text:
\section{Introduction}

Anomalous diffusion denominates deviations from the regular linear growth
of the mean squared displacement $\langle\mathbf{r}^2(t)\rangle\simeq K_1
t$ as a function of time $t$, where the proportionality factor $K_1$ is
the diffusion constant of dimension $\mathrm{cm}^2/\mathrm{sec}$. Often
these deviations are of power-law form, and in this case the mean squared
displacement in $d$ dimensions
\begin{equation}
\label{msd}
\langle\mathbf{r}^2(t)\rangle=\frac{2dK_{\alpha}}{\Gamma(1+\alpha)}t^{\alpha}
\end{equation}
describes subdiffusion when the anomalous diffusion exponent is in the range
$0<\alpha<1$, and superdiffusion for $\alpha>1$ \cite{report}. The generalized
diffusion constant $K_{\alpha}$ has dimension $\mathrm{cm}^2/\mathrm{sec}^{
\alpha}$. We here concentrate on subdiffusion phenomena. Power-law mean
squared displacements of the form (\ref{msd}) with $0<\alpha<1$ have been
observed in a multitude of systems such as amorphous semiconductors
\cite{scher}, subsurface tracer dispersion
\cite{grl}, or in financial market dynamics \cite{mainardi}. Subdiffusion
is quite abundant in small systems. These include the motion of small probe
beads in actin networks \cite{weitz}, local dynamics in polymer melts
\cite{kimmich}, or the motion of particles in colloidal glasses \cite{weeks}.
\emph{In vivo}, crowding-induced subdiffusion has been reported for RNA motion in \emph{E.coli}
cells \cite{golding}, the diffusion of lipid granules embedded in the
cytoplasm \cite{elbaum,lene}, the propagation of virus shells in cells
\cite{seisenhuber}, the motion of telomeres in mammalian cells \cite{garini},
as well as the diffusion of membrane proteins and of dextrane probes in
HeLa cells \cite{weiss}.

While the mean squared displacement Eq.~(\ref{msd}) is commonly used to classify
a process as subdiffusive, it does not provide any information on the physical
mechanism underlying this subdiffusion. In fact there are several pathways
along which this subdiffusion may emerge. The most common are:

(i) The continuous time random walk (CTRW) \cite{scher} in which the step length has
a finite variance $\langle\delta\mathbf{r}^2\rangle$ of jump lengths, but the
waiting time $\tau$ elapsing between successive jumps is distributed as a power
law $\psi(\tau)\simeq\tau_0^{\alpha}/\tau^{1+\alpha}$, with $0<\alpha<1$. The
diverging characteristic waiting time gives rise to subdiffusion of the form
(\ref{msd}) \cite{scher}. The subdiffusive CTRW in the
diffusion limit is equivalent to the fractional Fokker-Planck equation, that
directly shows the long-ranged memory intrinsic to the process \cite{report}.
Waiting times of the form $\psi(\tau)$ were, for instance, observed in the
motion of tracer beads in an actin network \cite{weitz}. We note that
recently subdiffusion was also demonstrated in a coupled CTRW model \cite{vincent}.

(ii) A random walk on a fractal support meets bottlenecks and dead ends on all
scales and is subdiffusive. The resulting subdiffusion is also of
the form (\ref{msd}), and the anomalous diffusion exponent is related to the
fractal and spectral dimensions, $d_f$ and $d_s$, characteristic of the
fractal, through $\alpha=d_s/d_f$ \cite{havlin}. A typical example is the
subdiffusion on a percolation cluster near criticality that was actually
verified experimentally \cite{kimmich1}.

(iii) Fractional Brownian motion (FBM) and the fractional Langevin
equation (FLE) that will be in the focus of this work and will be
defined in Sec.~\ref{sec2}. The understanding of these types of
stochastic motions is up to date somewhat fragmentary. Thus the
first passage behavior of FBM is known analytically only in one
dimension on a semi-infinite domain \cite{molchan}; the escape
from potential wells in the framework of FLE has been studied
analytically \cite{goychuck,chaudhury} and numerically
\cite{chechkin,goychuck2}; and a priori unexpected critical
exponents have been identified for the FLE \cite{stas1}. Here we
address a fundamental question related to FBM and FLE. Namely,
what is their behavior under confinement? Two main aspects of this
question will be addressed. One is the study of the relaxation
towards stationarity in a finite box by means of the ensemble
averaged mean squared displacement. We also investigate a new
quantity used to characterize the motion, the displacement
correlation function. For these aspects we also study the
dependence on the dimensionality of the motion.

The second aspect concerns how FBM and FLE motions under confinement behave
with respect to ergodicity.
Experimentally the recording of single particle trajectories has become a
standard tool, producing time series of data that are then analyzed by
time rather than ensemble averages. For subdiffusion processes both are
not necessarily identical. In fact for CTRW subdiffusion with an ensemble
averaged mean squared displacement of the form (\ref{msd}) the time averaged
mean squared displacement
\begin{equation}
\label{tamsd}
\overline{\delta^2(\Delta,T)}=\frac{1}{T-\Delta}\int_0^{T-\Delta}\big[x(t+
\Delta)-x(t)\big]^2dt
\end{equation}
on average scales like
$\left<\overline{\delta^2(\Delta,T)}\right>\simeq\Delta/T^{1-
\alpha}$ \cite{YHe,app}. That is, the anomalous diffusion is
manifested only in the dependence on the overall measurement time
$T$ and not in the lag time $\Delta$, that defines a window swept
along the time series. Thus time and ensemble averages are indeed
different. In contrast, for normal Brownian diffusion
$\left<\overline{\delta^2(\Delta,T)}\right>\simeq\Delta$ is
independent of $T$, and time and ensemble averages become
identical, i.e., the system is ergodic. Different from CTRW
subdiffusion, systems governed by FBM or FLE are ergodic. The
ergodicity breaking parameter measured from time averaged mean
squared displacements converges algebraically to zero [ergodic
behavior] as the measurement time increases, the convergence speed
depending on the Hurst exponent $H=\alpha/2$ \cite{deng}. For the
FLE case, however, it was also shown that the ergodicity measured
from the velocity variance can be broken for a class of colored
noises \cite{bao}.

One of the open questions in the context of ergodicity breaking for FBM and
FLE in the above sense is the influence of boundary conditions on the time
averages. It was shown for CTRW subdiffusion that confinement changes the
short time scaling $\left<\overline{\delta^2(\Delta,T)}\right>\simeq\Delta/
T^{1-\alpha}$ to the long time behavior $\left<\overline{\delta^2(\Delta,T)}
\right>\simeq(\Delta/T)^{1-\alpha}$ \cite{stas,thomas}. Although we expect that
FBM and FLE processes become stationary under confinement and, for instance,
attain the same long-time mean squared displacement dictated by the size of
the confinement volume, we investigate how fast this relaxation actually is,
and how it depends on the volume and the dimensionality. To this end we study
the ergodicity breaking parameter for the system. We find that for both FBM
and FLE confinement actually decreases the value of the ergodicity breaking
parameter with respect to unbounded motion, i.e., the process becomes more
ergodic. Ergodicity is also enhanced with increasing dimensionality. We also
discuss how FBM and FLE motions can be distinguished from time series from
single particle trajectories.

The paper is organized as follows. In Sec.~\ref{sec2}, we introduce FBM and
FLE motions, and review briefly their basic statistical properties. In
Sec.~\ref{sec3}, we describe the numerical scheme for simulating FBM and FLE
in confined space. Simulations results are presented in Sec.~\ref{sec4} and
\ref{sec5}, where we discuss the effects of confinement and dimensionality on
time-averaged mean squared displacement trajectory, ergodicity, and displacement correlation. We draw our Conclusions in Sec.~\ref{sec6}.

\section{Theoretical model}
\label{sec2}

We here define FBM and FLE. These two stochastic models share many common
features, however, their physical nature is different. In the following we
will see, in particular, how the two can be distinguished on the basis of
experimental or simulations data.

