented, taking one
sweep as eight consecutive cluster flips, to reduce autocorrelations caused by critical slowing down.
Here we quote the procedure followed in simulations of the 3DXY model [17]. For each disorder
realization, thermalization is reached after 2 × 104 sweeps, while 2 × 106 production sweeps are
taken at equilibrium for further statistical analysis. A suitable number of disorder realizations have
been taken, depending on lattice sizes, which run from L = 16 (256) to L = 64 (16). The method
is rather different in the disordered 3DIS [27], in that shorter time series (∼ 103MCS) are taken
in statistics, increasing the number of realizations at each size (∼ 2× 103). A detailed analysis on
self-averaging of thermodynamical quantities, performed for the 3DIS case, shows a lack of self-
averaging in the magnetization, suceptibility and specific heat, and on the other hand, the energy
is weakly self-averaged, with an exponent x ≈ 2.58. In that work [27], disorder distributions of
thermodynamic observables for simulations of the 3DIS in aerogels are shown to be more symetric
and narrower than for RDIS. Thus, good rough estimates could be found with not so much disorder
realizations for the aerogel case. For this reason, although such analysis has not been performed
for results presented here for the 3DXY case, an acceptable agreement with experimental results
on the SF transition of 4He is obtained [17].
4.1. Observables
The magnetization squared (order parameter) is calculated as,
M2 =
[
|〈~φ〉|2
]
=
[
|
∑
i
~φi/N |2
]
, (4)
where brackets represent canonical ensemble averages, while square brackets express averages over
disorder realizations, for fixed L and c. The energy per spin is calculated as the canonical ensemble
average:
E = [〈E〉] = [〈H〉/N ]. (5)
Moments for the energy per spin are estimated at the simulation coupling Ko, as En = [〈En〉],
as well as moments Mn for the magnetization are estimated for the corresponding energy bins.
Logarithmic derivatives of moments n = 1, 2, 4 of magnetization are then calculated using the
corresponding E −Mn covariance in the canonical ensemble:
Dn =
∂ ln(Mn)
∂K
= −〈M
nE〉 − 〈E〉〈Mn〉
〈Mn〉 . (6)
The specific heat is calculated through fluctuations of the energy per spin:
C = NK−2
(〈E2〉 − 〈E〉2) . (7)
310
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models
The helicity modulus [〈Υµ̂〉] has been determined using the Kubo formula [15,16], averaged over
the three directions µ̂ = x̂, ŷ, ẑ, assuming that the model and disorder distributions are isotropic.
This quantity is proportional to the superfluid density, as was derived by Fisher and colleagues
[39], so it is directly related to the order parameter of the SF transition. We use this relation to
show that the confined system (AEXY) acts in d = 3 external dimension, i. e., aerogel pores are
effectively three-dimensional [17].
These quantities have been stored from MC simulations at each realization of disorder and,
after the application of reweighting techniques, extrapolation of thermodynamical quantities of
interest are made at couplings K ≈ Ko [37,38].
4.2. Finite size scaling: thermal exponents
Maxima from reweighted curves for logarithmic derivatives of moments n = 1, 2, 4 of the mag-
netization (6), and the specific heat (7), as functions of the coupling K, are averaged over disorder
to estimate thermal critical exponents.
4.2.1. O(2)-vector models: 3DXY and AEXY
According to the finite-size scaling (FSS) ansatz [40] maxima of logarithmic derivatives scale
with lattice sizes L as:
[Dn]max ∼ L1/ν . (8)
Figure 4. FSS of maxima of logarithmic deriva-
tives Dn (n = 1, 2, 4) for bulk 3DXY and the
AEXY at concentration c = 0.05 of defects. Cor-
responding original plots have been vertically dis-
placed to coincide at L = 16. Dashed lines are fits
to the FSS expression (8) on collected data.
Respectively, data for the 3DXY and the
AEXY systems are depicted in figure 4, as func-
tion of lattice sizes L. Both groups of data
have been vertically displaced (multiplied by
a constant) to make corresponding plot coin-
cide at L = 16, which allows making a fit on
collected data. The FSS power-law fit (8) for
bulk 3DXY, using lattice sizes L > 24 gives
1/νpure = 1.483(3), which agrees well with pre-
viously reported results [41], and the exponent
ζ ≈ 0.67 reported for bulk λ-transition [19].
