The search for independence in chaotic systems
In this work we present a model for computation of random processes in digital computers which solves the problem of periodic sequences and hidden errors produced by correlations. We show that systems with non-invertible non-linearities can produce unpredictable sequences of independent random numbers. We illustrate our result with some numerical calculations related with random walks simulations.
Paper1.tex
\documentclass[showpacs,showkeywords,preprintnumbers,amsmath,amssymb,pre,12pt]{revtex4}
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\begin{document}
%\hyphenation{Post-Script}
%-----------------------------------------------Revtex4------------------------------------------------------%
%\title{The search for independence in chaotic systems}
\title{A recipe for an unpredictable random number generator}
%
\author{M\'onica A. Garc\'{\i}a-\~Nustes\footnote{E--mail: mogarcia@ivic.ve}, Leonardo Trujillo\footnote{E--mail: leo@ivic.ve}
and Jorge A. Gonz\'alez\footnote{E--mail: jorge@ivic.ve} }
%
\affiliation{Centro de F\'{\i}sica, Instituto Venezolano de
Investigaciones Cient\'{\i}ficas (IVIC), \\
A.P. 21827, Caracas 1020--A, Venezuela}
%
\date{\today}
%
%-------------------------------------------------------------------------------------------------%
\begin{abstract}
 In this work we present a model for computation of random processes in
digital computers which solves the problem of periodic sequences and
hidden errors produced by correlations. We show that systems with
non-invertible non-linearities can produce unpredictable sequences
of independent random numbers. We illustrate our result with some
numerical calculations related with random walks simulations.
\end{abstract}
\keywords{Random number generator, independent random numbers}
\pacs{05.10.-a, 05.40.a, 05.40.Fb, 07.05.Tp}
\maketitle

Many challenging problems in computational physics are associated
with reliable realizations of randomness (e.g Monte Carlo
simulations). In a typical 32-bit format a maximum of $2^{32}$
floating points numbers  can be represented. Therefore, a recursive
function $X_{n+1}=f(X_{n},X_{n-1},\ldots, X_{n-r+1})$ acting on
these numbers generates a sequence $X_{0},X_{1},X_{2},...,X_{N-1}$
which must repeat itself. It is known that for any recursive
function, a digital computer can only generate periodic sequences of
numbers \cite{Fer,Grass,Vatt,Souza,Nogues,Ecu,Mertens1}. These
generators are not unpredictable.

Definition of truly unpredictable: The next values are not
determined by the previous values. A process $X_n=P(\theta T Z^n)$
is said to be unpredictable if for any string of values
$X_{0},X_{1},X_{2},...,X_{m}$ of length $m+1$, generated using some
$\theta = \theta_1$, there are other values of $\theta$ for which
function $X_n=P(\theta T Z^n)$ generates exactly the same string of
numbers $X_{0},X_{1},X_{2},...,X_{m}$, but the next value $X_{m+1}$
is different, where $m$ is any integer. Note that this kind of
process cannot be expressed as a map of type
$X_{n+1}=f(X_n,X_{n-1},\ldots,X_{n-r+1})$
\cite{GroupPhysLettA,GroupPhysicaD,GroupEuroPL}

%There is a large literature dedicated to pseudorandom number
%generators (PNG's). A very fine theory has been developed in this
%area and this theory has produced many important results,.

%However, we should say here that the known PNG's are not supposed to
%generate truly random numbers. In fact, pseudorandom numbers are
%produced using recurrence relations $X_{n+1}=f(X_{n})$, and
%therefore not truly random \cite{James,James2,Souza,Nogues,Phatak}.


%Many known pseudorandom number generators are based on maps of type
%\begin{equation}
%\label{pseudofunction} X_{n+1}=f(X_{n},X_{n-1},...,X_{n-r+1}).
%\end{equation}

All known generators (in some specific physical calculations) give
rise to incorrect results because they deviate from randomness
\cite{Grass,Souza,Nogues,Mertens1}. It is trivial that any periodic
process is not unpredictable. Suppose $m_T$ is the period of the
generated sequence. Given any string of $m_T$ values:
$X_{s},X_{s+1},...,X_{m_T-1}$; the next value $X_{m_T}$ is always
known, because the process is periodic. On the other hand, for any
generator of type $X_{n+1}=f(X_n,X_{n-1},\ldots,X_{n-r+1})$, given
any string of $r$ values: $X_{s},X_{s+1},...,X_{s+r-1}$; the next
value $X_{s+r}$ is always determined by the previous $r$ values.
Thus it is not unpredictable. So the subsequences must be
correlated.

