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\markboth{}{\it Michael Blank ``Multiplicative cascade models and
multifractality''}
\begin{document}
\title{\bf Multiplicative cascade models and multifractality}
\author{{\Large Michael Blank}
\thanks{On leave from Russian Academy of Sciences, Inst. for
Information Transmission Problems,
Ermolovoy Str. 19, 101447, Moscow, Russia.}
\\ \\
C.N.R.S., Observatoire de Nice, BP 229, \\ 06304 Nice Cedex 4, France,
e-mail: blank@obs-nice.fr}
\date{March 16, 1995}
\maketitle
\n {\bf Abstract.} We construct a (chaotic) deterministic variant of
random multiplicative cascade models of turbulence. It preserves the
hierarchical tree structure, thanks to the addition of infinitesimal noise
or finite-state Markov approximations of chaotic maps. The zero-noise
limit can be handled by Perron-Frobenius theory, just as the
zero-diffusivity limit for the fast dynamo problem. We prove also the
absence of phase transitions in conservative random multiplicative cascade
models, corresponding to the non divergence of statistical moments.
\bigskip
\noindent{\bf Keywords.} Multiplicative cascade, chaotic map, phase
transition
\section{Introduction}
One popular way to describe the small scale activity of fully
developed turbulence is to suppose that energy is transferred from
the injection scale to the viscous scales, throughout a multi-steps
process along the inertial range. In spite of the fact that this idea
has been often used to predict many important features of turbulent
flows, its relations with the structure of Navier-Stokes equations
are still poorly understood. We construct a theory describing
{\em deterministic} multiplicative cascade models and prove the absence
of phase transitions in conservative {\em random} multiplicative cascade
models.
In order to construct the deterministic model for the multiplicative
cascade of energy we use shell models for fully developed turbulence and
we connect the well known case of random independent multiplicative
cascade model (the simplest model of the multifractality) and our case
(deterministic) in two steps through introducing at first Markov random
model and then by means of a special limit construction going to the
deterministic model. There are two ways of this construction. The first
one is to add a small amount of noise to the deterministic model and to
use it as a usual Markov chain in the Markov random multiplicative cascade
model. Then the deterministic model is the zero noise limit of this
construction. The second way is to use an approximation of a chaotic
dynamics by a finite-state random Markov chain, which states corresponds
to elements of a partition of the phase space. The deterministic model in
that case is obtained as a limit of the finer and finer partitions.
Random multiplicative models were introduced by Novikov and Stewart and by
Yaglom as a simple way to describe stochastic transfer of energy along the
inertial range. To define a random multiplicative model, a binary tree
structure, obtained by hierarchically partitioning the original volume of
size $\l_0$ in subvolumes of size $\l_n=2^{-n}\l_0$, they used to describe
fluctuations at different scales. The energy dissipation, $\ep_n$,
associated to a cube at scale $\l_n$, is multiplicatively linked to the
energy dissipation, $\ep_{n-1}$, at the larger scale, $\l_{n-1}$, through
a random variable $\xi_n$:
$$ \ep_n=\xi_n \ep_{n-1} = \xi_n\xi_{n-1}\xi_{n-2}....\xi_1 \bar{\ep}, $$
\n where $\{\xi_n\}$'s are identically and independently distributed
positive random variables. The structure functions are now defined as
\be S_n(q) := E\{\ep_n^{q/3}\}{\l}_n^{q/3}, \label{str.fun.phys} \ee
\n where $E\{.\}$ denotes the mathematical expectation.
\section{Construction of the deterministic multiplicative cascade}
Our aim now is to construct a deterministic variant of the previous model.
We shall do it in two steps. First, we consider, instead of independent
random variables $\{\xi_n\}$, successive points on the orbit of a Markov
process. This means that we consider a Markov process on a phase space $X$
with a transition probability operator $P$ and an observable $h: X \to
R_+^1$. The (generalized) structure function at the scale
$\l_n=2^{-n}\l_0$ is then defined as:
\be S_n(q) = S_n(q,h,P) := E\left\{\prod_{k=1}^n h^q(x_k)\right\},
\label{str.fun2} \ee
\n where $x_k \in X$ are points of an orbit of the Markov process.