\subsection{Fractional Brownian motion}

FBM was originally introduced by Kolmogorov in 1940 \cite{kolmogorov} and
further studied by Yaglom \cite{yaglom}. In a different context it was
introduced by Mandelbrot in 1965 \cite{mandelbrot1} and fully
described by Mandelbrot and van Ness in 1968 in terms of a stochastic
integral representation \cite{mandelbrot}. In the latter
reference the authors wrote that "We believe FBMs do provide useful models for
a host of natural time series". This study was motivated by Hurst's analysis
of annual river discharges \cite{hurst}, the observation that in economic time
series cycles of all orders of magnitude occur \cite{finance}, and that many
experimental studies exhibit the now famed $1/f$ noise. FBM by now is widely
used across fields. Among many others FBM has been identified as the underlying
stochastic process of the subdiffusion of large molecules in biological cells
\cite{weiss,weiss1,vincent1}. We note that FBM is
neither a semimartingale nor a Markov process, which makes it quite intricate
to study with the tools of stochastic calculus \cite{oksendal,weron}.

FBM, $x^H(t)$, is a Gaussian process with stationary increments which
satisfies the following statistical properties: the process is symmetric,
\begin{equation}
\langle x^H(t)\rangle=0,
\end{equation}
with $x^H(0)=0$; and the second moment scales like Eq.~(\ref{msd}):
\begin{equation}
\langle x^H(t)^2\rangle=2K_Ht^{2H}.
\end{equation}
For easier comparison with other literature we introduced the Hurst exponent
$H$ that is related to the anomalous diffusion exponent via $H=\alpha/2$. The
Hurst exponent may vary in the range $0<H<1$, such that FBM describes both
subdiffusion ($0<H<1/2$) and superdiffusion ($1/2<H<1$). The limits $H=1/2$
and $H=1$ correspond to Brownian and ballistic motion, respectively. Finally, the two
point correlation behaves as
\begin{equation}
\langle x^H(t_1)x^H(t_2)\rangle=K_H(t_1^{2H}+t_2^{2H}-|t_1-t_2|^{2H}).
\label{eq:covar-fbm}
\end{equation}
Here $\langle\cdot\rangle$ represents the ensemble average. It is convenient
to introduce fractional Gaussian noise (FGN), $\xi^H(t)$, from which the FBM
is generated by
\begin{equation}
x^H(t)=\int_0^tdt'\xi^H(t').
\end{equation}
FGN has the properties of zero mean
\begin{equation}
\langle \xi^H(t)\rangle=0
\end{equation}
and autocorrelation~\cite{qian2,xie}
\begin{eqnarray}
\label{xi}
\langle\xi^H(t_1)\xi^H(t_2)\rangle&=&2K_HH(2H-1)|t_1-t_2|^{2H-2}\nonumber \\
&+&4K_HH|t_1-t_2|^{2H-1}\delta(t_1-t_2),\nonumber \\
\end{eqnarray}
as can be seen by differentiation of Eq.~(\ref{eq:covar-fbm}) with
respect to $t_1$ and $t_2$. Here we see that $K_H$ plays the role
of a noise strength. For subdiffusion ($0<H<1/2$) the
autocorrelation is negative for $t_1\neq t_2$, i.e., the process
is anti-correlated or antipersistent~\cite{FGNcomment}. In
contrast, for $1/2<H<1$, the noise is positively correlated
(persistent) and the motion becomes superdiffusive. For normal
diffusion ($H=1/2)$, the noise is uncorrelated, i.e.,
$\langle\xi^H(t_1)\xi^H(t_2)\rangle=2K_H\delta(t_1-t_2)$. For
further details compare the discussions in
Ref.~\cite{mandelbrot,feder}.

We define $d$-dimensional FBM as a superposition of independent FBMs for each
Cartesian coordinate, such that
\begin{equation}
\label{eq:fbm-2d3d}
\vt{x}^H(t)=\sum_{i=1}^d \int_0^tdt'\xi_i^H(t')\hat{x}_i,
\end{equation}
where $\hat{x}_i$ is the Cartesian coordinate of the $i$th component and
$\xi_i^H$ is FGN which satisfies
\begin{equation}
\langle \xi_i^H(t)\rangle=0
\end{equation}
and
\begin{eqnarray}
\langle\xi_i^H(t_1)\xi_j^H(t_2)\rangle&=&2K_HH(2H-1)|t_1-t_2|^{2H-2}\delta_{ij}\nonumber
\\
&+&4K_HH|t_1-t_2|^{2H-1}\delta(t_1-t_2)\delta_{ij}.\nonumber \\
\end{eqnarray}
From this definition, $d$-dimensional FBM $\vt{x}^H(t)$ has the
properties of zero mean
\begin{equation}
\langle \vt{x}^H(t)\rangle=0,
\end{equation}
variance
\begin{equation}
\langle \vt{x}^H(t)^2\rangle=2dK_Ht^{2H},
\end{equation}
and autocorrelation
\begin{equation}
\langle\vt{x}^H(t_1)\cdot\vt{x}^H(t_2)\rangle=dK_H(t_1^{2H}+t_2^{2H}-|t_1
-t_2|^{2H}).
\end{equation}
Note that $|\vt{x}^H(t)|$ cannot satisfy these properties, thus it is not
an FBM.

A few remarks on this multidimensional extension of FBM are in
order. We note that, albeit intuitive due to the Gaussian nature
of FBM, this multidimensional extension is not necessarily unique.
In mathematical literature higher dimensional FBM in the above
sense was defined in Refs.~\cite{unterberger,qian}. In physics
literature an analogous extension to higher dimensions was used in
Ref.~\cite{weiss} based on the Weierstrass-Mandelbrot method. To
verify that this $d$-dimensional extension is meaningful, we
checked from our simulations of $d$-dimensional FBM that the
fractal dimension of FBM, $d_f=1/H$ for $H>1/d$ \cite{falconer}, is preserved in higher dimensions. Moreover we used an alternative method to create FBM in
$d$-dimensions, namely, to use 1D FBM to choose the length  of a
radius and then choose the space angle randomly. The results were
equivalent to the above definition to use independent FBMs for
every Cartesian coordinate. We are therefore confident that our
definition of FBM in $d$ dimensions is a proper extension of
regular 1D FBM.

\subsection{Fractional Langevin equation motion}

An alternative approach to Brownian motion is based on the Langevin equation
\cite{langevin,vankampen,coffey}
\begin{equation}
m\frac{d^2y(t)}{dt^2}=-\overline{\gamma}\frac{dy(t)}{dt}+\xi(t),
\end{equation}
where $\xi(t)$ corresponds to white Gaussian noise.

When the random noise $\xi(t)$ is non-white, the resulting motion is described
by the generalized Langevin equation (GLE)
\begin{eqnarray}
m\frac{d^2y(t)}{dt^2}=-\overline{\gamma}\int_0^t\mathscr{K}(t-t')\frac{dy}{dt'}
dt'+\xi(t),
\label{eq:GLE}
\end{eqnarray}
where $m$ is the test particle mass, $\mathscr{K}$ is the memory kernel
\cite{zwanzig,berne,kubo} which
satisfies the fluctuation-dissipation theorem $\langle\xi(t)\xi(t')\rangle=
k_BT\overline{\gamma}\mathscr{K}(t-t')$. When $\xi$ is the FGN introduced
above, $\mathscr{K}$ decays algebraically, and Eq.~(\ref{eq:GLE})
becomes the fractional Langevin equation (FLE)
\begin{eqnarray}
\nonumber
m\frac{d^2y(t)}{dt^2}&=&-\overline{\gamma}\int_0^t(t-t')^{2\overline{H}-2}\frac{dy}{dt'}
dt'+\eta\xi^{\overline{H}}(t)\\
&&\hspace*{-1.2cm}=-\overline{\gamma}\Gamma(2\overline{H}-1)\frac{d^{2-2\overline{H}}}
{dt^{2-2\overline{H}}}
y(t)+\eta\xi^{\overline{H}}(t).
\label{eq:FLE1}
\end{eqnarray}
Here, $\overline{\gamma}$ is a generalized friction coefficient. We also
define the coupling constant
\begin{equation}
\eta=\sqrt{\frac{k_BT\overline{\gamma}}{2K_H \overline{H}(2\overline{H}-1)}}
\end{equation}
imposed by the fluctuation dissipation theorem, and the Caputo fractional
derivative \cite{podlubny}
\begin{equation}
\frac{d^{2-2\overline{H}}}{dt^{2-2\overline{H}}}y(t)=\frac{1}{\Gamma(2\overline{H}-1)}\int_0^tdt'(t-t')^{2\overline{H}-2}
\frac{dy}{dt'}.
\end{equation}
Note that in Eq.~(\ref{eq:FLE1}) the memory integral diverges for
$\overline{H}$ smaller than $1/2$, such that the Hurst exponent in the FLE
is restricted to the range $1/2<\overline{H}<1$.

