%Sample file for writing articles to be included in SIMAI 2008 Proceedings
\documentclass[a4paper,leqno]{CSCsimai}
\newcommand{\DOI}{DOI: 10.1685/CSC09XXX}
\markboth{D.Jou, M.S. Mongiov\`{\i}}{\DOI}

\begin{document}
\title{Thermodynamics of superfluid vortex tangles}
\author{{David Jou$^1$ and Maria Stella Mongiov\`{\i}$^2$ }}

\address{$^1$ Departament de F\'{\i}sica,
Universitat Aut\`{o}noma de Barcelona, \\ 08193 Bellaterra,
Catalonia, Spain\\
david.jou@uab.cat}
\address{
$^2$ Dipartimento di Metodi e Modelli Matematici Universit\`a di
Palermo, \\ c/o Facolt\`{a} di Ingegneria, Viale delle Scienze,
90128 Palermo, Italy\\
mongiovi@unipa.it}

\begin{abstract}
We consider the main energetic and geometric aspects contributing
to the entropy of the vortex tangles appearing in superfluid
turbulence, and we propose a simplified model to evaluate them. We
consider the tangle and the underlying superfluid system as two
different systems, each one in internal quasi-equilibrium, but
exchanging heat with each other. The resulting thermodynamic
formalism is studied in two different significative situations.
\end{abstract}

\keywords{Superfluid turbulence. Nonequilibrium thermodynamics.}

\section{Introduction.}

One topic of interest arising in thermodynamics and hydrodynamics
of turbulent superfluids \cite{Do}--\cite{NemFisz95} is the form
of their thermodynamic functions, in terms of $L$, the average
vortex line density per unit volume, and of other relevant
geometrical features of the tangle, related, for instance, to the
orientation of the local unit tangent to the vortex line or to the
local curvature of the vortex lines. Thus, the thermodynamics of
turbulent superfluids is still open to many basic challenges, in
contrast with the thermodynamics of their laminar flows. A deeper
description of the tangle should also incorporate, for instance,
the length distribution of the vortex loops composing the tangle,
the number of their crossings, the spatial correlations between
the tangent vectors or the curvature vectors at different
positions, and so on \cite{NemPRB98}--\cite{Baren03}. Here, we
will try to identify some of the most relevant and basic features
and to provide a simple estimation for their contributions to the
entropy.

To this purpose in the paper we shall consider first the entropic
contribution to the entropy of the vortex-line density per unit
volume (briefly called line density) $L={\cal L}/V$,  where $\cal
L$ is the total vortex line length and $V$ the volume. Line
density $L$ may be directly measured -for instance, by means of
second-sound experiments- and is therefore the most used variable
describing the tangle\cite{Do}--\cite{NemFisz95}. The line density
$L$ has two kinds of contributions to the entropy: on the one
side, it carries out a well defined energy; on the other side, it
is a source of microscopic disorder -related to the length
distribution in vortex loops or, in simpler terms, to the average
number of vortex loops $N$- and, therefore, contributes to the
entropy also from this perspective.

In the paper, we will introduce several thermodynamic functions
for the tangle, entropy $S_{tangle}$, energy $U_{tangle}$,
chemical potential $\mu_{tangle}$, pressure $p_{tangle}$,
temperature $T_{tangle}$, ...; in the following, for the sake of
notation simplicity we will neglect in all these functions the
subindex $tangle$, writing simply $S$, $U$, $\mu$, $p$, $T$, ...

Concerning the energy contribution $U$ of the tangle we will
simply assume that it is
\begin{equation}\label{2.1}
U=N U_l, \end{equation}
where  $U_l$ is the energy of the loop of length $l$.

