Information-Entropic Signature of the Critical Point

Marcelo Gleiser1, and Damian Sowinski1, 1Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA

(Dated: January 26, 2015)

We investigate the critical behavior of continuous (second-order) phase transitions in the context of(2+1)-dimensional Ginzburg-Landau models with a double-well effective potential. In particular, weshow that the recently-proposed configurational entropy (CE)a measure of the spatial complexityof the order parameter based on its Fourier-mode decompositioncan be used to identify the criticalpoint. We compute the CE for different temperatures and show that large spatial fluctuations nearthe critical point (Tc)characterized by a divergent correlation lengthlead to a sharp decrease inthe associated configurational entropy. We further show that the CE density has a marked scalingbehavior near criticality, with the same power of Kolmogorov turbulence. We reproduce the behaviorof the CE at criticality with a percolating many-bubble model.

PACS numbers: 05.70.Jk,64.60.Bd,11.10.Lm,02.70.Hm

INTRODUCTION

From materials science [1] to the early universe [2],phase transitions offer a striking illustration of howchanging conditions can affect the physical properties ofmatter [3]. In very broad terms, and for the simplestsystems described by a single order parameter, it is cus-tomary to classify phase transitions as being either dis-continuous or continuous, or as first or second order, re-spectively. First-order phase transitions can be describedby an effective free-energy functional (or an effective po-tential in the language of field theory) where an energybarrier separates two or more phases available to the sys-tem. The system may transition from a higher to a lowerfree-energy state (or from a higher to a lower vacuumstate) either by a thermal fluctuation of sufficient size (acritical bubble) or, for low-temperatures, by a quantumfluctuation. Generally speaking, this description of ther-mal bubble nucleation is valid as long as F [b]/kBT 1,where F [b] is the 3d Euclidean action of the spherically-symmetric critical bubble or bounce b(r), kB is Boltz-manns constant, and T is the environmental tempera-ture. For quantum tunneling, one uses instead S4[b]/~,where S4[b] is the O(4)-symmetric Euclidean action ofthe 4d bounce.

For second-order transitions the order parameter variescontinuously as an external parameter such as the tem-perature is changed [3]. A well-known example is thatof an Ising ferromagnet, where the net magnetizationof a sample is zero above a critical temperature Tc, theCurie point, and non-zero below it. Below Tc the tran-sition unfolds via spinodal decomposition, whereby long-wavelength fluctuations become exponentially-unstableto growth. This growth is characterized by the appear-ance of domains with the same net magnetization whichcompete for dominance with their neighbors. In thecontinuum limit, systems in the Ising universality classcan be modeled by a Ginzburg-Landau (GL) free-energyfunctional with an order parameter (x) [4]. In the ab-

sence of an external source (or a magnetic field), the GLfree-energy functional is simply

E[] =

ddx

[

2()2 + a

2t2 +

b

44], (1)

where t (TTc), and , a, b are positive constants. ForT > Tc the system has a single free-energy minimum at = 0, while for T < Tc there are two degenerate minimaat 0 = (at/b)1/2. This mean-field theory descriptionworks well away from the critical point. In the neighbor-hood of Tc one uses perturbation theory and the renor-malization group to account for the divergent behaviorof the system. This behavior can be seen through thetwo-point correlation function G(r): away from Tc G(r)behaves as exp[r/(T )], where (T ) is the correla-tion length, a measure of the spatial extent of correlatedfluctuations of the order parameter. In mean-field the-ory, (T ) |T Tc| , where = 1/2, independently ofspatial dimensionality. In the neighborhood of Tc, wherethe mean-field description breaks down, the behavior ofspatial fluctuations is corrected using the renormaliza-tion group. Within the -expansion, one obtains, in 3d, = 1/2 + /12 + 72/162 ' 0.63 [4].

