![The pH (−Log[H + ]) of 120 samples of INW as a function of Log χ. Each point represented in the figure is obtained experimentally measuring the pH and the electrical conductivity χ (µS cm −1 ) of each of the 120 samples. 11](https://www.researchgate.net/profile/Vittorio-Elia/publication/272839310/figure/fig2/AS:669298532024376@1536584613993/The-pH-LogH-of-120-samples-of-INW-as-a-function-of-Log-ch-Each-point-represented_Q320.jpg)
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January 16, 2014 14:2 WSPC/Guidelines-IJMPB S0217979214500076
International Journal of Modern Physics B
Vol. 28, No. 3 (2014) 1450007 (20 pages)
c
World Scientific Publishing Company
DOI: 10.1142/S0217979214500076
SELF-SIMILARITY PROPERTIES OF NAFIONIZED AND
FILTERED WATER AND DEFORMED COHERENT STATES
A. CAPOLUPO∗, E. DEL GIUDICE†, V. ELIA‡,∗∗, R. GERMANO§, E. NAPOLI‡,††,
M. NICCOLI‡,‡‡, A. TEDESCHI¶and G. VITIELLOk
∗Dipartimento di Ingegneria Industriale, Universit´a di Salerno,
I-84084 Fisciano (Salerno), Italy
†Sezione INFN, I-20122 Milano, Italy (retired) and Centro Studi Eva Reich,
Via Orti, 5, I-20122 Milano, Italy
‡Dipartimento di Scienze Chimiche, Universit´a di Napoli “Federico II”,
I-80100 Napoli, Italy
§PROMETE Srl, CNR Spin off, via Buongiovanni,
49, I-80046 San Giorgio a Cremano (Napoli), Italy
¶WHITE Holographic Bioresonance, Via F. Petrarca, 16, I-20123 Milano, Italy
kDipartimento di Fisica “E. R. Caianiello” and Istituto Nazionale di Fisica Nucleare,
Universit´a di Salerno, I-84084 Fisciano (Salerno), Italy
∗capolupo@sa.infn.it
†delgiudice@mi.infn.it
∗∗vittorio.elia@unina.it
§germano@promete.it
††elena.napoli@unina.it
‡‡marcella.niccoli@unina.it
¶gowhite@usa.net
kvitiello@sa.infn.it
Received 31 August 2013
Revised 6 October 2013
Accepted 22 October 2013
Published 19 November 2013
By resorting to measurements of physically characterizing observables of water samples
perturbed by the presence of Nafion and by iterative filtration processes, we discuss their
scale free, self-similar fractal properties. By use of algebraic methods, the isomorphism is
proved between such self-similarity features and the deformed coherent state formalism.
Keywords: Self-similarity; fractals; squeezed coherent states; dissipation; iteratively
nafionized water; iteratively filtered water.
PACS numbers: 64.60.al, 03.70.+k, 03.65.Fd
1. Introduction
It has been shown recently1–4that in the framework of the theory of entire ana-
lytical functions the algebra of deformed (squeezed) coherent states provides the
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A. Capolupo et al.
functional realization of self-similarity properties of deterministic fractals. Such
a result has been obtained by using the quantum field theory (QFT) formalism
which describes topologically nontrivial “extended objects”, such as kinks, vor-
tices, monopoles, crystal dislocations, domain walls, and other so-called “defects”,
or macroscopic quantum systems, in condensed matter physics in terms of non-
homogeneous boson condensation5–7and which has been tested to be successful
in explaining many experimental observations in superconductors, crystals, ferro-
magnets, etc.7The observation8of defects in the lattice of crystals (dislocations)
submitted to stress actions, such as the bending of the crystal at low temperature,
may also be framed in the QFT description of defect formation. In such case, de-
fects appear to be the effect of nonhomogeneous coherent phonon condensation.
Remarkably, these lattice defects exhibit self-similar fractal patterns and provide
an example of “emergence of fractal dislocation structures”8in nonequilibrium dis-
sipative systems. These observations also provide an experimental support to the
above mentioned “theorem” on the isomorphism between the self-similar properties
of fractalsa,9,10 and the algebraic structure of deformed coherent states. Motivated
by such a scenario, our task in this paper is to show that such an isomorphism also
exists between the observed phenomenology11–14 of self-similar, scale free proper-
ties of water behavior in the specific conditions discussed below and the formalism
of deformed coherent states. Our discussion is limited to a general analysis based
on algebraic methods and aimed to account for the self-similarity properties. In
a subsequent paper, we will present a model on the dynamical molecular behav-
ior of water under the experimental condition specified below. A preliminary, brief
description of the model is anticipated in the Appendix B.
The experiments considered in this paper, published in Refs. 11–14, have been
motivated by those of the Pollack’s research group15–17 and are grounded on the
studies of the Elia’s group in the past couple of decades. We will summarize the
measurements in Sec. 2 and for the reader convenience we report full details of
the experimental protocol in Appendix A. The experiments essentially are aimed
to the measurements of physically characterizing observables of water samples per-
turbed by the presence of Nafion, a very hydrophilic polymer, and by filtration.
Pollack has indeed shown15–17 laboratory evidence that water in the presence of
Nafion acquires singular properties, such as, e.g., impenetrability by impurities in
a water stratum of ≈200 µm in proximity of the hydrophilic surface of Nafion.
As already mentioned, our discussion is limited essentially to the self-similarity
properties which appear from the log–log plots as a laboratory characterization of
all the measurements. Within such limits, without entering the specificity of the
molecular dynamics in terms of explicit dynamical modeling, we provide the proof
that the observed self-similarity phenomenology is mathematically isomorph to the
algebraic structure of deformed coherent states. As well-known, the isomorphism
aIn some sense self-similarity is considered to be the most important property of fractals (Ref. 9,
p. 150)
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Self-Similarity Properties of Nafionized and Filtered Water
between two systems or sets of elements does not mean “equality”, or analogy in
some generic terms, between the two systems. It has a mathematically well-defined
meaning, so that some difficult to study features or unknown properties of one of
the systems can be mapped, if an isomorphism has been found, to well-studied
features of the other system, for which a sound mathematical formalism has been
developed, thus moving from unexplored territories to more familiar ones, shar-
ing the same formal structure with the former ones. A remarkable example is the
Heisenberg discovery that nucleons share the same su(2) algebraic structure of the
spin quantum number, e.g. of the electron. This does not mean to propose the in-
terpretation of nucleons in terms of electrons. However, a new branch of physics,
i.e., nuclear physics, was born with the discovery of such an isomorphism (and the
nucleon isospin was discovered).