The energy (\ref{2.1}) will be identified with the energy of the
tangle, i.e., in contrast with other descriptions, where one
directly considers the global energy of the two-fluid helium plus
the tangle, here we give a separated consideration to the energies
of both subsystems. This distinction between both energies will
allow us to use a two-temperature model for the whole system,
namely one for the liquid helium and another one for the tangle.
Indeed, observing that the tangle is not a continuous line, but a
superposition of many closed vortex loops, one might consider the
tangle as an ideal gas of vortex loops and one may relate an
effective temperature of the vortex tangle  to the average energy
of the vortex loops, defining $T_{Tangle}=T$, as
   \begin{equation}\label{TamgleTemp}  k_BT= <U_l>   \end{equation}
 where $k_B$ is Boltzmann's constant.

Though this has not been done -to our knowledge- in this system,
it is a usual assumption in many other systems
\cite{JCLlibro}--\cite{CR-San06}. For instance, one speaks of two
temperature system in plasma physics, because ion and electron
temperatures may be very different, and in the study of glassy
systems where two different temperatures are defined: vibrational
temperature (linked to the fast degrees of freedom, kinetic and
vibrational ones, which is equal to the environment temperature)
and configurational temperature (linked to the slow degrees of
freedom, configurational ones) which turns out to be very high,
but that it is not felt because of the extremely slow rate of
exchange, which may be of the order of years, or even of
centuries.

The system as a whole (the helium background and the vortex
tangle) can be considered as a two-temperature system, where the
difference in temperatures is sustained by the external forcing.
This definition of temperature for the vortex tangle does not rely
on any particular distribution for $U_l$, as well as the kinetic
definition of temperature is used in kinetic theory even when the
velocity distribution function is very different from that of
Maxwell-Boltzmann. In the following sections, we will present two
statistical distribution functions for $l$, corresponding to two
different experimental situations.

\section{Polarized tangles: contributions of $L$.}\setcounter{equation}{2}
 In some regimes of superfluid turbulence produced
by classical means, as that generated by a towing or vibrating
grid, or in simultaneous presence of counterflow and rotation, the
vortex tangle results polarized, and the reconnection of vortices
is forbidden \cite{L'vov07}. In this case we can suppose that the
number $N$ of vortex loops is constant and that the energy of a
vortex loop is proportional to its length $U_l=\epsilon_V l$, with
$\epsilon_V$ the energy per unit length of vortex line
 \begin{equation}\epsilon_V = \rho_s \kappa\tilde\beta \hskip0.5in
\tilde\beta= { \kappa \over 4\pi} \ln\left( \delta\over
a_0 \right),% \approx {\rho_s \kappa^2 \over 4\pi}
\end{equation}
 where $\tilde\beta$ is the vortex tension parameter and $\delta\simeq L^{-1/2}$
is the average separation between vortices. Then, we assume
\cite{MJS-PRB07}
\begin{equation}\label{TangleTemp}  k_BT = \epsilon_V<l>,
\end{equation}
 while, for the distribution function of vortex loops with respect to
their length, we can choose \cite{MJS-PRB07}:
  \begin{equation} f(l)\propto   \exp\left[-{ \epsilon_V l\over k_BT }
 \right].\end{equation}
As we will see later, in counterflow, at extremely low
temperatures, the vortex lines form a very entangled structure and
possess fractal properties, so that $U_l$ becomes proportional to
some power of $l$.

We will concentrate our attention on the entropy of the vortex
tangle, which will be assumed to be a function of the total length
$\cal L$ of vortex lines in the tangle, of the average number $N$
of vortices and of a tensor $\bf\Pi$, describing the orientations
of the tangent vector $\bf s'$, which was introduced in a previous
work \cite{JM-Ani-PRB06} and will be specified in Section 3. Then,
we will search for
\begin{equation}\label{1.1}S_{tangle} = S ({\cal  L}, N,{\bf\Pi}). \end{equation}


Now, we will consider the purely entropic contributions of $L$,
i.e., the contributions related to the disorder. In this Section
we will not consider the orientational disorder, which will be
analyzed in the next Section, but another effect, related to the
fact that the tangle is not a single continuous line, but a
superposition of closed vortex loops. Therefore, there is a number
of possibilities of distributing the total length in several
closed loops of length $l_j$, each loop having the corresponding
energy $U_j = l_j\epsilon_V$.

