Entropic Measure for Localized Energy Configurations: Kinks, Bounces, and Bubbles
ABSTRACT We construct a configurational entropy measure in functional space. We apply
it to several nonlinear scalar field models featuring solutions with
spatiallylocalized energy, including solitons and bounces in one spatial
dimension, and critical bubbles in three spatial dimensions, typical of
firstorder phase transitions. Such field models are of widespread interest in
many areas of physics, from high energy and cosmology to condensed matter.
Using a variational approach, we show that the higher the energy of a trial
function that approximates the actual solution, the higher its relative
configurational entropy, defined as the absolute difference between the
configurational entropy of the actual solution and of the trial function.
Furthermore, we show that when different trial functions have degenerate
energies, the configurational entropy can be used to select the best fit to the
actual solution. The configurational entropy relates the dynamical and
informational content of physical models with localized energy configurations.
 Citations (1)
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Article: Variational Principles of mechanics
01/1986;
Page 1
Entropic Measure for Localized Energy Configurations: Kinks, Bounces, and Bubbles
Marcelo Gleiser1, ∗and Nikitas Stamatopoulos1, †
1Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA
(Dated: November 24, 2011)
We construct a configurational entropy measure in functional space.
nonlinear scalar field models featuring solutions with spatiallylocalized energy, including solitons
and bounces in one spatial dimension, and critical bubbles in three spatial dimensions, typical of
firstorder phase transitions. Such field models are of widespread interest in many areas of physics,
from high energy and cosmology to condensed matter. Using a variational approach, we show that
the higher the energy of a trial function that approximates the actual solution, the higher its relative
configurational entropy. Furthermore, we show that when different trial functions have degenerate
energies, the configurational entropy can be used to select the best fit to the actual solution. The
configurational entropy relates the dynamical and informational content of physical models and can
be applied to any nonlinear field model.
We apply it to several
PACS numbers: 11.10.Lm, 03.65.Ge,05.65.+b
INTRODUCTION
Hamilton’s principle of least action states that out of
the infinitely many paths x(t) connecting two fixed points
in time, nature always chooses the one that leaves the ac
tion S[x] stationary [1]. This path, as is well known, is
the solution to the EulerLagrange equation of motion,
obtained from imposing that the action’s first variation
vanishes, δS = 0. As we move from point particles to
continuous systems, say, as described by a scalar field
φ(x,t), Hamilton’s principle will generate the partial dif
ferential equation (or equations, for vector or tensor
valued fields) describing the motion that leaves the ac
tion functional S[φ] stationary. In this letter, we will
explore the relation between information and dynamics,
proposing an entropic measure that quantifies the infor
mational content of physical solutions to the equations
of motion and their approximations–the configurational
entropy in functional space. As we will see, this mea
sure may offer new insight into how spatiallylocalized
ordered structures such as topological defects emerge in a
widespread class of natural phenomena, from highenergy
physics and cosmology[2] to condensed matter [3].
To illustrate our point, we will investigate several non
linear scalar field models in one and three spatial di
mensions that have spatiallylocalized energy configura
tions such as solitons and critical bubbles. After defining
the configurational entropy, we will compute it for mod
els that admit spatiallylocalized energy solutions such
as kinks, bounces, and critical bubbles. We will then
compute the configurational entropy for several ansatze
that approximate the solutions to the eom with vary
ing degrees of coarseness. To compare solutions and ap
proximations, we will define the relative configurational
entropy–a measure of relative ordering in field configu
ration space–and show that it correlates well with the
energy: the coarser the approximation to the solution,
the higher its energy and its relative configurational en
tropy. On the basis of these results, we propose that
nature is the ultimate optimizer, not only in extremiz
ing energy through its myriad motions, but also from an
ordering perspective. We will also show how the configu
rational entropy can resolve ambivalent situations where
the energies of different trial functions are degenerate,
thus providing a useful criterion to obtain optimized an
alytical approximations to the solutions of the equations
of motion, as may be needed in different applications.