Our approach in the discussion presented in Sec. 2 is much similar to the one in
attempting to catch from the study of symmetry properties as much information
as possible, concerning some problem difficult to solve analytically and/or numeri-
cally (the extraordinary success of symmetry property studies in high energy physics
and condensed matter physics comes here to our minds). The non-negligible advan-
tage offered by such an approach (in our present case and in physics in general)
is that the conclusions one may reach do not depend on specific assumptions or
dynamical models. They are of general validity, not to be expected to decay with
a too short life-time implied sometimes by the heavy approximations which one
is forced to introduce in producing a molecular model or a numerical simulation.
For example, in the case of water studies, computational limits put strong con-
straints in modeling even moderate volumes of liquid water, e.g., V≈100 nm3, by
introducing classical limiting assumptions on the water quantum molecular struc-
ture.18 The consequent effect is the one of averaging out fluctuations which may
turn out in the system collective behavior, such as, e.g., self-similarity.19 On the
other hand, it is of course necessary that in addition to the analysis of the gen-
eral structure of the phenomenon, an accurate model be formulated by resorting to
conventional methods, such as those provided by statistical mechanics and molec-
ular dynamics, able to describe the dynamics at work at a microscopic level. As
a matter of fact, studies along such a direction are on the way and, although the
discussion of a dynamical model is out of the scope of this paper, in Appendix B we
present some preliminary modeling on which we are working. Section 3 is devoted to
conclusions.
2. Phenomenology and Algebraic Method Analysis of Nafionized
and Filtered Water
Here our aim is limited to analyze by use of algebraic methods the common
self-similarity features appearing in the results of three sets of measurements on
water that has been put in contact with Nafion, called Iteratively Nafionized Wa-
ter (INW), and water that has been iteratively filtered (Iteratively Filtered Water
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A. Capolupo et al.
(IFW)). We first describe these experimental results, published in Refs. 11–14, then
we present the algebraic method analysis of the common phenomenology of their
self-similarity properties.
From Pollack’s work,15–17 it is known that water in contact with Nafion mem-
branes presents peculiar behavior. We have measured the electrical conductivity
χof INW. Nafion membranes of given surface and width are placed in a capsule
made either of glass or plastic in contact with 10–20 ml of pure water. As described
in the Appendix A, manual agitation is performed repeatedly so that the liquid
laps against the membrane. Then we follow the evolution of χ, that systemati-
cally increases. The procedure is repeated after turning over the membrane. That
is iterated for some tens of times, each invariably producing a growth of electrical
conductivity. At intervals of few hours, the membrane is removed from the capsule
and left to dry in air. It is then placed back in the nafionized water where it came
from, and previous steps (manual agitation, measurement of conductivity, removal
of the membrane from the capsule, etc.) are repeated again and again. The mea-
sured very high increase of electrical conductivity χ(two- or about three-orders
of magnitude) excludes that the phenomenology depends on the impurity release.
The impurity release must be rapidly reduced to a null contribute as in a normal
washing procedure. When the pH and conductivity were measured for samples that
had changed their physical–chemical parameters (pH and conductivity) due to ag-
ing (15 and 30 days of aging), the measured values lie on the linear trend at the
place corresponding to the new coordinates.11 In Fig. 1, we report the plot of the
logarithm of heat of mixing, Log(−Qmix ), (with −Qmix >0) of INW with NaOH
solution 0.01 m (mol kg−1) as a function of logarithm of the electrical conductivity
χ. The pH (−Log[H+]) of samples of INW as a function of Log χis shown in Fig. 2.
All details on the measurement protocol are described in Appendix A.
Let us now consider the process of iteratively filtering a given volume of pure
liquid water. The liquid is first filtered in vacuum; the resultant filtrate is put
through the filtering step again; this filtration is repeated up to 250 times. It re-
sults12–14 that the qualitative effects on water are the same regardless of the filter
type, e.g., glass filter, disposable or ceramic filters. After filtration, electrical con-
ductivity increases by two-orders of magnitude, while density shows variations on
the fourth decimal digit. Approximately, 10–30% of the observed conductivity in-
creases can be attributed to impurities released by the glass filters. Therefore, we
paid careful attention to the impurities released by the glass filters. We found that
the main chemical impurities are derived from alkaline oxide (Na2O) released by
the glass. In contact with water, they transform into sodium hydroxide (NaOH) and
the last substance turns into sodium bicarbonate (NaHCO3) due to atmospheric
carbon dioxide (CO2). We therefore systematically determined the sodium con-
centration of the samples and subtracted the contribution of sodium bicarbonate
from the conductivity readings. The other components of the glass — SiO2, B2O3
and Al2O3— are very low compared to sodium bicarbonate and they do not con-
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Self-Similarity Properties of Nafionized and Filtered Water
tribute significantly to electrical conductivity neither at low alkalinity of the water
medium, nor do they affect the density, due to their low concentration (see Table 1
of the second quoted paper in Refs. 13 and 14 for their quantitative measurements).
Moreover, the IFW conductivity is not altered by very weak acids, such as H4SiO4,
H3BO3or by Al2O3, deriving from leaching from Pyrex glass filters, since they are
not dissociated in ions in low alkaline solutions, such as those of IFW, and thus
they do not contribute to electrical conductivity.13,14 We stress that using only
samples of pure water there was no possibility for contamination of the filters and
no extraneous chemical substances were introduced into the water other than those,
mentioned above, deriving from the partial dissolution of the glass solid support.
In Fig. 3, the logarithm of IFW density (ρ−ρ0) versus the logarithm of electrical
conductivity χis reported. ρ0is the density of pure untreated water.