A simple way to estimate the entropy would be to take into account
that $S=k_B \ln W$, with $W$ the number of available microstates,
which is proportional to the volume of phase space of the
vortices. This phase space is related to the position and the
length of the vortex loop. Therefore, the volume of the phase
space available to vortices will be proportional to $V{\cal L}$.
Since the different vortices are assumed to be independent, and
since the entropy $S({\cal L}, V,N)$ is a homogeneous function of
the first order, in such a way that $S({\cal L}, V,N)=NS[({\cal
L}/N),(V/N)] $ this leads to an expression for the entropy of the
form
\begin{equation}\label{2.5a} S=S
({\cal L},N,V)=S_0+k_BN\ln\left[ {{\cal L}\over N}
 {V\over N} \right].  \end{equation}
One could also write this expression in terms of the internal
energy $U$ rather than $\cal L$, to stress the analogy with ideal
gases, namely
\begin{equation}\label{2.5b}
S=S(U,V,N)=S_0+k_BN\ln\left[{U\over N}
 {V\over N} \right], \end{equation}
 In these equations $S_0$ is a reference constant.  From (\ref{2.5b}) one has
\begin{equation}\label{2.6}\left[ {\partial S\over\partial U}\right]_{V,N}
= {k_BN\over U}\equiv {1\over T} = {k_B\over <l> \epsilon_V }.
\end{equation}

This gives an entropic meaning to the effective temperature
introduced in (\ref{TamgleTemp}). In the thermodynamic analyses of
vortex tangles one is also especially interested in the quantity
$\partial S/\partial N$, which defines the chemical potential
conjugate to variations in $N$. This is obtained from (\ref{2.5b})
as
\begin{equation}\label{2.8}\left[{\partial S\over\partial N}\right]_{U,V} = -{k_B}\left(\ln {N^2\over UV}
+2\right) \equiv - {\mu \over T}, \end{equation}
 The chemical potential of vortex tangle is seen to be
\begin{equation}\label{2.9a}\mu (U, V, N)=
  {k_B T}\left(\ln {N^2\over
UV} +2\right) =\mu ^{(0)} +\epsilon_V <l> \ln { {\cal L}\over
{\cal L }^* }, \end{equation}
 with $\mu ^{(0)}
 = -\epsilon_V<l>\left[\ln (\epsilon_V <l>V)- 2\right]$ a term which
depends on temperature and volume but not on $N$. One could write
the chemical potential of vortex tangle in terms of the
temperature $T$ rather than the internal energy $U$, namely
\begin{equation}\label{2.9b}\mu (T, V, N)=
  {k_B T}\left(\ln {N \over
k_BTV} +2\right)= \mu ^{(0)}(T,V) +k_B T \ln { {\cal L}\over {\cal
L }^* }, \end{equation}
 with $\mu ^{(0)}=\mu ^{(0)}(T,V) = -k_B T\left[\ln (k_BTV)- 2\right]$.
Here we will take for the reference length $\cal L^*$ the value
${\cal L}^*= <l>$. In this way, the logarithm in (\ref{2.9b}) will
vanish when the tangle is composed of a single closed loop,
because in that case the average length $<l>$ will coincide with
the total vortex length $\cal L$ and there will be only a single
microstate for the vortex loop distribution.

The variation of the volume at constant total length allows us to
introduce the concept of tangle pressure, given in thermodynamic
terms as
\begin{equation}\label{2.10}{p \over T}=\left( {\partial S\over\partial V}\right)_{U,N}=
{k_BN\over V}.\end{equation}
 In view of our interpretation  (\ref{TamgleTemp})
of $T$ we would have
\begin{equation}\label{2.11} p =
{k_BN T \over V} ={\epsilon_V {\cal L} \over V} = \epsilon_V L .
\end{equation}
 The total pressure of the system will be the pressure of
the two-component helium fluid plus the contribution of the vortex
tangle given by (\ref{2.11}). This result has been also obtained
in Ref.\cite{MJ-Hydr-PRB07}, in a macroscopic thermodynamic
framework and in Ref.\cite{JMS-PRB07}, in a different simplified
microscopic description.

