Information-Entropic Signature of the Critical Point

1.6 MB·English

by  Marcelo Gleiser

Information-Entropic Signature of the Critical Point
Marcelo Gleiser1,∗ and Damian Sowinski1,†
1Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA
(Dated: June 4, 2015)
Weinvestigatethecriticalbehaviorofcontinuous(second-order)phasetransitionsinthecontextof
(2+1)-dimensionalGinzburg-Landaumodelswithadouble-welleffectivepotential. Inparticular,we
show that the recently-proposed configurational entropy (CE)–a measure of the spatial complexity
oftheorderparameterbasedonitsFourier-modedecomposition–canbeusedtoidentifythecritical
point. WecomputetheCEfordifferenttemperaturesandshowthatlargespatialfluctuationsnear
5 the critical point (Tc)–characterized by a divergent correlation length–lead to a sharp decrease in
1 the associated configurational entropy. We further show that the CE density has a marked scaling
0 behaviornearcriticality,withthesamepowerofKolmogorovturbulence. Wereproducethebehavior
2 of the CE at criticality with a percolating many-bubble model.

n
PACSnumbers: 05.70.Jk,64.60.Bd,11.10.Lm,02.70.Hm
u
J

3
INTRODUCTION sence of an external source (or a magnetic field), the GL

free-energy functional is simply
]
h
c From materials science [1] to the early universe [2], (cid:90) (cid:20)γ a b (cid:21)
e phase transitions offer a striking illustration of how E[φ]= ddx 2(∇φ)2+ 2tφ2+ 4φ4 , (1)
m changing conditions can affect the physical properties of
- matter [3]. In very broad terms, and for the simplest wheret≡(T−Tc),andγ,a,barepositiveconstants. For
at systems described by a single order parameter, it is cus- T > Tc the system has a single free-energy minimum at
t tomary to classify phase transitions as being either dis- φ=0,whileforT <T therearetwodegenerateminima
s c
. continuous or continuous, or as first or second order, re- atφ =±(−at/b)1/2. Thismean-fieldtheorydescription
t 0
a spectively. First-orderphasetransitionscanbedescribed workswellawayfromthecriticalpoint. Intheneighbor-
m by an effective free-energy functional (or an effective po- hood of T one uses perturbation theory and the renor-
c
- tential in the language of field theory) where an energy malization group to account for the divergent behavior
d
barrierseparatestwoormorephasesavailabletothesys- of the system. This behavior can be seen through the
n
o tem. Thesystemmaytransitionfromahighertoalower two-point correlation function G(r): away from Tc G(r)
c free-energy state (or from a higher to a lower vacuum behaves as ∼ exp[−r/ξ(T)], where ξ(T) is the correla-
[ state) either by a thermal fluctuation of sufficient size (a tion length, a measure of the spatial extent of correlated

2 critical bubble) or, for low-temperatures, by a quantum fluctuations of the order parameter. In mean-field the-
v fluctuation. Generally speaking, this description of ther- ory, ξ(T)∼|T −T |−ν, where ν =1/2, independently of
c
0 malbubblenucleationisvalidaslongasF[φb]/kBT (cid:29)1, spatial dimensionality. In the neighborhood of Tc, where
0 whereF[φb]isthe3dEuclideanactionofthespherically- the mean-field description breaks down, the behavior of
8
6 symmetric critical bubble or bounce φb(r), kB is Boltz- spatial fluctuations is corrected using the renormaliza-
0 mann’s constant, and T is the environmental tempera- tion group. Within the ε-expansion, one obtains, in 3d,
1. ture. For quantum tunneling, one uses instead S4[φb]/(cid:126), ν =1/2+ε/12+7ε2/162(cid:39)0.63 [4].
0 where S4[φb] is the O(4)-symmetric Euclidean action of For continuous transitions in GL-systems, the focus
5 the 4d bounce. of the present letter, the critical point is characterized
1 Forsecond-ordertransitionstheorderparametervaries by having fluctuations on all spatial scales. This means
:
v continuously as an external parameter such as the tem- that while away from Tc large spatial fluctuations are
i perature is changed [3]. A well-known example is that suppressed, near T they dominate over smaller ones.