An example of this can be found in Ref.\cite{Nogues}, where the
authors have shown that using common pseudo random number
generators, the produced random walks present symmetries, meaning
that the generated numbers are not independent. On the other hand,
the logarithmic plot of the mean distance $<d>$ versus the number of
steps $N$ is not a straight line (as expected theoretically,
$<d>\sim N^{1/2}$) after $N>10^{5}$ (in fact, it is a rapidly
decaying function). Here $d$ is defined as the end to end
mean-square distance from the origin of the random walk as a
function of the number of steps. Other papers on the influence of
the pseudorandom number generator on random walk simulations are the
following \cite{Grass93,Shchur97,Shchur99}
%Recently, the statistical
%properties of pseudo-random number generators have been discussed in
%refs. []. The authors of these papers have found that the failure of
%these generators in different simulations can be attributed to the
%low entropy of the production rule f() conditioned on the statistics
%of the input values $X_n,X_{n-1}, . . . , X_{n-r+1}$.

%In the following, we will show that using non-invertible nonlinear
%functions, we can create a truly random number generator that
%produces random walk simulations which possess the correct
%properties, including the mean distance behavior $\left<d^{2}\right>
%\sim N$, even for $N = 10^{9}$ steps.

In the following, we will show that using non-invertible nonlinear
functions, we can create an unpredictable random number generator
which does not contain visible correlations while simulating a
random walk with the length $10^9$.


%\section{Non-invertible Nonlinear Functions}
Let us investigate the following
function\cite{GroupPhysLettA,GroupPhysicaD,GroupEuroPL}:
\begin{equation}\label{Pfunction}
    X_{n}=P(\theta \; T Z^{n}),
\end{equation}
where $P(t)$ is a periodic function, $\theta$ is a real number, $T$
is the period of the function $P(t)$, and $Z$ is a noninteger real
number.

Let $Z$ be a rational number expressed as $Z=p/q$, where $p$ and $q$
are relative prime numbers. Now let us define the following family
of sequences
\begin{equation}
X_{n}^{(k,m,s)}=P \left[ T (\theta_{0} + q^{m}k)\left(
\frac{q}{p}\right)^{s}\left( \frac{p}{q}\right)^{n}\right],
\end{equation}
where $k$, $m$ and $s$ are non-negative integers. The parameter $k$
distinguishes the different sequences. For all sequences
parametrized by $k$, the strings of $m+1$ values
$X_{s},X_{s+1},X_{s+2},...,X_{s+m}$ are the same. This is so because
\begin{equation}
\nonumber X_{n}^{(k,m,s)}=P\left[T \theta_{0}
\left(\frac{q}{p}\right)^{s} \left(\frac{p}{q}\right)^{n} + T k
p^{n-s}q^{(m-n+s)} \right] =P\left[T \theta_{0}
\left(\frac{q}{p}\right)^{s} \left(\frac{p}{q}\right)^{n}\right]
%
\end{equation}
%\begin{equation}
%\nonumber
%\begin{split}
%X_{n}^{(k,m,s)}&=P\left[T\;\theta_{0} \left(\frac{q}{p}\right)^{s}
%\left(\frac{p}{q}\right)^{n} +
%T\; k \;p^{n-s}q^{(m-n+s)} \right],\\
%&=P\left[T\; \theta_{0} \left(\frac{q}{p}\right)^{s}
%\left(\frac{p}{q}\right)^{n}\right]
%\end{split}
%\end{equation}
%
for all $s\leq n \leq m+s$. Note that the number
$k\;p^{(n-s)}q^{(m-n+s)}$ is an integer for $s\leq n\leq m+s$. So we
can have an infinite number of sequences that share the same string
of $m+1$ values.

