%Anomalous fluctuations of time-averaged observables Distributional Behaviors of Time-averaged Observables in Langevin Equation with Fluctuating Diffusivity: Normal Diffusion but Anomalous Fluctuations
Takuma Akimoto
Eiji Yamamoto
Tomoshige Miyaguchi
We consider Langevin equation with dichotomously fluctuating diffusivity, where the diffusion coefficient changes dichotomously in time, in order to study fluctuations of time-averaged observables in temporary heterogeneous diffusion process. %Fluctuations of time-averaged observables such as diffusion coefficient and the ratio of occupation time are inevitable %in experiments at the single molecule levels. In some stochastic processes, %such fluctuations become intrinsic even when the measurement time goes to infinity. We find that occupation time statistics is a powerful tool for calculating the time-averaged mean square displacement in the model. We show that the time-averaged diffusion coefficients are intrinsically random when the mean sojourn time for one of the states diverges. %i.e., intrinsic non-equilibrium process. %(there are three different cases). %Thus, we find anomalous fluctuations in temporally heterogeneous diffusion processes. Our model provides anomalous fluctuations of time-averaged diffusivity, which have relevance to large fluctuations of the diffusion coefficient in single-particle-tracking experiments.
distributional_LEFD_arxiv.tex
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\begin {document}
%section {title}
%\preprint{APS/123-QED}

\title{%Anomalous fluctuations of time-averaged observables 
Distributional Behaviors of Time-averaged Observables 
in Langevin Equation with Fluctuating  Diffusivity: Normal Diffusion but Anomalous Fluctuations}% Force line breaks with \\
%\thanks{A footnote to the article title}%

\author{Takuma Akimoto}
\email{akimoto@keio.jp}
\affiliation{%
  Department of Mechanical Engineering, Keio University, Yokohama, 223-8522, Japan
}%

\author{Eiji Yamamoto}
\affiliation{%
  Department of Mechanical Engineering, Keio University, Yokohama, 223-8522, Japan
}%

\if0
\author{Tomoshige Miyaguchi}
%\email{tomo@a-phys.eng.osaka-cu.ac.jp}
\affiliation{%
Department of Mathematics Education, 
Naruto University of Education, Tokushima 772-8502, Japan}
\fi

%\collaboration{MUSO Collaboration}%\noaffiliation

\date{\today}% It is always \today, today,
%  but any date may be explicitly specified

\begin{abstract}
We consider Langevin equation with dichotomously fluctuating diffusivity, where the diffusion coefficient changes dichotomously in time, in order to study fluctuations of time-averaged observables in temporary heterogeneous diffusion 
process.  
%Fluctuations of time-averaged observables such as diffusion coefficient and the ratio of occupation time are inevitable
 %in experiments at the single molecule levels. In some stochastic processes,
%such fluctuations become intrinsic even when the measurement time goes to infinity. 
We find that occupation time statistics is a powerful tool for calculating the time-averaged mean square displacement 
in the model. We show that  the time-averaged diffusion coefficients are intrinsically random when the mean sojourn time for 
one of the states diverges. %i.e., intrinsic non-equilibrium process. %(there are three different cases). 
%Thus, we find  anomalous fluctuations in temporally heterogeneous diffusion processes.  
Our model provides anomalous fluctuations of time-averaged diffusivity, which have relevance to large fluctuations 
of the diffusion coefficient in single-particle-tracking experiments.
\end{abstract}

%\pacs{05.45.Ac, 05.40.Fb, 87.15.Vv}% PACS, the Physics and Astronomy
% Classification Scheme.
%\keywords{Suggested keywords}%Use showkeys class option if keyword
%display desired
\maketitle

%\tableofcontents

\section {Introduction}
Law of large numbers plays an important role in statistical physics. 
In stationary stochastic processes $\bm{X}_{t}$, law of large numbers or the
central limit theorem tells us that time-averaged observables such as
diffusivity and the ratio of occupation time converge to a constant when the
measurement time goes to infinity:
\begin{equation}
\int_0^t O( \bm{X}_{t'})dt'/t \to \langle O(\bm{X}) \rangle ~{\rm as}~ t\to \infty, 
\end{equation}
where the observable $O(\cdot)$ is a function of the stochastic process
$\bm{X}_{t}$. %and the ensemble average $\langle O(\bm{X}) \rangle$ must be finite. 
In experiments, time-averaged observables are not constant because of
finite measurement times. However, in some stochastic processes describing 
non-equilibrium phenomena, 
time-averaged observables are intrinsically random because of the breakdown of
law of large numbers or the central limit theorem \cite{Darling1957, Lamperti1958}.  In other words, they do not converge to a constant even when
the measurement time goes to infinity and the fluctuations never disappear.  
Such anomalous behavior has been studied
by infinite ergodic theory in dynamical systems \cite{Aaronson1997}.  Infinite
ergodic theory states that time-averaged observables converge in distribution,
and the distribution function depends on the invariant measure as well as a
class of the observation function \cite{Aaronson1981, Thaler1998, Akimoto2008,
  Akimoto2015}.

Continuous-time random walk (CTRW) is a model of anomalous diffusion, where the
mean square displacement (MSD) increases sub-linearly with time, and is extensively 
studied in disorder materials \cite{Scher1975} as well as biophysics \cite{He2008, Weigel2011}. 
In CTRW, a
random walker waits for the next jump and the waiting time is a random
variable whose probability density function (PDF) $\rho(\tau)$ follows a
power-law distribution:
\begin{equation}
\rho(\tau) \sim \frac{c_0}{|\Gamma(-\alpha)|}  \tau^{-1-\alpha}\quad (\tau \to \infty),
\label{power-law-pdf}
\end{equation}
where $c_0$ is a scale factor. When $\alpha \leq 1$, the mean waiting time diverges, thereby
causing a breakdown of law of large numbers and the central limit
theorem. In this case, it was shown that the time-averaged MSD (TMSD) for a fixed lag time $\Delta \ll t$, defined as
\begin{equation}
 \label{tamsd_definition}
  \overline{\delta^{2}(\Delta;t)} \equiv
  \frac{1}{t - \Delta} \int_{0}^{t - \Delta} dt' \,
  [\bm{r}(t' + \Delta) - \bm{r}(t')]^{2} ,
\end{equation}
does not converge to a constant but converges in distribution 
 as $t\to\infty$ \cite{Lubelski2008, He2008, Akimoto2010}.  Moreover, the
PDF of the normalized TMSD, i.e., $\overline{\delta^{2}(\Delta;t)}/ \langle
\overline{\delta^{2}(\Delta;t)} \rangle$, follows a universal distribution
called the Mittag-Leffler distribution, which is one of distributional limit
theorems in infinite ergodic theory \cite{Akimoto2010}.  This distributional
property for a time-averaged observable is called distributional ergodicity in
stochastic processes \cite{Miyaguchi2011a, Akimoto2013a}.

