TITLE: Tiling a minimal perimeter region with $n$ unit squares
QUESTION [2 upvotes]: Suppose I have $n$ identical unit squares and I want to use them all to tile a region with minimal perimeter $p(n)$. For instance I guess $p(n^2)=4n$, by arranging them im a $n\times n$ square.
Is there an explicit formula for $p(n)$? Or sharp bounds? How do one prove equalities or bounds for this type of quantities?
I would be happy also with a recursive formula for $p(n)$, like $p(n\cdot m) = f(p(m),p(n))$, but the factorization is not unique and I don't see how to get it easily.
My intuition is that given a certain number $n$, the minimal perimeter region is a rectangle with sides of integer lengths $l$ and $m$, where $l\cdot m=n$ and $(l,m)$ is the pair of integer numbers "closest" (in some sense) to $(\sqrt{n},\sqrt{n})$, that's where number theory could play a role.
I have no clue where to start proving something along these lines, so any hint, comment or reference is welcome!

REPLY [5 votes]: For every x in the range $n^2 \le x < (n+1)^2$ you have $4n \le p(x) \le 4n+4$, so $p(x) = \sqrt{x} + \operatorname{O}(1)$. If you want a closed formula I would try with (putting $t=n^2-x$)
$$ p(n^2+t) = \begin{cases}
4n \quad&\text{if }t=0\\4n+2 &\text{if }0 < t \le n\\ 4n+4 &\text{if }n < t \le 2n\end{cases}$$
which is the perimeter you obtain if you go from the  square $n\times n$ to the square $(n+1)\times(n+1)$ tile by tile.
EDIT (Proof sketch)
We can assume that the shape with least perimeter is connected (if the shape has two pieces, connecting them by a suitable edge gives a shape with strictly less perimeter than the original).
Now given $x = n^2+t$ with $0\le t \le 2n$ fix a shape that gives the least perimeter $p(x)$. Pick a rectangle with minimal dimensions $a\times b$ containing the shape. The perimeter is at least $2a+2b$, (as it goes all the way from left to right and from top to bottom and back by the other side).
On the other hand $ab \ge x$ because there are at least $x$ tiles inside the rectangle so
$$ p(x) \ge 2a + 2b \ge 2a + 2x/a $$
the right hand side has a single minimum at $a = \sqrt{x}$ but $a$ is an integer so we have
$$ p(x) \ge 2n + 2x/n = \frac{ 4n^2 + 2t}{n} $$
(it is easy to see that $a=n+1$ gives a larger value).  If $t = 0$ this gives $p(x) \ge  4n$, if $t \le n$  we have $p(x) > 4n$ but as $p(x)$ is even we have $p(x) \ge 4n+2$ and if $t > n$ we have  $p(x) > 4n+2$ so again $p(x) \ge 4n+4$. As we have examples obtaining this bounds we are finished.


TITLE: How do you find the one-sided limit of a particular function?
QUESTION [0 upvotes]: I'm having trouble understanding the solution to the following problem:

Given the function $f(x)=sgn(x)=\begin{equation}
    \begin{cases}
      1, x>0\\
      -1, x<0\\
      0, x=0
    \end{cases}\
\end{equation}$, determine the one sided limits of this function on $x_0=0$.

The solution I have in my notes is:
$\displaylines{\lim_{x\rightarrow {x_0}^-}sgn(x)=\begin{vmatrix}
{x_0}^-=x_0-\epsilon\\
0^-=0-\epsilon=-\epsilon, \epsilon\rightarrow 0^+
\end{vmatrix}=\lim_{\epsilon\rightarrow 0^+}sgn(-\epsilon)=-1}$
Similarly,
$\displaylines{\lim_{x\rightarrow {x_0}^+}sgn(x)=\lim_{\epsilon\rightarrow 0^+}sgn(x)=\lim_{\epsilon\rightarrow 0^+}sgn(0+\epsilon)=1}$.
Now, I don't know what the heck is going on in the solution. How is ${x_0}^-=x_0-\epsilon$ and $0^-=0-\epsilon=-\epsilon, \epsilon\rightarrow 0^+$. I just cannot see the relation between these equalities and the definiton of one-sided limits.

REPLY [0 votes]: $x_0^-$ is the left hand limit at $x_0$ (the superscript - indicates the left hand limit at a point).
Similarly $x_0^+$ is Right hand limit.
For left hand limit we can write x is smaller than $x_0$ by infinitesimal amount.
So $x = x_0 - \epsilon $ where $\epsilon > 0$ and $ \epsilon \rightarrow 0$


TITLE: Shows that $EFD$ is also equilateral.
QUESTION [1 upvotes]: In the following figure, it is known that ABC is equilateral and that $AE = BD = CF$, shows that $EFD$ is also equilateral.

Attemp:
I got a little lost in proving these triangles to be similar, but if you can do that, it's simple. Call the angles EAB, DBC and FCA x and the angles DBA, FCB and EAC y. (We know they are the same because we assume the triangles to be similar.) We know that x + y + x + y + x + y = 180 °, so x + y = 60 °. DEF is the outer angle of AEB, so DEF = x + y = 60 °. Similarly the other angles are 60 °, thus being DEF equilateral.
I also think this looks like Morley's triangle. But I don't know if DBC is 1/3 ABC:

REPLY [1 votes]: COMMENT:
1- Proof by reverse argument:
Consider equilateral triangle DEF and  construct triangle ABC as follows:
Extend EF to a certain value and mark it as A. Extend DE and FD to same value and mark them as B and C respectively. Join point A, B and C. Triangles AFC, AED and CBD are equal for two sides and angle between equal, therefore :
$AB=BC=AC$
That is triangle ABC is equilateral. Now we have a equilateral triangle in which, $AE=BD=FC$ and $DE=FE=DF$, that is triangle DEF is equilateral.
2-Proof by contradiction:
Suppose $DE≠FE≠DF$, then $AE≠EB≠CD$.
Therefore :
$\triangle AEB ≠ \triangle ACF ≠\triangle BCD$
And: $AB≠ BC≠ AC$ which contradicts $AB= BC= AC$.


TITLE: If $X$ is a metric space, $A \subset X$ y $r\in \mathbb{R}^+$, then $V_r(A)=\{x\in X: d(x,A)\leq r\}$ is closed.
QUESTION [0 upvotes]: If $X$ is a metric space, $A \subset X$ and $r\in \mathbb{R}^+$, then $V_r(A)=\{x\in X: d(x,A)\leq r\}$ is closed.
My attempt: let $x\in V_{r}(A)$, by definition, for all $\epsilon>0$, $B(x,\epsilon)\cap V_{r}(A)\not =  \emptyset$, then exist $y$ such that $y\in B(x,\epsilon)$ and $y\in V_{r}(A)$ then $d(y,A)\leq r$ and $d(y,x)\leq \epsilon$ then $d(x,A)\leq d(y,A)+d(y,x)\leq r+\epsilon$
But I need $d(x,A)\leq r$
please!
Can you help me?

REPLY [2 votes]: I often find it easier to prove that complements are open.  If $d(x, A) = r+\alpha$ with $\alpha \gt 0$, then by the triangle inequality, $B \left (x, \frac{\alpha}{2} \right) \cap V_r(A) = \varnothing$, so $x$ (an arbitrary point of the complement) is in the interior of the complement.  The complement is therefore open, so $V_r(A)$ is closed.