Introducing now:
\be \zeta_n(q)=\prod_{k=1}^n h^q(x_k), \label{str.fun3} \ee
\n and the conditional expectation of $\zeta_n(q)$ with respect to the
initial distribution density, $\rho(x)$, is given by:
$$ E\{\zeta_n(q)| \rho \} = \int_X h^q(x) \rho(x) P_{h,q}^n \,1(x) \, dx. $$
\n Here, the operator $P_{h,q}$ is defined by the relation $P_{h,q}\phi(x)
= h^q(x)P\phi(x)$. The second step is to change from a Markov {\it random}
dependence to a {\it deterministic} dependence by use of a chaotic map
$f:X \to X$. It seems that this can be done in an obvious way, replacing
the transition probability operator by the transfer-matrix
(Perron-Frobenius operator) $P_f$ of the chaotic map. However, with a
deterministic map, we should give the same value to the two off-springs of
the next generation, thereby trivializing the whole tree-structure.
To solve this problem, we build up this branching-deterministic process by
inserting a small amount of noise at each node and by considering the
total process as the superposition of the deterministic transfer along
consecutive levels plus the noise. The final map will be obtained by
taking the zero-noise limit. For a fixed value of $\ep>0$ it follows that
the large-$n$ limit of the structure functions is governed by the spectrum
of the operator:
$$ P_{f, h, q, \ep} = h^q Q_\ep P_f. $$
\noindent Here, $Q_\ep$ is the transition operator for the random
perturbation.
There is also another possibility to define a deterministic (chaotic)
process on a tree structure. Using a method of finite-state Markov
approximations of chaotic maps, proposed by Ulam, we obtain a Markov chain
with transition probabilities given by
$$ p_{ij} = \frac{|X_i \cap f^{-1}X_j |} {|X_i|}, $$
\n where the set of $X_i$'s defines a suitable finite partition of the
original phase space $X$. With this Markov process, we construct the
analog of the operator $P_{h,q}=h^qP$. (Here $P$ is the transition
operator with matrix elements $p_{ij}$.)
\section{Stability of statistics}
We now observe that the construction based on taking the zero-noise limit
appearing above is similar to that of the mathematical theory of {\it fast
dynamos}. Fast dynamo theory describes the phenomenon by which rapid
magnetic field growth can be sustained in the presence of a prescribed
velocity field, when taking the zero-diffusivity limit. From a formal
point of view, fast dynamo theory involves a combination of two
operators: a transfer-matrix for some deterministic map, associated to a
deterministic velocity field, and a diffusion-like operator, or
equivalently small-amplitude noise.
In our paper \cite{BBF}, assuming that the Perron-Frobenius operator,
corresponding to the map $f$ is stochastically stable, and the map is
chaotic and ergodic, we computed the structure functions as averages
along-the-orbit for the deterministic map, i.e.
$$ S_n(q)=\lim_{N \to \infty} \frac{1}{N}
\sum_{i=1}^N \prod_{k=1}^n h^q(x_{i+k}) $$
\n with $x_{i+1}=f(x_i)$. However, in the general case this is not true
and the aim of this paper is to give assumptions under which the structure
function could be calculated in the way above. Let $f$ be a piecewise
expanding (PE) one-dimensional map, i.e. the function $f$ is piecewise
monotonic and the derivative of some its power is strictly larger than $1$
(see for details \cite{Bl}). By singular points of the map $f$ we shall
mean all the points $x \in X$, where the derivative $f'(x)$ is not well
defined (for instance, all boundary points of its monotonicity intervals).
Denote by $q_\ep(x,y)$ the transition probability density of random
perturbations. Here $\ep$ is the perturbation ``amplitude''. We shall
suppose that var$(q_\ep(x,.)) < C/\ep$. This assumption is quite general
and clearly is valid for independent uniformly distributed perturbations.
\begin{theorem} \label{Theorem 1} Let a map $f$ be a PE map without
periodic singular points. Then the map $f$ is stochastically stable.