Other distributional behaviors have been found in other diffusion processes such
as a quenched trap model \cite{Miyaguchi2011, Miyaguchi2015} and
stored-energy-driven Levy flight (SEDLF) \cite{Akimoto2013a, Akimoto2014}, where the PDF
of the normalized TMSDs (time-averaged diffusion coefficients) follows other
distributions depending on the power-law exponent in the waiting time distribution,
the spatial dimension as well as parameters controlling jumps of a random
walker.  It is important to clarify whether fluctuations of
time-averaged observables are intrinsic or not, because diffusion coefficients
obtained by single-particle-tracking experiments in living cells exhibit large
fluctuations \cite{Golding2006, Jeon2011, Weigel2011, Tabei2013, Manzo2015}. 
Such large
fluctuations will have relevance to distributional behaviors in stochastic
models of anomalous diffusion.

\section {Langevin equation with dichotomously fluctuating Diffusivity}
%\subsection {Definition of the Model}

To investigate ergodic properties in heterogeneous diffusion processes, we
consider the following Langevin equation with fluctuating diffusivity (LEFD),
\begin{equation}
  \label{e.lefd}
 \frac{d\bm{r}(t)}{dt} = \sqrt{2 D(t)} \bm{w}(t),
\end{equation}
where $\bm{w}(t)$ is the $n$-dimensional white Gaussian noise with
%\begin{equation}
  $\left\langle \bm{w}(t) \right\rangle = 0$, and %\qquad
  $\left\langle w_i(t)w_j(t') \right\rangle = \delta_{ij}\delta(t-t')$.
%\end{equation}
On the other hand, the diffusion coefficient $D(t)$ can be a non-Markovian
stochastic process. We assume that $D(t)$ and $\bm{w}(t)$ are
statistically independent. Because the diffusion coefficient is determined by 
shape of the particle or surrounding environment, the LEFD can describe the dynamics of a particle with inner degree 
of freedom. In fact, 
this model can be utilized in the equation of motion
for the center-of-mass of entangled polymer in reptation model
\cite{Doi-Edwards-book} and is related to dynamic heterogeneity in supercooled
liquids \cite{Yamamoto-Onuki-1998, Yamamoto-Onuki-1998a, Richert-2002, Helfferich2014}.
Moreover, because the stochastic process $D(t)$ is generic, this system includes 
temporally heterogeneous diffusion models induced by spacial heterogeneity 
such as the ones studied in \cite{Massignan-Manzo-TorrenoPina-GarciaParajo-Lewenstein-Lapeyre-2014,Chubynsky2014,Akimoto2015b}

\begin{figure}
\includegraphics[width=.8\linewidth, angle=0]{traj_d.eps}
\caption{Trajectory of the Langevin equation with dichotomously fluctuating diffusivity. 
The lower inset represents the underlying diffusion coefficient. %of the Langevin equation.
 }
\label{traj_D}
\end{figure}

%\subsection {General Formula for the RSD}

In our previous study \cite{Uneyama2015}, we have obtained the relative standard deviation (RSD) of
the TMSD as a function of measurement time $t$
in LEFD when the stochastic process $D(t)$ is in equilibrium, where the RSD is
defined by
\begin{equation}
  \label{rsd_square_def}
  \Sigma(t;\Delta) \equiv \frac{\sqrt{\langle [\overline{\delta^{2}(\Delta;t)} -
  \langle \overline{\delta^{2}(\Delta;t)} \rangle]^{2} \rangle}}{\langle \overline{\delta^{2}(\Delta;t)} \rangle} .
\end{equation}
% If we assume $\Delta \ll t$, we obtain the RSD approximately ($t'>t''$):
%\begin{equation}
%\Sigma^2(t;\Delta) = \frac{\langle \overline{\delta^{2}(\Delta;t)}^2 \rangle}
 %{\langle \overline{\delta^{2}(\Delta;t)} \rangle^2} -1
 %\sim \frac{8n^2}{t^2 \langle \overline{\delta^{2}(\Delta;t)} \rangle^2} \int_0^t dt' \int_0^{t'} dt'' \int_{t'}^{t'+\Delta} ds \int_{t''}^{t''+\Delta} du
 %\langle D(s) D(u) \rangle -1
%\end{equation}
In equilibrium processes, the RSD becomes 
\begin{equation}
 \label{rsd_square_final}
  \Sigma^{2}(t;\Delta)
  \approx \frac{2}{t^{2}} \int_{0}^{t} ds 
  (t - s) \psi_1(s),
\end{equation}
where $\psi_1(t)$ is the normalized correlation function of diffusion
coefficients, i.e., $\psi_1(t) \equiv (\langle D(t) D(0)
\rangle - \langle D \rangle^2)/\langle D
\rangle^2$. Therefore, information on the underlying diffusion coefficient 
$D(t)$ can be extracted by the RSD analysis \cite{Uneyama2012,Uneyama2015}.
Here, we investigate ergodic properties of LEFD especially in non-equilibrium cases. In
particular, we consider two-state models for the stochastic process $D(t)$. When
the mean sojourn time of a state in $D(t)$ diverges, the stochastic process
becomes non-stationary, which implies that the system is intrinsically in
non-equilibrium. 
We show normal diffusion yet anomalous fluctuations of TMSD. 