TITLE: Is there a general name for matrices which only have zeros on their main diagonal?
QUESTION [1 upvotes]: A diagonal matrix is one where every component not on the main diagonal is zero. E.g.
$$
\begin{array}{cc}
12 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & -2
\end{array}
$$
Is there a term for matrices where every component on the main diagonal is zero? E.g.
$$
\begin{array}{cc}
0 & 3 & 5\\
-2 & 0 & 13\\
1 & 0 & 0
\end{array}
$$

REPLY [0 votes]: The precise definition is the hollow matrix.
https://en.wikipedia.org/wiki/Hollow_matrix


text:
\begin{definition}[Definition:Regular Value]
Let $X$ and $Y$ be [[Definition:Smooth Manifold|smooth manifolds]].
Let $f: X \to Y$ be a [[Definition:Smooth Mapping|smooth mapping]].
Then a point $y \in Y$ is called a '''regular value''' of $f$ {{iff}} the [[Definition:Pushforward|pushforward]] of $f$ at $x$:
: $f_* \vert_x: T_x X \to T_y Y$
{{explain|What do all the symbols mean in this context? Presume $\vert$ might mean restriction, but this is not obvious (if so then use $\restriction$); if "pushforward" actually means $f_* \vert_x: T_x X \to T_y Y$ then set up the page to define it, thus doing all the hard work of defining that concept all in one place.}}
is [[Definition:Surjection|surjective]] for every $x \in f^{-1} \left({y}\right) \subseteq X$.
\end{definition}

config: wiki


TITLE: How to get critical constants from Dieterci's real gas equation?
QUESTION [2 upvotes]: Please, before closing this question as off-topic, consider that this is completely a calculus problem, no chemistry involved

Dieterci's real gas equation is
  $$p = {RTe^{-a/(RTV)}\over V - b}$$

I need to find  critical constants of this equation.
To do that I need to equation first second derivative of this equation to 0. and find the expression for  $V$ and $T$ from them in terms of $a, b$. And then put those values in real equation to obtain a expression for $p$ in terms of $a, b$.
So differentiating the formula with respect to V,
$$p^\prime = RT\left[(e^{-a/(RTV)})^\prime (V -b)^{-1} + (-1)(1)(V -b)^{-2}e^{-a/(RTV)} \right] \tag{1}$$
$$p^\prime = -RTe^{-a/(RTV)}\left[ {a\over RT}(V^{-2})(V -b)^{-1} + (V - b)^{-2} \right] \tag{2}$$
$$p^\prime = -RTe^{-a/(RTV)}(V -b)^{-1}\left[ {a\over RT}(V^{-2}) + (V - b)^{-1} \right] \tag{3}$$
$$\bbox[5px,Border:2px solid black]{p^\prime = -p\left[ {a\over RT}(V^{-2}) + (V - b)^{-1} \right]} \tag{4}$$
And differentiating it again w.r.t to V
$$\bbox[5px,Border:2px solid black]{p^{\prime\prime} = p^\prime\left({a\over RT}(V^{-2} - (V - b)^{-1})\right) + \left[ {a \over RT}(-2)V^{-3} + (V- b)^{-2} \right]p} \tag {5}$$
Now equating $(5)$ and $(4)$ to zero I get,
For $(4)$
$$p^\prime = -p\left[ {a\over RT}(V^{-2}) + (V - b)^{-1} \right] = 0$$
$$\left[ {a\over RT}(V^{-2}) + (V - b)^{-1} \right] = 0$$
$$RTV^2 - aV + ab = 0 \tag{6}$$
For $(5)$
Since we know $p^\prime = 0$
$$0 = \left[ {a \over RT}(-2)V^{-3} + (V- b)^{-2} \right]p$$
$${a\over RTV^3} = \frac1{V^2 + b^2 -2Vb}$$
$${ RTV^3\over a} = {V^2 + b^2 -2Vb} \tag {7}$$
Now to get the critical constants I need to solve for $V$ and $T$ in equations $(6)$ and $(7)$.
I tried it but I ended up in a mess.
I need help as to how to solve these equations conviently. Thanks.

REPLY [5 votes]: This is too long for a comment.
You could have done the first steps faster using logarithmic differentiation $$P = {RT\over V - b}e^{-\frac a {RTV}}\implies \log(P)=\log(RT)-\log(V-b)-\frac a {RTV}$$ So, differentiate to get $$\frac{P'}{P}=\frac{a}{R T V^2}-\frac{1}{V-b}\implies P'=P\left(\frac{a}{R T V^2}-\frac{1}{V-b} \right)\tag 1$$ Differentiate a second time $$P''=P'\left(\frac{a}{R T V^2}-\frac{1}{V-b} \right)+P\left(\frac{a}{R T V^2}-\frac{1}{V-b} \right)'\tag 2$$ and since you will impose later $P'=P''=0$, the equations are $$\frac{a}{R T V^2}-\frac{1}{V-b}=0\tag 3$$ $$\left(\frac{a}{R T V^2}-\frac{1}{V-b} \right)'=\frac{1}{(V-b)^2}-\frac{2 a}{R T V^3}=0\tag 4$$ So $$\frac{1}{V-b}=\frac{a}{R T V^2}\tag 5$$ $$\frac{1}{(V-b)^2}=\frac{2 a}{R T V^3}\tag 6$$ Squaring $(5)$ and computing the ratio to $(6)$ leads to $a=2RTV$; using this result in $(5)$ gives$$\frac {1}{V-b}=\frac{2RTV}{RTV^2}=\frac 2 V\implies V=2b\implies T=\frac{a}{4 b R}$$ Back to the initial expression of $P$, the above give $$P=\frac{a}{4 e^2 b^2}$$ So, as  functions of parameters $a,b$, the critical coordinates of Dieterici equation of state are given by $$V_c=2b \qquad T_c=\frac{a}{4 b R} \qquad P_c=\frac{a}{4 e^2 b^2}\qquad Z_c=\frac{P_cV_c}{RT_c}=\frac{2}{e^2}$$
In practice, in this domain of equations of state, since, for components, critical properties are known, we use to compute the $a,b$ parameters from $T_c,P_c$ $$a=\frac{4 R^2 T_c^2}{e^2 P_c}\qquad b=\frac{R T_c}{e^2 P_c}$$


TITLE: IF a language L logspace reduces to SAT, does L
QUESTION [2 upvotes]: If a language L logspace reduces to SAT, does L also reduce to SAT in polynomial time?

REPLY [5 votes]: Shortest answer: Yes.
Short answer: Yes, for the same reason that  LOGSPACE $\subseteq$ P.
Less short: If the number of tape squares used is logarithmic, then the number of possible tape configurations is polynomial, and since the tape configurations mustn't repeat, the time taken is polynomial too.
In horrible detail:  Let $M$ be a TM that performs a logspace reduction of $L$ to SAT. Say that $M$ has $N$ states and a tape alphabet with $A$ symbols.  Since $M$ performs a logspace reduction of $L$, the space taken by $M$ in transforming any element $s$ of $L$ to a SAT instance must be bounded above by $c\cdot\log|s|$ for some constant $c$.
$\def\s#1{\langle T_{#1}, S_{#1}\rangle}$Consider a transformation by machine $M$ of a string $s$ to an instance of SAT.  At time $n$, $M$ is in state $S_n$ and its tape is in state $T_n$.  There can't be two different times at which both state and tape are the same, or $M$ would be in an infinite loop, which we know doesn't happen.  So the running time of $M$ is bounded above by the number of possible (state, tape configuration) pairs, which is $N$ times the total number of possible tape configurations. But at most $c\cdot\log|s|$ tape squares are used, so the total number of tape configurations is at most $A^{c\cdot\log|s|}$, and thus the total time is bounded by $$N\cdot A^{c\cdot\log|s|}.$$  This is a polynomial in $|s|$, so the whole computation is time-bounded by a polynomial in $|s|$, for any $s\in L$, and therefore $L$ is polynomial-time reducible to SAT.