Nevertheless, the next value
%\begin{equation}
$X_{s+1}^{(k,m,s)}= P \left[ T \; \theta_{0}
\left(\frac{q}{p}\right)^{s} \left(\frac{p}{q}\right)^{(s+1)} +
\frac{T\; k p^{(m+1)}}{q} \right]$
%\end{equation}
is uncertain. In general $X_{s+1}^{(k,m,s)}$ can take $q$ different
values. In addition, the value $X_{s-1}^{(k,m,s)}$,
%\begin{equation}
$ X_{s-1}^{(k,m,s)}= P \left[ T \; \theta_{0}
\left(\frac{q}{p}\right)^{s} \left(\frac{p}{q}\right)^{(s-1)} +
\frac{T\; k q^{(m+1)}}{p} \right]$,
%\end{equation}
is also undetermined from the values of the string
$X_{s},X_{s+1},X_{s+2},...,X_{s+m}$. There can be $p$ different
possible values.   In the case of a generic irrational $Z$, there
are infinite possibilities for the future and the past. From the
observation of the string $X_{s},X_{s+1},X_{s+2},...,X_{s+m}$, there
is no method for determining the next and the previous  values of
the sequence.

But this is not the only feature of these functions. It can be
demonstrated that there are no statistical correlations between
$X_{m}$ and $X_{n}$ if $m\neq n$, and that they are also independent
in the sense that their probability densities satisfy the
relationship $P(X_{n},X_{m})=P(X_{n})P(X_{m})$
\cite{GroupPolonica,GroupJap}.

Moreover, we will show that, given the function (\ref{Pfunction}),
any string of sequences $X_{s},X_{s+1},\ldots, X_{s+r}$ constitutes
a set of statistically independent random variables.

Without loss of generality, we assume that $P(t)$ has zero mean and
can be expressed using the following Fourier representation
%\begin{equation}
%
$ P(t)=\sum_{k=-\infty}^{\infty}a_k e^{i\pi k t}. $
%
%\label{fourier}
%
%\end{equation}

We can calculate the $r$--order correlation functions
\cite{GroupPolonica,GroupJap}:
\begin{eqnarray}
\label{EqA1}
E(X_{n_1}\cdots X_{n_r})&=&\int_X d\theta P(T\theta    Z^{n_1})\cdots P(T\theta Z^{n_r})\nonumber\\
&=&\sum_{k_1=-\infty}^{\infty}\cdots\sum_{k_r=-\infty}^{\infty}a_{k_1}\cdots a_{k_r}\int_{0}^{1}d\theta \exp\left\{ i\pi(k_1Z^{n_1}+\cdots +k_rZ^{n_r})T\theta\right\}\nonumber\\
%
&=&\sum_{k_1=-\infty}^{\infty}\cdots\sum_{k_r=-\infty}^{\infty}a_{k_1}\cdots
a_{k_r}\delta(k_1Z^{n_1}+\cdots +k_rZ^{n_r},0),
\end{eqnarray}
where the coefficients $k_{i}$ can be different integers, and
$\delta (n,m)=1$ if $n=m$ or $\delta(n,m)= 0$ if $n\neq m$.

When all $n_i$ are even, the following equation is satisfied
%
\begin{equation}
\label{EE}
%
E(X_{s}^{n_1}X_{s+1}^{n_2}\cdots
X_{s+r}^{n_r})=E(X_{s}^{n_1})E(X_{s+1}^{n_{2}})\cdots
E(X_{s+r}^{n_{r}}).
%
\end{equation}

The main problem in this equation occurs when one of the numbers
$n_i$ is odd. In this case, the correlations $E(X_{s}^{n_1}
X_{s+1}^{n_2}\cdots X_{s+r}^{n_r})$ must be zero. A nonzero
correlation in Eq.(\ref{EE}) exists only for the sets
$(n_1,n_2,\ldots,n_r)$ that satisfy the equation $k_1Z^{n_{1}} +
\cdots + k_{r}Z^{n_{r}}=0$. For a typical real number $Z$, this
equation is never satisfied.

%\begin{figure}[ht]
%\centerline{\includegraphics[width=0.36\textwidth]{SenZ4,3.ps}\hspace{0.5cm}\includegraphics[width=0.36\textwidth]{SenZPi,2.ps}}
%\caption{First-returned map constructed with the dynamics generated
%by function $X_{n}=\cos[2\pi\theta Z^{n}]$ with $Z=4/3$ (left) and
%$Z=\pi/2$ (right).}
%\end{figure}
%\begin{wrapfigure}{i}{0.4\textwidth}
%\centerline{\includegraphics[width=0.36\textwidth]{SenZ4,3.ps}\includegraphics[width=0.36\textwidth]{SenZPi,2.ps}}
%\caption{First-returned map constructed with the dynamics generated
%by function $X_{n}=\cos[2\pi\theta Z^{n}]$ with $Z=4/3$.}
%\label{fig-map}
%\end{wrapfigure}
%\begin{wrapfigure}{i}{0.4\textwidth}
%\centerline{\includegraphics[width=0.36\textwidth]{SenZPi,2.ps}}
%\caption{First-returned map constructed with the dynamics generated
%by function $X_{n}=\cos[2\pi\theta Z^{n}]$ with $Z=4/3$.}
%\label{fig-map}
%\end{wrapfigure}