%\section{correlation function of $D(t)$ and RSD}
%\subsection {Two-State System}
%{\it Langevin equation with dichotomously fluctuating diffusivity}.---
Here, we consider dichotomous processes for diffusivity $D(t)$ (see Fig.~1), i.e., %there are two states, $+$ state
%and $-$ state, and the diffusion coefficient is given by 
$D(t)=D_+$ if the state is $+$ and $D(t)=D_-$ otherwise ($-$ state). %We do not assume $D_- < D_+$.
Sojourn times for $+$ and $-$ states are random variables following different
probability density functions (PDFs), $\rho_+(\tau)$ and $\rho_-(\tau)$ for $+$ and
$-$ states, respectively. We assume that the one of the PDFs $\rho_+(\tau)$ follows either a narrow distribution
where all moments are finite or a broad distribution of power-law form [Eq.~(\ref{power-law-pdf})],
%$\rho_\pm (\tau) \sim c_0 \tau^{-1-\alpha_\pm}/|\Gamma(-\alpha_\pm)|$ for $\tau\to\infty$.  
and that the other PDF follows a power-law distribution, whose Laplace
transform is given by 
  $\hat{\rho}_-(s) = 1 - a_- s^{\alpha_-}   + o(s)$ 
  ($\alpha_-<1$).
  In particular, we consider three cases for $\rho_+(x)$:
%\begin{itemize}
  %\setlength{\leftskip}{4.5mm}
  %\item [(case 1)] $1<\alpha_- <2$: \quad
  %$\hat{\rho}_-(s) = 1 - \mu_- s + a_- s^{\alpha_-} + o(s^{\alpha_-})$
  %	\begin{itemize}
   % 	\setlength{\leftskip}{4.5mm}
   % 	%\item [(1a)] $\alpha_+ =2$: \hspace*{1.2cm}
    	%$\hat{\rho}_+(s) = 1 - \mu_+ s + a_+ s^{2} + o(s^{\alpha_+})$\\[-.2cm]
    %	\item [(1a)] $\alpha_- \leq \alpha_+ \leq 2$: \quad
    %	$\hat{\rho}_+(s) = 1 - \mu_+ s + a_+ s^{\alpha_+} + o(s^{\alpha_+})$\\[-.2cm]
    	%\item [(1b)] $\alpha_- =\alpha_+ $: \hspace*{.85cm}
    	%$\hat{\rho}_+(s) = 1 - \mu_+ s + a_+ s^{\alpha_+} + o(s^{\alpha_+})$
  %	\end{itemize}
  %\item [(case 2)] $\alpha_-<1$:\quad
  %$\hat{\rho}_-(s) = 1 - a_- s^{\alpha_-}  {\color{red}-A(\alpha_-) s} + o(s)$
  	\begin{itemize}
    	\setlength{\leftskip}{5.5mm}
   	% \item [(2a)] $\alpha_+ =2$: \hspace*{0.9cm}
    	%$\hat{\rho}_+(s) = 1 - \mu_+ s + a_+ s^{2} + o(s^{\alpha_+})$\\[-.2cm]
    	\item [(1)] narrow distribution: \hspace*{.5cm}
    	$\hat{\rho}_+(s) = \sum_{k=0}^\infty \frac{m_k}{k!} s^k$,\\[-.2cm]
    	\item [(2)] $\alpha_-<\alpha_+ <1$: \hspace*{0.25cm}
    	$\hat{\rho}_+(s) = 1 - a_+ s^{\alpha_+} + o(s^{\alpha_+})$,\\[-.2cm]
    	\item [(3)] $\alpha_- = \alpha_+$: \hspace*{0.9cm}
    	$\hat{\rho}_+(s) = 1 - a_+ s^{\alpha_+} + o(s^{\alpha_+})$,
  	\end{itemize}
%\end{itemize}
where $m_k$ is the $k$th moment of  sojourn times of the state $+$. 
In what follows, we set $\alpha_-=\alpha$. This kind of power-law behavior is observed in supercooled 
liquids \cite{Helfferich2014}. 

\if0
\subsection {Initial Ensembles}

In two-state process, we can consider equilibrium and non-equilibrium processes. 
The PDF of initial sojourn time denoted as $\tilde{\rho}_{\pm}^{0}(\tau)$, i.e., 
the probability that the initial sojourn time is in $[\tau,
\tau+d\tau]$ under the condition that the initial state is $+$ is given by
$\tilde{\rho}_{+}^{0}(\tau) d\tau$. For equilibrium initial ensemble, we put
$\tilde{\rho}_{\pm}^{0}(\tau) = \rho_{\pm}^{\mathrm{eq}}(\tau)$, whereas we use
$\tilde{\rho}_{\pm}^{0}(\tau) = \rho_{\pm}(\tau)$ for a typical non-equilibrium
initial ensemble. Here, the $\rho_{\pm}^{\mathrm{eq}}(\tau)$ is given by 
\begin{equation}
  \label{e.init-deinsity-equilibrium}
  \rho_{\pm}^{\mathrm{eq}}(\tau) = \frac {1 - \hat{\rho}_{\pm}(s)}{\mu_{\pm}s}.
\end{equation}
Thus, this PDF exists only if $\mu_{\pm}$ is finite.

\fi

%\section{distributional limit theorems}
%\subsection{TMSD and occupation time}

\section{ Representation of time-averaged mean square displacement}
%Since a particle undergoes Brownian motion in each state, the TMSD 
For $\Delta \ll t$, TMSD is represented by 
%$\overline{\delta^{2}(\Delta;t)} \approx \sum_{i=0}^{N_t-1} \{\int_{t_i}^{t_{i+1}} \delta \bm{r}^2(\Delta;t') dt' +\int_{t_{N_t}}^{t}\delta \bm{r}^2(\Delta;t') dt'\}/t$,
\begin{align}
  \label{e.0}
  \overline{\delta^{2}(\Delta;t)} %= 
  \underset{\Delta \ll t}{\approx}\!
  \frac{\displaystyle
    \sum_{i=0}^{N_t-1}\!\!
    \int_{t_i}^{t_{i+1}} \!\!\!\!\!\!\delta \bm{r}^2(\Delta;t') dt' +\!
    \int_{t_{N_t}}^{t}\!\!\!\!\! \delta \bm{r}^2(\Delta;t') dt'}{t },
\end{align}
where $\delta \bm{r}(\Delta;t')\equiv \bm{r}(t'+\Delta) - \bm{r}(t')$, $t_i$ is
the $i$th transition time from one state to the other state with $t_0=0$, $N_t$
is the number of transitions up to time $t$. Since a particle undergoes Brownian motion in each state, 
\begin{align}
  \sum_{i=0}^{N_t-1}&\int_{t_i}^{t_{i+1}} \delta \bm{r}^2(\Delta;t') dt' + \int_{t_{N_t}}^{t} \delta \bm{r}^2(\Delta;t') dt'\notag\\
  %\label{e.1}
  %=\,\,\,&
  %\sum_{i=0}^{N_t-1}\!
  %\int_{t_i}^{t_{i+1}-\Delta}\!\!\!\!\!\!\! \delta \bm{r}^2(\Delta;t') dt' + \!
  %\int_{t_{i+1}-\Delta}^{t_{i+1}}\!\!\!\!\!\!\!\delta \bm{r}^2(\Delta;t') dt'
  %\\[0.1cm]
  %\label{e.2}
  %\underset{\Delta \ll t}{\approx}& \left[
  %\sum_{\begin{subarray}{c}i=0\\i: \text{even}\end{subarray}}^{N_t-1}\!\!
  %\int_{t_i}^{t_{i+1}-\Delta}\!\!\!\!\!\!\!\!\!\!\! \delta \bm{r}_+^2(\Delta;t') dt'
  %+\!
  %\sum_{\begin{subarray}{c}i=0\\i: \text{odd}\end{subarray}}^{N_t-1}\!\!
  %\int_{t_i}^{t_{i+1}-\Delta}\!\!\!\!\!\!\!\!\!\!\! \delta \bm{r}_-^2(\Delta;t') dt'\right]/t
  %\\[0.1cm]
  \label{e.3}
  \underset{\Delta\ll \tau_0}{\approx}&
  \int_{0}^{T_+(t_{N_t})} \!\!\!\delta \bm{r}_+^2 (\Delta; t') dt'
  +\!
  \int_{0}^{T_-(t_{N_t})} \!\!\!\delta \bm{r}_-^2 (\Delta; t') dt',
\end{align}
where $\delta \bm{r}_{\pm}(\Delta; t') \equiv \int_{t'}^{t'+\Delta} dt''
\sqrt{2D_{\pm}} \bm{w}(t'')$, $T_{\pm}(t)$ is the occupation time of the
state $\pm$ up to time $t$ [Thus, $T_+(t) + T_-(t) = t$], and $\tau_0$ is a characteristic time 
for the transitions of $D(t)$. The condition of $\Delta \ll \tau_0$ validates the approximation 
 that the state in $[t_i, t_i + \Delta]$ does not change. 
%The second term in Eq.~(\ref{e.1}) is asymptotically unimportant, because
%$\delta\bm{r}^2(\Delta;t') \sim \Delta$.  In addition, we assume in
%Eq.~(\ref{e.2}) that the initial state is $+$, but the final results,
%Eq.~(\ref{e.3}), is valid for the case started form the $-$ state.
%
%Combining Eqs.~(\ref{e.0}) and (\ref{e.3}) yields
We have
\begin{align}
  \overline{\delta^{2}(\Delta;t)}
  &\approx
  %\frac{\displaystyle
  %\int_{0}^{T_+(t)} \delta \bm{r}_+^2 (\Delta; t') dt'
  %+
  %\int_{0}^{T_-(t)} \delta \bm{r}_-^2 (\Delta; t') dt'
  %}{t}
  %\notag\\[0.1cm]
  \label{e.tmsd.approx}
  %&=
  2n 
  \frac{
    \overline{D_+(t)}\, T_+(t) +
    \overline{D_-(t)}\, T_-(t)}{t} \Delta,
\end{align}
where we define a time-averaged diffusion coefficient of each state as
$
\overline{D_\pm(t)} \equiv 
\int_{0}^{T_\pm(t) } \delta \bm{r}^2_\pm(\Delta;t') dt'/2nT_\pm(t).  
$
Therefore, TMSDs always show normal diffusion and the time-averaged diffusion
coefficient defined as $\overline{D(t)}\equiv \overline{\delta^{2}(\Delta;t)}/(2n\Delta)$ is given by
\begin{equation}
  \overline{D(t)} \approx
  \overline{D_-(t)}  +
  \left[\overline{D_+(t)} - \overline{D_-(t)}\right]\frac{T_+(t)}{t}.
  \label{D_t_TAMSD}
\end{equation}
Using Eqs.~(\ref{e.tmsd.approx}) and (\ref{D_t_TAMSD}), we have the RSD
[Eq.~(\ref{rsd_square_def})]:
$%\begin{equation}
  \Sigma^2(t; \Delta)
  \approx
  \langle [\overline{D(t)} -
      \langle \overline{D(t)} \rangle ]^{2} \rangle /
  \langle \overline{D(t)} \rangle^2 .
$%\end{equation}