It can be shown that the relaxation dynamics governed by the FLE
(\ref{eq:FLE1}) follows the form
\begin{equation}
\langle y(t)\rangle=v_0t E_{2\overline{H},2}\left(-\gamma t^{2\overline{H}}\right)
\end{equation}
for the first moment, where $v_0$ is the initial particle
velocity. The rescaled friction coefficient is
$\gamma=\overline{\gamma}\Gamma(2\overline{H}-1)/m$. The coordinate
variance behaves as
\begin{equation}
\langle y^2(t)\rangle=2\frac{k_BT}{m}t^2E_{2\overline{H},3}\left(-\gamma t^{2\overline{H}}\right)
\label{eq:msd-FLE}
\end{equation}
where $\langle v_0^2\rangle=k_BT/m$ is assumed, and we employed
the generalized Mittag-Leffler function \cite{bateman}
\begin{equation}
E_{\alpha,\beta}(z)=\sum_{n=0}^{\infty}\frac{z^n}{\Gamma(\alpha n+\beta)}
\end{equation}
whose asymptotic behavior for large $z$ is
\begin{equation}
E_{\alpha,\beta}(z)=-\sum_{n=1}^{\infty}\frac{z^{-n}}{\Gamma(\beta-\alpha n)}.
\end{equation}
Thus the mean squared displacement shows a turnover from short
time ballistic motion to long time anomalous diffusion of the
form~\cite{lutz,pottier}
\begin{equation}
\langle y^2(t)\rangle\sim\left\{\begin{array}{ll} t^2, & t\to0\\
t^{2-2\overline{H}}, & t\to\infty\end{array}\right..\label{FLEscaling}
\end{equation}
Therefore, for persistent noise with $1/2<\overline{H}<1$ the resulting
motion is in fact subdiffusive, i.e., the persistence of the noise
has the opposite effect than in FBM.

In analogy to our discussion of FBM in a $d$-dimensional embedding the FLE
is generalized to
\begin{eqnarray}
m\frac{d^2\vt{y}(t)}{dt^2}=-\overline{\gamma}\int_0^t\Vec{\Vec{\mathscr{K}}}
\cdot\frac{d\vt{y}}{dt'}dt'+\mbox{\boldmath${\xi}$}(t),
\end{eqnarray}
where $\vt{y}(t)=\sum_iy_i(t)\hat{x}_i$, $\mbox{\boldmath${\xi}$}(t)=\sum_{i}
\xi_i^{H}(t)\hat{x}_i$, and $\Vec{\Vec{\mathscr{K}}}$ is the memory tensor
which is in diagonal form (i.e., $\mathscr{K}_{ij}=\mathscr{K}(t-t')
\delta_{ij}$) in the absence of motional coupling between different coordinates.

\section{Simulations scheme}
\label{sec3}

We here briefly review the simulations scheme used to produce time series for
FBM and FLE motion.

\subsection{Fractional Brownian motion}

$d$-dimensional FBM is simulated via Eq.~(\ref{eq:fbm-2d3d}) by
numerical integration of $\xi_i^H(t)$. The underlying FGN was
generated by the Hosking method which is known to be an exact but
time-consuming algorithm \cite{hosking}. We checked that in the
one-dimensional case the generated FBM in free space successfully
reproduces the theoretically expected behavior, the mean squared
displacement (\ref{msd}), the fractal dimension $d_f=2-\alpha/2$ of
the resulting trajectory, and the first passage time distribution.
To simulate the confined motion, reflecting walls were considered
at locations $\pm L$ for each coordinate. For instance in the  1D
case, if $|x^H(t)|>L$ at some time $t$, the particle bounces back
to the position $x^H(t)-2|x^H(t)-\mathrm{sign}(x^H)L|$. Similar
reflecting conditions were taken into account in the
multi-dimensional case.

\subsection{Fractional Langevin equation motion}

In simulating FLE motion, we follow the numerical method presented
by Deng and Barkai~\cite{deng}. First, integrating
Eq.~(\ref{eq:FLE1}) from 0 to $t$, we obtain the Volterra integral
equation for velocity field $v(t)=dy(t)/dt$
\begin{eqnarray}
\nonumber
v(t)&=&-\frac{\overline{\gamma}}{(2\overline{H}-1)m}\int_0^t(t-t')^{2\overline{H}-1}v(t')dt'\\
&&+v_0+\frac{\eta}{m}x^{\overline{H}}(t),
\end{eqnarray}
where $v(t=0)=v_0$. This stochastic integral equation can be
evaluated by the predictor-corrector algorithm presented in
Ref.~\cite{kai} with the FBM $x^{\overline{H}}(t)$ independently obtained
by the Hosking method. We calculated $y(t)=y_0+\int_0^t v(t')dt'$
by the trapezoidal algorithm. For discrete time steps, the
equation of motion is given by
\begin{eqnarray}
\nonumber
y_{n+1}&=&y_0+\frac{dh}{2}(v_0+v_{n+1})+dh\sum_{i=1}^{n}v_i,\\
&=&\frac{dh}{2}(v_n+v_{n+1})+y_n,
\label{eq:GLE-numerical}
\end{eqnarray}
where $dh$ is the time increment. When evaluating
Eq.~(\ref{eq:GLE-numerical}), a reflecting boundary condition was
considered in the sense that $y_n\rightarrow
y_n-2|y_n-\hbox{sign}(y_n)L|$ if $|y_n|>L$.

\begin{figure}
\includegraphics[width=0.50\textwidth]{Fig1b.eps}
\caption{(Color online) The mean squared displacements (MSD) for FLE motion in free space.
The ensemble averaged and time averaged MSDs, $\langle y^2(t)\rangle$ and
$\overline{\delta^2(\Delta,T)}$, obtained from simulation are compared with
the exact solution $2t^2E_{5/4,3}[-10\Gamma(5/4-1)t^{5/4}]$
given by Eq.~(\ref{eq:msd-FLE}). The ensemble averaged value was obtained from 200 simulated trajectories.
In the simulation, we chose the Hurst exponent $\overline{H}=5/8$, time increment $dh=0.01$, particle mass $m=1$, initial velocity
$v_0=1$, initial position $y_0=0$, friction coefficient $\overline{\gamma}=10$,
and $k_BT=1$.}
\label{f:nerror}
\end{figure}
To show the reliability of our simulation, we compare the
simulation result with the well-known solution for free space
motion. In Fig. 1, we simulate the subdiffusion case with the parameters
values, $\overline{H}=5/8$, $m=1$, $v_0=1$, $y_0=0$,
$\overline{\gamma}=10$, $k_BT=1$, and $dh=0.01$. From 200
simulated trajectories, we obtain the ensemble averaged and time
averaged mean squared displacements, $\langle y^2(t)\rangle$ and
$\overline{\delta^2(\Delta,T)}$, and compare them with the exact
solution Eq.~(\ref{eq:msd-FLE}). Note that $\langle y^2(t)\rangle$
should be identical to $\overline{\delta^2(\Delta,T)}$,
Eq.~(\ref{eq:tmsd-FLE}), with $t$ regarded as the lag time
$\Delta$ due to the ergodicity of the FLE motion in free
space~\cite{deng}. The deviation from the exact solution is
markedly reduced with decreasing time increment $dh$. With our
chosen value $dh=0.01$ the mean squared displacements obtained
from simulation appear to be in good agreement with the theory.

\section{Fractional Brownian Motion in confined space}
\label{sec4}

We now turn to the investigation of the behavior of FBM under
confinement, analyzing the mean squared displacement and potential
ergodicity breaking. We then define the displacement correlation
function, and finally study the influence of dimensionality.