The same fit to data for the AEXY, at con-
centration c = 0.05, gives 1/νAEXY = 1.29(2),
clearly different from the bulk value. Fits on
figure 4 (dashed lines) have been vertically di-
splaced to facilitate comparison between the
slopes.
The correlation length exponent νAEXY ≈
0.77 obtained for this confined system agrees
well with the exponent ζ ≈ 0.76 obtained in
experiments [9]. Both results, numerical and experimental, suggest that a LRC distribution of
defects must be affecting the critical behaviour of the 3DXY universality class, when confined
in aerogel-like structures. After the extended criterion (2), this structure should have a decaying
exponent a < 2/νpure ≈ 2.98. Analysis on the DLCA structure resumed above gives a ≈ 2.6 for
gelling clusters of DLCA aerogels at c = 0.05, so the WH condition is fulfilled and the distribution
GC of disorder could be relevant. Nevertheless, there are other correlated structures within DLCA
aerogels, the backbones, but given that a > 3 for these structures, they do not satisfy the WH
condition. If these structures have a decaying exponent smaller than d, figure 2 shows that this
exponent is already greater than the corresponding exponent for the GC. The analysis resumed by
WH [2] when multiple exponents are present on LRC strucutures, i. e., when g(r) =
∑
i gir
−ai , it
is the term with the smallest exponent which rules the critical behaviour for the impure system.
Indeed, Prudnikov et al. [5] predict that for a decaying exponent similar to that reported above
311
C.Vásquez, R.Paredes
for the GC (a ≈ 2.6), the correlation length exponent should be νLRC ≈ 0.73, so both experimental
and numerical results roughly confirm this prediction. It is worth recalling that the whole DLCA
structure itself is not LRC, but it is a mixture of SRC structures with a clearly LRC structure,
the GC. As follows from WH results, SRC subsets are irrelevant. Thus, evident changes observed
in critical exponents for this univerality class suggest that only the LRC subset rules the critical
behaviour of both systems, SF 4He and 3DXY interacting spins.
Moreover, we have further determined critical exponents for the specific heat and the helicity
modulus as well. Maxima for the specific heat, assuming that α < 0, scale as:
[C]max ∼ C∞ + ALα/ν . (9)
Figure 5. FSS of maxima of the specific heat for bulk
3DXY (circles) and confined AEXY at concentration
c = 0.05 of defects (squares). Lines are fits to the
FSS expression (9).
These maxima have been depicted in figure 5
as function of L, normalized to c(L = 64) ≡
1, for bulk 3DXY (circles) and the confined
AEXY system at c = 0.05 (squares). Non-
linear fits of corresponding points to the FSS
expression (9), using data for L > 28 to avoid
corrections to scaling, are shown in figure 5
by corresponding lines. The method was de-
veloped by Schultka and Manousakis [42], to
determine the specific heat exponent for the
pure 3DXY system. The corresponding best
fits give α/ν ≈ −0.015 for the pure system,
and α/ν ≈ −0.38 for the confined system.
These results agree well with our results on cor-
relation length exponents, shown above, after
the Josephson hyperscaling relation [18] in the
form 2/ν − α/ν = d∗, which gives d∗ ≈ 3.0
for both systems. Using corresponding results
on the correlation length exponents, we ob-
tain α ≈ −0.010 for bulk 3DXY, which agrees well with calorimetric studies on SF 4He
[19]. The result α = −0.29 for the AEXY is lower than the reported experimental value
α − 0.57 for 4He confined in aerogel pores at the same volume fraction c = 0.05 [11]. How-
ever, in numerical simulations reported here, the question posed by authors about the appar-
ent violation of hyperscaling seems not to emerge from our results on exponents ν and α.
Figure 6. FSS plots of the helicity modulus at Kc.
Bulk 3DXY (circles) and confined AEXY at con-
centration c = 0.05 of defects (squares).
The critical coupling Kc has been deter-
mined using the crossing of reweighted plots of
the Binder fourth cumulant for the magnetiza-
tion U4 = 1−〈M4〉/3〈M2〉2 at different lattice
sizes L, which is universal at Kc (independent
of L) [43]. This analysis yields Kc ≈ 0.45416
for the pure system, while the crossing of aver-
age reweighted curves in the AEXY case gives
Kc ≈ 0.46495.