In Eq.~(\ref{D_t_TAMSD}), the time-averaged diffusion coefficient
$\overline{D(t)}$ is controlled by three stochastic variables,
$\overline{D_{\pm}(t)}$, and $T_+(t)$. As shown below, the RSD of $T_+(t)$
decays slowly $t^{-\beta}$ with $\beta < 1/2$ in the limit $t \to \infty$, while 
those of $\overline{D_{\pm}(t)}$ decay as $t^{-0.5}$. Therefore, in the long
time limit, the fluctuations of $T_+(t)$ is dominant over those of
$\overline{D_{\pm}(t)}$, and thus we can approximate as $\overline{D_\pm(t)}
\simeq D_\pm$. Under this approximation, we have an asymptotic behavior of the RSD:
\begin{equation}
\Sigma^{2} (t; \Delta) \sim %\frac{(D_+-D_-)^2 \left( \left\langle T_+(t)^2 \right \rangle  - \left\langle T_+(t) \right \rangle ^2\right)}{\langle D_t \rangle^2 t^2}.
\frac{
    \left\langle T_+^2(t) \right\rangle  -
    \left\langle T_+(t)   \right\rangle^2
  }
  {\frac {D_-}{(D_+ - D_-)^2}t^2 +  \langle T_+(t) \rangle^2}.
\label{rsd_occupation_time}
\end{equation}
This is another representation of the RSD by the occupation time in LEFD with two-state diffusivity. We confirmed that 
 the asymptotic behavior is the same as the RSD (\ref{rsd_square_final}) in equilibrium processes. 
Since we neglect fluctuations of $\overline{D_\pm (t)}$, this expression for the RSD is valid only when the right-hand side of 
Eq.~(\ref{rsd_occupation_time}) decays slower than $t^{-0.5}$. Otherwise, the asymptotic behavior of the RSD is the same 
as that in Brownian motion (see Appendix.~A): 
\begin{equation}
\Sigma^{2} (t; \Delta) \sim \frac{\langle (\overline{D_-(t)}-D_-)^2\rangle}{D_-^2} \sim \frac{4\Delta }{3nt}.
\label{rsd_occupation_time2}
\end{equation}