X c
of an Ising ferromagnet, where the net magnetization In this letter, we will explore this fact to obtain a new
r
of a sample is zero above a critical temperature T , the measure of the critical point based on the information
a c
Curie point, and non-zero below it. Below T the tran- content carried by fluctuations at different momentum
c
sitionunfoldsviaspinodaldecomposition, wherebylong- scales. For this purpose we will use a measure of spatial
wavelength fluctuations become exponentially-unstable complexity known as configurational entropy (CE), re-
to growth. This growth is characterized by the appear- cently proposed by Gleiser and Stamatopoulos [5]. The
ance of domains with the same net magnetization which CE has been used to characterize the information con-
compete for dominance with their neighbors. In the tent [5] and stability of solitonic solutions of field theo-
continuum limit, systems in the Ising universality class ries [5, 6], to obtain the Chandrasekhar limit of compact
can be modeled by a Ginzburg-Landau (GL) free-energy Newtonian stars [6], the emergence of nonperturbative
functional with an order parameter φ(x) [4]. In the ab- configurations in the context of spontaneous symmetry
2
breaking [7] and in post-inflationary reheating [8] in the need here is to simulate a phase transition with enough
contextofatop-hillmodelofinflation[9]. (Notethatour accuracy, we leave such technical issues of lattice im-
usage of the name “configurational entropy” differs from plementation of effective field theories aside. We follow
otherinstancesintheliterature,asforexampleinprotein Ref. [12] for the implementation of the stochastic dy-
folding [10].) Here we show that in the context of con- namics, so that the noise is drawn from a unit Gaussian
tinuous phase transitions the CE provides a very precise scaled by a temperature dependent standard deviation,
(cid:112)
signature and a marked scaling behavior at criticality. ξ = 2ηθ/∆th2. To satisfy the Courant condition for
stable evolution we used a time-spacing of ∆t = h/4.
The discrete Laplacian is implemented via a maximally
MODEL AND NUMERICAL IMPLEMENTATION rotationally invariant convolution kernel [15] with error
of O(h2).
Weconsidera(2+1)-dimensionalGLmodelwherethe We start the field X at the minimum at X = −1 and
system is in contact with an ideal thermal bath at tem- thebathatlowtemperature,θ =0.01. Wewaituntilthe
perature T. The role of the bath is to drive the sys- field equilibrates, checked using the equipartition theo-
tem into thermal equilibrium. This can be simulated rem: in equilibrium, the average kinetic energy per de-
as a temperature-independent GL functional (so, setting gree of freedom of the lattice field, (cid:104)X˙2/2(cid:105), is (cid:104)X˙2(cid:105)=θ.
ij ij
t=−1 in Eq. 1) with a stochastic Langevin equation Oncethefieldisthermalized,whichtypicallytakesabout
1,000 time steps, we use ergodicity to take 200 readings
X¨ +ηX˙ −∇2X−X+X3+ξ =0, (2)
separatedby50timestepseachtoconstructanensemble
average. Wethenincreasethetemperatureinincrements
where we have introduced dimensionless variables xµ =
√1 yµ and φ = (cid:112)aX, and we took γ = 1. In Eq. of0.01andrepeattheentireprocedureuntilwecoverthe
a b interval θ ∈[0.01,0.83].
2, η is the viscosity and ξ(x,t) is the stochastic driv-
Wethenobtaintheensemble-averaged(cid:104)X(cid:105)vs. θ. The
ing noise of zero mean, (cid:104)ξ(cid:105) = 0. The two are related
couplingtothebathinducestemperature-dependentfluc-
bythefluctuation-dissipationrelation,(cid:104)ξ(x,t)ξ(x(cid:48),t(cid:48))(cid:105)=
tuations which, away from the critical point, can be de-
2ηθδ(x−x(cid:48))δ(t−t(cid:48)). θ ≡T/a is the dimensionless tem-
scribedbyaneffectivetemperature-dependentpotential,
perature and we take k =(cid:126)=1.
B as in the Hartree approximation [13]. The critical point
We use a staggered leapfrog method and periodic occurs as (cid:104)X(cid:105)→0, when the Z symmetry is restored.