MATHEMATICAL PRELIMINARIES: DEFINING
THE CONFIGURATIONAL ENTROPY
Since we are interested in spatiallylocalized structures
(and/or their related energy densities), consider the set
of squareintegrable functions f(x) ∈ L2(R) and their
Fourier transforms F(k).Plancherel’s theorem states
that [4]
?∞
Now define the modal fraction f(k),
−∞
f(x)2dx =
?∞
−∞
F(k)2dk.(1)
f(k) =
F(k)2
?F(k)2ddk,
(2)
where the integration is over all k where F(k) is defined
and d is the number of spatial dimensions. f(k) mea
sures the relative weight of a given mode k. For periodic
functions where a Fourier series is defined, f(k) → fn=
An2/?An2, where An is the coefficient of the nth
Let us the define the configurational entropy SC[f] as
?
We call the integrand f(k)ln[f(k)] the configura
tional entropy density.We note that SC[f] is sim
Fourier mode.
SC[f] = −
f(k)ln[f(k)]ddk.(3)
arXiv:1111.5597v1 [hepth] 23 Nov 2011
Page 2
2
ilar to the Gibbs entropy of nonequilibrium thermo
dynamics, although the latter is defined for a statis
tical ensemble with microstates with probability pi =
exp(−Ei/kBT)/?exp(−Ei/kBT), where Ei is the en
stant [5]. Note also that the modal fraction has physi
cal dimensions of m−d, where m is the relevant mass or
energy scale of the system. Thus, when computing the
configurational entropy in physical systems, the entropy
density will be calculated using f(k)ln[mdf(k)]. SC[f]
is dimensionless. For a discrete set of amplitudes, the
configurational entropy is
?
In analogy with Shannon’s information entropy, SS =
−?pilog2pi, which represents an absolute limit on the
[6], we can think of the configurational entropy as provid
ing the informational content of configurations compat
ible with the particular constraints of a given physical
system. When all N modes k carry the same weight,
fn = 1/N and the discrete configurational entropy has
a maximum at SC= lnN. If only one mode is present,
SC= 0.
As an example, consider a Gaussian in d dimensions,
f(r) = N exp(−αr2), and its Fourier transform, F(k) =
N exp(−k2/4α)
[2α](d/2)
. Using the definition of Eq. 2,
ergy of the ith microstate and kB is Boltzmann’s con
SC[f] = −
fnln(fn).(4)
best possible lossless compression of any communication
f(k) =exp(−k2/2α)
[2πα]d/2
.(5)
Note that the zero mode carries the most weight. Impos
ing that f(k) ≤ 1 for all modes, we obtain the consistency
condition 2πα ≥ 1. Similar conditions can be imposed
for other functions. Eq. 3 gives
SC(α) =d
2[1 + ln(2πα)].(6)
As we go up in dimensions, SC(α) increases in multiples
of [1 + ln(2πα)]/2. The consistency condition for f(k)
implies in a lower bound for the configurational entropy
of Gaussians, SC≥ d/2. As we shall see, Gaussians that
violate this lower bound don’t seem to play a role on
physically interesting solutions. Defining the dispersion
of a function f(x) as
?∞
we know that for f(x) and F(k) normalized to unity, they
satisfy the minimum uncertainty relation D0(f)D0(F) =
1. We thus expect the configurational entropy of a Gaus
sian to represent an absolute minimum for spatially
localized functions f(x) parameterized by the same spa
tial dispersion with one (or more) maxima for x ∈
(−∞,∞). In this case, we can write, SC[f] ≥ SC(α).
D0(f) =
−∞
x2f(x)2dx,(7)
In physical applications where the function f(x) rep
resents a spatiallyconfined structure (e.g. a φ4or a sine
Gordon kink [7]), the parameter α or its equivalent de
termining the spatial extent of the structure will be re
stricted by the relevant interactions or constraints. For
example, a free particle of mass m confined to a rigidwall
box of length 2R has momentum pn= n?π/R. If mod
eled by a Gaussian with dispersion σ ≡ (2α)−1/2centered
at R, the uncertainty relation gives α ≤ π2/2R2. On
the other hand, σ2= (1/2α) < R2and so, α ≥ 1/2R2.