%\subsection{Limit distributions of occupation times}
\section{ Occupation time statics} 
Here, we consider the occupation time statics for three cases. 
%provide distributional limit theorems on occupation times for three cases. 
We define the joint probability
distribution, $g^{\pm}_n(y;t)$, of the occupation time $T_+(t)=y$ and the number
of renewal $N_t = n$ up to time $t$ under the condition that the initial state is
$\pm$, given by
\begin{equation}
  g^{\pm}_n(y;t) =
  \left\langle
  \delta\left(y-T_+(t)\right) I\left(t_n\leq t<t_{n+1}\right)
  \right\rangle_\pm.
\end{equation}
The Laplace transform of $g^{\pm}_n(y;t)$ with respect to $y$ and $t$ is given
by
\begin{equation}
  \hat{g}^{\pm}_n(u;s)
  %&=& \int_0^\infty dt e^{-st} \int_0^\infty dy e^{-uy} \langle \delta(y-T_+(t)) I(t_n\leq t<t_{n+1}\rangle\\
  = \left\langle \int_{t_n}^{t_{n+1}} e^{-st} e^{-u T_+(t)} dt\right\rangle_{\pm},
\end{equation}
where $n=1,2,\dots$. For example, if the initial state is $+$ and $n=2k$ or
$2k+1$, it can be represented as
\begin{align}
  \hat{g}^{\pm}_{2k}(u;s)
  &=
  \left\langle
  \int_{t_{2k}}^{t_{2k+1}} \!\!\! e^{-st} e^{-u [\tau_1 + \tau_3 + \dots + \tau_{2k-1}+ (t-t_{2k})]}dt
  \right\rangle,
  \\[-0.1cm]
  \hat{g}^{\pm}_{2k+1}(u;s)
  &=
  \left\langle
  \int_{t_{2k+1}}^{t_{2k+2}}\!\!\! e^{-st} e^{-u (\tau_1 + \tau_3 + \dots + \tau_{2k+1})}dt 
  \right\rangle,
\end{align}
where $\tau_k$ is the $k$th sojourn time, and thus $t_k = \sum_{i=1}^{k} \tau_i$.
Integrating the above equations, and using interindependence of $\tau_k$ and
$\tau_l$ ($k\neq l$), we have %\cite{Cox}
\begin{align}
  \hat{g}^{+}_{2k+1}(u;s)
  &=
  \frac{1- \hat{\rho}_-(s)}{s} \hat{\rho}_-^k(s) \hat{\rho}_+^{k+1}(s+u),
  \\[0.1cm]
  \hat{g}^{+}_{2k}(u;s) 
  &= \frac{1- \hat{\rho}_+(s+u)}{s+u} \hat{\rho}_-^k(s) \hat{\rho}_+^{k}(s+u).
\end{align}
The cases in which the system starts from $-$ state can be calculated in 
the similar way. 
%Similarly, for the cases in which the system starts from $-$ state, we have
%\begin{align}
%  \hat{g}^{-}_{2k}(u;s)
%  &=
%  \frac{1- \hat{\rho}_-(s)}{s} \hat{\rho}_+^k(s+u) \hat{\rho}_-^{k}(s),
%  \\[0.1cm]
%  \hat{g}^{-}_{2k+1}(u;s) 
%  &=
%  \frac{1- \hat{\rho}_+(s+u)}{s+u} \hat{\rho}_+^k(s+u) \hat{\rho}_-^{k+1}(s).
%\end{align}
Then, the the PDF of $T_+(t)$ is obtained by summing up $g^{\pm}_n(y;t)$ in
terms of $n$: ${g}^\pm (y;t) = \sum_{n=0}^\infty g^\pm_n (y;t)$, and thus we
have
\begin{align}
  \hat{g}^+ (u;s)
  &=
  \frac{1-\hat{\rho}_-(s)}{s \hat{\rho}(s,u)} \hat{\rho}_+(s+u) +
  \frac{1-\hat{\rho}_+(s+u)}{(s+u)\hat{\rho}(s,u)},
  \\[0.1cm] 
  \hat{g}^-(u;s) 
  %&=& \sum_{n=0}^\infty \{\hat{g}^- (u;s, 2n) + \hat{g}^- (u;s, 2n+1)\}  \\
  &=
  \frac{1-\hat{\rho}_+(s+u)}{(s+u)\hat{\rho}(s,u)} \hat{\rho}_-(s) +
  \frac{1-\hat{\rho}_-(s)}{s\hat{\rho}(s,u)},  
\end{align}
where $\rho(s,u)\equiv 1- \hat{\rho}_+(s+u)\hat{\rho}_-(s)$.
In the small $s$ and $u$ limit, 
\begin{align}
  \hat{g}^{\pm}(u;s)  
  &\sim
  \frac{1-\hat{\rho}_-(s)}{s \hat{\rho}(s,u)}  +
  \frac{1-\hat{\rho}_+(s+u)}{(s+u)\hat{\rho}(s,u)}.
  %\notag\\[0.1cm]
  %&\sim
  %\frac {a_-s^{\alpha-1} + \frac {1-\hat{\rho}_+(s+u)}{s+u}}
  %{1 - \hat{\rho}_+(s+u) + a_-s^{\alpha}}.
  \label{Laplace_pdf_occupation}
\end{align}



\section {Distributional limit theorems}


%% In this case, the stochastic process $D(t)$ has the equilibrium state, because
%% the sojourn times for the two states have finite means.  This process is called
%% an equilibrium renewal process in renewal theory \cite{Cox}. In an equilibrium
%% process, the correlation function of $D(t)$ is defined by Because the sojourn
%% times for two states have finite means in this case,


%
% case (2a)
%
%\subsection {Case 1}

%Using Eq.~(\ref{Laplace_transform_MSD}), we have
%\begin{equation}
%\mathcal{L}
 % \left[\left\langle \left[ \bm{r}(t) - \bm{r}(0) \right]^2 \right\rangle_{\pm}\right](s)
 % = 2n \left[ \frac{D_-}{s^2} + \frac{\mu_+(D_+-D_-)}{a_-s^{1+\alpha}} - \frac{D_+\mu_+ \{A(\alpha) + \mu_+\}}{a_-^2s^{2\alpha}}\right],
%\end{equation}
%where we used $s \ll [\{A(\alpha) + \mu_+\}/a_-]^{1/(1-\alpha)}$. For $t \gg [a_-/\{A(\alpha) + \mu_+\}]^{1/(1-\alpha)}$,
%\begin{equation}
  %\left\langle \left[ \bm{r}(t) - \bm{r}(0) \right]^2 \right\rangle_{\pm}
  %= 2n \left[ D_- t + \frac{\mu_+(D_+-D_-)}{a_- \Gamma(1+\alpha)}t^\alpha - \frac{D_+\mu_+ \{A(\alpha) + \mu_+\}}{a_-^2 \Gamma(2\alpha)} t^{2\alpha-1}\right].
%\end{equation}
\subsection{ Case (1)} 
From Eq.~(\ref{Laplace_pdf_occupation}), the Laplace transform
of the PDF of $T_+(t)$ for the case (1) is given by
\begin{equation}
  \hat{g}^\pm(u;s) 
  \sim
  \frac{a_-s^{\alpha-1} + \mu_+}{a_-s^{\alpha} + \mu_+(s+u)}.
\end{equation}
Using the relation between the moments of $T_+(t)$ and $\hat{g}^\pm(u;s)$, we
have the asymptotic behavior of the $n$th moment of $T_+(t)$
\begin{align}
  \label{moments_case1}
  \left\langle T_+^n(t) \right\rangle_{\pm}
  %&=
  %\mathcal{L}^{-1}\!
  %\left[
  %(-1)^n \frac{\partial^n \hat{g}^{\pm} }{\partial u^n} (0;s)
  %\right](t)
  %\notag\\[0.1cm]
  &\sim
  \left(\frac{\mu }{a_-}\right)^{\!\!n}\!\!
  \frac{n! \,t^{n\alpha}}{\Gamma(1 + n\alpha)},
\end{align}
where $\mu=m_1$. % and $\mathcal{L}^{-1}$ is the inverse Laplace transform. 
%Because the TMSD can be represented by $T_+(t)/t$ through Eq.~(\ref{e.tmsd.approx}), 
It follows that the ETMSD shows normal diffusion:
\begin{equation}
\langle \overline{\delta^{2}(\Delta;t)}  \rangle %\sim 2n \left[D_- + (D_+ - D_-) \left\langle \frac{T_+(t)}{t} \right\rangle \right] \Delta
\sim 2n \left[D_- + \frac{\mu (D_+ - D_-)}{a_-\Gamma(1+\alpha)} \frac{1}{t^{1-\alpha}} \right] \Delta,
\end{equation}
where we used $\langle \overline{D_\pm(t)}\rangle \sim D_\pm$ and Eq.~(\ref{moments_case1}). 
Because TMSD converges to $2nD_- \Delta$ as $t\to \infty$, this process seems to be normal diffusion.