In this Section, we have neglected the configurational entropy,
which will be considered in the next Section.

\section{Geometric contributions: orientation of vortex lines.}\setcounter{equation}{14}

The geometric characteristics of the tangle, beside its length,
also contribute to the entropy. An essential geometric
characteristic is  the vector field ${\bf s'}(\xi)$, which gives
the unit tangent vector along the vortex lines. The first and the
second moments  of the orientational distribution of $\bf s'$
(namely polarization and anisotropy of the tangle, respectively)
can be determined through measurements of the second sound
attenuation, when the wave is propagating in some given direction,
as for instance along the rotation vector or the counterflow
velocity.


To recall the physical relevance of these vectors, we must have in
mind that the friction force per unit length exerted by the normal
fluid flowing with velocity ${\bf v}_n$ on the superfluid is
\cite{Do}:
\begin{equation}\label{3.1} {\bf f}= -\alpha\rho_s\kappa {\bf s'}\times[{\bf s'}\times({\bf
v}_n- {\bf v}_{sl})] - \alpha'\rho_s\kappa {\bf s'}\times({\bf
v}_n- {\bf v}_{sl}), \end{equation}
 where ${\bf v }_{sl }$ is any externally  applied superfluid
velocity ${\bf v}_s$ plus the self-induced velocity of the vortex
line, $\bf v_i$. In the usual local-induction approximation, $\bf
v_i$ is given by
 ${\bf v_i} \simeq \tilde\beta {\bf s'}\times {\bf s''}$
 \cite{Do}.
It is evident that $\bf s'$ and $\bf s''$ (and also their
vectorial product $\bf s'\times s''$), play a relevant mechanical
role in the dynamics of the vortex lines. Here, we will discuss
the  role of $\bf s'$ in thermodynamics.

In a previous analysis of the structure of the vortex tangle
\cite{JM-Ani-PRB06}, we have proposed to use for the description
of the effects of $\bf s'$ the tensorial variables
\begin{equation}\label{3.3} {\bf \Pi}^s \equiv {3\over 2}<{\bf U} -{\bf s'\bf s'}>,
\hskip0.4in {\bf\Pi}^a \equiv{3\over 2} {\alpha'\over\alpha} <{\bf
W}\cdot {\bf s'}>, \end{equation}
 where $\bf U$ is the unit tensor, ${\bf s'\bf s'}={\bf s' \otimes  s'}$
the diadic product, and the angular brackets stand for average
over the total line length, whereas {\bf W} is the Ricci tensor, a
completely antisymmetric third-order tensor; $\alpha'$ and
$\alpha$ are dimensionless friction coefficients appearing in
equation (\ref{3.1}). The antisymmetric tensor ${\bf\Pi}^a$ is
related to the polarization. In a purely isotropic tangle, ${\bf
\Pi}^a$ vanishes and ${\bf \Pi}^s$ is equal to $\bf U$. The latter
definition deserves some explanation. Indeed, since the average
value of $\bf s'$ is zero along any closed curve, the polarization
related to the average value of $\bf s'$ will be zero when the
tangle is composed entirely of vortex loops. The use of the term
"polarization" is analogous to that used in magnetic of dielectric
dipole systems, where the polarization is related to the net value
of the magnetic or electric moment, rather than with the fact of
the dipoles being oriented along an axis without taking into
account their sign. In this setting, polarized tangles would
require the existence of open vortex lines, which are also
possible provided they being and end on the walls. This is the
case, for instance, of the vortex lines produced in rotating
superfluids, which form an array of straight vortex lines parallel
to the rotation axis in absence of counterflow. A detailed
analysis of the dynamical role of (\ref{3.3}) may be found in
Ref.\cite{JM-Ani-PRB06}. Here we will focus our attention on
purely thermodynamic aspects of $\bf s'$.