The fitting by a straight line of the results of the measurements in Figs. 1–3
show that we are in the presence of a scale free, self-similar phenomenon in the three
cases. This result is reproducible by the use of a detailed experimental protocol
presented in Appendix A and to our knowledge it is not described by existing
conventional methods of statistical mechanics and molecular modeling. Therefore,
we first analyze the phenomenology by use of algebraic methods without proposing
any model of the dynamical molecular behavior. Then, in a future publication, we
will present a molecular model. Although this is beyond the scope of this paper,
nevertheless a preliminary description of a molecular model on which we are working
is anticipated in Appendix B.
Let us thus proceed in analyzing by algebraic methods the self-similarity (power
law) phenomenology of INW and IFW as described above (Figs. 1–3). The straight
Fig. 1. Logarithm of heat of mixing, Log(−Qmix ), (with −Qmix >0) of INW with a NaOH
solution 0.01 m (mol kg−1) as a function of logarithm of the electrical conductivity χfor INW.
Each point represented in the figure is obtained experimentally measuring the Qmix (J kg−1) and
the electrical conductivity χ, (µS cm−1) of each sample.11
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A. Capolupo et al.
Fig. 2. The pH (−Log[H+]) of 120 samples of INW as a function of Log χ. Each point repre-
sented in the figure is obtained experimentally measuring the pH and the electrical conductivity
χ(µS cm−1) of each of the 120 samples.11
Fig. 3. Logarithm of IFW density (ρ−ρ0)×105g cm−1, versus logarithm of electrical conductivity
χ(µS cm−1) for Pyrex glass and Millipore filters, irrespective to the number of filtration or the
filter porosity. Each point represented in the figure is obtained experimentally measuring the
density, ρ(g cm−1) and the electrical conductivity of each samples.12 ρ0is the density of pure
Milli Quntreated water.
line fitting the data in each of Figs. 1–3is generically represented by the equation
d=Log α
Log β,(2.1)
where the ordinate and the abscissa have been denoted by Log αand Log β, respec-
tively, with the specific meaning they assume in each of the figures and the reference
frame has been translated conveniently, so that the straight line crosses the zero in
each of the three cases (such a translation is equivalent to divide the ordinate (or
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Self-Similarity Properties of Nafionized and Filtered Water
multiply the abscissa) by t, with t= 10c, where cis the point intercepted by the
line on the abscissa axis). It is also understood that the angular coefficient dhas
the proper ±sign in each of the three cases. In the following, it is convenient to
switch from common logarithms (to base 10) to the natural ones since the ratio d
in Eq. (2.1) does not depend on the chosen base. Equation (2.1) is equivalent to
un,q(α)≡(qα)n= 1 ,for any n∈N+,with q≡1
βd,(2.2)
where we have used the notation (qα)n≡un,q(α) and as customary N+denotes
positive integers. The constancy of the angular coefficient din the plot expresses the
scale free character of the relations among physical quantities represented in (each
of) the figures and, together with the independence of nof Eq. (2.1), it also expresses
their self-similarity properties, namely the ratio d= ln α/ ln βis independent of the
order nof the power to which αand βare simultaneously elevated. Notice that
self-similarity is properly defined only in the n→ ∞ limit.
The above remarks are the ones which are done in the standard analysis of
fractal structures.10 As already observed in Sec. 1 (see footnote a), self-similarity is
considered to be the most important property of fractals.9The angular coefficient d
in the Figs. 1–3is called the self-similarity dimension, or also the fractal dimension.9
Now, the functions un,q(α) in Eq. (2.2), representing, for any n∈N+, the nth
power component of the self-similarity relation (the nth stage of the fractal), are
readily recognized to be, apart of the normalization factor 1/√n!, nothing but the
restriction to real qα of the entire analytic functions in the complex α-plane
˜un,q(α) = (qα)n
√n!, n ∈N+.(2.3)
They form in the space Fof the entire analytic functions a basis which is orthonor-
mal under the Gaussian measure dµ(qα) = (1/π)e−|qα|2dqαdqα. The factor 1/√n!
ensures the normalization condition with respect to the Gaussian measure. This
means that, to the extent in which fractals are considered under the point of view
of self-similarity, the study of the fractal properties may be carried on in the space
Fof the entire analytic functions, by restricting, at the end, the conclusions to real
qα,qα →Re(qα).2,3In other words, a mathematical isomorphism is recognized to
exist between the observed self-similarity properties (Figs. 1,2and 3) and the de-
formed coherent states. In order to prove this, by closely following Ref. 1, we remark
that Fis the vector space providing the representation of the Weyl–Heisenberg al-
gebra of elements {a, a†,1}with number operator N=a†a.20,21 Fis in fact the
Fock–Bargmann representation22,23 (FBR) of the (Glauber) coherent states with
identification:
N→αd
dα , a†→α , a →d
dα .(2.4)
In explicit terms, we recall2,3that FBR is the Hilbert space Kgenerated by the
basis ˜un(α)≡˜un,q(α)|q=1 , i.e., the space Fof entire analytic functions. A one-to-
one correspondence exists between any vector |ψiin Kand a function ψ(α)∈ F.
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A. Capolupo et al.
The vector |ψiis then described by the set {cn;cn∈ C,P∞
n=0 |cn|2= 1}defined by
its expansion in the complete orthonormal set of eigenkets {|ni} of N:
|ψi=
∞
X
n=0
cn|ni → ψ(α) =
∞
X
n=0
cn˜un(α),(2.5)
hψ|ψi=
∞
X
n=0 |cn|2=Z|ψ(α)|2dµ(α) = kψk2= 1 ,(2.6)
|ni=1
√n!(a†)n|0i,(2.7)
where |0idenotes the vacuum vector, a|0i= 0, h0|0i= 1. The condition
P∞
n=0 |cn|2= 1 [cf. Eq. (2.6)] ensures that the series expressing ψ(α) in Eq. (2.5)
converges uniformly in any compact domain of the α-plane, confirming that ψ(α)
is an entire analytic function, indeed. The explicit expression of the (Glauber) co-
herent state |αiis20,21
|αi= exp −|α|2
2∞
X
n=0
αn
√n!|ni.(2.8)
It is convenient now to put q=eζ,ζ∈C. The q-deformed algebraic structure
is obtained then by introducing the finite difference operator Dq, also called the q-
derivative operator. For brevity we do not comment more on this point, see Refs. 24–
30 for details. Then, one can show24–26 that the q-deformed coherent state |qαiis
obtained by applying qNto |αi
qN|αi=|qαi= exp −|qα|2
2∞
X
n=0
(qα)n
√n!|ni.(2.9)
As observed in Refs. 1–4, the nth power component un,q(α) of the (fractal) self-
similarity Eq. (2.2) is “seen” by applying (a)nto |qαiand restricting to real qα
hqα|(a)n|qαi= (qα)n=un,q(α), qα →Re(qα).(2.10)
In other words, the operator (a)nacts as a “magnifying lens”2,3,10 whose application
picks up the nth component of the q-deformed coherent state series representing
the nth power component un,q(α).