TITLE: Calculation the reflection coefficient of a mutlilayer material
QUESTION [2 upvotes]: For our project we have to study an infrared filter. This filter is composed of glass and several layers (nanolayers of titanium oxide, silver and cupper deposited on one side of the glass).
Now we have to determine the reflection coefficient of the coating (i.e. the nanolayers alone) given that of the glass and that of the filter as a whole.
Now our supervisor claims that this is calculated by the formula: $R_{filter} = R_{coating}R_{glass}$. This formula seems a bit odd to us though, f.i. because it doesn't hold in general (one can imagine a material composed of two materials, where one has 100% reflectance and the other 50%, so this material should have 100% reflectance, while according to the formula it has 50%). I wonder if somebody maybe knows if this formula holds, and maybe can explain which assumptions one could make under which one can derive the formula.

REPLY [1 votes]: Yes... your supervisors formula is incorrect. Smart people make mistakes -- it happens. If your supervisor isn't open to criticism, then I suggest that you give him or her a few simple examples that illustrate the problem, and then immediately present a solution. Your example of a mirror coated over a non-mirror is a great place to start.
In order to solve your multiple reflection problem, you can use the transfer-matrix method. Unfortunately you'll need to invert the result, because you want to calculate the properties of one layer given the properties of several, which is the opposite of what the method is usually set up to do. Whether or not you can do this depends on if you have enough information to separate the effects of the different layers.
Also, note that the reflectance of one layer is actually undefined, because reflection is a property of an interface between two materials, not a property of a single material. Water in water doesn't reflect, and air in air doesn't reflect, but there is reflection at air-water interfaces. Your coatings will reflect different amounts if they are in contact with one another as compared to if they're in contact with air.


text:
\begin{document}

\begin{frontmatter}

\title{Centre-of-mass like superposition of Ornstein--Uhlenbeck processes:
a pathway\\ to non-autonomous stochastic differential equations\\
and to fractional diffusion}

\author[lasapienza]{Mirko D'Ovidio}
\author[unibo]{Silvia Vitali}
\author[upotsdam,bcam]{Vittoria Sposini}
\author[bcam]{Oleksii~Sliusarenko}
\author[bcam,isti]{Paolo Paradisi}
\author[unibo]{Gastone Castellani}
\author[bcam,iker]{Gianni Pagnini\corref{cor}}
\cortext[cor]{Corresponding author}
\ead{gpagnini@bcamath.org}

\address[unibo]{DIFA - Department of Physics and Astronomy,
University of Bologna, Viale B. Pichat 6/2, I-40127 Bologna, Italy}
\address[lasapienza]{Department of Basic and Applied Science for Engineering,
Sapienza University of Rome, Via Antonio Scarpa 16, I-00161 Rome, Italy}
\address[bcam]{BCAM -- Basque Center for Applied Mathematics,
Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country, Spain}
\address[iker]{Ikerbasque -- Basque Foundation for Science,
Calle de Mar\'ia D\'iaz de Haro 3, E-48013 Bilbao,  Basque Country, Spain}
\address[isti]{ISTI-CNR, Institute of Information Science and Technology "A. Faedo",
via Moruzzi 1, I-56124, Pisa, Italy}
\address[upotsdam]{Institute for Physics and Astronomy, University of Potsdam,
Karl-Liebknecht-Strasse 24/25, D-14476 Potsdam-Golm, Germany}

\begin{abstract}
We consider an ensemble of Ornstein--Uhlenbeck processes featuring
a population of relaxation times and a population of noise amplitudes
that characterize the heterogeneity of the ensemble.
We show that the centre-of-mass like variable corresponding to this ensemble
is statistically equivalent to a process driven by a
non-autonomous stochastic differential equation
with time-dependent drift and a white noise.
In particular, the time scaling and the density function of such variable
are driven by the population of timescales and of noise amplitudes, respectively.
Moreover, we show that this variable is equivalent in distribution
to a randomly-scaled Gaussian process,
i.e., a process built by the product of a Gaussian process times
a non-negative independent random variable.
This last result establishes a connection with the so-called
generalized gray Brownian motion and suggests application
to model fractional anomalous diffusion in biological systems.
\end{abstract}

\begin{keyword}

Ornstein--Uhlenbeck process \sep
Heterogeneous ensemble \sep
Superposition \sep
Center of mass \sep
Non-autonomous stochastic differential equation \sep
Randomly-scaled Gaussian process \sep
Generalized grey Brownian motion \sep
Anomalous diffusion \sep
Fractional diffusion.

\end{keyword}

\end{frontmatter}

\section{Introduction}
\label{intro}
Let $W^\kappa_t$, $\kappa \in \{1,2, \ldots, N\}$ be a sequence of Wiener processes
$W_t^\kappa: [0,\infty) \to \R$ with $\E_x [W_t^\kappa]=0$ and
$\E_x [W_t^\kappa W_s^\kappa]=\min(t,s)$.
As usual, we denote by $\E_x$ the expectation with respect to $\mathbb{P}_x$ where $x$
is the starting point for the process under consideration.

In this paper we consider the ensemble $\{\Ukappa\}_{\kappa}$
of the Ornstein--Uhlenbeck (OU) processes $\Ukappa: [0,\infty)  \to \R$
satisfying the stochastic differential equations (SDE)
\be
d\Ukappa = - \frac{\Ukappa}{\taukappa} dt + \sqrt{2 \, \Dkappa} \, dW_t^\kappa \,,
\quad X^\kappa_0=x \,,
\label{OU}
\ee
where, $\forall\, \kappa$, $\taukappa > 0$ and $\Dkappa >0$.
We call $\taukappa$ relaxation times and $\Dkappa$ noise amplitudes.
In particular, the following equivalence in law holds true
\be
\Ukappa = e^{-\frac{t}{\taukappa}} ( x +  W_{\varphi(t)}^\kappa),
\quad \kappa =1,2, \ldots, N \,,
\ee
where
\be
\varphi(t) = \Dkappa \taukappa \left(e^{2t / \taukappa} -1 \right) \,.
\ee
Hence, each $\Ukappa$ is a real-valued Gaussian process
with $\E_x (\Ukappa)=x e^{-t /\taukappa}$.
The correlation function
\be
\E_x [X_t^\kappa X_s^\kappa] = e^{-(t+s)/\taukappa} \min(\varphi(t), \varphi(s))
\label{correlationOU}
\ee
gives the covariance function
\be
C(t,s) = \Dkappa \taukappa \,
\e^{-|t-s|/\taukappa} - (\Dkappa \taukappa  + x^2) \e^{-(t+s)/\taukappa} \,.
\ee
Without loss of generality, we assume that $x=0$ and therefore
\be
C(t,s) = \Dkappa \taukappa \, \left( \e^{-|t-s|/\taukappa} - \e^{-(t+s)/\taukappa} \right) \,.
\label{covarianceOU}
\ee
For a given $\kappa$, the infinitesimal generator $A$ of $\Ukappa$ is therefore
\be
Au = - \frac{x}{\taukappa} \frac{du}{dx} + \Dkappa \frac{d^2 u}{d x^2},
\quad u \in C_c(\mathbb{R}, \mathbb{R}) \,,
\ee
where $C_c$ is the set of smooth and compactly supported functions on $\mathbb{R}$.
In view of the applications in diffusion problems,
in analogy with the Langevin equation (for $\gamma, m>0$)

we introduce the mass-like variable $\mkappa > 0$ such that
\be
\frac{1}{\taukappa} = \frac{\gammakappa}{\mkappa} \,, \quad
\sqrt{\Dkappa} = \frac{\sqrt{\sigma_0}}{\mkappa} \,,
\label{def}
\ee
where $\sigma_0> 0$ is a fixed parameter independent of $\kappa$,
and, $\forall\, \kappa$,
the pair of independent parameters $(\taukappa, \Dkappa)$ in the SDE (\ref{OU})
is now replaced by the pair $(\gammakappa, \mkappa)$.