For continuous transitions in GL-systems, the focusof the present letter, the critical point is characterizedby having fluctuations on all spatial scales. This meansthat while away from Tc large spatial fluctuations aresuppressed, near Tc they dominate over smaller ones.In this letter, we will explore this fact to obtain a newmeasure of the critical point based on the informationcontent carried by fluctuations at different momentumscales. For this purpose we will use a measure of spatialcomplexity known as configurational entropy (CE), re-cently proposed by Gleiser and Stamatopoulos [5]. TheCE has been used to characterize the information con-tent [5] and stability of solitonic solutions of field theo-ries [5, 6], to obtain the Chandrasekhar limit of compactNewtonian stars [6], the emergence of nonperturbativeconfigurations in the context of spontaneous symmetry

2breaking [7] and in post-inflationary reheating [8] in thecontext of a top-hill model of inflation [9]. (Note that ourusage of the name configurational entropy differs fromother instances in the literature, as for example in proteinfolding [10].) Here we show that in the context of con-tinuous phase transitions the CE provides a very precisesignature and a marked scaling behavior at criticality.

MODEL AND NUMERICAL IMPLEMENTATION

We consider a (2+1)-dimensional GL model where thesystem is in contact with an ideal thermal bath at tem-perature T . The role of the bath is to drive the sys-tem into thermal equilibrium. This can be simulatedas a temperature-independent GL functional (so, settingt = 1 in Eq. 1) with a stochastic Langevin equation

X + X 2X X +X3 + = 0, (2)where we have introduced dimensionless variables x =1ay and =

abX, and we took = 1. In Eq.

2, is the viscosity and (x, t) is the stochastic driv-ing noise of zero mean, = 0. The two are relatedby the fluctuation-dissipation relation, (x, t)(x, t) =2(x x)(t t). T/a is the dimensionless tem-perature and we take kB = ~ = 1.

We use a staggered leapfrog method and periodicboundary conditions to implement the simulation in asquare lattice of size L and spacing h. We used L = 100and h = 0.25. Simulations with larger values of L pro-duce essentially similar results, apart from typical finite-size scaling effects [4]. Different values of h can be renor-malized with the addition of proper counter terms, ashas been discussed in Refs. [11] and [12]. Since all we

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Temperature

1.0

0.8

0.6

0.4

0.2

0.0

0.2

X

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Volume Fraction

FIG. 1: (Color online.) The order parameter X [top (blue)line] and the volume fraction (pV ) occupied by the X > 0phase [bottom (green) line] vs. temperature. The shadedregions correspond to 1 deviations from the mean. Withinthe accuracy of our simulation, the critical temperature isc ' 0.43 .01, marked by the vertical band.

need here is to simulate a phase transition with enoughaccuracy, we leave such technical issues of lattice im-plementation of effective field theories aside. We followRef. [12] for the implementation of the stochastic dy-namics, so that the noise is drawn from a unit Gaussianscaled by a temperature dependent standard deviation, =

2/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 maximallyrotationally invariant convolution kernel [15] with errorof O(h2).

We start the field X at the minimum at X = 1 andthe bath at low temperature, = 0.01. We wait until thefield equilibrates, checked using the equipartition theo-rem: in equilibrium, the average kinetic energy per de-gree of freedom of the lattice field, X2ij/2, is X2ij = .Once the field is thermalized, which typically takes about1, 000 time steps, we use ergodicity to take 200 readingsseparated by 50 time steps each to construct an ensembleaverage. We then increase the temperature in incrementsof 0.01 and repeat the entire procedure until we cover theinterval [0.01, 0.83].

We then obtain the ensemble-averaged X vs. . Thecoupling to the bath induces temperature-dependent fluc-tuations which, away from the critical point, can be de-scribed by an effective temperature-dependent potential,as in the Hartree approximation [13]. The critical pointoccurs as X 0, when the Z2 symmetry is restored.

In Fig. 1 we plot the results. The top (blue) line is X,while the bottom (green) line is the ensemble-averagedfraction of the volume occupied by X > 0 (pV ). Symme-try restoration corresponds to this fraction approaching0.5. Shadowed regions correspond to 1 deviation fromthe mean. Within the accuracy of our simulation, thecritical temperature is c ' 0.43 .01. In the top rowof Fig. 2 we show the field at different temperatures, in-cluding at Tc, where large-size fluctuating domains areapparent, indicative of the divergent correlation lengthat c.