Thus, as a result, we have formally established the one-to-one correspondence
between the nth power component un,q(α) (the nth fractal stage of iteration), with
n= 0,1,2,...,∞, and the nth term in the q-deformed coherent state series. The
operator qNapplied to |αi“produces” the fractal in the functional form of the
coherent state |qαiand therefore it has been called the fractal operator.2,3Note
that |qαiis actually a squeezed coherent state24–26,31 with ζ= ln qthe squeezing
parameter. Thus, qNacts in Fas the squeezing operator. The proof of the iso-
morphism between self-similarity properties of the observed phenomenology and
the q-deformed algebra of the squeezed coherent states is formally provided by
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Self-Similarity Properties of Nafionized and Filtered Water
Eqs. (2.9) and (2.10).1–3As already said, in a future work we will complement the
present result obtained by use of algebraic methods with the formulation of a model
of the molecular dynamics. See Appendix B for its brief, preliminary presentation.
3. Conclusions
The conclusion of our discussion is that an isomorphism exists between the ob-
served scale free, self-similar properties of INW and IFW and the deformed coherent
state formalism. The fractal dimension dhas been shown to be related to the q-
deformation parameter,1–3to squeezing and dissipation.24–26 The relation between
d,qand the squeezing parameter ζis given by −dln β= ln q=ζ[cf. Eq. (2.2)].
There are however further questions which need to be asked in order to make our
analysis more complete and to make it more clear. Among them, the most urgent
is perhaps the one related with the identification of the system variables involved
in the formation of the coherent states and of their deformation. It might be thus
helpful to introduce some clarification for each of the measurements discussed above.
Let us start with the case of Fig. 2. In this case, the system variable involved in
coherence is the (proton) charge density distribution (represented in terms of pH)
and it is “deformed” by its interaction with the Nafion surface.bWe may write the
(proton) charge density wavefunction σ(r, t) as
σ(r, t) = pρ(r, t)eiθ(r,t),(3.1)
with real ρ(r, t) and θ(r, t). The Nafion action is responsible of the spontaneous
breakdown of the system U(1) symmetry and nonvanishing |σ(r, t)|2=ρ(r, t) de-
notes the expectation value of the charge density operator in the system ground
state. One may show5,6,32,33 that θ(r, t) represents the Nambu–Goldstone (NG)
field and that the (space component of the) current is given by
J(r, t) = 1
mρ(r, t)(∇θ(r, t)−qA(r, t)).(3.2)
where Adenotes the electromagnetic (e.m.) vector field. On the other hand, the
current density is also defined to be proportional to the conductivity χ. This and
Eq. (3.2) gives the relation between ρand χ. The experiment shows that as an
effect of the presence of the Nafion, the conductivity changes. We then assume
that χ→χ′=eζχwith ζa small (ζ < 1) deformation parameter. It is then easy
to show that the linearity and the coherent state properties lead to the plot of
Fig. 2. The fitting is indeed obtained by tuning the deformation parameter such
that ζ=δln χ, with 1 + δ=d. The fractal dimension dthus provides a measure
of such a dynamical “deformation”, so that the observed scale free law relating pH
bIndeed Pollack’s EZ water is observed15–17 to be arranged in ordered (coherent state) patterns
and pH measurements show a peculiar gradient of the proton concentration (orthogonal to the
Nafion surface in the geometry of Pollack’s experiments15–17).
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A. Capolupo et al.
and conductivity appear to be the macroscopic manifestations of dissipative local
deformations at a microscopic level.
In Fig. 3, the underlying molecular dynamics of IFW manifests itself in the labo-
ratory observations in terms of the molecular density variations ρ−ρ0as a function
of the electrical conductivity with self-similarity properties. This means that the
filtering process to which water has been undergoing in the described experimen-
tal protocol produces molecular rearrangements and displacements. As well-known
from the study of many-body ordered pattern formation (e.g., as in crystal for-
mation), density behaves as an order parameter and condensation of long range
correlation modes is responsible of the dynamical occurrence of coherent states5
(for a formal treatment in a specific model, see Appendix B and the comments
between Eqs. (B.11) and (B.12) and between Eqs. (B.13) and (B.14)). By following
a derivation similar to the one presented above for Fig. 2, the variations in the
system density (nonhomogeneous condensation) are related to the concurrent vari-
ations of the conductivity in the squeezing process characterized by the scale free
exponent d.
Finally, Fig. 1gives for INW the exchanges of mixing heat, Qmix , in function
of the conductivity at constant pressure and temperature fixed at 25.00◦C±0.001.
Since T∆S= ∆Q, with Sdenoting the entropy, measurement of heat exchange
provides the entropy variations in the water molecular configurations as an effect of
the nafionization. In such case, the coherent state is of thermal origin, as it is ob-
tained for example in the thermo field dynamics (TFD) formalism.5We have then
a two-mode SU (1,1) coherent state representation related to dissipative (thermal)
processes and we can show5,6,24–26 that minimization of the free energy F,dF = 0,
gives, at constant pressure and temperature, dE =PkEk˙
Nkdt =T dS, where
˙
Nkdenotes time derivative of the long range correlation modes (the NG modes).
This equation specifically shows that heat exchange is related to variations of long
range correlations out of which molecular coherent patterns emerge. Thus, again
we have coherent state deformations (squeezing). The scale free power law relating
heat exchanges and the conductivity changes is then obtained. Summing up, in the
present case the self-similarity properties observed in the experiments reflect the
self-similarity properties of the SU (1,1) thermal states. In this connection, we ob-
serve that a SU (1,1) coherent state representation related to dissipative processes
can also be exhibited1where the notion of topologically nontrivial dissipative phase
is introduced and the dynamics is characterized by noncommutative geometry in the
plane. Here, we omit details on these last issues since they are out of the tasks of this
paper. We only remark that a number of specific characterizations of the molecular
coherent dynamics can be derived by the above discussed laboratory observations
and their understanding in terms of coherent state self-similar properties. Perhaps,
of general interest is the possibility, suggested by our approach, of “extracting re-
liable information from noisy experiments”,csuch as those described in this paper.
cWe are grateful to the anonymous referee for such an observation.