We thus have, 1/2 ≤ αR2≤ π2/2. Since 2R cannot be
smaller than the reduced Compton wavelength of the par
ticle, 2R ≥ 2π/m, we get, 1/2π2≤ α/m2≤ 1/2.
FREE FIELD IN A BOX
As an application of the configurational entropy for
periodic boundary conditions, consider a free massive
scalar field in a rigidwall box of size L.
φ(0,t) = φ(L,t) = 0, we obtain the familiar quanti
zation of momentum modes, kn = nπ/L, and disper
sion relation ω2
Further imposing that˙φ(x,0) = 0, the general solu
tion is φ(x,t) =
ficients An are given in terms of the initial configura
tion φ0(x) = φ(x,0) as An= (2/L)?L
butions from different normal modes to a given precision
of the series expansion. The configurational entropy of a
general field profile, obtained from Eq. 4, is given by
??(An2)(An2)
Imposing
n= (nπc/L)2+ m2c2, with n = 1,2....
?Ansinknxcosωnt, where the coef
0φ0(x)sin[knx]dx.
Thus, the Fourier amplitudes {An} determine the contri
SC[φ] = −
1
?An2ln
(?An2)(?An2)
?
.(8)
If An= A for n = 1,...,N, then SC[φ] = lnN, and the
configuration entropy is maximal [5]. Also, if only a single
mode is present, SC[φ] = 0: the most ordered state is a
pure state. From an initial configuration with a certain
SC[φ0], the configurational entropy will evolve in time as
the Fourier coefficients An(t) = Ancosωnt change.
CONFIGURATIONAL ENTROPY OF KINKS
Consider a scalar field model in d = 1 with energy
density ρV[φ] = (˙φ)2/2 + (φ?)2/2 + V (φ), where the dot
and the prime denote time and spatial derivative, respec
tively. (We set c = ? = 1). We are interested in situ
ations where the EulerLagrange equations admit static
solutions with localized energy density. (So, the energy
density, and not necessarily the field, must be square
integrable.) This is the case, for example, for the kink
solution for the doublewell and for the sineGordon po
tentials [7], which we now examine in turn.
Page 3
3
Case 1: Symmetric doublewell potential
If V (φ) = (λ/4)(φ2− m2/λ)2, the kink (or antikink)
is the static solution interpolating the two minima of the
potential at φmin= ±(m/√λ) as x → ±∞, and is given
by φk(x) = ±(m/√λ)tanh(mx/√2). The kink’s energy
density is ρ[φk](x) = (m4/2√λ)sech4(mx/√2) and its
energy is Ek=√8/3m3/λ ? 0.9428m3/λ. From the en
ergy density and its Fourier transform, Eq. 2 gives the
modal fraction
f(k) =35πk2(4α2+ k2)2
2304α7
csch2
?kπ
2α
?
,(9)
where α ≡ m/√2. Using f(k) into Eq. 3, we obtain the
configurational entropy for the φ4kink, SC[φk] = 1.3588.
How does this value compare to other configurations
satisfying the same boundary conditions that approxi
mate the kink? To check this, we use a variational ap
proach. As an illustration, consider the function h(x) =
(2/π)tan−1(αx) expressed in units of m/√λ. The energy
E[h] is minimized for αc= [24ζ(3)/π2]1/2m ? 1.7097m.
With this value, the energy is E[h] = 1.0884m3/λ and the
configurational entropy–computed, as for the kink, from
the energy density–is SC[h] = 1.8828 > SC[φk]. Hence,
for this first example, we see that the trial function h(x)
has both larger energy (as it should) and larger entropy
than the kink solution.
We would like to have a more general measure to com
pare the configurational entropy of trial functions to that
of the solution of the eom which is also insensitive to the
rare cases when f(k) > 1. Writing the modal fraction for
the trial function (or for its energy density, when appro
priate) as f(k) and the modal fraction for the solution
of the eom (or its energy density) as g(k), we define the
relative configurational entropy
?