CONFIGURATIONAL ENTROPY OF THECRITICAL POINT

Consider the set of square-integrable bounded periodicfunctions with period L in d spatial dimensions, f(x) L2(Rd), and their Fourier series decomposition, f(x) =

knF (kn)e

iknx, with kn = 2pi(n1/L, . . . , nd/L), andni integers. Now define the modal fraction fkn =|F (kn)|2/

|F (kn)|2. (For details and the extension tononperiodic functions see [5].) The configurational en-tropy for the function f(x), SC [f ], is defined as

SC [f ] =

fkn ln[fkn ]. (3)

The quantity (kn) fkn ln[fkn ] gives the relative en-tropic contribution of mode kn. In the spirit of Shan-

3T=.20 T=.43 T=.45 T=.80

10-1 100

k

10-3

10-2

10-1

(k)

10-1 100

k

10-3

10-2

10-1

10-1 100

k

10-3

10-2

10-1

10-1 100

k

10-3

10-2

10-1

k5/3

1.5

1.2

0.9

0.6

0.3

0.0

0.3

0.6

0.9

1.2

1.5

FIG. 2: (Color online.) Top Row: Snapshots of the equilibrium field X(x, y) for different temperatures. The bar on the rightdenotes the field magnitude. Bottom Row: The corresponding mode distribution of the CE density. Far from the criticaltemperature, the CE density is scale-invariant for low k, while close to the critical temperature flow into IR produces scalingsuggestive of turbulent behavior, with power-law k5/3 as in Kolmogorov turbulence [16].

nons information entropy [14], SC [f ] gives an informa-tional measure of the relative weights of different k-modescomposing the configuration: it is maximized when allN modes carry the same weight, the mode equipartitionlimit, fkn = 1/N for any kn, with SC [f ] = lnN . If onlya single mode is present, SC [f ] = 0. For the lattice usedhere, with N = 4002 points, the maximum entropy isSmaxC = 11.98.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Temperature

7.5

8.0

8.5

9.0

9.5

10.0

SC (T)

SC (Tc )8.0.3

FIG. 3: (Color online.) Configurational entropy vs. for theensemble-averaged field X. The minimum at c is apparent.The lighter shade of grey corresponds to a 1 deviation fromthe mean.

Consider what a field profile looks like for the aboveexamples. Plane waves in momentum space have equallydistributed modal fractions, and their position space rep-resentations are highly localized. Conversely, singularmodes in momentum space have plane wave representa-tions in position space which are maximally delocalized.Localized distributions in position space maximize CE

(many momentum modes contribute), while delocalizeddistributions minimize it. SC [f ] is, in a sense, an entropyof shape, an informational measure of the complexity ofa given spatial profile in terms of its momentum modes.The lower SC [f ], the less information (in terms of con-tributing momentum modes) is needed to characterizethe shape. In the context of phase transitions, we shouldexpect SC [f ] to vary at different temperatures, as dif-ferent modes become active. In particular, given thatnear Tc the average field distribution is dominated bya few long-wavelength modes, we should expect a sharpdecrease in the CE.

We use Eq. 3 to compute the configuration entropy of(x, y). The discrete Fourier transform is

(kx, ky) = Nnx=0

ny=0

(nxh, nyh)eih(nxkx+nyky),

(4)where kx(y) = 2pimx(y)/L, and mx(y) are integers. Wedisplay the CE density as a function of mode magnitude|k| in the bottom row of Fig. 2. For temperatures farfrom the critical value, we find a scale-invariant CE den-sity for long wavelength modes, and a tapering distribu-tion for short wavelengths. As the critical temperatureis approached, the field configurations are dominated bylarge wavelength modes, and the CE density flows towardlower values of |k|, disrupting the scale invariance of thespectrum. This is expected from the diverging correla-tion length that characterizes the critical point. We finda scaling behavior near Tc with slope k

5/3, the sameas Kolmogorov turbulence in fluids [16]. Following theanalogy, we call this critical scaling information-entropicturbulence, here with modes flowing from the UV intothe IR.

The flow of CE-density into IR modes illustrates how

4lower-k modes occupy a progressively larger volume ofmomentum space as the critical point is approached.Hence the sharp decline in CE near Tc. In fact, a simplemean-field estimate predicts that the CE will approachzero at Tc in the infinite-volume limit: using that inmean-field theory (T ) |TTc|1/2 and considering thedominant fluctuations far away from Tc as being Gaus-sians with radius (T ), we can use the continuum limitto compute the CE of a single Gaussian in d = 2 as [5]:SC [T ] = 2pi/

2 = 4pi(1 T/Tc). Although even far awayfrom Tc we shouldnt expect this simple approximationto match the behavior of Fig. 3, the general trend isfor CE to vanish at Tc in the infinite volume limit. Ina finite lattice of length L, there is going to be a mini-mum value for CE given by the largest average fluctua-tion within the lattice. Indeed, in Fig. 3 we see that thecritical point is characterized by a minimum of the CEat SC(c) ' 0.80 0.30. We found that as a function of| c| the CE drops super-exponentially as criticality isapproached from below, whereas from above we obtainthe approximate scaling behavior SC []() | c| 14 .