In Brownian motion, $\overline{D(t)}$ converges to a constant and the distribution follows Gaussian. Thus, deviation from Gaussian detects anomaly 
of the process. Since $\overline{D(t)}$ is given by Eq.~(\ref{D_t_TAMSD}) and $\overline{D(t)} \to D_-$,
 we consider the deviation, i.e., $\delta D_t \equiv \overline{D(t)} - D_-$.
%TMSD is represented by a random variable $\delta D_t$:
%\begin{equation}
 %\overline{\delta^{2}(\Delta;t)} = 2n(D_- + \delta D_t)\Delta.
%\end{equation}
%because $T_+(t)/t \to 0$ as $t\to \infty$. 
%which is consistent with a previous study. 
%To investigate deviations of the time-averaged diffusion coefficient from $D_-$, i.e., $\delta D_t \equiv \overline{D(t)} - D_-$, 
 %we consider the limit distribution of $\delta D_t/\langle \delta D_t\rangle$. 
 By  Eq.~(\ref{D_t_TAMSD}), we have 
 \begin{equation}
\frac{\delta D_t}{\langle \delta D_t\rangle} \cong \frac{(\overline{D_-(t)} - D_-)t}{(D_+ - D_-)\langle  T_+(t)\rangle} +  \frac{T_+(t)}{\langle  T_+(t)\rangle}.
\label{relation_delta_Dt_and_Tt}
 \end{equation} 
 Here, the first term in the right-hand side can be neglected if $\langle (\overline{D_-(t)} - D_-)^2t^2 \rangle 
 =o(\langle  T_+(t)^2\rangle - \langle T_+(t)\rangle^2$). Note that this condition is satisfied when $\alpha >0.5$ [see Eq.~(\ref{moments_case1})]. 
By Eq.~(\ref{moments_case1}), moments of the normalized occupation time defined by $T_\alpha(t) \equiv T_+(t) /\langle T_+(t) \rangle$ 
%has a non-trivial limit distribution because the $n$-th moment of the normalized occupation time converges to a non-zero constant:
becomes
\begin{equation}
\langle T_\alpha(t) ^n \rangle \sim \frac{n! \Gamma(1+\alpha)^n}{\Gamma(1 + n\alpha)}\quad (t\to\infty). 
\end{equation}
When the PDF of a random variable $M_\alpha$ follows the Mittag-Leffler distribution of order $\alpha$, 
the Laplace transform is given by
$%\begin{equation}
\langle e^{-z M_\alpha} \rangle = \sum_{k=0}^\infty \frac{\Gamma(1+\alpha)^k z^k}{\Gamma(1+k\alpha)}.
%\label{ML_moments}
$%\end{equation}
 Therefore, %the deviations of time-averaged diffusion coefficient, $\delta D_t/\langle \delta D_t\rangle$, are intrinsic, and 
 the distribution of  $\delta D_t/\langle \delta D_t\rangle$ is not Gaussian but 
 converges  to the Mittag-Leffler distribution when $\alpha>0.5$ (see Fig.~\ref{limit_dist_case2}a). For $\alpha<0.5$, the first term in 
 Eq.~(\ref{relation_delta_Dt_and_Tt}) becomes the leading term and the distribution of $\delta D_t/\langle \delta D_t\rangle$ becomes Gaussian 
 with the mean 0 and the variance $2 \Delta D_-^2 a_-^2\Gamma(1+\alpha)^2 t^{1-2\alpha}/\{n(D_+ - D_-)^2\mu^2\}$. 
 For $\alpha > 0.5$, using Eq.~(\ref{rsd_occupation_time}) yields
 \begin{equation}
 \Sigma^2 (t;\Delta) \sim \frac{\mu^2 (D_+ - D_-)^2 A(\alpha)}{a_-^2 D_-^2 \Gamma(1+\alpha)^2}
  t^{-(1-\alpha)},
 \end{equation}
 where $A(\alpha)=\frac{2\Gamma(\alpha+1)^2}{\Gamma(2\alpha+1)} -1$.
 For $D_-=0$, the result is exactly the 
same as that in CTRW \cite{He2008}:  
%In fact, using the moments in (\ref{ML_moments}), we have the RSD, which is the same as in CTRW,  
$
\Sigma(t;\Delta) \sim \sqrt{A(\alpha)}.
$

%
% case (2b)
%
%\subsection {Case 2}

%Using Eq.~(\ref{Laplace_transform_MSD}), we have
%\begin{equation}
%\mathcal{L}
%  \left[\left\langle \left[ \bm{r}(t) - \bm{r}(0) \right]^2 \right\rangle_{\pm}\right](s)
%  = 2n \left( \frac{D_-}{s^2} + \frac{a_+ (D_+ - D_-)}{a_- s^{2+\alpha -\alpha_+}} - \frac{a_+^2 D_+ }{a_-^2 s^{2+2(\alpha -\alpha_+)}}\right) ,
%\end{equation}
%where we used $s \ll (a_-/a_+)^{1/(\alpha_+-\alpha)}$. For $t \gg (a_+/a_-)^{1/(\alpha_+-\alpha)}$,
%\begin{equation}
%  \left\langle \left[ \bm{r}(t) - \bm{r}(0) \right]^2 \right\rangle_{\pm}
%  = 2n \left( D_- t + \frac{a_+ (D_+ - D_-)}{a_- \Gamma(2+\alpha -\alpha_+)} t^{1-(\alpha_+-\alpha)}
 % - \frac{a_+^2 D_+ }{a_-^2 \Gamma(2+2\alpha -2\alpha_+)} t^{1-2(\alpha_+-\alpha)}\right) .
%\end{equation}
%Moreover, 
\subsection{ Case (2)} 
%which is consistent with a previous study. 
In this case, Eq.~(\ref{Laplace_pdf_occupation}) yields the Laplace transform of the PDF of $T_+(t)$: 
\begin{equation}
\hat{g}^\pm(u;s) 
\sim \frac{a_+(s+u)^{\alpha_+-1}+a_-s^{\alpha-1}}{a_+(s+u)^{\alpha_+} +a_-s^{\alpha}}.
\end{equation}
The Laplace transform of the first moment $\langle T_+(t)\rangle$ is scaled as 
\begin{equation}
\langle \hat{T}_+(s)\rangle = \left. \frac{\partial \hat{g}^\pm(u;s)}{\partial u}\right|_{u=0} \sim 
- \frac{a_+ }{a_-} \frac{1}{s^{2 - \delta-\alpha}},  
\end{equation}
where $\delta \alpha = \alpha_+ - \alpha$. Thus, 
The asymptotic behavior of $\langle T_+(t)\rangle$ becomes 
\begin{equation}
\langle T_+(t)\rangle  \sim  \frac{a_+}{a_-\Gamma(2 -\delta \alpha)} t^{1-\delta \alpha}, 
\label{mean_occupation_2b}
\end{equation}
%where $\delta \alpha = \alpha_+ - \alpha$. %which is small order of $t$,  i.e., $T_+(t)/t \to 0$ as $t\to\infty$. 
Moreover, the second moment of $T_+(t)$ is scaled as 
\begin{equation}
\langle T_+(t)^2 \rangle \sim \frac{2a_+ (1-\alpha_+)}{a_- \Gamma(3- \delta \alpha)} t^{2- \delta \alpha}.
\end{equation}
It follows that the second moment of $T_+(t)/\langle T_+(t) \rangle$
diverges for $t\to \infty$. 
Using Eqs.~(\ref{D_t_TAMSD}) and (\ref{mean_occupation_2b}) yields the ETMSD:
\begin{equation}
\langle \overline{\delta^{2}(\Delta;t)}  \rangle 
\sim 2n \left[D_- + \frac{a_+ (D_+ - D_-)}{a_-\Gamma(2-\alpha_+ + \alpha)} \frac{1}{t^{\alpha_+ -\alpha}} \right] \Delta.
\end{equation}
As in the previous case, TMSD converges to $2nD_- \Delta$ as $t\to \infty$. By  Eq.~(\ref{rsd_occupation_time}), 
the RSD decays as 
\begin{equation}
\Sigma^2(t;\Delta) \sim \frac{2a_+(D_+ - D_-)^2 (1-\alpha_+)}{a_- D_-^2 \Gamma(3-\delta \alpha)} t^{- \delta \alpha}. 
\end{equation}