In the following, we study the centre-of-mass like process
\be
\Vcm=\sum_\kappa \frac{\mkappa}{M} \Ukappa \,,
\label{MOD1}
\ee
with $M=\sum_\kappa \mkappa$.

We highlight that, because of the center-of-mass like formulation,
the process $\Vcm$ here considered differs
from the superposition of OU processes considered for example in
Refs. \cite{csorgo_etal-cjm-1990,lin-spa-1995}
or from the superposition of OU-type processes considered in Refs.
\cite{leonenko_etal-s-2005,barndorffnielsen_etal-mcap-2005,grahovac_etal-jsp-2016},
and also the type of provided results is different.
In particular, in this paper we derive the non-autonomous SDE
satisfied by a process equivalent in distribution to the process $\Vcm$
and more
we show also that the process $\Vcm$ is equivalent in distribution to
a randomly-scaled Guassian process, i.e.,
a process built by
the product of a non-negative random variable times a Gaussian process.
This last result can be understood in the framework of the so-called
generalized grey Brownian motion (ggBm), which can be recovered when the Gaussian
process is the fractional Brownian motion.
The ggBm is a generalization of the theory of the white noise analysis
by introducing non-Gaussian measures of Mittag--Leffler type
\cite{mura-phd-2008,mura_etal-jpa-2008,mura_etal-itsf-2009,
grothaus_etal-jfa-2015,grothaus_etal-jfa-2016}.

Under the physical point of view,
the considered system can be understood as a
{\it heterogeneous ensemble of Brownian particles},
i.e., a system composed
of non-identical Brownian particles that differ in their density
(mass devided by volume).
The study of the centre of mass allows for estimating the average concentration and
the momentum of inertia of the ensemble by computing the mean
and the mean square displacement,
respectively.
Such a system can be related to the so-called
{\it anomalous diffusion} which refers to diffusion processes that,
in opposition to Brownian motion,
do not show a Gaussian density function of particle displacements and
neither a linear growth in time of the mean square displacement.
Anomalous diffusion is a widespread phenomenon
\cite{anomaloustransport,klm}
that requires specific statistical tools \cite{meroz_etal-phyrep-2015},
that is sometimes related to fractional diffusion \cite{metzler_etal-jpa-2004},
and that can be generated by
the polydispersity when classical thermodynamics holds
\cite{gheorghiu_etal-pnas-2004},
i.e., with Gaussian noise as in the present case,
or by noises with long-range spatiotemporal correlations
with even "anomalous" thermodynamics
\cite{gheorghiu_etal-pnas-2004}, i.e., with non-Gaussian noise.
In this respect,
the derived link between the process $\Vcm$ and the ggBm
allows for perspective applications of the present results
to model fractional diffusion in biological systems
in view of the promising application of the ggBm in these systems
\cite{molina_etal-pre-2016,sposini_etal-njp-2018}.

In the next Section \ref{main}, the main theorems are stated and
their proofs are given in Section \ref{proofs}.
In Section \ref{simulations},
the numerical simulations related to the main results of Section \ref{main} are shown.
In Section \ref{conclusions} the conclusions are reported and the
application to model anomalous diffusion in biological systems is discussed.

\section{Main results}
\label{main}
In this Section we present the two main results,
i.e., Theorem \ref{Th:nonautonomous} and Theorem \ref{Th:ggBm2}, that we obtain
for the process $\Vcm$ defined in (\ref{MOD1}).
The first concerns the determination of the non-autonomous SDE satisfied by
a process $\Vcm^H$ that is equivalent in distribution to $\Vcm$,
and the second concerns the equivalence in distribution of the process $\Vcm$
with a randomly-scaled Gaussian process,
i.e., a process defined by the product of a non-negative random variable and
a Gaussian process.

We introduce  the scaled process
\be
\Ykappa=\frac{\Ukappa}{\sqrt{\Dkappa}} \,,
\label{YOU}
\ee
for which
\be
\label{PYp}
\mathbb{P}_0(\Ykappa \in dy) = \int_{\Omega_\tau} p^\kappa(y;t|\tau) q(d\tau) dy
\,,
\ee
where the Gaussian density $p^\kappa$ is conditioned to $\taukappa$ and
\be
\mathbb{P}(\tau^\kappa \in \Omega_\tau) = \int_{\Omega_\tau} q(d\tau) =1.
\ee
We assume that $q(d\tau)= q(\tau) d\tau$, that is $\taukappa$ has population density $q : \mathbb{R}^+ \supseteq \Omega_\tau \mapsto \mathbb{R}^+$.
We notice that $\taukappa \stackrel{d}{=} \tau$, $\forall\, \kappa$,
where $\tau$ is distributed according with $q$.
We denote by \lq\lq$\stackrel{d}{=}$" the equality in distribution for random variables.

We maintain the superscript $\kappa$ in \eqref{PYp} although the sequence
$\Ukappa$ (and therefore, the sequence \eqref{YOU}) are identically distributed.
Indeed, we have that $\Ukappa \stackrel{d}{=} X_t$
and $p^\kappa(y;t|\tau) = \sqrt{\sigma^\kappa} v(\sqrt{\sigma^\kappa} y; t | \tau)$
where $v(x;t| \tau) dx = \mathbb{P}_0(X_t \in dx)$.

\newtheorem{theorem}{Theorem}
\begin{theorem}
\label{Th:nonautonomous}

Let $\Vcm:[0,\infty) \to \R$ be the center-of-mass like stochastic process in \eqref{MOD1}.

Then, $\Vcm \stackrel{d}{=} Z^*_t$ where $Z^*_t$ satisfies the following non-autonomous SDE
\be
d\Vcm^*= - \frac{\Vcm^*}{\taueff(t)}dt + \frac{\sqrt{2 \sigma_0}}{M} \, dW_t^{\rm{eff}} \,,
\quad Z_0^*=z \,,
\label{langevintype}
\ee
with $\sqrt{\sigma_0}=\mkappa \sqrt{\Dkappa}$,
$dW_t^{\rm{eff}}=\sum_\kappa dW_t^\kappa$ such that
$\E [(dW_t^{\rm{eff}})^2]=N \E [(dW_t^\kappa)^2]$,
and where $\tau_{\rm{eff}}(t)$ is defined as
\be
\taueff(t)=
\left[
\frac
{\displaystyle{\int_{-\infty}^{+\infty} \int_{\Omega_\tau}
y^2 p^\kappa(y;t|\tau) q(\tau) \, d\tau dy}}
{\displaystyle{\int_{-\infty}^{+\infty} \int_{\Omega_\tau}
\frac{y^2}{\tau^2} p^\kappa(y;t|\tau) q(\tau) \, d\tau dy}}
\right]^{1/2} \,.
\ee
\end{theorem}

We observe that the SDE (\ref{langevintype}) is non-autonomous because of the dependence
on $t$ of the drift coefficient through the function $\taueff(t)$,
and that the noise amplitude $\sqrt{2 \sigma_0}/M$ is not a process but
a non-negative random variable. The opposite situation with a drift
independent of $t$ and the noise amplitude given in terms of a stochastic process has been
recently studied in Ref. \cite{benth_etal-spa-2018}.
By setting
$$
\Lambda=\sigma_0/M^2 \,,
$$
from (\ref{def}), we have that $\Lambda$ is an independent non-negative random variable.

\begin{theorem}
\label{Th:ggBm1}
Let $\{\Ukappa\}_\kappa$ be the ensemble defined in (\ref{OU}) and (\ref{def}).
Let $\Vcm:[0,\infty) \to \R$ be the center-of-mass like process defined in (\ref{MOD1}).