\subsection{Mean squared displacement}

For FBM in free space $\langle x^H(t)^2\rangle$ can be estimated by the time
averaged mean squared displacement Eq.~(\ref{tamsd}) via the exact relation
\cite{deng}
\begin{equation}
\left<\overline{\delta^2(\Delta,T)}\right>=2K_H\Delta^{2H}.
\end{equation}
Here $\langle\cdot\rangle$ denotes the ensemble average. In contrast to
CTRW subdiffusion, in FBM this quantity is ergodic. However, as mentioned
above, the approach to ergodicity is algebraically slow, and we want to
explore here whether boundary conditions have an impact on the ergodic
behavior. Let us now analyze the behavior in a box of size $2L$.

\begin{figure}
\includegraphics[width=0.50\textwidth]{Fig2.eps}
\caption{(Color online) Time-averaged mean squared displacement (MSD) versus the lag time
$\Delta$ for given values of $L$. The drawn line has slope $1/2$,
corresponding to the expected short lag time behavior for the used value
$H=1/4$ of the Hurst exponent. For $L=1$, 2, and 3, five different
trajectories each are drawn to be able to see whether the trajectories
scatter. The simulation time is $T=2^{17}\approx1.3\cdot10^5$.}
\label{f:msd-fbm}
\end{figure}

In Fig.~\ref{f:msd-fbm} we show typical curves for the time averaged mean
squared displacement for Hurst exponent $H=1/4$ and three different interval
lengths $L$. Regardless of the size of $L$, the confined environment does not
affect the power law with exponent $2H$ for short lag times. Moreover at long
lag times we observe saturation of the curves to a value that depends on $L$.
This behavior is distinct from that of the CTRW case where $\left<\overline{
\delta^2(\Delta,T)}\right>$ shows a power law with slope $1-\alpha$ \cite{stas,thomas}.

One can estimate the saturated value as a function of $L$. For long $\Delta$
and measurement time $T$, the probability $p(x)$ to find the particle located
at $x$ is independent of $x$ due to the equilibration between the reflecting
walls, and thus $\int_{-L}^{L} x^2p(x)dx=L^2/3$. The dotted lines in
Fig.~\ref{f:msd-fbm} represent these values.

We observe that the scatter between different single trajectories becomes more
pronounced when the interval length is increased. In fact the scatter is
negligible for $L=1$ while it is quite appreciable for $L=3$, even though
the slope of all curves at finite $L$ converges to a horizontal slope, with
an amplitude close to the predicted value $L^2/3$. We also note that the
scatter depends on the total measurement time $T$. For given $L$ it tends to
be reduced as we increase $T$. This effect will be discussed quantitatively
in detail using the ergodicity breaking parameter.

\subsection{Ergodicity breaking parameter}

In contrast to CTRW subdiffusion, FBM in free space is known to be ergodic
\cite{deng}. The time averaged mean squared displacement traces displayed
in Fig.~\ref{f:msd-fbm} exhibit no extreme scatter as known from the CTRW
case. This implies that ergodicity is indeed preserved for confined FBM. We
quantify this statement more precisely in terms of the ergodicity breaking
parameter \cite{YHe}
\begin{eqnarray}
E_B(\Delta,T)=\frac{\left<\left(\overline{\delta^2(\Delta,T)}\right)^2\right>
-\left<\overline{\delta^2(\Delta,T)}\right>^2}{\left<\overline{\delta^2(\Delta
,T)}\right>^2},
\label{eq:EB}
\end{eqnarray}
where $\lim_{T\rightarrow\infty}E_B(T)=0$ is expected for ergodic
systems. For the case of free FBM, Deng and Barkai analytically
derived that $E_B$ decays to zero as
\begin{eqnarray}
E_B(\Delta,T)\sim\left\{
\begin{array}{ll}\displaystyle\frac{\Delta}{T} & \mbox{for}~ 0<H<\frac{3}{4},\\[0.4cm]
\displaystyle\frac{\Delta}{T}\log T & \mbox{for}~ H=\frac{3}{4}, \\[0.4cm]
\displaystyle  \left(\frac{\Delta}{T}\right)^{4-4H} & \mbox{for}~\frac{3}{4}<H<1,
\end{array}\right.
\end{eqnarray}
for long measurement time $T$~\cite{deng}.

We numerically investigate the boundary effects on the ergodicity
breaking parameter. First, in Fig.~\ref{f:EB-fbm-lag} we evaluate
$E_B$ as function of the lag time $\Delta$ from 200 FBM
simulations for each given $L$. The dotted line represents the
expected free space behavior $E_B\sim\Delta$, which is nicely
fulfilled by the data at shorter times and sufficiently large $L$.
At longer times or small $L$ the results show that $E_B$ behaves
very differently when confinement effects are present. The plateau
in $E_B$ is related to the saturation of the curves for the mean
squared displacement, Fig.~\ref{f:msd-fbm}. As the motion is
restricted by the walls roughly above a crossover lag time
$\Delta_{cr}=(L^2/2K_H)^{1/2H}$, the ergodicity breaking parameter
$E_B$ levels off at $\Delta\gtrsim \Delta_{cr}$. The sharp
increase at the end of the curve is due to the singularity when
the lag time reaches the size of the overall measurement time $T$,
which would disappear in the infinite measurement time.

\begin{figure}
\includegraphics[width=0.50\textwidth]{Fig3.eps}
\caption{(Color online) Ergodicity breaking parameter $E_B$ versus lag time $\Delta$ for
$L=1$, 3, 5, and 10 (from bottom to top) with Hurst exponent $H=1/4$. The
overall measurement time is $T=2^{14}\approx1.6\cdot10^4$. The dotted line with slope 1
represents the theoretical expectation $E_B\simeq\Delta$ in free space. For
each $L$ the curve was obtained from 200 single trajectories. The vertical lines show the
crossover lag time $\Delta_{cr}=(L^2/2K_H)^{1/2H}$ for $L=1,~3$, and 5. }
\label{f:EB-fbm-lag}
\end{figure}

\begin{figure}
\includegraphics[width=0.50\textwidth]{Fig4.eps}
\caption{(Color online) Ergodicity breaking parameter $E_B$ versus overall measurement time
$T$ at given lag time $\Delta=1$, 100, and 1000. The dotted line depicts a
power law with slope -1, representing the analytic behavior $E_B\simeq T^{
-1}$ in free space. For each $\Delta$, the $E_B$ curves are drawn for different
interval lengths $L=1$, 3, 5, and 10. Each curve was obtained from 200 single
trajectories, and the Hurst exponent was $H=1/4$.}
\label{f:EBt-fbm}
\end{figure}

In Fig.~\ref{f:EBt-fbm} we show $E_B$ for given $\Delta$ as
function of the measurement time $T$ for the same choice of
interval lengths, $L=1,3,5$, and 10. For short lag times $\Delta$,
all $E_B$ curves coincide and decay as $T^{ -1}$, in complete
analogy to the free space motion (dotted line). In the case of
longer $\Delta$ the general trend is that $E_B$ decays like
$T^{-1}$, unaltered with respect to the free case. However, there
is a sudden decrease in $E_B$ for the smallest interval size, for
$L=1$. One can understand this behavior by observing the $E_B$
curve for $L=1$ in Fig.~\ref{f:EB-fbm-lag}; as the fluctuations of
the mean squared displacement are strongly suppressed due to the
tight confinement in this case, $E_B$ has almost no dependence on
$\Delta$ for $\Delta_{cr}\lesssim\Delta \lesssim T$ and the
saturated value is quite small compared to those for other cases.
Therefore, the curves for $L=1$ appear disconnected from the other
curves. Corresponding to the approximate independence of the $L=1$
curve for $\Delta \gtrsim10$ in Fig.~\ref{f:EB-fbm-lag}, we
observe in Fig.~\ref{f:EBt-fbm} that at longer times $T$ the $L=1$
curves approach each other. Only at $T\approx\Delta$ these curves
separate, as then
$\overline{\delta_i^2(T,T)}=[x^H_i(t+T)-x^H_i(t)]^2$, and $E_B$ is
evaluated with the same small number of squared displacement data.
Note that the splitting of the $E_B$ curve can be also observed
for larger $L$ at $\Delta$s larger than $\Delta_{cr}$ under longer
total measurement time $T$ as other $E_B$ curves also have
corresponding constant saturation values for $\Delta
\gtrsim\Delta_{cr}(=(L^2/2K_H)^{1/2H})$ which increases with the
size $L$.