%\section{Random Number Generator}
If we use non-invertible nonlinear functions, type of
(\ref{Pfunction}), we can implement a Truly Random Number Generator
(TRNG). In this case, we propose the following function
\begin{equation}
\label{generatingfunction} X_{n}=[\theta_{s} Z^{n}] \quad mod \;1
\end{equation}

Function (\ref{generatingfunction}) is an example of the general
case $X_n=P[\theta T Z^n]$ studied in this paper. We have shown that
the subsequences $X_s,X_{s+1},\ldots , X_{s+r}$ constitutes a set of
statistically independent random variables. The particular case of
function (\ref{generatingfunction}) is well--known to produce
uniformly distributed numbers
\cite{GroupPhysLettA,GroupPhysicaD,GroupEuroPL}.

Now we will formulate a central limit theorem. Using theorems proved
in previous studies \cite{GroupPolonica,GroupJap,r16,r17,r18,r19}
and the results obtained from this paper, we obtain the following
formula: If $Z$ is a generic real number and $X_n=2(Y_n-1/2)$,
$Y_n=[\theta Z^n]mod 1$, then

\begin{equation}\label{Eq30}
    \lim_{r\rightarrow\infty}P\left\{ \alpha<\frac{X_1+X_2+\cdots X_r}{\sqrt{r}}<\beta
    \right\}=\frac{1}{\sqrt{\pi}}\int_{\alpha}^{\beta}\mathrm{e}^{-\xi^2}\mathrm{d}\xi.
\end{equation}
%

The Gaussian distribution of the sums is correct even for other
functions $X_n=P[\theta Z^n]$, where $P(t)$ is periodic. This has
been shown in numerical simulations \cite{GroupPolonica}.

The numbers $X_n=[\theta Z^n]\mod 1$ are uniformly
distributed\cite{GroupPhysLettA,GroupPhysicaD,GroupEuroPL}. We can
simulate different stochastic processes (with different
distributions) using different functions $X_n=P[\theta T Z^n]$. As
$\rho (X_n)=1$, $\rho(X_{n+1})=1$, $\rho(X_{n},X_{n+1})=1$, it is
trivial that they are independent.

It is interesting to check the theoretical predictions using
numerical simulations of the behavior of different stochastic
processes.

For instance, let us study the function
\begin{equation}\label{EqU}
    U_n = \cos[2\pi\theta Z^n].
\end{equation}

All the moments and higher--order correlations can be calculated
exactly \cite{GroupPolonica,GroupJap}:

For odd m:
\begin{equation}\label{Eq15}
    E(U_{n}^{m})=0.
\end{equation}

If any $n_i$ is odd, then
\begin{equation}\label{Eq16}
    E(U_{s}^{n_0}U_{s+1}^{n_1}\cdots U_{s+r}^{n_r})=0
\end{equation}
%

Suppose now that all $n_i$ are even:
%
\begin{eqnarray}\label{Eq18}
 E(U_s^{n_0}U_{s+1}^{n_1}\cdots U_{s+r}^{n_r})= 2^{-(n_0+n_1+\cdots +n_r)} \left(
\begin{array}{c}
  n_0 \\
  \frac{n_0}{2} \\
\end{array}
\right) \left(
\begin{array}{c}
  n_1 \\
  \frac{n_1}{2} \\
\end{array}
\right)
%
\cdots
%
\left(
\begin{array}{c}
  n_r \\
  \frac{n_r}{2} \\
\end{array}
\right),
%\label{relation4}
%\end{equation}
\end{eqnarray}
%------------------------------------------------------------------------
\begin{eqnarray}\label{Eq19}
 E(U_s^{n_0})= 2^{-n_0} \left(
\begin{array}{c}
  n_0 \\
  \frac{n_0}{2} \\
\end{array}
\right),
\end{eqnarray}
%------------------------------------------------------------------------
\begin{eqnarray}\label{Eq20}
 E(U_{s+1}^{n_1})= 2^{-n_1} \left(
\begin{array}{c}
  n_1 \\
  \frac{n_1}{2} \\
\end{array}
\right),\ldots,
\end{eqnarray}
%------------------------------------------------------------------------
\begin{eqnarray}\label{Eq21}
 E(U_{s+r}^{n_r})= 2^{-n_r} \left(
\begin{array}{c}
  n_r \\
  \frac{n_r}{2} \\
\end{array}
\right).
\end{eqnarray}
%