Although we do not have the limit distribution of $T_+(t)/ \langle T_+(t) \rangle$, the tail should be a heavy tail (power-law 
distribution) because the second moment of  $T_+(t)/ \langle T_+(t) \rangle$ diverges. By the relation between 
$\delta D_t$ and $T_+(t)$, i.e., Eq.~(\ref{relation_delta_Dt_and_Tt}), we find that  
the deviations of time-averaged diffusion coefficient, $\delta D_t/\langle \delta D_t\rangle$, are random and the 
distribution is a non-trivial distribution characterized by a power law (see Fig.~\ref{limit_dist_case2}b).
This is a similar situation for the PDF of time-averaged diffusion coefficients in some parameter region of SEDLF \cite{Akimoto2014}.

%
% case (2c)
%
%\subsection {Case 3}
%For $t\gg 1$, the EMSD is given by
%\begin{equation}
%\left\langle \left[ \bm{r}(t) - \bm{r}(0) \right]^2 \right\rangle_{\pm} \sim 
% \frac{2n(D_+a_+ + D_- a_-)}{a_++a_-} t,
%\end{equation}
%and the ETMSD is also given by 
%\begin{equation}
%\left\langle \left[ \bm{r}(t) - \bm{r}(0) \right]^2 \right\rangle_{\pm} 
% \frac{2n(D_+a_+ + D_- a_-)}{a_++a_-} \Delta.
%\end{equation}
%Therefore, the ETMSD is equivalent to the EMSD.

\begin{figure}
\includegraphics[width=1.\linewidth, angle=0]{case12.eps}
%\includegraphics[width=.8\linewidth, angle=0]{dist_case2c_asym.eps}
\caption{Distribution of deviations of the time-averaged diffusivity, $\delta D\equiv \delta D_t / \langle D_t \rangle$, 
in the cases (1) and (2), corresponding to (a) and (b), respectively ($D_-=1$, $D_+=10$, and $t=10^4$).
In Fig.~(a), the Mittag-Leffler distributions are drawn by solid lines. 
In Fig.~(b), power-law distribution with exponent $-3$ is drawn for reference.  
 Squares with colors are the results of numerical 
simulations. 
%(\ref{pdf_D_case2c}). 
In numerical simulations, we used the following power-law distribution 
for sojourn time distribution: $\rho_\pm (\tau) = c_\pm \tau^{-1-\alpha_\pm}$ for $\tau\geq \tau_0^\pm$ 
and $0$ for $\tau<\tau_0^\pm$, where $c_\pm$ is the normalization constant. In case (1), we used the 
exponential distribution for $\rho_+(\tau)$.
%We use $\tau_0^+=\tau_0^- = 10^{-3}$.} % for a), and $\tau_0^+=10^{-3}$ and $\tau_0^- = 10^{-2}$ for b)}.
 }
\label{limit_dist_case2}
\end{figure}

\subsection{ Case (3)}
Contrary to the previous two cases, TMSDs do not converge to a constant in the case (3), whereas 
TMSD shows normal diffusion [see Eq.~(\ref{e.tmsd.approx})].
Eq.~(\ref{Laplace_pdf_occupation}) yields the Laplace transform of the PDF of $T_+(t)$: 
\begin{equation}
\hat{g}^\pm(u;s)  
\sim \frac{a_+(s+u)^{\alpha-1}+a_-s^{\alpha-1}}{a_+(s+u)^{\alpha} +a_-s^{\alpha}}.
%=\frac{1}{s} \frac{a_+(1+u/s)^{\alpha-1}+a_-}{a_+(1+u/s)^{\alpha} +a_-}.
\label{Laplace_case3}
\end{equation}
By Appendix~B in \cite{God2001}, Eq.~(\ref{Laplace_case3}) implies that the limit distribution of $T^+(t)/t$ 
exists:  
\begin{equation}
\lim_{t\to\infty}  g_{T^+/t}(x) = g_{\alpha,\beta}(x), 
\end{equation}
and the distribution is given by 
\begin{equation}
g_{\alpha,\beta}(x) =
 \frac{(a \sin \pi \alpha/\pi) x^{\alpha-1} (1-x)^{\alpha-1}}{a^2 x^{2\alpha} +2a \cos \pi \alpha (1-x)^\alpha x^\alpha + (1-x)^{2\alpha}},
\label{T+_limit_dist}
\end{equation}
where $g_{T^+/t}(x)$ is the PDF of $T_+(t)/t$, $a=a_-/a_+$ and $\beta\equiv 1/(1+a)$. 
This is the Lamperti's generalized arcsine law \cite{Lamperti1958}, which is observed for time-averaged drift in superdiffusion \cite{Akimoto2012}. 
By Eq.~(\ref{D_t_TAMSD}),  the distribution of the time-averaged diffusion coefficient is given by that of $T_+(t)/t$:
\begin{eqnarray}
\Pr \left(\overline{D(t)}\leq x\right) %&=& \Pr \left( (D_+-D_-)\frac{T_+(t)}{t} +D_- \leq x\right)\\
&=& \Pr \left(  \frac{T_+(t)}{t} \leq  \frac{x-D_-}{D_+-D_-}\right). 
\end{eqnarray}
Because the PDF of $T_+(t)/t$ follows the Lamperti's generalized arcsine law, Eq.~(\ref{T+_limit_dist}), the PDF of $\overline{D(t)}$ is given by
%\begin{equation}
%g_{\alpha,\beta}(x) = \frac{a\sin {\color{red}\pi}\alpha}{\pi} \frac{x^{\alpha-1}(1-x)^{\alpha-1}}{a^2x^{2\alpha} + 2a x^\alpha (1-x)^\alpha \cos \pi \alpha + (1-x)^{2\alpha}},
%\end{equation}
%where {\color{red}$a=a_-/a_+$} and we use $\beta\equiv 1/(1+a)$. Using this PDF yields the PDF of $D_t$: 
$%\begin{equation}
P_D(x) = g_{\alpha,\beta} \left(\frac{x-D_-}{D_d}\right)/D_d,
$ %\label{pdf_D_case2c}
%\end{equation}
 where $D_d=D_+ - D_-$.
Because the mean and second moment of $T_+(t)/t$ are given by $\langle T_+(t)/t \rangle = \beta$ and 
$\langle (T_+/t)^2 \rangle = m(\alpha,\beta)\equiv \beta (\alpha \beta +1 - \alpha)$, respectively \cite{God2001}, we have the RSD
\begin{equation}
\Sigma(t;\Delta) \sim \sqrt{\frac{D_-^2 + 2D_- D_d\beta+D_d^2 m(\alpha,\beta)}{(D_- + D_d\beta)^2} -1}.
\end{equation}
As shown in Fig.~\ref{limit_dist_case3}, theory is in good agreement with numerical results.