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Self-Similarity Properties of Nafionized and Filtered Water
This leads us to one further observation, namely that in the above discussion coher-
ent state formation coexists with noncoherent molecular dynamics, so that one has
a two-component system (the coherent and the non-coherent component), with con-
tinual migration of molecules from the coherent component to the noncoherent one
and vice-versa in an overall stationary regime at fixed temperature. Recent experi-
mental observation points, indeed, to the existence of a two-component structure of
water from ambient temperature to supercooled conditions34 (see also Refs. 35–38,
40).
In order to analyze the self-similarity phenomenology in terms of a specific
molecular dynamics, we need to consider an explicit dynamical model, thus going
beyond the limits of the algebraic method analysis to which this paper is devoted.
Preliminary results of the dynamical analysis are presented in Appendix B. In our
model, the molecule dynamics is assumed to be ruled by the interaction of the
molecule electrical dipole moment of magnitude Dwith the radiative e.m. field,
thus disregarding the static dipole–dipole interaction. The Nmolecule system is
collectively described by the properly normalized complex dipole wave field χ(x, t)
and, by resorting to the analysis of Refs. 39 and 40, we restrict ourselves to the
resonant radiative e.m. modes with k= 2π/λ ≡ω0. The field equations are41,42:
i∂χ(x, t)
∂t =L2
2Iχ(x, t)
−iX
k,r
D√ρrk
2(ǫr·x)[ur(k, t)e−ikt −u†
r(k, t)eikt]χ(x, t),
i∂ur(k, t)
∂t =iD√ρqk
2eikt ZdΩ(ǫr·x)|χ(x, t)|2,
(3.3)
where ur(k, t) denotes the radiative e.m. field operator with polarization r,ρ≡
N/V ,Vis the volume and ǫris the polarization vector of the e.m. mode, for which
the condition k·ǫr= 0 is assumed to hold. We use natural units ~= 1 = cand
the dipole approximation exp(ik·x)≈1. We have ω0≡1/I, where Idenotes the
moment of inertia of the molecule; L2is the squared angular momentum operator.
The system of Nwater molecules is assumed to be spatially homogeneous and in
a thermal bath kept at a nonvanishing temperature T. For further details on the
model see the Appendix B. Preliminary results seem to suggest that the observed
self-similarity may occur.
Appendix A. The Experimental Protocol
For the reader convenience, we present here the details of the protocol of three sets
of measurements on water under two different physical treatments11–14: water that
has been put in contact with Nafion, that we call INW and water that has been
iteratively filtered (IFW).
Background and procedure for INW. In order to prepare some water perturbed
by the presence of Nafion INW, we followed these steps:
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•Step 1. Initially, the pristine membrane is washed five times using 20 ml of “ultra-
pure” (Milli QTM) water.
•Step 2. Nafion membranes with a surface of 60–120 cm2and a width of 50–
180 µm, were placed in a Petri capsule (made either of Pyrex glass or plas-
tic) in contact with 10–20 ml of Milli Qwater (electrical conductivity χ=
1−2µS cm−1). Manual agitation is performed repeatedly so that the liquid
laps against the membrane. Then we follow the evolution of χ(µS cm−1), that
systematically increases. The procedure is repeated after turning over the mem-
brane. That is iterated for some tens of times, each invariably producing a growth
of electrical conductivity.
•Step 3. At intervals of few hours, (from 3 to 12) the membrane is removed from
the Petri capsule and left to dry in air (1–24 h). It is then placed back in the
nafionized water it came from, and steps 2 and 3 are repeated again.
To obtain a sufficiently high conductivity, i.e., 50–100 µS cm−1, about 10–20
iterations of the last two steps are needed. Even though successive iterations invari-
ably determine a growth of electrical conductivity, it has not yet been possible to
link quantitatively the number of iterations with the increment of χ. Intuitively, the
procedure is akin to a sort of “washing”, iterated hundreds of times. The measured
continuous increase of conductivity is such that it is not consistent with the hy-
pothesis of impurity release. In fact, the phenomenology takes places independently
of the number of steps 2 and 3 of the protocol or of whether the membrane is a
pristine one. Membranes used for prolonged periods (months) and for hundreds of
steps 2 and 3 behave just like as a new membrane, namely the conductivity always
increases at increasing number of steps. It appears, though, that this capability
improves with the use of the membrane. In any case if the liquid obtained is com-
pletely consumed for experimental measures, and we begin a new procedure using
the same membrane with pure water (10–30 ml) the conductivity increases but does
not start at the value obtained in the previous procedure. Because the high number
of steps 2 or 3 and the very high increase of electrical conductivity χ(two or about
three orders of magnitude), we can exclude that the phenomenology depends on the
impurity release. The impurity release must be rapidly reduced to a null contribute
as in a normal washing procedure. Notice that the measures of physical–chemical
parameters here reported are obtained after the removal of Nafion membrane from
the liquid water. In such a way, we obtain information on the effect induced by
Nafion on water.
Moreover, we have tested the effect of aging on the samples and we found that
when the pH and conductivity are measured for samples that had changed their
physical–chemical parameters (pH and conductivity) due to aging over 15 and 30
days, the measured values lie on the linear trend at the place corresponding to the
new coordinates. The results for samples aged in polyethylene or polypropylene
containers are reported in Table 1 in Ref. 11, from which one can see that after
30 days from the sample preparation there is an increase in the conductivity for
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the majority of the samples, stability in several of them and a strong decrement
in one sample. It has been observed also that if the aging is performed in presence
of small quantity of Nafion membranes, the variation of conductivity with time is
practically reduced to zero.11
Background and procedure for IFW. The process of iteratively filtering a given
volume (1–10 ml) of Milli Qwater consists in: filtering the liquid in vacuum; taking
the resultant filtrate and putting it through the filtering step again; repeating this
filtration up to 250 times. The following filters were used: Millipore filters made of
cellulose nitrate, with porosities of 450, 200, 100 and 25 nm and Pyrex glass filters
having mean porosity of 120, 65, 27.5, 10 and 2.5 µm. One observes12–14 that, re-
gardless of the filter type, e.g., Pyrex glass filter (B¨uchner), disposable Millipore or
ceramic filters, the qualitative effects on water are the same. Upon examining the re-
peatability of the phenomenon, we decided to first use Pyrex glass filters (B¨uchner).