Note that the condition for independence is satisfied
\begin{equation}\label{Eq22}
    E(U_{s}^{n_0}U_{s+1}^{n_1}\cdots
    U_{s+r}^{n_r})=E(U_{s}^{n_0})E(U_{s+1}^{n_1})\cdots
    E(U_{s+r}^{n_r}),
\end{equation}
%
for all integer $n_{0},n_{1},\ldots n_{r}$.

We have performed extensive numerical simulations that confirm the
values of these moments and the independent conditions.

An additional checking is the following.

The probability density of $U_n$ is
$\rho(U)=\frac{1}{\pi\sqrt{1-U^2}}$. Define $V_n=U_{n+1}$. The
probability density of $V_n$ is $\rho(V)=\frac{1}{\pi\sqrt{1-V^2}}$.
We have checked both theoretically and numerically that
$\rho(U,V)=\frac{1}{\pi^2\sqrt{(1-U^2)(1-V^2)}}$, that is
$\rho(U,V)=\rho(U)\rho(V)$. This can be observed in Fig. 1 and Fig.
2.
\begin{figure}[ht]
\centerline{\includegraphics[height=6.0cm]{grafica1A}\hspace{0.5cm}}
\caption{Probability densities for random variables
$U_{n}=\cos[2\pi\theta Z^{n}]$ and $V_{n}=U_{n+1}$.
a)$\rho(U)=\frac{1}{\pi \sqrt{1-U^{2}}}$; b)$\rho(V)=\frac{1}{\pi
\sqrt{1-V^{2}}}$}
\end{figure}
\begin{figure}[ht]
\centerline{\includegraphics[height=6.0cm]{grafica2B}\hspace{0.5cm}}
\caption{Probability density $\rho(U_{n},U_{n+1})$, when
$U_{n}=\cos[2\pi\theta Z^{n}]$. Here $\rho(U,V)=\frac{1}{\pi^{2}
\sqrt{(1-U^{2})(1-V^{2})}}$}
\end{figure}
In order to avoid computation problems, we have used the following
procedure.
%We should say that, if we are using a single-precision machine, the
%straightforward implementation of (\ref{generatingfunction}) can
%present a machine-precision problem. For a large $Z$ the argument of
%the function increases so fast that singularities can occur. To
%solve this problem,
%
We change parameter $\theta$ after each set of $M$ values of
$X_{n}$, where $M$ is the maximum number for which there are not
overflow problems, in such a way that the next value of $X_{n+1}$ is
obtained with the new $\theta$. Let us define
\begin{equation}
\theta_{s}=A (C_{s}+X_{s}) + 0.1
\end{equation}
%$\theta_{s}$ changes after a fixed number $n=M$ of sequences values
%$X_{n}$. This is due that (\ref{Pfunction}) can present a machine
%precision problem. For large $Z$, the argument of the function
%increases so fast that singularities can occur.
%$M$ is the maximum
%number for which there are no calculating problems.
%problems
%A solution to these
%problems is to change parameter $\theta$ after a fixed number $n=M$
%of sequences values $X_{n}$.
%Suppose $M$ is the maximum number for which there are no calculating
%problems. We can determine a new value for the parameter $\theta$
%after each set of $M$ values of $X_{n}$, in such a way that the last
%value of $X_{n}$ is obtained with the new $\theta$. For producing
%the new set of values of $X_{n}$ we start with $n=0$. This procedure
%can be repeated the desired number of times.
%In this case we use the following functions in order to generate
%random numbers, the parameter $\theta$ changes after each set of $M$
%sequences values
where $C_{s}$ is a  sequence obtained using the digits of the
Champernowne's number \cite{Champ} (i.e., $0.1234567891011...$):
$C_{0}=0.123456$ , $C_{1}=0.234567$, $C_{2}=0.345678$,
$C_{3}=0.456789$, $C_{4}=0.567891$, and so on. This sequence is
nonperiodic. Index $s$ is the order number of $\theta$ in such a way
that $s=0$ corresponds to the $\theta$ used for the first set of $M$
sequence values $X_{1}, X_{2},...,X_{M}$; $s=1$ for the second set
$X_{M+1}, X_{M+2},...,X_{2M}$, and so on. $X_{0}$ represents the
TRNG's seed.
%With  $s=0$ we have $\theta_{0}(C_{0},X_{0})$ this is used to
%generate the first set of $M$ sequence values $X_{1},
%X_{2},...,X_{M}$; with $s=1$ we can calculated the second set
%$X_{M+1}, X_{M+2},...,X_{2M}$, and so on.