2
boundary conditions to implement the simulation in a
InFig. 1weplottheresults. Thetop(blue)lineis(cid:104)X(cid:105),
square lattice of size L and spacing h. We used L=100
while the bottom (green) line is the ensemble-averaged
and h = 0.25. Simulations with larger values of L pro-
fraction of the volume occupied by X >0 (p ). Symme-
V
duce essentially similar results, apart from typical finite-
try restoration corresponds to this fraction approaching
sizescalingeffects[4]. Differentvaluesofhcanberenor-
0.5. Shadowed regions correspond to 1σ deviation from
malized with the addition of proper counter terms, as
the mean. Within the accuracy of our simulation, the
has been discussed in Refs. [11] and [12]. Since all we
critical temperature is θ (cid:39) 0.43±.01. In the top row
c
of Fig. 2 we show the field at different temperatures, in-
cludingat∼T ,wherelarge-sizefluctuatingdomainsare
c
apparent, indicative of the divergent correlation length
0.2 0.6
at θ .
c
0.0 0.5
n
0.2 0.4ctio CONFIGURATIONAL ENTROPY OF THE
a CRITICAL POINT
X ›fi0.4 0.3e Fr
m
u Considerthesetofsquare-integrableboundedperiodic
0.6 0.2ol
V functions with period L in d spatial dimensions, f(x) ∈
0.8 0.1 L2(Rd), and their Fourier series decomposition, f(x) =
1.00.0 0.1 0.2 0.3Temp0.e4ratur0e.5 0.6 0.7 0.80.0 (cid:80)n|Fik(kninFt)e|(g2ke/nr(cid:80)s).ei|kFnN·(xko,ww)|2idt.hefi(Fknoenrt=dheet2aπiml(snoad1n/adLl ,tf.rh.ae.c,tenixodtn/enLfs)ik,onnant=do
n n
nonperiodic functions see [5].) The configurational en-
FIG. 1: (Color online.) The order parameter (cid:104)X(cid:105) [top (blue) tropy for the function f(x), S [f], is defined as
C
line] and the volume fraction (p ) occupied by the X > 0
V
(cid:88)
phase [bottom (green) line] vs. temperature. The shaded S [f]=− f ln[f ]. (3)
regions correspond to 1σ deviations from the mean. Within C kn kn
the accuracy of our simulation, the critical temperature is The quantity σ(k )≡−f ln[f ] gives the relative en-
θc (cid:39)0.43±.01, marked by the vertical band. tropic contributionn of mokdne k k.nIn the spirit of Shan-
n
3
T=.20 T=.43 T=.45 T=.80 1.5
1.2
0.9
0.6
0.3
0.0
0.3
10-1 10-1 10-1 10-1 0.6
σ(k)10-2 10-2 ∝k−5/3 10-2 10-2 0.9
10-3 10-3 10-3 10-3 1.2
1.5
10-1 100 10-1 100 10-1 100 10-1 100
k k k k
FIG. 2: (Color online.) Top Row: Snapshots of the equilibrium field X(x,y) for different temperatures. The bar on the right
denotes the field magnitude. Bottom Row: The corresponding mode distribution of the CE density. Far from the critical
temperature, the CE density is scale-invariant for low k, while close to the critical temperature flow into IR produces scaling
suggestive of turbulent behavior, with power-law k−5/3 as in Kolmogorov turbulence [16].
non’s information entropy [14], S [f] gives an informa- (many momentum modes contribute), while delocalized
C
tionalmeasureoftherelativeweightsofdifferentk-modes distributionsminimizeit. S [f]is,inasense,anentropy
C
composing the configuration: it is maximized when all of shape, an informational measure of the complexity of
N modes carry the same weight, the mode equipartition a given spatial profile in terms of its momentum modes.
limit, f = 1/N for any k , with S [f] = lnN. If only The lower S [f], the less information (in terms of con-
kn n C C
a single mode is present, S [f]=0. For the lattice used tributing momentum modes) is needed to characterize
C
here, with N = 4002 points, the maximum entropy is theshape. Inthecontextofphasetransitions,weshould
Smax =11.98. expect S [f] to vary at different temperatures, as dif-
C C
ferent modes become active. In particular, given that
near T the average field distribution is dominated by
c
10.0 a few long-wavelength modes, we should expect a sharp
decrease in the CE.