We paid careful attention to the impurities released by the glass filters which might
affect electrical conductivity and density. The main chemical impurities that we
found are derived from alkaline oxide (Na2O) released by the glass. In contact with
water, they transform into sodium hydroxide (NaOH) and the last substance, due to
atmospheric carbon dioxide (CO2), turns into sodium bicarbonate (NaHCO3). We
therefore systematically determined the sodium concentration of the samples, and
subtracted the contribution of sodium bicarbonate from the conductivity readings.
The concentrations of impurities deriving from the other components of the glass
— SiO2, B2O3and Al2O3— are very low compared to sodium bicarbonate (for
numerical values of measured concentrations see Table 1 of the second quoted paper
in Refs. 13 and 14). We stress that the IFW conductivity is not altered by very
weak acids such as H4SiO4, H3BO3, or by Al2O3, derived from leaching of Pyrex
glass filters, since they are not dissociated in ions in low alkaline solutions, such as
those of IFW, and thus they do not contribute to electrical conductivity.13,14
Using Millipore filters for the iterative vacuum filtration process requires use of
a sintered glass filter as support. It is observed13,14 that, after filtration, electrical
conductivity increases by two-orders of magnitude, while density shows variations
on the fourth decimal digit. Approximately 10–30% of the observed increases can
be attributed to impurities released by the glass filters. At the porosity of Pirex
glass filters (R1, mean porosity 120 µm) used as a solid support for the Millipore
membrane filter, the contribution to electrical conductivity is so light that its effect
does not need to be taken into account. To exclude the contribution of chemical
impurities from inside the Millipore filters, they were rinsed with abundant water
until they produced a filtrate with electrical conductivity of 1.2–2.0 µS cm−1. This
procedure is equivalent or better than the sometimes suggested soaking. In fact
soaking tends to diminishes the release of impurities, while our goal is to remove
all the impurities of electrolyte nature that can increase the electrical conductivity.
After rinsing, we can be sure that all the soluble impurities were removed from the
filters. We found that the quantitative reproducibility was improved by rinsing the
filter with Milli Qwater after each iterative filtration step, rather than replacing
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the filter with a new one. It is in fact worth repeating that during the experiment
no extraneous chemical substances were introduced into the water other than those
deriving from the partial dissolution of the glass solid support. In other words,
there was no possibility for contamination of the filters when using only samples of
Milli Qwater.
Electrical conductivity measurements. Systematic measurements of specific elec-
trical conductivity were performed on the samples (INW and IFW), using an YSI
3200 conductometer with an electrical conductivity cell constant of 1.0 cm−1. Be-
fore measuring the electrical conductivity of a sample, the cell was calibrated by
determining the cell constant K(cm−1). The specific conductivity χ(µS cm−1)
was then obtained as the product of the cell constant and the conductivity of the
solution. For a given conductivity measuring cell, the cell constant was determined
by measuring the conductivity of a KCl solution having a specific conductivity
known to high accuracy, at several concentrations and temperatures. All electrical
conductivities were temperature-corrected to 25◦C, using a pre-stored temperature
compensation for pure water.
Calorimetry. The heat of mixing, Qmix (−Qmix >0), of NaOH solution with
IFW samples was monitored using a Thermal Activity Monitor (TAM) model 2227,
by Thermometric (Sweden) equipped with a flow mixing vessel. A P3 peristaltic
pump (by Pharmacia) envoys the solutions (of the solutions of NaOH and of the
samples of IFW) into the calorimeter, through Teflon tubes. The flow rates of the
two liquids are the same, and are constant in the inlet tubes, so that the solution
coming out of the calorimeter has a concentration half the initial one. The mass
flow-rate, constant within 1%, amounts to 3 ×10−3g s−1: it was the same for all
the experiments. The values of the mixing enthalpies, ∆Hmix, were obtained using
the following formula:
∆Hmix mi
x, mi
y→mf
x, mf
y=dQ
dt Pw,(A.1)
where (dQ/dt) is the heat flux (W), Pwis the total mass flow-rate of the solvent
(kg s−1) and mi
x,mi
yand mf
x,mf
y, are the initial and final molalities. ∆Hmix is
given in J kg−1of solvent in the final solution. For our control, our Qmix represents
the difference between the heat of mixing of NaOH solution with the samples of
IFW or INW, minus the one with pure untreated Milli Qwater (heat of dilution of
NaOH solution).
Density measurements. The solution densities were measured using a vibrating-
tube digital density meter (model DMA 5000 by Anton Paar, Austria) with a pre-
cision of ±1×10−6g cm−3and an accuracy of ±5×10−6g cm−3. The temperature
of the water around the densitometer cell was controlled to ±0.001 K. The densit-
ometer was calibrated periodically with dry air and pure water. As a control for
our measurements, we use the difference ρ−ρ0between the density of the samples
minus the density of pure untreated Milli Qwater.
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Self-Similarity Properties of Nafionized and Filtered Water
pH measurements. The pH were monitored using a pH-meter model micropH
2002 by Crison, equipped with a pH electrode for micro-samples, model 5209. The
electrode specification is: asymmetry potential <±15 mV, pH sensitivity 4–7 (at
25◦C) >98%.
Results. The results of the measurements of heat of mixing and pH for INW
and density for IFW in function of the electrical conductivity are reported in the
log–log plots in Figs. 1–3, respectively. Their fitting by a straight line shows that
we are in the presence of a scale free, self-similar phenomenon in the three cases.