For a more realistic estimate of the minimum value ofCE for a finite lattice near criticality, we model a largefluctuation as a domain of radius R and a kink-like func-tional profile

(r, ) =1

2

[1 + tanh

(R a()r

d

)], (5)

where d measures the thickness of the domain wall inunits of lattice length L and a is a random perturbationdefined as

a() = 1 +

10n=3

nn

cos (n+ n) , (6)

with n and n being uniformly-distributed randomnumbers defined in the intervals (0, 1) and (0, 2pi), re-spectively. We took d = 0.01L so that the domainwall thickness matches the zero-temperature correlationlength. Eqs. 5 and 6 define a fluctuation interpolatingbetween = 0 and near = 1. In Fig. 4 we showthe results for an ensemble of such fluctuations occupy-ing different volume fractions. Note how the symmetricfluctuation (dashed line, obtained by setting a() = 1 inEq. 5) has lower CE than the asymmetric fluctuations, asone should expect given that CE measures spatial com-plexity. Fluctuations with R L/4 (volume fraction& 0.2) begin to have boundary issues due to the tailof the configuration. If we thus take a fluctuation withR/L = 0.25 to represent large fluctuations near Tc, fromFig. 4 its ensemble-averaged CE is CE 3.72 0.13. Ofcourse, a single bubble doesnt match the complexity ofthe percolating behavior at Tc. To simulate the perco-lating transition, we placed different numbers of bubblesrandomly on the lattice with initial radius R/L = 0.001.We then let their radius R grow to R/L = 0.1, measuring

the volume fraction (pV ) they occupy, while computingthe CE as their radii increase. In Fig. 5 we show theCE as a function of pV . From bottom up, the numberof bubbles is 5, 10, 25, 50, 100, and 150. The dashedline corresponds to a symmetric bubble. As the numberof bubbles increases, the CE at pV = 0.5 approaches thenumerical value at Tc (see Fig. 3). The results showa simple behavior, SC(N, pV ) B(N) log(pV /L2), withB(N) having a weak N dependence.

We have presented a new diagnostic tool to studycritical phenomena based on the configurational entropy(CE). We have shown how criticality implies in a sharpminimum of CE, and identified a transition from scale-free to scaling behavior at criticality, with the samepower law as Kolmogorovs turbulence. We introduceda percolating-bubble model to describe our numerical re-sults. We are currently exploring the notion of informa-tional turbulence and its relation to more complex per-colation models with varying bubble sizes and hope toreport on our results soon.

10-2 10-1

Volume Fraction

3

4

5

6

7

8

SC

FIG. 4: (Color online.) Configurational entropy vs. the logof volume fraction occupied by a single fluctuation with tanhprofile defined in Eqs. 5 and 6. The shadowed area corre-sponds to 1 deviation from mean. The dashed line corre-sponds to a symmetric bubble.

The authors thank Adam Frank for useful discussions.MG and DS were supported in part by a Departmentof Energy grant DE-SC0010386. MG also acknowledgessupport from the John Templeton Foundation grant no.48038.

Electronic address: mgleiser@dartmouth.edu Electronic address: Damian.Sowinski.GR@dartmouth.edu

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510-2 10-1

Volume Fraction

3

4

5

6

7

8

9

10

11

SC

FIG. 5: (Color online.) Configurational entropy vs. the logof volume fraction for ensembles of bubbles placed at randompositions in lattice of side L = 100. From bottom up, thenumber of bubbles is 5, 10, 25, 50, 100, and 150. The shad-owed areas corresponds to 1 deviation from the mean. Thedashed line corresponds to a single bubble with tanh profiledefined in Eqs. 5 and 6.

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IntroductionModel and Numerical ImplementationConfigurational Entropy of the Critical PointAcknowledgmentsReferences