9.5 We use Eq. 3 to compute the configuration entropy of
φ(x,y). The discrete Fourier transform is
9.0 (cid:88) (cid:88)
SC(T) Φ(kx,ky)=N φ(nxh,nyh)e−ih(nxkx+nyky),
8.5 nx=0ny=0
(4)
where k = 2πm /L, and m are integers. We
8.0 SC(Tc)≈8.0±.3 x(y) x(y) x(y)
display the CE density as a function of mode magnitude
|k| in the bottom row of Fig. 2. For temperatures far
7.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 from the critical value, we find a scale-invariant CE den-
Temperature
sity for long wavelength modes, and a tapering distribu-
tion for short wavelengths. As the critical temperature
FIG. 3: (Color online.) Configurational entropy vs. θ for the
is approached, the field configurations are dominated by
ensemble-averagedfield(cid:104)X(cid:105). Theminimumatθ isapparent.
c
largewavelengthmodes,andtheCEdensityflowstoward
Thelightershadeofgreycorrespondstoa1σ deviationfrom
the mean. lower values of |k|, disrupting the scale invariance of the
spectrum. This is expected from the diverging correla-
Consider what a field profile looks like for the above tion length that characterizes the critical point. We find
examples. Planewavesinmomentumspacehaveequally a scaling behavior near T with slope k−5/3, the same
c
distributedmodalfractions,andtheirpositionspacerep- as Kolmogorov turbulence in fluids [16]. Following the
resentations are highly localized. Conversely, singular analogy, wecall thiscriticalscaling information-entropic
modes in momentum space have plane wave representa- turbulence, here with modes flowing from the UV into
tions in position space which are maximally delocalized. the IR.
Localized distributions in position space maximize CE The flow of CE-density into IR modes illustrates how
4
lower-k modes occupy a progressively larger volume of the volume fraction (p ) they occupy, while computing
V
momentum space as the critical point is approached. the CE as their radii increase. In Fig. 5 we show the
Hence the sharp decline in CE near T . In fact, a simple CE as a function of p . From bottom up, the number
c V
mean-field estimate predicts that the CE will approach of bubbles is 5, 10, 25, 50, 100, and 150. The dashed
zero at T in the infinite-volume limit: using that in line corresponds to a symmetric bubble. As the number
c
mean-fieldtheoryξ(T)∼|T−T |−1/2andconsideringthe of bubbles increases, the CE at p =0.5 approaches the
c V
dominant fluctuations far away from T as being Gaus- numerical value at T (see Fig. 3). The results show
c c
sians with radius ξ(T), we can use the continuum limit a simple behavior, S (N,p ) ∼ B(N)log(p /L2), with
C V V
to compute the CE of a single Gaussian in d = 2 as [5]: B(N) having a weak N dependence.
S [T]=2π/ξ2 =4π(1−T/T ). Although even far away We have presented a new diagnostic tool to study
C c
from T we shouldn’t expect this simple approximation critical phenomena based on the configurational entropy
c
to match the behavior of Fig. 3, the general trend is (CE). We have shown how criticality implies in a sharp
for CE to vanish at T in the infinite volume limit. In minimum of CE, and identified a transition from scale-
c
a finite lattice of length L, there is going to be a mini- free to scaling behavior at criticality, with the same
mum value for CE given by the largest average fluctua- power law as Kolmogorov’s turbulence. We introduced
tion within the lattice. Indeed, in Fig. 3 we see that the apercolating-bubblemodeltodescribeournumericalre-
critical point is characterized by a minimum of the CE sults. We are currently exploring the notion of informa-
at S (θ )(cid:39)0.80±0.30. We found that as a function of tional turbulence and its relation to more complex per-
C c
|θ−θ | the CE dropssuper-exponentially as criticality is colation models with varying bubble sizes and hope to
c
approached from below, whereas from above we obtain report on our results soon.
the approximate scaling behavior SC[φ](θ)∝|θ−θc|−14.