Let $\Ykappa$ be the scaled process in \eqref{YOU}.

Denote by $B_t^H:[0,\infty) \to \R$ the Gaussian process
$B_t^H \stackrel{d}{=} \sqrt{\E_0 [(\Ykappa)^2]} \, W_1$, $\forall\, \kappa$.
Then we have that , $\forall\, t>0$, as $N\to \infty$,
\be
\frac{1}{\sqrt{N}} \Vcm \stackrel{d}{\to} \sqrt{\Lambda} \, B_t^H \,.
\label{ggBmMURA0}
\ee
\end{theorem}

\begin{theorem}
\label{Th:ggBm2}
Let $\{\Ukappa\}_\kappa$ be the ensemble defined in (\ref{OU}) and (\ref{def}).
Let $\Vcm:[0,\infty) \to \R$ be the center-of-mass like process defined in (\ref{MOD1}).

Let $\Ykappa$ be the scaled process in \eqref{YOU}.

Denote by $B_t^H:[0,\infty) \to \R$ the Gaussian process
$B_t^H \stackrel{d}{=} \sqrt{\E_0 [(\Ykappa)^2]} \, W_1$, $\forall\, \kappa$.
Then we have that , $\forall\, t>0$, $\forall\, N>0$,
\be
\Vcm \stackrel{d}{=} \sqrt{N\Lambda} \, B_t^H=\Vcm^H \,.
\label{ggBmMURA}
\ee
\end{theorem}

We observe that since $\Lambda$ is a random variable that is
different for any realization of the process $B_t^H$ and
it is independent of $t$,
the process $\Vcm^H$ is not the same process studied
in Ref. \cite{garet-spa-2000}.

\newtheorem{lemma}{Lemma}
\begin{lemma}
The stochastic process $\Vcm$ has a non-exponential correlation
controlled by the population of $\taukappa$ distributed according to the density $q(\tau)$.
The exponential correlation is recovered in the case $q(\tau)=\delta(\tau-\tau_0)$.
\end{lemma}
\begin{proof}
Consider the scaled process
\be
\Ykappa = \frac{\Ukappa}{\sqrt{\Dkappa}} \,,
\ee
which satisfies the SDE
\be
d\Ykappa = - \frac{\Ykappa}{\taukappa} dt + \sqrt{2} \, dW^\kappa_t \,,
\label{eqOU0}
\ee
then it holds
\be
\Vcm=
\frac{\sqrt{\sigma_0}}{M} \sum_\kappa \Ykappa \,.
\label{ggBm}
\ee
Each stochastic process $\Ykappa$ is an OU process
with exponential correlation (\ref{correlationOU}), i.e.,
$\displaystyle{\E [\Ykappa Y_s^\kappa | \tau^\kappa] = \tau^\kappa \, \left( \e^{-|t-s|/ \tau^\kappa} - \e^{-(t+s)/ \tau^\kappa} \right)}$, $\forall\, \kappa$, according to (\ref{eqOU0}).
The correlation of the process $\Vcm$ is
\begin{eqnarray}
\E [\Vcm Z_s]
&=& \E \left[\frac{\sigma_0}{M^2} \, \E \left[\sum_\kappa \Ykappa \sum_\kappa Y_s^\kappa \bigg| \taukappa \right] \right] \nonumber \\
&=& \E \left[\frac{\sigma_0}{M^2}\, \sum_\kappa \E \left[\Ykappa Y_s^\kappa \big| \taukappa \right] \right] \nonumber \\
&=& \E \left[\frac{\sigma_0}{M^2} \sum_\kappa \taukappa \, \left( \e^{-|t-s|/ \tau^\kappa} - \e^{-(t+s)/ \tau^\kappa} \right) \right] \nonumber \\
&=& \E \left[\frac{\sigma_0}{M^2} \right]  \sum_\kappa \E \left[ \taukappa \, \left( \e^{-|t-s|/ \tau^\kappa} - \e^{-(t+s)/ \tau^\kappa} \right) \right] \nonumber \\
&=& N \, \E \left[\frac{\sigma_0}{M^2}\right]
\int_{\Omega_\tau} \tau \, \left( \e^{-|t-s|/ \tau} - \e^{-(t+s)/ \tau} \right) \, q(d \tau),
\label{correlazione}
\end{eqnarray}
where, we recall that $\Lambda=\sigma_0/M^2$ is an independent random variable and $\taukappa
$ are independent and identically distributed. The distribution $q(\tau)$ modifies the classical correlation displayed, $\forall\, \kappa$, by $\E [\Ykappa Y_s^\kappa \big| \tau]$. When $q(\tau)=\delta(\tau-\tau_0)$, the exponential correlation follows.
\end{proof}

From Theorem \ref{Th:ggBm2}, we have that
\be
\E [\Vcm Z_s]  = N \E [\Lambda] \, \E [B_t^H B_s^H]= N \E [\Lambda] \, R(t,s) \,,
\ee
and then, after re-scaling $N\Lambda \to \Lambda$, the following holds

\begin{lemma}
Let $f(\lambda)$, $\lambda \in \Omega_\lambda \subseteq \R^+$,
be the density of the non-negative random variable $\Lambda$,
then the $n$-dimensional density of $\Vcm$ is
\be
\p(\bz;R)=\frac{1}{\sqrt{(2\pi\lambda)^n \, {\rm det} \, R}}
\int_{\Omega_\lambda} \exp\left\{-\frac{1}{2\lambda} \, \bz^T R^{-1} \bz\right\}
\, f(\lambda) \, d\lambda \,,
\label{ggBmpdfn}
\ee
where $\bz=(z_1,\dots,z_n)$ and $R=R(t_i,t_j)$, $i,j=1,2,\dots,n$,
is the covariance matrix of $B_t^H$.
\end{lemma}

\newtheorem{corollary}{Corollary}
\begin{corollary}
Let $R(t,t)=t^{2H}$,
with $0 < H < 1$, and $\Lambda \sim f(\lambda)$,
then the one-point one-time density function of $\Vcm$ is
\be
\p(z;t)
= \frac{1}{\sqrt{4 \pi \lambda \, t^{2H}}} \int_{\Omega_\lambda}
\exp\left\{-\frac{z^2}{4 \lambda \, t^{2H}}\right\}
\, f(\lambda) \, d\lambda \,.
\ee
\end{corollary}