Using these values, we estimate the helicity
modulus at Kc, averaged over disorder realiza-
tions:
[〈Υµ̂〉] ∼ L−υ/ν . (10)
Figure 6 shows corresponding data for the he-
licity modulus in the pure 3DXY case (circles),
and the AEXY case (squares). Lines represent
fits to the FSS expression 10. The exponent
312
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models
υ/ν = 1 confirms that the effective external dimension for the system is d = 3, through the hy-
peruniversality relation υ = (d − 2)ν [39]. This result also confirms that the correlation length
exponent we have estimated here corresponds to the exponent ζ for the SF transition of 4He in
aerogels.
Results reported here seem to answer the question quoted above from [6,7]: numerical evidence
has been presented here (section 3) that a physically well defined hidden LRC distribution of defects
exists in aerogels, which explains changes on the critical behaviour of SF 4He when confined in
those structures.
4.2.2. O(1)-vector models: 3DIS and AEIS
We turn now to a brief discussion on changes of the correlation length exponent for the 3DIS,
when confined in DLCA aerogels at concentration c = 0.2 of defects. A detailed analysis is reported
elsewhere [27]. The method, as stated above, is quite different from that used in the AEXY case
in that shorter MC time-series are used for extrapolation, but total numbers of realizations have
been incremented in two orders of magnitude.
1/ν ≅ 1.41
0 0.002 0.004 0.006L
−1/ν0.2578
0.2580
0.2582
0.2584
0.2586
0.2588
Kc
∗
χ
D1
D2
D4
Figure 7. FSS plots of pseudocritical couplings K∗
c
for the AEIS at c = 0.2, taken from positions of
maxima for the susceptibility (χ), and logarithmic
derivatives (Dn). The horizontal axes have been
rescaled to L−1/ν , taking 1/ν = 1.41 (estimated
in the insert).
Shorter CPU times, in this case, permit to
make simulations at greater lattice sizes, being
L = 8 − 96 in this case. After extrapolating
by reweighting, curves have been averaged over
disorder for each K, so a unique curve is ob-
tained for each set of data. Position of max-
ima (pseudocritical couplings K∗
c (L)) for the
magnetic susceptibility (χ), and logarithmic
derivatives Dn of moments of the magnetiza-
tion (n = 1, 2, 4) have been estimated through
the averaged curves. According to the finite size
scaling theory [43], maxima for a given observ-
able O scale as
K∗
c (O, L) = Kc + a(O)L−1/ν + c. s. (11)
Kc being the critical coupling at the thermo-
dynamical limit, L →∞. Coefficients a(O) are
non-universal, i. e. , dependent on the observ-
able O and on details of the system. Correcti-
ons to scaling (c. s. ) are avoided in the analysis
reported below, taking data for large enough lattice sizes (L > 40). Points depicted on the insert
of figure 7, have been calculated through local differences K∗
c (Oi, L) − K∗
c (Oj , L) between data
from two different observables Oi and Oj . This transformation of data permits to eliminate the
non-singular term (Kc) from equation (11), common by hypothesis to all observables, allowing for
an easier estimation of the exponent on the leading term aL−1/ν .
Power-law fits have been made to these transformed data, giving a rough estimate of the
(common) FSS exponent 1/ν ≈ 1.41. This approximate has been used to rescale the horizontal
axis to L−1/ν , as indicated in figure 7. As expected, a linear behaviour is observed for all data,
and linear fits to the expression (11) give just one value Kc = 0.25857(1) for all plots. A more
detailed analysis on the scaling of logarithmic derivatives at the critical point reveals, however,
that the correlation length exponent tends to a value closer to 3/2 [27]. The correlation length
exponent νAEIS & 0.67 obtained is clearly greater than that for the pure 3DIS and the LV critical
point (ν ≈ 0.63). These results point to a negative or zero value for the specific heat exponent α.
Experiments on the LV critical point of 4He in aerogels [25] report specific heat curves that seem
to have a finite peak at Tc, but authors do not estimate the critical exponent α which may either
be negative or zero as well. After Josephson hyperscaling, this is consistent with our rough result
1/ν . 1.5, tending to values closer to that predicted for the SRC Ising universality class than that
predicted for the LRC fixed point [5].