To deal with the geometric effects related to the orientation of
$\bf s'$, we make a parallelism with an analogous problem in
polymer physics or in liquid crystals, where the geometrical
configurational characteristics are described by the so-called
configuration tensor $\bf c$ \cite{Ottinger-05}--\cite{Bird-87},
defined as ${\bf c}=< {\bf RR}>$, where {\bf R} is the end to end
vector (in polymers) or the orientational vector (in nematic
liquid crystals or in rigid dumbbell molecular models). The
corresponding expression for the configurational entropy used in
polymer physics is \cite{Ottinger-05}--\cite{Bird-87}:
\begin{equation}\label{3.5}
s({\bf c})=k_B \left[{1\over2}Tr ({\bf U}-{\bf c})+ \ln det({\bf
c}) \right]. \end{equation}
 In the situation considered in this paper, we would have, in view of
the definition (\ref{3.3}) of ${\bf \Pi}^s$:
\begin{equation}\label{3.6} s_{anisotr}({\bf \Pi'}^s)=k_B \left[{1\over2}Tr{\bf \Pi'}^s
+ \ln det({\bf U}-{\bf \Pi'}^s) \right], \end{equation}
 where we use ${\bf \Pi'}^s=(2/3){\bf \Pi}^s= <\bf U-s's'>$.
Indeed, $\bf<s's'>$ plays an analogous role to that of $\bf c$ in
polymer physics.
 Introducing here the form of ${\bf \Pi'}^s$ we would obtain an
explicit expression of the configurational geometrical entropy
$s_{anisotr}({\bf \Pi}^s)$.

To give an explicit example we may consider the vortex tangles
under the simultaneous presence of counterflow ${\bf V}_{ns}$ and
rotation $\bf \Omega$, assumed parallel to each other
\cite{SBD-PRB07}. In this case, in Ref. \cite{JM-Ani-PRB06} for
the vortex orientational distribution was proposed the Langevin
expression:
\begin{equation}\label{4.2} Pr({\bf s'}) \sim \exp\left[-{x}{\bf\widehat\Omega} \cdot{\bf  s'}
\right]=\exp\left[-{a}\cos\theta \right],
\end{equation}
 where $\theta$ is the angle between $\bf s'$ and $\bf \Omega$.
The distribution (\ref{4.2}) reflect the competition between the
orienting effects of $\bf\Omega$ and the randomizing effects of
${\bf V}_{ns}$, respectively analogous to the orienting effects of
a magnetic field $H$ on magnetic dipoles $\mu$ and the randomizing
effects of temperature, in such a way that in a magnetic systems
$x$ would be $x = {\mu H\over kT}$. The value of $x$ was found to
be $x= 11 {L_R\over L_H}$, where $L_R$ would be the line density
if we only had rotation and $L_H$ would be the line density if we
only had counterflow.

In this case, by assuming that the vortex tangle is isotropic in
the plane orthogonal to the rotation first axis, we have
\cite{JM-Ani-PRB06}:
\begin{equation}\label{Pis-Pia-matrix}%
{\bf\Pi}^s =\left(
                \begin{array}{ccc}
                  1-b& 0& 0 \\
                  0&1+{b\over 2}&0 \\
                  0&0&1+{b\over 2} \\
                \end{array}
              \right), \hskip0.3in {\bf\Pi}^a
              ={3\over2}{\alpha'\over\alpha}
\left(
  \begin{array}{ccc}
    0& 0& 0  \\
    0&0&c \\
    0&-c&0 \\
  \end{array}
\right)  \end{equation} where coefficients $b$ and $c$ are linked
to moments of the tangent vector $\bf s'$ by:
\begin{equation}\label{polar}   <{s'_x}>=c=\coth x-{1\over x} ,\end{equation}
\begin{equation}<{s'_x}^2>={1+2b\over 3}= 1+{2\over x}\left[ {1\over x}- \coth x
\right],
 \end{equation}

 With respect to the form of the contribution of the
polarity to the entropy it may be obtained implicitly. Note in
fact that equation (\ref{polar}) relates $x$ with the average
polarity $<\cos\theta>$. The entropy related to (\ref{4.2}) turns
out to be \cite{JM-Ani-PRB06}, in terms of $x$,
\begin{equation}\label{4.8}  S_{polarity}(\Omega,V_{ns}) = k_B \left[ 1+\ln
4\pi-{x} \coth {x} +\ln \left( {1\over  x} \sinh {x}\right)
\right]. \end{equation}
 Combination of the two above equations
would allow one to express the polarity contribution to the
entropy in terms of the polarity itself.