\newtheorem{remark}{Remark}
\begin{remark}
If $f(\lambda)$ is the density function of $\Lambda$, then from the relation
$\Lambda=\sigma_0/M^2$ we have that the density function of $M$ is
$$
g(M)= \frac{2 \sigma_0}{M^2} \, f\!\left(\frac{\sigma_0}{M^2}\right) \,.
$$
\end{remark}

\begin{remark}
If $\mkappa$, $\kappa \in \N$, are independent identically distributed variables
according to the distribution $\rho(m)$,
then $M=\sum \mkappa$ follows the same distribution, i.e.,
$$
g(M) \equiv \rho(m) \,.
$$
\end{remark}

\begin{corollary}
[See Theorem 2.1 and Corollary 2.1 in Ref. \cite{pagnini_etal-fcaa-2016}]
Let $R(t,t)=t^{2\beta/\alpha}$ and
$f(\lambda)=K^{-\alpha/2}_{\alpha/2,\beta}(\lambda)$, with
\cite{mainardi_etal-fcaa-2001}
\be
K^{-\theta}_{\alpha,\beta}(\lambda)=
\frac{1}{2 \pi} \int_{-\infty}^{+\infty} \e^{- i \omega \lambda}
\left\{
\frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty}
\e^{pt} \frac{p^{\beta-1}}{p^\beta + \Psi_\alpha^\theta(\omega)} \, dp
\right\} \, d\omega \,,
\ee
where $0 < \beta < 1$, $0 < \alpha < 2$, $\theta=\min\{\alpha,2-\alpha\}$ and
$\Psi_\alpha^\theta(\omega)=|\omega|^\alpha \e^{i ({\rm{sgn}} \omega)\theta \pi/2}$,
then the one-point one-time density function of $\Vcm$ is
\cite{pagnini_etal-fcaa-2016}
\begin{eqnarray}
\p(z;t)
&=& \frac{1}{\sqrt{4 \pi \lambda \, t^{2\beta/\alpha}}}\int_0^{+\infty}
\exp\left\{-\frac{z^2}{4 \lambda \, t^{2\beta/\alpha}}\right\}
\, K^{-\alpha/2}_{\alpha/2,\beta}(\lambda) \, d\lambda
\nonumber \\
&=& \frac{1}{t^{\beta/\alpha}} \, K^0_{\alpha,\beta}\left(\frac{|z|}{t^{\beta/\alpha}}\right) \,.
\label{stpdf}
\end{eqnarray}
\end{corollary}

The density function $\p(z;t)$ defined in (\ref{stpdf}) is the
kernel of the following space-time fractional diffusion equation
\cite{mainardi_etal-fcaa-2001,pagnini_etal-fcaa-2016}
\be
\left\{
\begin{array}{lr}
_tD_*^\beta \, u = {_zD^\alpha_0} \, u \,, & {\rm{in}} \,\, \Omega \,, \\
\\
u(z,0)=u_0(z) \,, & {\rm{in}} \,\, \R \,,
\end{array}
\right.
\label{stfde}
\ee
with $(z,t) \in \Omega = \R \times (0,\infty)$,
$0 < \alpha < 2$, $0 < \beta < 1$,
and where $_tD_*^β$ is the Caputo time-fractional derivative of order $\beta$
defined through the Laplace transfor pair
$$
\int_0^{+\infty} \e^{-st} \{{_tD_*^\beta} u(z,t)\} \, dt=
s^\beta \, \widetilde{u}(z,s) -\sum_{j=0}^{m-1} s^{\beta-1-j} \, u^{(j)}(z,0^+) \,,
$$
with $m-1< \beta < m$, $m \in \N$,
and $_zD_\theta^\alpha$ is the Riesz--Feller space-fractional derivative
of order $\alpha$, and asymmetry parameter $\theta=\min\{\alpha,2-\alpha\}$,
defined through the Fourier transform pair
$$
\int_{-\infty}^{+\infty}
\e^{+ i \xi z}
\{{_zD_\theta^\alpha} u(z,t)\} \, dz
= -|\xi|^\alpha \, \e^{i (\rm{sign} \, \xi)\theta\pi/2} \, \widehat{u}(\xi,t) \,.
$$

\begin{corollary}
The density function $\p(z;t)$ in (\ref{stpdf})
solves the space-time fractional diffusion equation (\ref{stfde}),
whose special cases are
the time-fractional diffusion ($\alpha=2$),
the space-fractional diffusion ($\beta=1$),
the neutral fractional diffusion ($\alpha=\beta$)
and the classical diffusion ($\alpha=2$, $\beta = 1$).
\end{corollary}

The stochastic process $\Vcm^H$ defined in (\ref{ggBmMURA})
is built with the same constructive approach adopted
by Mura \cite{mura-phd-2008}
to build up the ggBm
\cite{mura-phd-2008,mura_etal-jpa-2008,mura_etal-itsf-2009},
i.e., a Gaussian process times a non-negative random variable,
and here we call such type processes:
randomly-scaled Gaussian processes.
The ggBm proposed by Mura is recovered from (\ref{ggBmMURA})
in the case $B_t^H$ is the fractional Brownian motion
and $\Lambda \sim M_\beta(\lambda)$ where $M_\beta(\lambda)$, $0 < \beta < 1$,
is the M-Wright/Mainardi function \cite{mainardi_etal-ijde-2010,pagnini-fcaa-2013},
i.e., $M_\beta(\lambda)=K^{-1}_{1,\beta}(\lambda)$.
Then we have the following

\begin{corollary}
Let $R(t,t)=t^{2H}$
and $f(\lambda)= M_\beta(\lambda)$,
with $0 < H < 1$ and $0 < \beta < 1$,
then the one-point one-time density function of $\Vcm$ is
\cite{pagnini-fcaa-2012}
\begin{eqnarray}
\p(z;t)
&=& \frac{1}{\sqrt{4 \pi \lambda \, t^{2H}}}\int_0^{+\infty}
\exp\left\{-\frac{z^2}{4 \lambda \, t^{2H}}\right\}
\, M_\beta(\lambda) \, d\lambda
\nonumber \\
&=& \frac{1}{2 \, t^H} \, M_{\beta/2}\left(\frac{|z|}{t^H}\right) \,.
\label{fbmpdf}
\end{eqnarray}
\end{corollary}

The density function $\p(z;t)$ defined in (\ref{fbmpdf}) is the
kernel of the following Erd\'elyi--Kober fractional diffusion equation
\cite{pagnini-fcaa-2012}
\be
\left\{
\begin{array}{lr}
\displaystyle
\frac{\partial u}{\partial t} =
\frac{2H}{\beta} t^{2H-1} \, \mathcal{D}_{2H/\beta}^{\beta-1,1-\beta}
\frac{\partial^2 u}{\partial x^2} \,, & {\rm{in}} \,\, \Omega \,, \\
\\
u(z,0)=u_0(z) \,, & {\rm{in}} \,\, \R \,,
\end{array}
\right.
\label{ekfde}
\ee
with $(z,t) \in \Omega = \R \times (0,\infty)$,
$0 < H < 1$, $0 < \beta < 1$,
where $\mathcal{D}_{\eta}^{\sigma,\mu}$ is
the Erd\'elyi--Kober fractional derivative defined as follows.
Let $n-1 < \mu \le n$, $n \in \N$, $\eta > 0$, $\gamma \in \R$,
the Erd\'elyi--Kober fractional derivative is defined as
$$
D^{\gamma,\mu}_\eta \, \varphi(t)=\prod_{j=1}^n
\left(\gamma+j+\frac{1}{\eta}t\frac{d}{dt}\right)(I^{\gamma+\mu,n-\mu}_\eta \, \varphi(t)) \,,
$$
where
$$
I^{\gamma,\mu}_{\eta} \, \varphi(t)
=\frac{\eta}{\Gamma(\mu)} t^{-\eta(\mu+\gamma)}
\int_0^t \tau^{\eta(\gamma+1)-1} (t^\eta-\tau^\eta)^{\mu-1} \varphi(\tau) \, d\tau \,.
$$

\begin{corollary}
The density function $\p(z;t)$ in (\ref{fbmpdf})
solves a fractional diffusion equation
in the Erd\'elyi--Kober sense (\ref{ekfde}),
whose special cases are
the time-fractional diffusion ($\beta = 2H$),
the Brownian non-Gaussian motion ($2H=1$),
the Gaussian non-Brownian motion ($\beta = 1$)
and the classical diffusion ($\beta = 2H = 1$).
\end{corollary}

\section{Proofs of Theorems \ref{Th:nonautonomous}, \ref{Th:ggBm1} and \ref{Th:ggBm2}}
\label{proofs}