313
C.Vásquez, R.Paredes
We should then conclude, in this case, that the FSS method applied here may not be precise
enough to determine really asymptotic values for critical exponents, but a remarkable difference
exists between the results approached here and the exponents for a pure system, as expected after
Harris criterion.
5. Concluding remarks
Here we have reported on the effects of a specific type of disordered distributions of defects on
the critical behaviour of O(m)-vector models, using the O(1) and the O(2) models, represented
by the 3DIS and the 3DXY models, respectively. A detailed scaling analysis has been made to
structures created through the DLCA on-lattice algorithm, which well reproduces the geometrical
tasks of real silica aerogels. This analysis shows that the whole structure is not, strictly speaking,
a fractal: the power-law decaying regime is preserved only through a few lattice constants, and
even more, it does not scale with lattice sizes. A more detailed dissection shows the existence of
physically well defined subsets within the whole strucuture, gelling clusters, that keep the algebraic
decaying regime through lengths comparable to the lattice size, with a cutoff that scales with
the system size. In the light of WH theory, the existence of this fractal (LRC) subset already
explains the changes on critical exponents of the 3DXY model, when confined in these aerogel-like
simulated structures, with respect to those of the pure system. Results for the confined (AEXY)
system roughly confirm those observed for the SF transition of 4He within aerogels, and explain
why a crossover to bulk values is not observed in critical exponents reported from experiments,
as T → Tc, when the correlation length diverges [11]. Even more intriguing are the results from
the 3DIS confined in the same type of disorder distribution, as exponents seem to be affected
by the LRC fixed point to finally tend values far from those predicted by theory [5], and closer
to those of the SRC fixed point. The question about the critical behaviour of the AEIS rests
open as conclusive results on asymptotic values for critical exponents have not yet been obtained.
Previous renormalization group flows to fixed points, sketched by Weinrib and Halperin [2], and
the analysis made by Prudnikov et al. [5], point out that the approach to the LRC fixed point
may be oscillating, which can make it very dificult to enter a really asymptotic regime applying
conventional finite size scaling.
Acknowledgements
Authors thank CNRS and FONACIT (PI2004000007) for their support. Invaluable discussions
with A. Hasmy and R. Jullien have improved our understanding on aerogel structure. C. Vásquez
kindly acknowledges the collaboration and technical support received by the personnel of the LCVN
at Montpellier, France.
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Effects of aerogel-like disorder on the critical behavior of O(m)-vector models
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315
C.Vásquez, R.Paredes
Questions and answers
Q (Rafael Rangel): What are the theoretical predictions for the (possible) change of
exponents by disorder?
A Harris criterion predicts that a weak uncorrelated disorder is not relevant if the correlation
length exponent ν is larger than 2/d for the pure system, or after Josephson hyperscaling,
α = 2 − dν < 0. For 3DIS case, α > 0, and then short-range-correlated disorder could be
relevant. Weinrib and Halperin extend this criterion to cases where there exist long-range
correlations (LRC) in the distribution of defects, as expressed by equations (1) and (2). It
is this extension that permits to explain explain the changes on the critical behaviour of
the 3DXY model when confined in aerogel-like distributions of defects, and the changes on
critical exponents of the SF transition of 4He in aerogels as well.
Q (Dragi Karevski): How does your FSS technique work if there are different corre-
lations?
A Weinrib and Halperin [2] have already considered the case when there exist terms with
different decaying exponents in the correlation function g(r), as was quoted in Subsection
4.2.1: the critical behaviour couples to the term with the lowest exponent. Indeed, in the
model disorder we are considering in simulations presented here, DLCA, there exist several
subsets of the whole cluster which present different decaying regimes. Results point to state
that, in the 3DXY confined case (AEXY), it is the gelling cluster (GC) that rules the critical
behaviour, as it is the LRC subset with the lowest exponent (see figure 2). The AEIS case
is by far more interesting in that competing effects are present, as SRC disorder is also
relevant. Being the LRC fixed point marginal, it seems that it is the SRC fixed point that
finally governs the critical behaviour, i. e., the effective exponent 1/ν turns from being
attracted to 1/ν ≈ 1.4 (predicted by Prudnikov et al. [5] for the LRC fixed point) to
1/ν . 3/2, closer to the SRC fixed point value. Nevertheless, even if only one model LRC
distribution of defects were present, the application of FSS techniques should consider that
the convergence of the correlation function to the final decaying regime L−(d−df) will depend
on the lattice size L, so the convergence of effective exponents to critical values will be ruled
by an exponent that asymptotically tends to a stable value. This task of the correlation
function may cause succesive crossover regimes which appear in the convergence of effective
exponents to asymptotic critical exponents.