 \section{The fractal model.}\setcounter{equation}{23}

In pure counterflow, an interesting case to study is when the
temperature $T_0$ of the helium background is so small ($T_0 < 1
\rm K$) that the normal fluid fraction of helium~II is negligible.
Hence viscous dissipation and mutual friction are absent. In this
low temperature range, superfluid turbulence takes its purest
form: a tangle of reconnecting vortex filaments which move under
the velocity field induced by the presence of the other vortices.
In this case the tangle can be supposed isotropic, so that
${\bf\Pi}^s={\bf U}$ and  ${\bf\Pi}^a=0$.

The vortex breaking and reconnections are the prevalent phenomena;
they randomize the vortex tangle and cause the formation of
helical Kelvin waves
\cite{Kivotides-cascade}--\cite{Vinen-cascade}. A process of
generation of smaller and smaller scales proceeds, without loss of
kinetic energy, to the smallest scales in which the kinetic energy
of the highly curved and cusped fragments of the vortices is
radiated away as sound. If the above processes act in a
self-similar way on several orders of spatial lengths, one expects
the vortex tangle to exhibit fractal features. In this case, we
have led to a probability distribution function of the length of
loops $n(l)$ which obeys a power law of the form
  {\begin{equation} n(l) =c{ l^{-a} \over (l_{min})^{4-a}} , \end{equation}}
where $c$ is a positive dimensionless constant, $a$ an exponent
related to the fractal dimension of the vortex tangle, and
$l_{min}$ the minimum length of the vortex loops. The values of
$N$ and $L$ may be immediately obtained as
  {\begin{equation} N
= \int_{l_{min}}^{\infty} c{ l^{-a} \over (l_{min})^{4-a}} dl = {
c \over a -1} (l_{min})^{-3},
    \end{equation}}
  {\begin{equation} L =
\int_{l_{min}}^{\infty}  { l^{-a+1} \over (l_{min})^{4-a}}dl =  {
c \over a -2} (l_{min})^{-2}.
 \end{equation}}
Putting $\lambda={c^{1/2}(a-1)\over  (a-2)^{3/2}}$, one has also:
   \begin{equation} <l>={L\over N}={a-1\over a-2 }l_{min}=\lambda
  L^{-1/2},
\hskip0.5in  N={1\over \lambda}L^{3/2} .  \end{equation}
 Result $<l>=\lambda L^{-1/2}\simeq \lambda\delta$ is in accord with some
microscopic analyses of vortex tangles which strongly suggest that
the average length of the vortex loops is of the order of the
intervortex spacing.

The corresponding behavior of the entropy is
  {\begin{equation} {S  \over k_B} = -  \int_{l_{min}}^{\infty}  n(l)
\log n(l) dl = a c \left[1-{a-1\over a}\log (l_{min}^{-4}c)
\right] \end{equation}}
 and in  terms of $L$ and $\lambda$
  {\begin{equation} {S  \over k_B}={1\over \lambda } L^{3/2}
\left[{a\over a-1} + \log {(a-2)\lambda^2 L^{-2}\over (a-1)^2}
\right].  \end{equation}} As one sees, the leading term in this
expression is in $L^{3/2}$.