SR[f] provides a measure of entropic distance in func
tional space. In information theory, the KullbackLeibler
divergence offers a similar measure [8]. In Table I we
show the results for SRfor the trial function h(x) above
and two other approximations to the kink solution. We
define ∆E = [Ea− Ek]/Ek as the relative difference
in energy between the kink solution and the various
ansatze approximating it, where the subscript a refers
to “ansatz”. In all three cases, the coarser the ansatz–in
the sense of having larger energy than the kink–the larger
its relative configurational entropy.
SR[f] = ddkf(k)ln[f(k)/g(k)]. (10)
Case 2: sineGordon potential
For another example, varying the 1d action with
the sineGordon potential, VsG(φ) = [1 − cos(aφ)]/a2,
Ansatzαc[m]
1.1647
∆ESR
4tan−1[exp(αx)]/π − 1
exp[αx] − 1,x < 0
−exp[−αx] + 1,x ≥ 0
2tan−1[αx]/π
1.3 × 10−3
1.55 × 10−2
0.0354
0.95740.2304
1.70970.14470.6956
TABLE I: Comparison of energy and relative configurational
entropy of the φ4kink and several trial functions.
gives the static kink (antikink) solution φsG(x)
(4/a)tan−1exp[±x/√a], with energy density ρsG[φ] =
2VsG(φ). Integrating over space we get EsG = 8/a2=
8m3/λ.We will set a = 1 from now on.
energy density, we obtain the modal fraction f(k) =
(3/8)πk2csch2(kπ/2), and Eq. 3 gives the configurational
entropy for the sineGordon kink, SC[φsG] = 1.3029. We
now proceed as before and compare these values to those
for trial functions that approximate φsG(x), computing
the relative configurational entropy SR. Results are dis
played in Table II:
=
From the
Ansatzαc[m]
0.6087
∆ESR
π tanh[αx] + π
π exp[αx],x < 0
−π exp[−αx] + 2π,x ≥ 0
2tan−1[αx] + π
1.2 × 10−3
8.3 × 10−3
0.0312
0.81730.1580
1.4136 0.10240.4678
TABLE II: Comparison of energy and relative configurational
entropy of the sineGordon kink and several trial functions.
Again, we see that as the energy of the approxima
tion to the solution to the eom increases, so does its rela
tive configurational entropy. Although we haven’t proved
that this correlation between lower energy and lower rel
ative configurational entropy holds for all possible trial
functions approximating the solutions, the examples an
alyzed here make for a compelling case. The configura
tional entropy offers an efficient measure of the ordering
associated with localized energy configurations. Next, we
extend our approach to bounces and critical bubbles as
sociated with metastable decay in one and three spatial
dimensions, respectively. We will see that not only the
same correlation holds, but that the relative configura
tional entropy provides a way to break possible energy
degeneracies between two or more trial functions.
Page 4
4
CONFIGURATIONAL ENTROPY OF BOUNCES
AND BUBBLES
Consider a onedimensional scalar field model with po
tential [9]
V (φ) =λ
4φ2(φ − 2φ0)2− αφ0φ3, (11)
where λ and α are positive constants. For α = 0, the
potential is a symmetric doublewell as in the φ4kink
but shifted so that its minima are at φ = 0 and φ = 2φ0.
Defining˜φ = φ/φ0, ? = α/λ and dropping the tildes, the
potential can be written as
V (φ) =λφ4
0
4
φ2?φ2− 4φ(1 + ?) + 4?.(12)
As is well known [10], the bounce [φbo(x)] is a static solu
tion to the equation of motion that vanishes as x → ±∞
and reaches a maximum at x = 0 at the classical turn
ing point φtp= 2[(1 + ?) − (2? + ?2)1/2]. So, differently
from kinks, here it is more convenient to define the con
figurational entropy of the field itself, as opposed to that
of the energy density. Also differently from kinks, so
lutions for different values of ? must be found numeri
cally. The procedure, however, is the same: first, solu
tions to the eom and their Fourier transforms are found.