Appendix B. Molecular Dynamical Model. Preliminary Analysis
We present a preliminary analysis of the water molecular dynamical model described
by Eqs. (3.3). Since, as observed at the end of Sec. 3, the molecule density is assumed
to be spatially uniform, the only relevant variables are the angular ones. In full
generality, we may expand the field χ(x, t) in the unit sphere in terms of spherical
harmonics: χ(x, t) = Pl,m αl,m(t)Ym
l(θ, φ). By setting αl,m (t) = 0 for l6= 0, 1, this
reduces to the expansion in the four levels (l, m) = (0,0) and (1, m), m = 0,±1. The
populations of these levels are given by N|αl,m(t)|2and at thermal equilibrium, in
the absence of interaction, they follow the Boltzmann distribution. The three levels
(1, m), m= 0,±1 are in the average equally populated under normal conditions
and we can safely write Pm|α1,m(t)|2= 3|a1(t)|2, with normalization condition
|α0,0(t)|2+Pm|α1,m(t)|2= 1. The system is invariant under (molecular) dipole
rotations, which means that the amplitude of α1,m(t) does not depend on m, and
that the time average of the polarization Pnalong any direction nmust vanish in
such conditions. It is useful to write39
α0,0(t)≡a0(t)≡A0(t)eiδ0(t),
α1,m(t)≡A1(t)eiδ1,m (t)e−iω0t≡a1,m(t)e−iω0t,
um(t)≡U(t)eiϕm(t),
(B.1)
where a1,m(t)≡A1(t)eiδ1,m (t).A0(t), A1(t), U(t), δ0(t), δ1,m(t) and ϕm(t) are real
quantities.
Due to the rotational invariance, the rate of change of the population in each
of the levels (1, m), m= 0,±1, equally contributes, in the average, to the rate of
change in the population of the level (0,0), at each time t. In full generality, we can
set the initial conditions at t= 0 as
|a0(0)|2= cos2θ0,|a1(0)|2=1
3sin2θ0,0< θ0<π
2,(B.2)
|u(0)|2= 0 .(B.3)
By properly tuning the parameter θ0, in its range of definition one can adequately
describe the physical initial conditions (e.g., θ0=π/3 describes the equipartition
of the field modes of energy E(k) among the four levels (0,0) and (1, m), |a0(0)|2≃
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|a1,m(0)|2, m = 0,±1, as typically given by the Boltzmann distribution when the
temperature Tis high enough, kBT≫E(k)). The values zero and π/2 are excluded
since they correspond to the physically unrealistic conditions for the state (0,0) of
being completely filled or completely empty, respectively.
From Eqs. (3.3) one may study the ground state of the system for each of the
modes a0(t), a1(t) and u(t). Without reporting the details of the derivation one
finds in the mean-field approximation39:
¨a0(t) = 4Ω2γ2
0(θ0)a0(t)−4Ω2|a0(t)|2a0(t),(B.4)
¨a1(t) = −σ2a1(t) + 12Ω2|a1(t)|2a1(t),(B.5)
¨u(t) = −µ2u(t)−6Ω2|u(t)|2u(t),(B.6)
respectively, where γ2
0(θ0)≡(1/2)(1 + cos2θ0), σ2= 2Ω2(1 + sin2θ0) and µ2=
2Ω2cos 2θ0, with Ω ≡(2D/√3)pρ/2ω0ω0. We see that Eq. (B.4) can be written
in the form
¨a0(t) = −δ
δa∗
0
V0[a0(t), a∗
0(t)] ,(B.7)
where the potential V0[a0(t), a∗
0(t)] is
V0[a0(t), a∗
0(t)] = 2Ω2(|a0(t)|2−γ2
0(θ0))2.(B.8)
Similarly, the potentials from which the right-hand side of Eqs. (B.5) and (B.6) are
derivable are
V1[a1(t), a∗
1(t)] = σ2|a1(t)|2−6Ω2(|a1(t)|2)2,(B.9)
Vu[u(t), u∗(t)] = 3Ω2(|u(t)|2+1
3cos 2θ0)2,(B.10)
respectively. As usual, in order to study the ground state of the theory, we search for
the minima of the potentials V. For V0, let a0,R(t) and a0,I (t) denote the real and
the imaginary component, respectively, of a0(t): |a0(t)|2=A2
0(t) = a2
0,R(t) + a2
0,I (t).
One finds a relative maximum of V0at a0= 0 and a (continuum) set of minima
given by the points on the circle of squared radius γ2
0(θ0) in the (a0,R(t), a0,I (t))
plane:
|a0(t)|2=1
2(1 + cos2θ0) = γ2
0(θ0),(B.11)
We are thus in the familiar case where the cylindrical SO(2) symmetry (the phase
symmetry) around an axis orthogonal to the plane (a0,R(t), a0,I (t)) is spontaneously
broken. The points on the circle represent (infinitely many) possible vacua for the
system and they transform into each other under shifts of the field δ0:δ0→δ0+α
(SO(2) rotations in the (a0,R (t), a0,I (t)) plane). The phase symmetry is broken
when one specific ground state is singled out by fixing the value of the δ0field. As
usual,43 we transform to new fields: A0(t)→A′
0(t)≡A0(t)−γ0(θ0) and δ′
0(t)→
δ0(t), so that A′
0(t) = 0 in the ground state for which A0(t) = γ0(θ0). Use of these
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Self-Similarity Properties of Nafionized and Filtered Water
new variables in V0shows that the amplitude A′
0(t) describes a (massive) mode with
pulsation m0= 2ωp(1 + cos2θ0) and that the field δ′
0(t) corresponds to a zero-
frequency (massless) mode playing the rˆole of the so-called NG collective mode,
implied by the spontaneous breakdown of symmetry. Our assumption is that the
perturbations to the water molecular dynamics induced by iterated interaction with
the highly hydrophilic Nafion polymers and the iterated filtration processes are the
responsible for the breakdown of the symmetry described in our dynamical molecular
model.
The value a0= 0, which we have excluded in our initial conditions, cf. Eq. (B.2),
on the basis of physical considerations, consistently appears to be the relative maxi-
mum for the potential, and therefore an instability point out of which the perturbed
system runs away. One can also show that |u(t)|2= (2/3)(|a0(t)|2−cos2θ0), which
implies that |u(t)|moves away from its vanishing value at t= 0 (the initial condi-
tion Eq. (B.3)) as soon as |a0(t)|reaches its minima on the circle or squared radius
γ2
0(θ0), considering that θ06= 0,±π, etc. as indeed it is since 0 < θ0< π/2.