Using this method we have generated a very long sequence of random
numbers without computational problems.

%\begin{wrapfigure}{i}{0.5\textwidth}
%\centerline{\includegraphics[width=0.5\textwidth]{rw1RNG1E9box}}
%\caption{Logarithmic plot of the mean distance $<d>$ versus the
%number of steps N with A=6.9109366. The red line represents the
%theoretical line. The blue line represents numerical results using
%function (\ref{generatingfunction}) as TRNG.} \label{fig-map}
%\end{wrapfigure}

To test function (\ref{generatingfunction}) as a truly random number
generator, we have implemented a random walk simulation program in
C++. We have made a sampling test of a random walk with $N=10^{9}$
steps with $100$ realizations with different initial seeds. The mean
distance $\left<d\right>$ was calculated every $1000$ steps of the
random walk.

The Champernowne sequence of numbers used in the generator  was
produced previously by a short C++ program, who created a sequence
of a maximum of 40000 Champernowne's numbers. If a larger amount of
values to $C_{s}$ is necessary, it can be obtained using a segment
code that uses the 40 thousand values already stored in $C_{s}$ and
mixing them, e.g the algorithm takes the first value of the series
$C_{1}$,  the third $C_{3}$ and so on, and adds them at the end of
the series, obtaining that $C_{s+1}=C_{1}$, $C_{s+2}=C_{3}$,...; if
more values are necessary, this procedure or cycle is repeated but
now skipping two values $C_{1}$, $C_{3}$, $C_{5}$,... three values
$C_{1}$, $C_{4}$, $C_{7}$ and so on. In this way, we can make the
$C_{s}$ sequence as large as we wish.

%The simulation time  of each realization was around of $10$ minutes,
%using  intel\textregistered  C++ compiler for linux, being the total
%simulation time around of $16$ hours of simulation.

We present  a logarithmic plot of the mean distance $<d>$ versus the
number of steps $N$ with $N=10^{9}$ steps with $A=6.9109366$ and
$Z=\pi/2$ (See Fig. 3). We can verify that there is no deviation
from the theoretical straight line, even for $N>> 10^{5}$ steps,
which is a very good test of the reliability of the Random Number
Generator used in the random walk simulations.
%Also we show a random
%walk graphics in 2D with $N=10^{5}$ steps created using the TRNG. As
%we can appreciate there is no apparent symmetries in this graphics
%which can



\begin{figure}[ht]
%\centerline{\includegraphics[height=6.0cm]{rw1RNG1E9boxGris4}\hspace{0.5cm}}
\begin{center}
\includegraphics[height=5.5cm]{grafica3}\hspace{0.5cm}
\includegraphics[height=5.5cm]{grafica4}\hspace{0.5cm}
\end{center}
\caption{Logarithmic plot of the mean distance $<d>$ versus the
number of steps $N=10^{9}$ steps. a) for generator
(\ref{generatingfunction}), b) the same simulation for a generator
of type $X_{n+1}=aX_n\mod T$.}
\end{figure}

%We have presented a random number generator, which can produce truly
%unpredictable and uncorrelated sequences of random numbers. We have
%applied successfully this generator in a random walk simulation in
%C++.
%\begin{figure}[ht]
%\centerline{\includegraphics[height=6.5cm]{Hist3DFunDiscretaGris3}\hspace{0.5cm}}
%\caption{Logarithmic plot of the mean distance $<d>$ versus the
%number of steps $N=10^{9}$ steps.}
%\end{figure}
%This a advance to find useful random number generators for
%physics simulation problems.

We have presented a random number generator based on the properties
of non--invertible transformations of truncated exponential
functions. The obtained random process is unpredictable in the sense
that the next values are not determined by the previous values. We
have applied this generator to the numerical simulation of
statistically independent random variables. In the simulation of a
random walk with the length $10^9$, the random process does not
contain visible correlations.



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