\subsection{Displacement correlation function}

As explained for the stochastic properties of FBM in
Sec.~\ref{sec2}, the position autocorrelation $\langle
x^H(t_1)x^H(t_2)\rangle$ explicitly depends on $t_1$ and $t_2$ as
well as their difference, $|t_1-t_2|$. It is therefore not an
efficient quantity to estimate directly from experimental or
simulations data. However, the correlation function of the
displacements
\begin{equation}
\delta x^H(t,\Delta)=x^H(t+\Delta)-x^H(t)
\end{equation}
depends only on the time interval $\Delta$ of the displacement,
\begin{eqnarray}
\label{eq:jump-fbm}
\langle\delta x^H(t,\Delta)\delta x^H(t-\Delta,\Delta)\rangle=K_H(2^{2H}-2)
\Delta^{2H}
\end{eqnarray}
for free FBM. This relation is derived in App.~\ref{app1}. This quantity is
anticorrelated for $0<H<1/2$ (subdiffusive motion), uncorrelated for $H=1/2$
(normal Brownian motion), and positively correlated for $1/2<H<1$
(superdiffusion). As Eq.~(\ref{eq:jump-fbm}) does not depend on the measurement
time $T$ the value of the ensemble averaged value is identical to the
corresponding time average.

\begin{figure}
\includegraphics[width=0.50\textwidth]{Fig5.eps}
\caption{(Color online) Absolute value of the displacement correlation (DC) versus
lag time $\Delta$ for $L=1$, 2, and 3 (from bottom to top) with a slope
proportional to $\Delta^{2H}$ (dotted line). Total measurement time is $T=
2^{17}\approx1.3\cdot10^5$, and the Hurst exponent is $H=1/4$.
Each curve was obtained via time averaging from a single particle trajectory. }
\label{f:jump-fbm}
\end{figure}

We present the time averaged displacement correlation for confined
subdiffusive motion ($H=1/4$) in Fig.~\ref{f:jump-fbm}. Because of
the negativity of expression (\ref{eq:jump-fbm}) the absolute
value of the displacement correlation is drawn in the log-log
representation. At short lag times the slope of the correlation
functions is proportional to $2H$ as expected from
Eq.~(\ref{eq:jump-fbm}). However, at long lag times, we
interestingly observe fluctuations of the correlations around a
constant value, reflecting the confinement of the motion.

\subsection{Dimensionality}

To mimic the anomalous diffusion of particles inside biological cells, we
also simulate two- and three-dimensional FBM based on Eq.~(\ref{eq:fbm-2d3d})
in the presence of reflecting walls. In free space, the ensemble average of
the time averaged mean squared displacement is simply given by
\begin{eqnarray}
\nonumber
\left<\overline{\delta^2(\Delta,T)}\right>&=&\frac{1}{T-\Delta}\int_0^{T-
\Delta}\left<[\vt{x}^H(t+\Delta)-\vt{x}^H(t)]^2\right>dt,\\
&=&2dK_H\Delta^{2H},
\label{eq:msdfbm3D}
\end{eqnarray}
i.e., it is additive as for the ensemble average. This behavior is indeed
observed in Fig.~\ref{f:msd3D-fbm1} where five different mean squared
displacement curves are drawn for $L=1$ in 1D, 2D, and 3D, respectively. Only
the height of
the curves are affected by the dimensionality. There is no noticeable
difference in the scatter of the curves.

\begin{figure}[tb]
\includegraphics[width=0.50\textwidth]{Fig6.eps}
\caption{(Color online) Time-averaged mean squared displacement (MSD) curves versus lag
time $\Delta$ for $L=1$ in 1D, 2D, and 3D space (from bottom to top), with
a slope $\Delta^{2H}$ (dotted line). For each dimension, 5 trajectories
were drawn with total measurement time $T=2^{17}$ and $H=1/4$.}
\label{f:msd3D-fbm1}
\end{figure}

We further investigate the effects of dimensionality on the
scatter of the mean squared displacement curves, as possibly the
strong scatter observed in experiments \cite{golding,lene,elbaum}
may also occur for FBM in higher dimensions. To see the effect of
dimensionality on the ergodicity behavior we measure $E_B$ versus
lag time for one-, two-, and three-dimensional embedding dimension
for the same values of $L$ and $H$. Interestingly, the result
shows that $E_B$ tends to decrease with increasing dimensionality
$d$, meaning that for FBM big scatter is not caused by higher
dimensions in presence of reflecting walls. In fact, from
Eqs.~(\ref{eq:EB}) and (\ref{eq:msdfbm3D}), we can analytically
derive the relation
\begin{equation}
\label{eq:EBD}
E_B(d)=\frac{E_B(d=1)}{d},
\end{equation}
which still holds in the case of confined motion (see appendix B
for the derivation). This relation is numerically confirmed in
Fig.~\ref{f:EB3D-fbm2} where three $E_B$ curves collapse upon
rescaling by $dE_B(d)$. According to this relation, we expect that
ergodic behavior obtained in one-dimensional confined motion
(Figs.~\ref{f:EB-fbm-lag} and \ref{f:EBt-fbm}) will also be
present in multiple dimensions with a factor of $1/d$.

\begin{figure}
\includegraphics[width=0.50\textwidth]{Fig7.eps}
\caption{(Color online) Rescaled ergodicity breaking parameter $dE_B$ versus lag time $\Delta$ for
interval size $L=5$ for 1D, 2D, and 3D space (from top to bottom), with
Hurst exponent $H=1/4$ and measurement time $T=2^{14}$. $E_B$ was evaluated from 200 trajectories with
different initial positions.}
\label{f:EB3D-fbm2}
\end{figure}

\section{Fractional Langevin equation motion in confined space}
\label{sec5}

In this section we analyze FLE motion under confinement. Due to the different
physical basis compared to FBM, in particular, the occurrence of inertia, we
observe interesting variations on the properties studied in the previous
section.

\subsection{Mean squared displacement}

Using the correlation function \cite{pottier}
\begin{eqnarray}
\nonumber \langle
y(t_1)y(t_2)\rangle&=&\frac{k_BT}{m}[t_1^2E_{2\overline{H},3}(-\gamma
t_1^{2\overline{H}})+t_2^2E_{2\overline{H},3}(-\gamma t_2^{2\overline{H}})\nonumber
\\
&&\hspace*{-0.8cm}-(t_2-t_1)^2E_{2\overline{H},3}(-\gamma
|t_2-t_1|^{2\overline{H}})],
\end{eqnarray
one can show analytically that, similarly to the FBM, the ensemble
averaged second moment $\langle y^2(t)\rangle$ is identical to its
time averaged analog $\left<\overline{\delta^2(\Delta)}\right>$,
for all $\Delta$ in free space, namely
\begin{equation}
\left<\overline{\delta^2(\Delta,T)}\right>=2\frac{k_BT}{m}\Delta^2E_{2\overline{H},3}(-\gamma
\Delta^{2\overline{H}}).\label{eq:tmsd-FLE}
\end{equation}
Thus, the time averaged mean squared displacement turns over from a ballistic
motion
\begin{equation}
\left<\overline{\delta^2(\Delta,T)}\right>\simeq\Delta^2
\end{equation}
at short lag time to the subdiffusive behavior
\begin{equation}
\left<\overline{\delta^2(\Delta,T)}\right>\simeq\Delta^{2-2\overline{H}}
\end{equation}
at long lag times, in free space.

\begin{figure}
\includegraphics[width=0.50\textwidth]{Fig8.eps}
\caption{(Color online) Time-averaged mean squared displacement (MSD) versus lag time
$\Delta$. The two dashed lines represent the two asymptotic scaling
behaviors $\left<\overline{\delta^2(\Delta,T)}\right>\simeq\Delta^2$ and
$\left<\overline{\delta^2(\Delta,T)}\right>\simeq\Delta^{2-2\overline{H}}$. For each
given $L=1/2$, 1, and 3, five different trajectories are drawn to visualize
the scatter. As a reference curve for motion in free space, the curve for
$L=100$ (line) is drawn. In the
simulation, we chose the Hurst exponent $\overline{H}=5/8$ [NB: for the FLE this means
subdiffusion], time increment $dh=0.01$, particle mass $m=1$, initial velocity
$v_0=1$, initial position $y_0=0$, friction coefficient $\overline{\gamma}=10$,
and $k_BT=1$.}
\label{f:msd-gle}
\end{figure}

We numerically study how this scaling behavior is affected by the
confinement. Figure~\ref{f:msd-gle} shows typical curves for the
time averaged mean squared displacement, for interval sizes
$L=1/2$, 1, 3, and 100 (regarded as free space motion) with
identical initial conditions and Hurst exponent $\overline{H}=5/8$. The
results are summarized as follows: (1) We observe both scaling
behaviors,
$\left<\overline{\delta^2(\Delta,T)}\right>\simeq\Delta^2$ turning
over to $\simeq\Delta^{2-2\overline{H}}$, for confined FLE motions. (2) For
narrow intervals, the curves eventually reach the saturation
plateau within the chosen total measurement time $T$. The
saturation values are approximately $L^2/3$. For interval size
$L=1/2$, the saturated value is noticeably larger than $L^2/3$,
which appears to be caused by multiple reflection events on the
walls. The same behavior is observed in the FBM case when
considering a large value of $H\geq1/2$, or very narrow intervals
for the given $H=1/4$. (3) As in the case of FBM, the scatter
becomes pronounced as the interval length increases.