Q (Dragi Karevski): How does the FSS with V = Ld (and not with L) hold?
A The FSS theory is based on the homogeneity of the free energy density, which permits to
write it in terms of the scaling variable L/ξ = Ltν , which at its time may be redimensioned to
tL−1/ν . The external dimension d of the system appears only in corresponding hyperscaling
relations, such as those of Josephson 2/ν − α/ν = d and Rushbrooke γ/ν + 2β/ν = d.
Q (Wolfhard Janke): What is the dimension df of your fractal?
A The whole DLCA clusters in our model disorder are not fractal, as they enter homogeneous
regime at very few lattice constants, but the well defined GC have correlations that seem
to span all the scales at thermodynamical limit. The best estimate we have of the fractal
dimension for these objects is df ' 1.7.
Q (Yurij Holovatch): How is c related to a?
A As the concentration of defects increases, the fractal dimension of the GC increases as well,
as stated in [13], so the exponent a decreases.
316
Effects of aerogel-like disorder on the critical behavior of O(m)-vector models
Âïëèâ áåçëàäó òèïó àåðîãåëþ íà êðèòè÷íó ïîâåäiíêó
O(m)-âåêòîðíèõ ìîäåëåé. Íåäàâíi ñèìóëÿöi¿ òà
åêñïåðèìåíòàëüíi ïðîÿâè
Ê.Âàñêåñ1, Ð.Ïàðåäåñ2,3
1 Ôiçè÷íèé ôàêóëüòåò, Óíiâåðñèòåò Ñiìîíà Áîëiâàðà,
ïîøò. ñêðèíüêà 89000, Êàðàêàñ 1080–À, Âåíåñóåëà
2 Öåíòð ôiçèêè, Âåíåñóåëüñüêèé iíñòèòóò íàóêîâèõ äîñëiäæåíü,
ïîøò. ñêðèíüêà 21827, Êàðàêàñ 1020–À, Âåíåñóåëà
3 Ãðóïà òåõíîëîãi¿ ÷àñòèíîê, Òåõíîëîãi÷íèé óíiâåðñèòåò,
Äåëüôò, 2629 BL, Íiäåðëàíäè
Îòðèìàíî 10 êâiòíÿ 2006 ð., â îñòàòî÷íîìó âèãëÿäi – 12 òðàâíÿ 2006 ð.
Ïðîâåäåíî îãëÿä íåäàâíiõ ðåçóëüòàòiâ äîñëiäæåííÿ âïëèâó ïåâíîãî òèïó çàìîðîæåíîãî áåçëàäó íà
äîáðå âiäîìi O(m)-âåêòîðíi ìîäåëi ó òðüîõ âèìiðàõ: XY ìîäåëü (3DXY, m = 2) i ìîäåëü Içèíãà (3DIS,
m = 1). Êîðîòêî çãàäóþòüñÿ ïðîÿâè çìií â êðèòè÷íîñòi îáîõ ñèñòåì ó ïîðàõ àåðîãåëiâ. 3DXY ìîäåëü
º ïðåäñòàâíèêîì êëàñó óíiâåðñàëüíîñòi, äî ÿêîãî íàëåæèòü λ-ïåðåõiä íàäïëèííîãî 4He. Åêñïåðèìåí-