As well-known, the generator of the transformation δ0→δ0+αis the generator
of coherent states 5,6,20 : the infinitely many unitarily inequivalent (i.e., physically
inequivalent) vacua, among themselves related by such a transformation, are coher-
ent condensates of the NG modes δ0. The family of such coherent states includes
squeezed coherent states5,6,20 parametrized by the q-deformation (or squeezing)
parameter through the (“form”) factors (qα)n, for any integer n, thus susceptible
to be represented by a straight line in a log–log plot, which is the wanted self-
similarity resulting from the (perturbed) dynamical interaction between molecules
and radiative e.m. field.
In the case of V1,a1= 0 is a relative minimum and a set of relative maxima is
on the circle of squared radius
|a1(t)|2=1
6(1 + sin2θ0)≡γ2
1(θ0).(B.12)
For |a1(t)|2=γ2
1(θ0), U2=−(1/3) cos2θ0<0, which is not acceptable since Uis
real. Thus, the amplitude A1cannot assume the values on the circle of radius γ1(θ0),
which is consistent with the intrinsic instability of the excited levels (1, m). One can
also show that the conservation laws in the model (here not reported for brevity)
and the reality condition for Urequire that |a1(t)|2≤(1/3) sin2θ0which lies indeed
below γ2
1(θ0), and the value (1/6) sin2θ0taken by A2
1when |a0(t)|2=γ2
0(θ0) also
lies below the bound. The potential V1thus must be lower than (1/3) sin2θ. These
observations show that the consistency between Eqs. (B.4) and (B.5) is satisfied and
the field a1(t) described by Eq. (B.5) is a massive field with (real) mass (pulsation)
σ2= 2Ω2(1 + sin2θ0).
For Vu, we see that µ2≥0 for θ0≤π/4 and the only minimum is at u0= 0.
This solution describes the system when the initial condition, Eq. (B.3), holds at
any time. However, as mentioned above this is not consistent with the dynami-
cal evolution of the system moving away from the initial conditions exhibited by
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A. Capolupo et al.
Eq. (B.4). Consistency is recovered provided θ0> π/4. Then, µ2= 2Ω2cos 2θ0<0
and a relative maximum of the potential is at u0= 0. A set of minima is given by
the points of the circle of nonvanishing squared radius v2(θ0) in the (uR(t), uI(t))
plane:
|u(t)|2=−1
3cos 2θ0=−µ2
6Ω2≡v2(θ0), θ0>π
4.(B.13)
These minima represent (infinitely many) possible vacua for the system and they
transform into each other under shifts of the field ϕ:ϕ→ϕ+α. The phase symmetry
is broken when one specific ground state is singled out by fixing the value of the
ϕfield. The fact that u0= 0 is now maximum for the potential means that the
system evolves away from it, consistently with the similar situation for the a0mode.
The symmetric solution at u0= 0 is thus excluded for internal consistency and the
lower bound π/4 for θ0guarantees dynamical self-consistency.
We transform now to new fields: U(t)→U′(t)≡U(t)−v(θ0) and ϕ′(t)→ϕ(t)
and we find that U′(t) describes a “massive” mode with real mass p2|µ2|=
2Ωp|cos 2θ0|(a quasi-periodic mode), as indeed expected according to the
Anderson–Higgs–Kibble mechanism,5,6,44–46 and that ϕ′(t) is a zero-frequency mode
(a massless mode), also called the “phason” field.47 ϕ′(t) plays the rˆole of the NG
collective mode. Again, as in the case of the V0potential, the generator of the trans-
formation ϕ(t)→ϕ(t) + αis the generator of coherent states 5,6,20: the infinitely
many unitarily inequivalent vacua are coherent condensates of the NG modes ϕ(t),
whose family includes q-deformed (squeezed) coherent states,5,6,20 also they are
susceptible to be represented by a straight line in a log–log plot and thus lead-
ing us again to the wanted self-similarity resulting from the (perturbed) molecular
dynamics.
As a further step, one can show39 that, provided θ0> π/4, which we assume
our system is forced to reach under the Nafion and filtering perturbing effects,
˙
U(t) = 2ΩA0(t)A1(t) cos α(t),(B.14)
˙ϕ(t) = 2Ω A0(t)A1(t)
U(t)sin α(t),(B.15)
where α≡δ1(t)−δ0(t)−ϕ(t). We thus see that ˙
U(t) = 0, i.e., a time-independent
amplitude ¯
U= const.exists, if and only if the phase locking relation
α=δ1(t)−δ0(t)−ϕ(t) = π
2,(B.16)
holds. In such a case, ˙ϕ(t) = ˙
δ1(t)−˙
δ0(t) = ω: any change in time of the difference
between the phases of the amplitudes a1(t) and a0(t) is compensated by the change
of the phase of the e.m. field. The phase locking relation (B.16) expresses nothing
but the gauge invariance of the theory. Since δ0and ϕare the NG modes, Eqs. (B.16)
also exhibit the coherent feature of the collective dynamical regime, the “in phase
locked” dynamics of δ0and ϕcoherent condensates, resulting in definitive in the in
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Self-Similarity Properties of Nafionized and Filtered Water
phase coherence between the system of Ndipoles and of the e.m. radiative field. In
such a regime we also have ¯
A2
0−¯
A2
16= 0 to be compared with A2
0(t)−A2
1(t)≈0 at
the thermal equilibrium in the absence of the collective coherent dynamics.
A final remark concern the finite temperature effects which have not been con-
sidered in the above discussion. Also on such a problem we will focus our study
in the planned developments. Here we observe that the √N(appearing in √ρ) in
Eqs. (3.3) signals strong coupling, namely for large Nthe interaction time scale is
much shorter (by the factor 1/√N) than typical short range interactions among
molecules. Thus, for large Nthe collective interaction is expected to be protected
against thermal fluctuations.
Further work is still necessary and some aspects of the model may need much
refinement. As said in the text, the discussion of such a model is out of the scope
of this paper. This Appendix is only the anticipation of a preliminary, rudimentary
modeling whose final version will be published in a forthcoming paper. There we
will also consider the specific spectral analysis on the line of the quantitative results
presented in Ref. 40.
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