\subsection{Ergodicity breaking parameter}

From a simple argument and simulations it was shown in Ref.~\cite{deng}
that the FLE and FBM mean squared displacements are asymptotically equal,
$\left<\overline{\delta^2(y)}\right>\sim\left<\overline{\delta^2(x^H)}\right>$,
similarly for the ergodicity breaking parameter, $E_B(y)\sim E_B(x^H)$. Here
the asymptotic equivalence is valid at long measurement times $T$, and the
derivation holds for motion in free space. From 200 trajectories of the mean
squared displacement we measure the ergodicity breaking parameter $E_B$ as
function of lag time $\Delta$ for
interval lengths $L=1/2$, 1, 3, and 100 in Fig.~\ref{f:EB-gle}. The behavior
is similar to the corresponding curves for FBM, displayed in
Fig.~\ref{f:EB-fbm-lag}: $E_B$ significantly deviates from the reference curve
for free space motion (i.e., the longer $\Delta$ behavior for $L=100$ and the
drawn power law $\simeq\Delta$), due to the confinement effect. $E_B$ tends to
decrease with smaller $L$ for the same value of $\Delta$. However, the plateau
at short lag times that is still observed for $L=100$ (regarded as free space
motion), is due to the initial ballistic motion of FLE. In that regime the
initially directed motion renders the random noise effect negligible.

\begin{figure}
\includegraphics[width=0.50\textwidth]{Fig9.eps}
\caption{(Color online) Ergodicity breaking parameter $E_B$ versus lag time $\Delta$ for
interval sizes $L=1/2$, 1, 3, and 100. The straight line
corresponds to the free space behavior $E_B\simeq\Delta$. Each curve was
obtained from 200 single trajectories, with the same parameter values used
in Fig.~\ref{f:msd-gle}.}
\label{f:EB-gle}
\end{figure}

\subsection{Displacement correlation function}

Using the correlation function $\langle y(t_1)y(t_2)\rangle$, we
analytically obtain the displacement correlation function in free
space in the form (refer to App.~\ref{app1} for the derivation)
\begin{eqnarray}
\nonumber
\langle\delta y(t,\Delta)\delta y(t-\Delta,\Delta)\rangle&=&4\frac{k_BT}{m}\Delta^2E_{
2\overline{H},3}[-\gamma(2\Delta)^{2\overline{H}}]\\
&&\hspace*{-0.8cm}-2\frac{k_BT}{m}\Delta^2E_{2\overline{H},3}[-\gamma\Delta^{2\overline{H}}],
\label{eq:jump_gle}
\end{eqnarray}
so that we observe the following asymptotic behavior
\begin{eqnarray}
\nonumber
\langle\delta y(t,\Delta)\delta y(t-\Delta,\Delta)\rangle&&\\[0.2cm]
&&\hspace*{-2.8cm}\sim\left\{
\begin{array}{ll}\displaystyle\frac{2k_BT}{m\Gamma(3)}\Delta^2 & \mbox{for }
\Delta\rightarrow 0,\\[0.4cm]
\displaystyle\frac{(2^{2-2H}-2)k_BT}{m\gamma \Gamma(3-2\overline{H})}\Delta^{2-2\overline{H}} &
\mbox{for }\Delta\rightarrow\infty,
\end{array}\right.
\label{eq:jump_gle2}
\end{eqnarray}
where $\delta y(t,\Delta)=y(t+\Delta)-y(t)$. Above expression
shows that the displacement correlation has two distinct scaling
behaviors. At short lag times, it grows like $\Delta^2$ and is
positive, due to the ballistic motion. At long lag times, it is
negative in the domain $1/2<\overline{H}<1$, exhibiting the same
subdiffusive behavior as observed for FBM
[cf.~Eq.~(\ref{eq:jump-fbm})] when we replace
$H\rightarrow2-2\overline{H}$. Note that to bridge these two scaling
behaviors the displacement correlation passes the zero axis at
$\Delta= \Delta_c$ that satisfies
$2E_{2\overline{H},3}\left[-\gamma(2\Delta_c)^{2\overline{H}}\right]=E_{2\overline{H},
3}\left[-\gamma\Delta_c^{2\overline{H}}\right]$ in free space. For small
$\gamma$, we find approximately
\begin{equation}
\Delta_c\approx\left(\frac{\Gamma(2\overline{H}+3)}{2\Gamma(2\overline{H}-1)(2^{2\overline{H}+1}-1)}\frac{m}{
\overline{\gamma}}\right)^{1/2\overline{H}},
\end{equation}
such that it becomes exactly the momentum relaxation time
$m/\overline{ \gamma}$ for normal Brownian motion ($\overline{H}=1/2$). In
the limit $\overline{H}\rightarrow1$, $\Delta_c$ goes to infinity to
satisfy the equality $2E_{2\overline{H},3}\left[-\gamma(
2\Delta_c)^{2\overline{H}}\right]=E_{2\overline{H},3}\left[-\gamma\Delta_c^{2\overline{H}}\right]$.
Thus, $\Delta_c$ can be interpreted as the typical timescale for
the persistence of the ballistic motion.

Figure~\ref{f:jump-gle} shows (a) the displacement correlation
versus lag time $\Delta$, and (b) the absolute value of the
displacement correlation as function of $\Delta$, for $L=1/2$, 1,
3, and 100. The scaling properties derived in
Eq.~(\ref{eq:jump_gle2}) are indeed observed. At short lag times
all curves are positive and scale like $\sim\Delta^2$, before
decreasing to zero. In the long lag time regime the displacement
correlation becomes negative and the predicted scaling behavior
$\simeq \Delta^{2-2\overline{H}}$ is observed. For small intervals ($L=1/2$
and 1), it is saturated due to the confinement effect as seen in
the case of FBM.

\begin{figure}
\includegraphics[width=0.50\textwidth]{Fig10.eps}
\caption{(Color online) (a) Displacement correlation (DC) versus lag time $\Delta$
for $L=1/2$, 1, 3, and 100 (from bottom to top). (b) Absolute value of the
displacement correlation as a function of $\Delta$ for the given values
of $L$. The two slopes correspond to the limiting behavior $\Delta^{2}$ and
$\Delta^{2-2\overline{H}}$. In (a) and (b), the ensemble averaged curves were obtained
from 200 different time averaged displacement correlation curves. Same parameter
values as in Fig.~\ref{f:msd-gle}.}
\label{f:jump-gle}
\end{figure}

\subsection{Dimensionality}

In the case when the memory tensor is diagonalized, each coordinate motion is
independent and FLE motion exhibits qualitatively the same behavior as shown
in the case of FBM with increasing dimensionality. From the mean squared
displacement curves, the same scaling behavior is expected with more
elevated amplitude for
higher dimensionality. In fact, when each coordinate motion is decoupled, a
$d$-dimensional motion effectively increases the number of
single trajectories $d$ times compared to the one-dimensional case. Therefore,
the scatter in the mean square displacement curves decreases with increasing
dimensionality and the ergodicity breaking parameter $E_B$ is expected to
follow the relation Eq.~(\ref{eq:EBD}).