òè ñâiä÷àòü ïðî öiêàâi çìiíè â êðèòè÷íèõ ïîêàçíèêàõ öüîãî ïåðåõîäó, êîëè íàäïëèííèé 4He ìiñòèòüñÿ
â àåðîãåëÿõ. Áóëè çðîáëåíi òàêîæ ÷àñîâi ñèìóëÿöi¿, ÿêi ïiäòâåðäæóþòü òå, ùî 3DXY ìîäåëü íà ãðàòöi
iç ñòðóêòóðîþ äåôåêòiâ, ïîäiáíèõ äî ñòðóêòóðè àåðîãåëþ, õàðàêòåðèçóºòüñÿ êðèòè÷íèìè ïîêàçíè-
êàìè, âiäìiííèìè âiä ÷èñòîãî âèïàäêó. Íà ïåðøèé ïîãëÿä âèäàºòüñÿ, ùî îáèäâà ðåçóëüòàòè ñóïåðå-
÷àòü êðèòåðiþ Ãàððiñà. Àëå ó âèïàäêó, êîëè êðèòè÷íèé ïîêàçíèê ïèòîìî¿ òåïëîºìíîñòi ÷èñòî¿ ñèñòåìè
âiä’ºìíèé, (α3DXY ' −0.011 < 0), çìiíè ïîâèííi ïîÿñíþâàòèñÿ óçàãàëüíåíèì êðèòåðiºì Âàéíðiáà òà
Ãàëïåðiíà, ÿêèé ñòîñóºòüñÿ äàëåêîñÿæíî-ñêîðåëüîâàíîãî íà âñiõ ìàñøòàáàõ áåçëàäó. Ó ÷èñåëüíèõ
ñèìóëÿöiÿõ àåðîãåëi ìîäåëþþòüñÿ àëãîðèòìîì êåðîâàíî¿ äèôóçiºþ êëàñòåð-êëàñòåðíî¿ àãðåãàöi¿.
Âiäîìî, ùî òàêèé àëãîðèòì âiäòâîðþº ïåâíi ãåîìåòðè÷íi õàðàêòåðèñòèêè àåðîãåëiâ. Öi îá’ºêòè, ÷è
ðåàëüíi, ÷è ñèìóëüîâàíi, º ôðàêòàëàìè ëèøå íà ïåâíèõ ìàñøòàáàõ. Iç çáiëüøåííÿì ìàñøòàáó âiäáó-
âàºòüñÿ ïåðåõiä äî îäíîðiäíîãî ðåæèìó.  ñâîþ ÷åðãó, öå ñïðè÷èíÿº ïîðóøåííÿ êðèòåðiþ Ãàððiñà.
Òàêå ïîðóøåííÿ áóëî ïîÿñíåíå ïðèñóòíiñòþ â àåðîãåëÿõ ïðèõîâàíèõ äàëåêîñÿæíî-ñêîðåëüîâàíèõ
âêëþ÷åíü [Phys. Rev. Lett., 2003, 90, 170602]. Ç iíøîãî áîêó, ïiä÷àñ åêñïåðèìåíòàëüíèõ äîñëiäæåíü
ïåðåõîäó ðiäèíà-ïàðà â 4He i N2 â àåðîãåëÿõ òàêîæ ñïîñòåðiãàëàñü çìiíà ïîêàçíèêiâ ó êðèòè÷íié
òî÷öi. ßêáè öåé ïåðåõiä áóâ ó êëàñi óíiâåðñàëüíîñòi O(1), òî íà êðèòè÷íó ïîâåäiíêó âïëèâàâ áè i
êîðîòêîñÿæíî-, i äàëåêîñÿæíî-ñêîðåëüîâàíèé áåçëàä. Ñèìóëÿöi¿ 3DIS â àåðîãåëÿõ, ìîäåëüîâàíèõ
àëãîðèòìîì êîðåëüîâàíî¿ äèôóçiºþ êëàñòåð-êëàñòåðíî¿ àãðåãàöi¿, ïiäòâåðäæóþòü åêñïåðèìåíòàëüíi
ðåçóëüòàòè. I åêñïåðèìåíòè, i ñèìóëÿöi¿ ñâiä÷àòü ïðî òå, ùî çíà÷åííÿ êðèòè÷íèõ ïîêàçíèêiâ ñòàþòü
áëèæ÷èìè äî çíà÷åíü ó êîðîòêîñÿæíî-ñêîðåëüîâàíié, à íå â äàëåêîñÿæíî-ñêîðåëüîâàíié íåðóõîìié
òî÷öi.
Êëþ÷îâi ñëîâà: ôàçîâi ïåðåõîäè, âåêòîðíi ìîäåëi, ñêîðåëüîâàíèé áåçëàä, àåðîãåëi
PACS: 64.60.Cn, 64.60.Fr, 64.70.-p
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