\subsection{Proof of Theorem \ref{Th:nonautonomous}}
The stochastic dynamics of the process $\Vcm$ is governed by
$$
d\Vcm = \sum_\kappa \frac{\mkappa}{M} d\Ukappa \,.
\label{MOD10}
$$
By remembering the definitions in (\ref{def}),
if we plug (\ref{OU}) into (\ref{MOD1}),
we can derive the following Langevin-type dynamics:
\begin{eqnarray*}
d\Vcm
&=& \frac{1}{M} \sum_\kappa \left\{ - \gammakappa \Ukappa dt
+ \sqrt{2 \sigma_0} \, dW_t^\kappa\right\} \\
&=& - \frac{1}{M} \left\{\sum_\kappa \gammakappa \Ukappa\right\} dt
+ \frac{\sqrt{2 \sigma_0}}{M} \sum_\kappa dW_t^\kappa \,.
\end{eqnarray*}
We introduce now the scaled process $\Ykappa = \Ukappa/\sqrt{\Dkappa}$,
then we have
\begin{eqnarray}
\sum_\kappa \gammakappa \, \Ukappa
&=& \sum_\kappa \gammakappa \sqrt{\Dkappa} \, \Ykappa
= \sqrt{\sigma_0} \sum_\kappa \frac{\Ykappa}{\taukappa} \nonumber \\
& \stackrel{d}{=} & \sqrt{\sigma_0} \, \sqrt{\sum_\kappa
\E \left[\left(\frac{\Ykappa}{\taukappa}\right)^2\right]} \, W_1 \nonumber \\
&=& \sqrt{\sigma_0} \,
\frac{\sqrt{\sum_\kappa \E \left[\left(\frac{\Ykappa}{\taukappa}\right)^2\right]}}
{\sqrt{\sum_\kappa \E [(\Ykappa)^2]}}
\sqrt{\sum_\kappa \E [(\Ykappa)^2]} \, W_1 \nonumber \\
&=& \frac{1}{\taueff} \, \sqrt{\sum_\kappa \E [(\sqrt{\sigma_0} \Ykappa)^2]} \, W_1 \nonumber \\
&=& \frac{1}{\taueff} \, \sqrt{\sum_\kappa \E [(\mkappa \Ukappa)^2]} \, W_1 \nonumber \\
& \stackrel{d}{=} & \frac{1}{\taueff} \sum_\kappa \mkappa \Ukappa = M \, \frac{\Vcm}{\taueff} \,,
\label{drift}
\end{eqnarray}
where
\be
\frac{1}{\taueff}
=
\frac{\sqrt{\sum_\kappa \E \left[\left(\frac{\Ykappa}{\taukappa}\right)^2\right]}}
{\sqrt{\sum_\kappa \E [(\Ykappa)^2]}}
=
\left[
\frac{\displaystyle{\int_{-\infty}^{+\infty} \int_{\Omega_\tau}
\frac{y^2}{\tau^2} p^\kappa(y;t|\tau) q(\tau) \, d\tau dy}}
{\displaystyle{\int_{-\infty}^{+\infty} \int_{\Omega_\tau}
y^2 p^\kappa(y;t|\tau) q(\tau) \, d\tau dy}}
\right]^{1/2}
\,,
\label{gammaeff}
\ee
which is a function of time, i.e., $\tau_{\rm{eff}}=\tau_{\rm{eff}}(t)$,
with $p^\kappa(y;t|\tau)$ the Gaussian densities corresponding to the SDE (\ref{eqOU0}).

In the derivation of (\ref{drift}),
the first line follows from (\ref{def})
and the second from the application of the rule of the sum of Gaussian variables
reminding that the process $\Ykappa$ is Gaussian, see (\ref{eqOU0}).
The third line contains the multiplication and division by $\sqrt{\sum_\kappa \E [(\Ykappa)^2]}$
and in the fourth $\tau_{\rm{eff}}$ is introduced
and the parameter $\sigma_0$ is moved below the square root.
The fifth line follows from (\ref{def}) and, finally,
in the sixth line the rule of the sum of Gaussian variables is used again
in the first equality and (\ref{MOD1}) is used to have the last equality.

To conclude, by setting $\sum_\kappa dW_t^\kappa = dW_t^{\rm{eff}}$,
the stochastic process $\Vcm$ satisfies the following non-autonomous SDE
$$
d\Vcm= - \frac{\Vcm}{\taueff} dt + \frac{\sqrt{2 \sigma_0}}{M} \, dW_t^{\rm{eff}} \,.
$$

\subsection{Proof of Theorem \ref{Th:ggBm1}}
By considering the scaled process $\Ykappa=\Ukappa/\sqrt{\Dkappa}$,
the stochastic process $\Vcm$ results to be
$$
\Vcm= \frac{\sqrt{\sigma_0}}{M} \sum_{\kappa} \Ykappa \,.
$$
Let us write
\be
\overline{Y}_t^N = \frac{1}{N} \sum_{\kappa=1}^N \Ykappa.
\ee
Recall that $Y^\kappa_t \stackrel{d}{=} Y^1_t$, $\forall\, \kappa$. For $x=0$,
\be
\mathbb{E}_x[\overline{Y}^N_t] = x \,
\mathbb{E}_0\left[ \frac{e^{-t/\tau}}{\sqrt{\sigma}} \right] = 0 \,,
\ee
and
\be
Var_0[\overline{Y}^N_t] = \frac{1}{N} \,
\mathbb{E}_0 [\tau (1- e^{-2t/\tau})] = \frac{1}{N} \, \E_0[(Y^1_t)^2] \,.
\ee
From the Central Limit Theorem, as $N\to \infty$
$$
\frac{\overline{Y}^N_t - \mathbb{E}_0[\overline{Y}^N_t]}{\sqrt{Var_0[\overline{Y}^N_t]}} =  \sqrt{N} \frac{\overline{Y}_t^N}{\sqrt{\E_0[(Y^1_t)^2]}} = \frac{1}{\sqrt{\E_0[(Y^1_t)^2]}} \frac{1}{\sqrt{N}} \sum_{\kappa} \Ykappa \stackrel{d}{\to} W_1 \,,
$$
that is, as $N\to \infty$,
$$
\frac{1}{\sqrt{N}} \sum_{\kappa=1}^N \Ykappa \stackrel{d}{\to}
\sqrt{\E_0 [(Y^1_t)^2]} \, W_1 = B^H_t \,, \quad \forall \, t > 0 \,.
$$
Finally, by multiplying both side by $\sqrt{\sigma_0}/M$,
we obtain the following equivalence in distribution
\be
\forall\, t>0 \,, \quad
\frac{1}{\sqrt{N}} Z_t \stackrel{d}{\to} \frac{\sqrt{\sigma_0}}{M} B^H_t \,,
\quad N \to \infty \,.
\ee
Moreover,  by setting $\sqrt{\sigma_0}/M=\sqrt{\Lambda}$,
the stochastic process $N^{-\frac{1}{2}} \Vcm$
is statistically equivalent in the sense of
the Central Limit Theorem to the process $\sqrt{\Lambda} \, B_t^H$, i.e.,
$$
\frac{1}{\sqrt{N}} \Vcm \to \sqrt{\Lambda} \, B_t^H \,.
$$

\subsection{Proof of Theorem \ref{Th:ggBm2}}
The proof of Theorem \ref{Th:ggBm2}
follows by considering the previous theorem.
Let us consider the Fourier transforms of both densities.
We have that
\begin{align*}
\mathbb{E}_0[e^{i \xi Z_t}] = & \int e^{i \xi z} \frac{d}{dz} \mathbb{P}_0(Z_t < z) \, dz\\
= & \int e^{i \xi z} \frac{d}{dz} \mathbb{P}_0(\sqrt{\Lambda} B^H_t < \frac{z}{\sqrt{N}}) \, dz\\
= & \int e^{i \xi \sqrt{N} z} \frac{d}{dz} \mathbb{P}_0(\sqrt{\Lambda} B^H_t < z) \, dz\\
= & \mathbb{E}_0[e^{i \xi \sqrt{N\Lambda} B^H_t}] \,.
\end{align*}

\section{Numerical simulations}
\label{simulations}
In this Section we show by numerical simulations the equivalence in distribution of
the three processes $\Vcm$ (\ref{MOD1}), $\Vcm^*$ (\ref{langevintype})
and $\Vcm^H$ (\ref{ggBmMURA}) as proved
in Theorems \ref{Th:nonautonomous} and \ref{Th:ggBm2}.
The results of the simulations are shown in Fig.~\ref{figure1}
by plotting the density functions of the
three processes at different times.