\section{Conclusion}
\label{sec6}

Motivated by recent single particle tracking experiments in biological cells,
in which confinement due to the rather small cell size becomes relevant, we
studied FBM and FLE motions in confined space. In particular we analyzed the
effects of confinement and dimensionality on the stochastic and ergodic
properties of the two processes. Interestingly for both stochastic models,
the confinement tends to decrease the value of the ergodicity breaking
parameter $E_B$ compared to that in free space. The same trend is observed
for increasing dimensionality. Correspondingly the scatter of time averaged
quantities between individual trajectories is quite small, apart from regimes
when the lag time $\Delta$ becomes close to the overall measurement time $T$
and the sampling statistics for the corresponding time average become poor.
The relaxation of the ergodicity breaking parameter as function of measurement
time is quite similar to previous results in free space. We conclude that
neither confinement nor dimensionality effects lead to the appearance of
significant ergodicity breaking or scatter between single trajectories.

The displacement correlation function introduced here is a useful
quantity that can be easily obtained from single particle
trajectories. It can be used as a tool to discriminate one
stochastic model from another. For subdiffusive motion governed by
FBM and FLE motion, the displacement correlation should be
negative and saturate in the long measurement time limit due to
the confinement. Notably, the negative decrease
($\sim-\Delta^{\alpha}$) with lag time $\Delta$ and anomalous
diffusion exponent $\alpha$ is an intrinsic property of FBM and
FLE displacement correlations which is clearly distinguished from
that of CTRW subdiffusion. In the latter case, the subdiffusive
motion occurs due to the long waiting time distribution between
successive jumps and there is no spatial correlation between them,
so that displacement correlation only fluctuates around zero
with time. FLE motion can be distinguished from FBM motion since
the displacement correlation has a positive value at short times
due to the ballistic motion in the FLE model.

\acknowledgments

We thank Stas Burov and Eli Barkai for helpful and enjoyable discussions.
Financial support from the DFG is acknowledged.