For a more realistic estimate of the minimum value of
CE for a finite lattice near criticality, we model a large 8
fluctuationasadomainofradiusR andakink-likefunc-
tional profile 7
(cid:20) (cid:18) (cid:19)(cid:21)
1 R−a(ϕ)r
φ(r,ϕ)= 1+tanh , (5) 6
2 d SC
5
where d measures the thickness of the domain wall in
units of lattice length L and a is a random perturbation
4
defined as
a(ϕ)=1+(cid:88)10 αn cos(nϕ+β ), (6) 3 10-2 10-1
n n Volume Fraction
n=3
with α and β being uniformly-distributed random
n n FIG. 4: (Color online.) Configurational entropy vs. the log
numbers defined in the intervals (0,1) and (0,2π), re-
ofvolumefractionoccupiedbyasinglefluctuationwithtanh
spectively. We took d = 0.01L so that the domain profile defined in Eqs. 5 and 6. The shadowed area corre-
wall thickness matches the zero-temperature correlation sponds to 1σ deviation from mean. The dashed line corre-
length. Eqs. 5 and 6 define a fluctuation interpolating sponds to a symmetric bubble.
between φ = 0 and near φ = 1. In Fig. 4 we show
the results for an ensemble of such fluctuations occupy- The authors thank Adam Frank for useful discussions.
ing different volume fractions. Note how the symmetric MG and DS were supported in part by a Department
fluctuation (dashed line, obtained by setting a(ϕ)=1 in of Energy grant DE-SC0010386. MG also acknowledges
Eq. 5)haslowerCEthantheasymmetricfluctuations,as support from the John Templeton Foundation grant no.
one should expect given that CE measures spatial com- 48038.
plexity. Fluctuations with R ≥ L/4 (volume fraction
(cid:38) 0.2) begin to have boundary issues due to the tail
of the configuration. If we thus take a fluctuation with
R/L=0.25 to represent large fluctuations near T , from
c
Fig. 4 its ensemble-averaged CE is CE∼3.72±0.13. Of ∗ Electronic address: [email protected]
† Electronicaddress: [email protected]
course, a single bubble doesn’t match the complexity of
[1] B. Fultz, Phase Transitions in Materials (Cambridge
the percolating behavior at T . To simulate the perco-
c University Press eBook, Cambridge, UK, 2014).
lating transition, we placed different numbers of bubbles
[2] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and
randomly on the lattice with initial radius R/L=0.001. Other Topological Defects (Cambridge University Press,
WethenlettheirradiusRgrowtoR/L=0.1,measuring Cambridge, UK, 1994).
5
ed. (Pergamon, New York, 1980), Vol. 1.
11 [4] N. Goldenfeld, Lectures on Phase Transitions and The
10 Renormalization Group, Frontiers in Physics, Vol. 85
(Addison-Wesley, New York, 1992).
9
[5] M.Gleiser, N.Stamatopoulos,Phys.Lett.B713(2012)
8 304.
SC7 [6] M. Gleiser, D. Sowinski, Phys. Lett. B 727 (2013) 272.
[7] M. Gleiser, N. Stamatopoulos, Phys. Rev. D 86 (2012)
6
045004.
5 [8] M.Amin,M.P.Hertzberg,D.I.Kaiser,andJ.Karouby,
4 Int. J. Mod. Phys. D 24 (2015) 1530003.
[9] M.Gleiser,N.Graham,Phys.Rev.D89(2014)083502.
3
10-2 10-1 [10] J.H.Missimer, et al., Protein Sci. 16 (2007) 1349.
Volume Fraction
[11] G.Parisi,StatisticalFieldTheory(Addison-Wesley,New
York, 1988).
[12] J.BorrillandM.Gleiser,Nucl.Phys.B483(1997)416.
FIG. 5: (Color online.) Configurational entropy vs. the log
[13] G. Aarts, G. F. Bonini, and C. Wetterich, Phys. Rev. D
ofvolumefractionforensemblesofbubblesplacedatrandom
63, 025012 (2000); Nucl. Phys. B 587, 403 (2000).
positions in lattice of side L = 100. From bottom up, the
[14] C. E. Shannon, The Bell System Technical J. 27, 379
number of bubbles is 5, 10, 25, 50, 100, and 150. The shad-
(1948); ibid. 623 (1948).
owed areas corresponds to 1σ deviation from the mean. The
[15] T.Lindeberg, Scale-space for discrete signalsPAMI(12),
dashed line corresponds to a single bubble with tanh profile
No. 3, March 1990, pp. 234-254.
defined in Eqs. 5 and 6.
[16] J. C. R. Hunt, O. M. Philips, D. Williams, eds., Tur-
bulence and Stochastic Processes: Kolmogorov’s Ideas 50
Years On, Proc. Roy. Soc. London 434 (1991) 1 - 240.
[3] L.D.LandauandE.M.Lifshitz,Statistical Physics,3rd