\begin{figure}
\includegraphics[width=1.\linewidth, angle=0]{case3.eps}
%\includegraphics[width=.8\linewidth, angle=0]{dist_case2c_asym.eps}
\caption{Anomalous fluctuations of time-averaged diffusivity in the case 3 ($D_-=1$ and $D_+=10$).
(a) TMSDs for 10 different realizations ($\alpha=0.5$ and $t=10^4$).
(b) and (c) Distribution of time-averaged diffusion coefficients ($t=10^3$).
Symbols are the results of numerical simulations and solid curves are the theoretical ones. 
(d) RSD as a function of $\alpha$ and $\beta$ ($t=10^5$). Squares with colors are the results of numerical 
simulations. In numerical simulations, we used the same power-law distribution as in Fig.~\ref{limit_dist_case2}.
%(\ref{pdf_D_case2c}). 
%In numerical simulations, {\color{red}we use $\tau_0^+=\tau_0^- = 
%10^{-3}$ for a), and $\tau_0^+=10^{-3}$ and $\tau_0^- = 10^{-2}$ for b)}.
 }
\label{limit_dist_case3}
\end{figure}

%\section{conclusion}
\section{ Conclusion}
We have shown three distributional limit theorems for time-averaged observables related to diffusivity in 
the Langevin equation with dichotomously fluctuating diffusivity. When one of the states is zero ($D_-=0$)
 in the case (1), statistical properties of TMSD are exactly the same as those in CTRW. 
Therefore, this model is a generalization of CTRW. When both diffusion coefficients are not zero, 
the TMSD asymptotically show normal diffusion in all cases, whereas fluctuations of  TMSD
(deviations of time-averaged diffusion coefficients) are intrinsically random. 
Especially in case (3), time-averaged diffusion coefficients are intrinsically random and the distribution 
follows the generalized arcsine law. 
%while the deviations are random in other cases. 
As a result, we have found anomalous fluctuations in apparently normal diffusion processes.

%appendix
\section*{ Acknowledgments}
We thank T. Miyaguchi and T. Uneyama for discussions on these issues. 
T. A. was partially supported by a Grant-in-Aid for Young Scientists (B) (26800204). 
E. Y. was funded by MEXT Grant-in-Aid for the ``Program for Leading Graduate Schools."


%section {bibliography}
%\begin{thebibliography}{24}%

%\end{thebibliography}%


\appendix
\section{Derivation of Eq.~(\ref{rsd_occupation_time2})}
Here, we derive the RSD in Brownian motion with the diffusion coefficient $D_-$. Since this process is described by Brownian motion, displacement 
  $\delta \bm{r}(\Delta;t) \equiv \bm{r}(\Delta + t) - \bm{r}(t)  $ follows a Gaussian distribution with the mean 0 and the variance $2nD_- \Delta$. 
  The mean TMSD is straightforwardly calculated as $\langle \{\overline{\delta^{2}(\Delta;t)}\} \rangle= 2nD_- \Delta$.
The second moment of TMSD can be calculated as follows:
\begin{widetext}
\begin{align}
\langle \{\overline{\delta^{2}(\Delta;t)}\}^2 \rangle
&\sim \frac{2}{t^2} \int_0^t dt' \int_{t'}^t dt'' \langle \delta \bm{r}^2(\Delta;t') \delta \bm{r}^2(\Delta;t'') \rangle\\
&= \frac{2}{t^2} \int_0^t dt' \int_{t'}^{t'+\Delta} dt'' \langle \delta \bm{r}^2(\Delta;t') \delta \bm{r}^2(\Delta;t'')  \rangle
+ \frac{2}{t^2} \int_0^t dt' \int_{t'+\Delta}^t dt'' \langle \delta \bm{r}^2(\Delta;t')  \rangle \langle  \delta \bm{r}^2(\Delta;t'')  \rangle\\
&= \frac{2}{t^2} \int_0^t dt' \int_{t'}^{t'+\Delta} dt''\left\{ \langle \delta \bm{r}^2(t'' -t';t') \rangle \langle \delta \bm{r}^2(\Delta;t'') \rangle 
+ \langle \delta \bm{r}^4(t'+\Delta - t'';t'') \rangle \right.\\
&+ \langle  \delta \bm{r}^2(t'+\Delta-t'';t'') \rangle \langle \delta \bm{r}^2(t''-t';t'+\Delta) \rangle \}
+\frac{2}{t^2} \int_0^t dt' (t-t'-\Delta) (2nD\Delta)^2\\
%&= \frac{2(2nD)^2}{t^2} \left( \Delta^3 t + \frac{2\Delta^3 t}{3n} + \frac{\Delta^2 t^2}{2} -\Delta^3t\right)\\
&=(2nD \Delta)^2 \left( 1 + \frac{4\Delta}{3nt}  \right).
\end{align}
\end{widetext}
It follows that the RSD decays as
\begin{equation}
\Sigma^2(t;\Delta) %\equiv \frac{\langle \{\overline{\delta^{2}(\Delta;t)}\}^2 \rangle_{\rm eq}  - \langle \overline{\delta^{2}(\Delta;t)} \rangle_{\rm eq}^2}
%{\langle \overline{\delta^{2}(\Delta;t)} \rangle_{\rm eq}^2 } 
\sim \frac{4\Delta}{3nt} \quad (t \to \infty ).
\label{EB}
\end{equation}
 
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