The temperature of the vortex tangle is obtained as the average
energy of the vortex loops:
  {\begin{equation}\label{Temp} k_B T =<U_l> ={1\over N}\int_{l_{min}}^{\infty}  n(l)
U_l(l) dl  \end{equation}}
 We assume that $U_l$ is proportional to a power of $l$
  {\begin{equation} U_l=d \left({l\over l_{min}}\right)^{\chi} \epsilon_V l, \end{equation}}
 with the exponent $\chi$ linked to the fractal dimension of the
tangle, and $d$ a dimensionless coefficient. It is, for
$a>2+\chi$:
  {\begin{equation} k_BT = {(a-1) d\over a-(2+\chi)} \epsilon_Vl_{min} =d
   {(a-2) \lambda \over a-(2+\chi)}\epsilon_VL^{-1/2} . \end{equation}}
The  {energy per unit volume} is
  {\begin{equation} U=
   {cd\over a-(2+\chi)}\epsilon_Vl_{min}^{-2}=
   {(a-2)d \lambda \over a-(2+\chi)}\epsilon_VL  . \end{equation}}
   A potential distribution function $n(l)\propto l^{-a}$ for the
length of the loops was proposed by Nemirovskii
\cite{Nemir-PRB08}, which found $a = 5/2$ as a stationary solution
of a master equation describing the evolution of the length
distribution function.


The main results of the previous analysis are $<l>\propto
L^{-1/2}$, $N\propto L^{3/2}$, $T \propto L^{-1/2}$ and $S$ with a
leading term in $L^{3/2}$, or, in terms of the tangle temperature:
  {\begin{equation} <l>\propto
T  \hskip0.3in N\propto T ^{-3 } \hskip0.3in U\propto T ^{-2}
 \end{equation} and $S$ with a leading term in $T ^{-3}$}.

Note that, owing to the relation $U\propto T ^{-2}$, the decay of
the vortex tangle leads to a decrease in $L$ and, therefore, an
increase in $<l>$. Thus, when the system loses energy its
temperature increases. Furthermore, this behavior implies that the
heat capacity of the vortex tangle is negative: $\partial
U/\partial T  \propto -T ^{-3} < 0$.


These features share some analogies with the thermodynamics of
black holes \cite{Page05}, for which it is known that $U \propto
T ^{-1}$ and ${\cal S} \propto T^{-2}$. Since the entropy
increases faster that energy, the heat capacity of the black holes
is negative. Thus, fractal vortex tangles and black holes are
thermodynamically unstable by themselves, and they must be
accompanied by sufficient electromagnetic radiation (Hawking
radiation) or by a superfluid in the presence of a heat flux.


 \section{Conclusions}\setcounter{equation}{0}

We have proposed a thermodynamic analysis of superfluid vortex
tangles by focusing our attention on the explicit form of several
different contributions to the entropy.   We have examined two
kinds of contributions: on the one side, the energetic and
entropic contributions of $L$, which has led us to introduce the
vortex temperature, in terms of the average energy of the closed
vortex loops constituting the tangle, and on the other side, the
contributions arising from the geometrical details of the vortex
lines which describe the entropy associated to the anisotropy and
the polarization of the tangle. Our approach may be taken as a
first exploration, with the aim of clarifying the origin of the
different contributions to the entropy of the tangle. The model
borrows ideas and methods of classical mechanics for gases,
polymers and magnetic systems, related, respectively to the
entropy of the length distribution of loops, of the orientation
and curvature of loops, and of the orientation of vortex lines in
the presence of external rotations. Future analyses will lead to
more elaborate expressions for these contributions, based on a
deeper understanding of the required nonequilibrium distribution
functions.


\section*{Acknowledgements.}
The authors acknowledge the support of the Universit\`{a} of
Palermo (under grant 2006 ORPA0642ZR and Progetto CoRI 2007,
Azione D, cap. B.U. 9.3.0002.0001.0001) and the collaboration
agreement between Universit\`{a} di Palermo and Universit\`{a}t
Aut\`{o}noma de Barcelona. DJ acknowledges the financial support
from the Direcci\'{o}n General de Investigaci\'{o}n of the Spanish
Ministry of Education under grant FIS2006-12296-C02-01 and of the
Direcci\'{o} General de Recerca of the Generalitat of Catalonia,
under grant 2005 SGR-00087.

\addcontentsline{toc}{section}{\refname}

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