Moreover, the insets show the corresponding variances.

The centre-of-mass like process $\Vcm$ is simulated
by using its definition~\eqref{MOD1}
through the process $X_t^\kappa$, which obeys Eq.~\eqref{OU}.
We remind, that in~\eqref{OU}
the parameters $\tau^\kappa$ and $\sigma^\kappa$, see Eq.~\eqref{def},
follow the corresponding distributions $q(\tau)$ and $g(\sigma)$.
In particular, in \textit{Panel a)}, these distributions are
\begin{equation}
\label{eq:par_a}
g(\sigma) =
-\frac{3}{8} \frac{\sigma^{-3/4}}{\Gamma(-2/3)} \exp\left[-\sigma^{-3/8}\right] \,,
\qquad q(\tau)=\exp(-\tau) \,,
\end{equation}
where we remind that $\Gamma(-2/3) < 0$,
and in \textit{Panel b)}
\begin{equation}
\label{eq:par_b}
g(\sigma)=\eta \frac{\sigma^{\nu - 1}}{\Gamma(\nu/\eta)} \exp(-\sigma^\eta) \,,
\qquad q(\tau)=\frac{\alpha}{\Gamma(1/\alpha)}\frac{1}{\tau}L^{-\alpha}_\alpha\left(\tau\right) \,,
\end{equation}
where $L^{-\alpha}_\alpha\left(\tau\right)$ is the L\'evy extremal density and $\nu=0.5$, $\eta=1.3$, $\gamma_0=10^{-5}$, $\alpha=0.75$.

The process $\Vcm^H$ is simulated by
\be
Z_t^H=\sqrt{N\Lambda} \, B_t^H=\sqrt{N\Lambda}\sum_\kappa{Y_t^\kappa}=
\frac{\sqrt{N\sigma_0}}{M} \sum_\kappa{Y_t^\kappa} \,,
\ee
where $Y_t^\kappa$ obeys Eq.~\eqref{YOU}.

Finally, the process $Z_t^*$ is simulated by the SDE
\[
dZ_t^*= - \frac{Z_t^*}{\taueff(t)}
dt + \frac{\sqrt{2 N\sigma_0}}{M} \, dW_t \,,
\]
where, in analogy with previous cases,
we use the properties of the sum of Gaussian variables to set
$dW_t^{\mathrm{eff}}=\sum_\kappa dW_t^\kappa=\sqrt{N} dW_t$.
Since we already have the trajectories $Y_t^\kappa$ that
we simulated for the process $\Vcm^H$,
for numerical convenience, the computation of $\tau_\mathrm{eff}$
is performed according to definition in terms of series given in Eq.~\eqref{gammaeff}.
Moreover,
we set $\sigma_0 = 1/N$ because in the simulations we sum $N$ trajectories.

Simulations support the equivalence in distribution among the three processes,
at least for the chosen distributions of the parameters.

\begin{figure}[!h]
\includegraphics[width=0.48\linewidth]{pdfsVouc}
\includegraphics[width=0.48\linewidth]{pdfsVc}
\caption{(colour online) Equivalence in distribution of the processes $Z_t$ ($\blacksquare$), $Z_t^H$ ($\vartriangle$) and $Z_t^*$ (solid lines). Different colours represent different times.
Insets: Log-log plots of the
variances of the corresponding processes as functions of time.
The distributions of the parameters are given with Eq.~\eqref{eq:par_a} for Panel a) and Eq.~\eqref{eq:par_b} for Panel b).
}
\label{figure1}
\end{figure}

\section{Conclusions and perspectives for applications}
\label{conclusions}
In this paper we have considered an ensemble of OU processes characterized
by a population of relaxation times and of noise amplitudes.
In the framework of the Brownian motion, the OU process can be viewed as a
stochastic dynamical equation and is called Langevin equation.
In this sense, we have considered
a heterogeneous ensemble of Brownian particles.

In particular, we have studied
the stochastic dynamics of the centre-of-mass like process of this ensemble
and proved that this process
is equivalent in distribution to a process generated by a non-autonomous SDE
with time-dependent drift term and uncorrelated white noise, i.e.,
no-memory effects emerge in opposition
to approaches based on the generalized Langevin equation
or on Langevin equations with coloured noises,
and it is equivalent in distribution also to a randomly-scaled Gaussian process,
i.e., the product of a Gaussian process times a non-negative random variable.

Moreover we showed that
{\it anomalous diffusion} emerges as a consequence of the heterogeneity.
In fact, we found that
the population of the relaxation times contributes
to the emergence of an anomalous scaling in time, as shown in (\ref{correlazione}),
and the population of the noise amplitudes contributes
to have a non-Gaussian density function (\ref{ggBmpdfn}).

The present study is a further step towards the interpretation
of anomalous diffusion as a consequence of a complex environment
\cite{pagnini-pa-2014,molina_etal-pre-2016,pagnini_etal-fcaa-2016}
that is here represented by the heterogeneity of the ensemble of particles.
The improvement provided with this paper lays in the fact
by using the OU process we establish a relation
between the anomalous diffusion and the Brownian motion modelled by the Langevin equation,
which is a dynamical equation, so put the emergence of anomalous diffusion
on physical basis.
Then, future developments will concern the study of the motion
and the diffusion properties
of a test-particle immersed in a complex environment
embodied by such heterogeneous ensemble.
This has a direct application for modelling anomalous diffusion
\cite{hofling_etal-rpp-2013}
and collective motion \cite{mayor_etal-nr-2016} in biological systems.
In fact, anomalous diffusion in biological systems is largely observed in experiments

and it is due to macromolecular crowding in the interior of cells
and in cellular membranes, because of their densely packed and heterogeneous structures
\cite{hofling_etal-rpp-2013,regner_etal-bj-2013}.
The proposed modelling approach can be applied for modelling
diffusion of meso-scopic test-particle in an environment also composed
of meso-scopic particles embodying the heterogeneous and
crowded environment where the test-particle moves.
The study of diffusion of a test-particle in a heterogeneous and crowed
environment motivates future developments of the present research.

\section*{Acknowledgments}
This research is supported by the Basque Government through the
BERC 2014-2017 program and the BERC 2018-2021 program,
and by the Spanish Ministry of Economy and Competitiveness MINECO
through BCAM Severo Ochoa excellence accreditation SEV-2013-0323
and through project MTM2016-76016-R "MIP".
VS acknowledges BCAM, Bilbao, for the financial support to her internship research period,
and SV acknowledges the University of Bologna for the financial support through
the "Marco Polo Programme" for funding her PhD research period abroad spent at BCAM, Bilbao.

\section*{References}

config: arxiv