\end{theorem}
\begin{theorem}\label{Theorem 2} Let a map $f$ be an arbitrary PE map. If
random perturbations satisfy the condition
$$ \int_{-\infty}^{x-C\ep} q_\ep(x,y) \, dy > \delta \quad {\rm and } \quad
\int_{x+C\ep}^{\infty} q_\ep(x,y) \, dy > \delta $$
\n for some positive $C, \delta$ and any $\ep > 0$, then the map $f$ is
stochastically stable. If this condition is not satisfied, then there
exists a family of small random perturbations, stabilizing unstable
periodic orbits of the map. \end{theorem}
\begin{theorem}\label{Theorem 3} Finite-state Markov approximations of PE
maps are stochastically stable (even for the case of periodic singular
points). \end{theorem}
\section{Absence of phase transitions in conservative multiplicative
cascade models}
The shortcoming of our construction is that we, following the tradition
in the physical literature, consider the mathematical expectation of the
structure function (\ref{str.fun.phys},\ref{str.fun2}), rather than the
random structure function itself (\ref{str.fun3}). We discuss now this
question for random measures constructed by means of multiplicative
cascade models. Consider a sequence of nonnegative random values
$\{\xi_{ij}\}$ with
$E\{\xi_{n,2k+1}^q\}=x_q, \; E\{\xi_{n,2k}^q\}=y_q, \;
E\{\xi_{n,2k+1}^q \xi_{n,2k+2}^q\}=z_{2q}, \; X_q=x_q+y_q$,
distributed on nodes of the binary tree:
\be 1 \label{bin.tree} \CR
\xi_{11} \qquad \xi_{12} \CR
\xi_{21} \;\; \xi_{22} \quad \xi_{23} \;\; \xi_{24} \ee
$$ \xi_{31} \;\; \xi_{32} \quad \dots \quad \xi_{37} \;\; \xi_{38} \CR
\dots \dots \dots \dots \dots \dots \dots \dots $$
\n Our assumptions correspond to the nature of the cascade process,
because subtrees of the binary tree (\ref{bin.tree}) of the same volume
should have the same statistical properties. However, in the general case
there is no need to suppose that these random values are uniquely
identically distributed, or independent.
On the $n$-th level of the tree (\ref{bin.tree}) there are $2^n$ nodes. We
correspond these nodes to elements of the hierarchical partition of the
initial volume $\Delta$ into $2^n$ disjoint subvolumes $\Delta_i$ of equal
size. Consider a random measure $\mu_n$, such that for any $i$ the value
$\mu_n(\Delta_i)$ is equal to the multiplication of all random values on
the nodes along the branch from the root of the tree to the considered
node. Under some weak assumptions (see, for example, \cite{CK}) these
random measures $\mu_n$ converge (in the weak topology) to a
deterministic limit measure $\mu_\infty$. This construction is known as
a {\em multiplicative cascade model}.
Consider now the following functionals depending on a variable $q$:
\be \zeta_n(q) := \zeta_n(\mu_n, \, q) = \sum_i \mu_n^q(\Delta_i), \quad
\zeta_\infty(q) := \zeta_n(\mu_\infty, \, q) \ee
\be \Phi(q) := \lim_{n \to \infty} \frac{\log(\zeta_n(q))}{\log(2^n)},
\quad \Phi_\infty(q) := \lim_{n \to \infty}
\frac{\log(\zeta_n(\mu_\infty, \, q))}{\log(2^n)}.
\label{lim.str} \ee
\n Our aim is to investigate the existence of the limit above and the
properties of these functionals as functions of the variable $q$. We shall
consider phase transitions in the weak sense as the divergence of the
series (\ref{lim.str}), while in the strict sense the latter means that
the function $\Phi(q)$ differs from $\Phi_\infty(q)$ (see \cite{CK}) and
thus the statistics of the random measures $\mu_n$ do not converge to the
corresponding statistics of the limit measure $\mu_\infty$. In \cite{CK}
it was shown that the problem under consideration may be considered as a
problem of statistical mechanics, where the measure $\mu_\infty$ plays the
role of the {\em free energy} and the parameter $q$ corresponds to the
{\em inverse temperature}. The absence of phase transitions corresponds to
the self-averaging property of the free energy, which characterizes the
unique random state.
The multiplicative model is well known for the case of independent
uniquely distributed random values $\xi_{ij}$ with $E\{\xi_{ij}\}=1/2$ and
$x_q=y_q, \, z_{2q}=x_q^2$, and it is known \cite{CK} that there exists a
critical value $q_c$ (may be infinite), such that only while $q1$ we have
\be X_q = x_q+y_q = E\{\xi^q\} + E\{(1-\xi)^q\} < 1, \ee
\n which gives the convergence of the second central moments to zero, and
thus, by Chebishev's inequality, the convergence of $\zeta_n(q)$ to its
mathematical expectation, and thus leads to the absence of phase
transitions.
\smallskip The author thanks Uriel Frisch for very useful discussions.
%\newpage
\begin{thebibliography}{99}
\bibitem{BBF} Biferale L., Blank M., Frisch U. Chaotic cascades with
Kolmogorov 1941 scaling, {\it J. Stat. Phys.} {\bf 75}:5-6(1994), 781-795.
\bibitem{Bl} Blank M. Singular phenomena in chaotic dynamical systems,
{\it Doklady Akad. Nauk (Russia)}, {\bf 328}:1(1993), 7-11.
\bibitem{CK} Collet P. and Koukiou F. Large deviations for multiplicative
chaos, {\it Commun. Math. Phys.} {\bf 147} (1992), 329-342.
\end{thebibliography}
\end{document}