Then, the modal fraction and the configurational en
tropy SC[φbo] are computed. In order to compare the
bounce solution with trial functions that approximate
it, we compute the relative configurational entropy from
Eq. 10. For example, a useful trial function is the Gaus
sian f(x) = φtpexp[−α(?)x2], where α(?) is the varia
tional parameter. Minimizing the energy with respect to
α we find αc(?) = 2√2[φ2
Note that the consistency condition we imposed above
for Gaussians, α > 1/2π, implies that the Gaussian ap
proximation is only valid for ? > 3.28 × 10−4, not very
stringent. This is consistent with the fact that for very
small asymmetries (thin wall approximation) the Gaus
sian is not a good trial function for the bounce.
Forasecondtrialfunction
φtpsech[α(?)x].In Fig. 1 we compare the energy (in
sert top left) and relative configurational entropy as a
function of ? for the two trial functions. Notice that the
energy is degenerate at ?d= 0.46 and that for ? < [>]?d
the Gaussian [sech(x)] has smaller energy. On the other
hand, the Gaussian has lower relative configurational en
tropy for all ?. At the degenerate point, the variational
approach is ambiguous: energy considerations alone are
insufficient to pick the best trial function there. The de
generacy is broken by using the relative configurational
entropy, which clearly picks the Gaussian as the closest to
the bounce solution. In the top right insert we plot the
functional distance measure ∆f(?) ≡
for both trial functions as a function of ?.
the Gaussian is clearly a better approximation for the
tp/8−φtp(1+?)√3/3+√2/2)].
we usef(x)=
?dxf(x) − φbo
For low ?
bounce, including at ?d. As ? increases, the two trial
functions have comparable values of ∆f(?).
0.00.20.40.60.81.0
?
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
SR(?)
0.00.20.4 0.60.81.0
0.01
0.02
0.03
∆E(?)
0.00.2 0.40.60.81.0
0.2
0.4
0.6
0.8
∆f(?)
FIG. 1: Relative configurational entropy SR(?), energy differ
ence ∆E(?) and functional distance ∆f(?) for two trial func
tions approximating the 1d bounce solution. Continuous line
denotes the Gaussian and dashed line the Sech. The energy
difference is degenerate at ?d = 0.46, while the Gaussian al
ways has lower relative configurational entropy.
We repeat the computations for a 3d scalar field model
with the same potential as in Eq. 11. Now, the eom is
φ??+ 2φ?/r = ∂V/∂φ. The sphericallysymmetric critical
bubble solution has φ(r = 0) = φ0, φ?(r = 0) = 0 and
φ(r) = 0 as r → ∞ [11]. As with the 1d case, we use the
field to compute the modal fraction and relative config
urational entropy as a function of the tilt ?. The results
are shown in Fig. 2. Here, there is no energy degeneracy
and the Gaussian has both lower energy and lower rela
tive configurational entropy for all ? studied. For ? ? 0.3,
we can’t extremize the energy to find αcsince the critical
bubble has a flatter profile near the origin. Other trial
functions could be used to explore this range without
difficulty.
SUMMARY AND OUTLOOK
We have proposed an entropic measure in field configu
ration space for nonlinear models with spatiallylocalized
energy solutions. We computed the relative configura
tional entropy SRfor several trial functions approximat
ing solutions to the eom for different models, showing
that higher SR correlates with higher energy. In cases
where there is a degeneracy in the energy of trial func
tions, SR can be used as a discriminant. In forthcom
ing papers we will extend our approach to nonequilib
rium field theory and cosmology. Our measure can be
extended to models including gravity and gauge fields,
and could potentially be used to discriminate between
Page 5
5
0.30.40.50.60.70.80.9 1.0
?
0.0
0.5
1.0
1.5
2.0
2.5
SR(?)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.2
0.6
1.0
1.4
∆E(?)
FIG. 2: Relative configurational entropy SR(?) and energy
difference ∆E(?) for two trial functions approximating the 3d
critical bubble solution. Continuous line denotes the Gaussian
and dashed line the Sech. The Gaussian has lower relative
configurational entropy and energy for all ? probed.
solutions in the superstring landscape [12]. We hope the
present work will spearhead research in these topics.
MG is supported in part by a National Science Founda
tion grant PHY1068027. NS is a Gordon F. Hull Fellow
at Dartmouth College.
∗Electronic address: mgleiser@dartmouth.edu
†Electronic address: nstamato@dartmouth.edu
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