redpajama_set_name: RedPajamaArXiv


text:
Amory H. Danek
Heidelberg University, Germany
Mark Beeman
Northwestern University, United States
Saskia Jaarsveld
University of Kaiserslautern, Germany
Front. Psychol., 30 July 2018
Sec. Cognition
Volume 9 - 2018 | https://doi.org/10.3389/fpsyg.2018.01311
A Growth-Curve Analysis of the Effects of Future-Thought Priming on Insight and Analytical Problem-Solving
Monica Truelove-Hill*, Brian A. Erickson, Julia Anderson, Mary Kossoyan and John Kounios
Department of Psychology, Drexel University, Philadelphia, PA, United States
Research based on construal level theory (CLT) suggests that thinking about the distant future can prime people to solve problems by insight (i.e., an "aha" moment) while thinking about the near future can prime them to solve problems analytically. In this study, we used a novel method to elucidate the time-course of temporal priming effects on creative problem solving. Specifically, we used growth-curve analysis (GCA) to examine the time-course of priming while participants solved a series of brief verbal problems. Participants were tested in two counterbalanced sessions in a within-subject experimental design; one session featured near-future priming and the other featured far-future priming. Our results suggest high-level construal may temporarily enhance analytical thinking; far-future priming caused transient facilitation of analytical solving while near-future priming induced weaker, transient facilitation of insightful solving. However, this effect is short-lived; priming produced no significant differences in the total number of insights and analytical solutions. Given the fleeting nature of these effects, future studies should consider implementing methodology that allows for aspects of the time-course of priming effects to be examined. A method such as GCA may reveal mild effects that would be otherwise missed using other types of analyses.
Construal level theory (CLT) proposes that psychological distance from the self determines the way that one represents an object or event through mental construal (Trope and Liberman, 2003, 2010). High-level construals encompass the abstract, general features of an event or object. They omit the fine details about an object in favor of a broader representation of the object's features (Trope and Liberman, 2010). Conversely, low-level construals include the context-dependent, concrete features of events or objects. For example, moving from 'animal' to 'mammal' to 'canine' to 'dog' represents a gradual shift from high-level to low-level construal. According to CLT, events that are psychologically distant will be represented by high-level, abstract construals, while those that are psychologically proximal will be represented by low-level concrete construals. Temporal distance reflects psychological distance in time of an event from the individual.
In line with CLT, thinking about the distant future requires more high-level construals than thinking about the near future, the latter requiring more low-level construals. In other words, individuals will form a more abstract mental representation of an event in the distant future than of an event in the near future. Because the near future is relatively proximal to the present, one has a more concrete idea of what to expect of events that occur in this time period. The distant future, on the other hand, requires more imagination—the context is unknown, and factors that are relevant to the present may change in the meantime. For example, when someone is planning a trip in the near future, there are very specific deadlines that must be met. Tickets must be booked, accommodations must be arranged, and even minor details such as the upcoming weather are known and may be incorporated in one's decisions. If a trip is taking place in the distant future, the planning is much more abstract. General ideas such as where to go and what to do may be identified, but the concrete details cannot be considered until the trip is much closer.
Research by Liberman et al. (2002) supports this idea. In one study, participants were asked to think about completing everyday life tasks in either the distant future (1 year from the present) or the near future (1 week from the present). Participants in the distant future condition rated their ability to cope with a wide variety of everyday life tasks more similarly than those in the near future condition, suggesting less nuance in the way that distant-future tasks are conceptualized compared to near future tasks. Additionally, participants who underwent distant future priming implemented broader categories when sorting objects than those who underwent near future priming, suggesting that the more abstract mindset promoted by distant future thought can be generalized to other tasks. Other research has substantiated this idea—inducing a more abstract mindset may influence, for example, how consumers perceive advertisements (Martin et al., 2009) and how individuals deploy self-presentation strategies (Carter and Sanna, 2008). Another area which may be influenced by temporal construal priming is the method by which someone solves problems.
One of the methods people commonly use when confronted with a problem is to consciously manipulate the elements of the problem until a solution is derived. In this analytical approach, one works through a problem, step by step, and gradually comes to a solution. For example, one typically uses analytical problem solving when faced with an arithmetic problem. Another method by which one may solve a problem is through insight, commonly considered a form of creative cognition (for a discussion of the relationship between creativity and insight, see Kounios and Beeman, 2015). To solve by insight involves a sudden restructuring of the problem so that the solution is immediately clear. Unlike analytical solving (DeWall et al., 2008), insight solving is largely the result of unconscious processing (Kounios and Beeman, 2014); one's subjective experience is that the solution came from nowhere (Schooler and Melcher, 1995). Indeed, research has shown that participants are able to rate their nearness to a solution in the case of analytical solving, but not for insight solving (Metcalfe and Wiebe, 1987). In this study, we tested whether these two problem-solving styles would be differentially affected by temporal construal priming.
Research has already shown that problem-solving style may be affected by a person's prior internal state (see review by Kounios and Beeman, 2014). For example, neural activity immediately preceding the presentation of a problem predicts whether participants will solve that problem insightfully or analytically (Kounios et al., 2006). Subramaniam et al. (2009) showed that mood may also influence one's brain state; in their study, a positive mood facilitated insightful solving, while an anxious mood enhanced analytical solving. Furthermore, resting-state brain activity predicts individual differences in problem-solving strategies: Participants who tend to rely more on insight exhibit different patterns of prior resting-state electroencephalogram (EEG) brain activity than those who tend to rely on analysis (Kounios et al., 2008). In sum, neuroimaging findings are consistent with the idea that mindset changes via temporal construal priming could have a significant influence on cognitive style.
A behavioral study by Förster et al. (2004) suggested that temporal construal priming influences problem-solving style. Specifically, they hypothesized that high-level construals utilized to imagine the distant future would promote insightful problem solving and that low-level construals utilized to imagine the near future would promote analytical solving. In a series of experiments, participants were asked to both imagine their life in general and imagine solving the subsequent task either in the distant future (1 year from the present day) or the near future (the next day). They reported that individuals asked to think about the distant future solved more insight problems, performed better on a creativity task, and performed worse on an analytical task. Often in creative problem-solving, one must overcome a cognitive fixation on how they assume the problem should be solved to restructure the problem in a novel manner (Smith, 1995). This fixation would be more difficult to overcome if a problem is presented in a greater level of detail, as might be expected for concrete, low-level construal. Research has supported this—when individuals were given examples on how to solve a problem, they were less likely to produce novel solutions than participants who were not provided with examples (Marsh et al., 1999). Therefore, it seems intuitive that high-level, abstract construal would benefit insight, as Förster and colleagues hypothesized. Indeed, previous research has shown that approaching a problem in a more abstract manner leads to more novel solutions than when the task is approached more concretely (Ward et al., 2004).
However, studies in which a specific mindset is primed in order to observe its effect on subsequent behavior have proven difficult to replicate (e.g., Gong and Medin, 2012; Pashler et al., 2012; Shanks et al., 2013). This report examines the consequences of mindset priming for problem-solving style. In particular, we applied a new analytic approach to investigate the time-course of the effects of future thought priming on analytical solving versus solving by insight. The present study had two main goals. The first was to test whether distant prospection benefits insight while more proximal prospection benefits analytical thinking, as suggested by Förster et al. (2004). Because of recent concerns about the replicability of social priming studies (Kahneman, 2012) we deemed it worthwhile to examine this issue.
Second, we implemented several methodological refinements to better isolate and elucidate the effects of priming. Förster et al. (2004) tasked their participants with solving both verbal and visual insight problems but did not verify whether their participants actually solved these problems with insight. Insight research has shown that just because a person has solved a so-called "insight problem" does not mean that he or she solved it with insight (Kounios and Beeman, 2014; Danek et al., 2016). We used the insight judgment procedure developed by Bowden et al. (2005) to determine which problems were solved insightfully and which were solved analytically. Instead of using classic insight problems which take participants a considerable amount of time to solve (when they are able to solve them), we used compound remote associates (CRA) problems, verbal puzzles which can be solved in less than 15 s and which have a long history of use in studying creativity and insight (Bowden and Jung-Beeman, 2003). CRAs are well-defined, convergent problems. Each CRA problem consists of 3 stimulus words that can be combined with a single solution word to form 3 individual compound words or phrases (e.g., horse, plant, over; solution = power: horsepower, power plant, overpower). Importantly, CRA problems can be solved either by insight or analysis. Based on an individual participant's trial-by-trial reports of their solution strategy, insightful and analytical solutions can be sorted and compared. One of the major benefits of this approach is that it allows the experimenter to compare solving strategies while holding constant the type of problem.
Another benefit of using short puzzles is that it allows researchers to trace the time-course of priming effects on solving strategy. One reason that priming effects are difficult to replicate may be because these effects are too short-lived to reliably influence a subsequent task. We were able to assess this possibility by adapting growth curve analysis (GCA) to examine the time-course of temporal construal priming. GCA is a type of multilevel regression that allows for the analysis of the trajectory of time-course data (Mirman, 2014) so that one can examine change in the data over time. Using more traditional statistical methods (e.g., t-tests), one can compare between individual time points. However, these methods provide no information about what is happening across those time points. Using GCA, one can observe the patterns of change that occur across time points.
Growth curve analysis models are developed based on the shape of the data, fixed effects (group-level predictor variables), and random effects (variables that represent individual variability). In many cases, a linear model is a suitable reflection of time-course data. Indeed, if the priming effect persists throughout the experiment, we would expect that a linear model would best fit the data as the primed behavior would remain relatively stable. However, a linear model would not accurately identify the deterioration of priming effects over the course of an experiment. Rather, a quadratic model would successfully reveal this pattern, as one would expect an initial increase in the primed behavior, followed by a decline as the effect decays. Therefore, GCA is a useful analysis that allows for the examination of the nature of the priming effect. If these effects deteriorate over the course of a short experiment, priming researchers should take that into account during future study development. This is particularly important for those who utilize classic insight problems in their research, as these problems may require extensive solving time. Depending upon the number of problems used, the effect of priming may deteriorate before all problems have been solved.
Additionally, to maximize statistical power, we used a within-subject experimental design in which each participant was included in both a near-future and far-future thought condition (in separate counterbalanced sessions). This contrasts with the lower-power between-group design of Förster et al. (2004) and most other social priming studies.
Finally, given that one's brain activity before a problem is presented is known to influence the strategy with which one solves the problem (Kounios and Beeman, 2014), we also measured participants' resting-state electroencephalograms (EEG) between priming phases in order to ascertain how such priming affects ongoing brain activity.
In sum, we tested the effects of temporal-construal priming on problem-solving style (insightful versus analytic) and examined the time-course and neural correlates of the resulting effects.
This study was carried out in accordance with the recommendations of the Drexel University IRB with written informed consent from all the subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the Drexel University IRB. The data are available for download at: https://figshare.com/articles/Temporal_Priming_Creative_Insight/4007745.
Förster et al. (2004) reported large effects of temporal priming. Furthermore, based on past EEG studies with the insight judgment procedure and a within-subject design (e.g., Kounios et al., 2008), we expected that approximately 25 participants would yield good statistical power for analyses of both the behavioral and EEG data. Given expected participant exclusions due to EEG artifacts, low problem-solving accuracy, failure to follow instructions, and participant withdrawals, we recruited 38 participants.
All participants were right-handed, had no self-reported neurological disorders or psychiatric conditions, and refrained from taking substances that might affect cognition (i.e., alcohol, psychoactive medications, or recreational drugs) for 24 h prior to the experiment. We excluded 2 participants who did not produce at least 1 solution of each type (insight and analytic) because this suggested that they were responding stereotypically or were not following instructions. We also excluded 2 subjects who did not achieve an accuracy lower than 1.5 standard deviations below the sample mean (∼15% accuracy) in solving the problems, 3 due to equipment problems, and 4 who chose not to complete the study. After these exclusions, our final sample included 27 Drexel University students ages 18-30 (M = 22.15, SD = 3.28, 13 females, 13 males, 1 declined to report) who were paid $30 to participate.
Participants completed 2 2-h experimental sessions on different days. During the first session, participants filled out demographic and handedness questionnaires and watched an instructional video during which the experimental procedure was explained and the differences between analytical and insightful problem solving were described. We recorded 5 min of eyes-closed baseline resting-state EEG data during which participants were instructed to let their minds wander. Then, participants were presented with 1 of 4 possible priming scenarios (2 in the near condition and 2 in the far condition, as described below) and asked to write about that scenario for 5 min. After this priming, we recorded 5 min of eyes-closed resting-state EEG data. Because of the documented effects of mood on insight (Subramaniam et al., 2009), participants then completed the Positive and Negative Affect Scale (PANAS). Participants completed another priming scenario (same time-frame) for 5 min to refresh the priming after the EEG recording and PANAS. Following this, participants attempted 72 CRA problems while recording EEG. The second session used the same procedure (Figure 1). Participants who received far-future priming scenarios in the first session received near-future scenarios in the second session, and vice versa.
FIGURE 1. (A) Experimental procedure timeline and (B) CRA procedure.
Priming Scenarios
We used 4 priming scenarios, differing both in content and temporal proximity. The scenarios were restricted to the Philadelphia area to control for potential spatial-distance priming effects. The scenarios are as follows:
• "Imagine that you will be finding a place to live in Philadelphia next week. You have 5 min to write about whatever comes to mind about this."
• "Imagine that you will be finding a new job in Philadelphia next week…"
• "Imagine that you will be finding a place to live in Philadelphia in 10 years…"
• "Imagine that you will be finding a new job in Philadelphia in 10 years…"
Compound Remote Associates
Participants were presented with 144 CRA problems over the course of the study. The assignment of CRA problems to sets was randomized and the sets were counterbalanced between groups. Each set of CRAs was presented to participants in a single random order. The problems were presented using e-Prime 2.0. Eight practice trials were presented before each session. Participants held a mouse in both hands with left and right thumbs placed on the corresponding buttons. A fixation cross was displayed in the center of the screen until participants initiated the presentation of a problem with a bimanual button press. Once participants initiated the problem, crosshairs appeared around the fixation cross for 1000 ms after which the problem appeared. The 3 words of each problem were displayed in a column for 15 s. If a participant was unable to reach a solution, the screen returned to the fixation cross and the trial was terminated. If a participant reached a solution, she or he indicated this with a bimanual button-press. Then, a prompt appeared on the screen, participants verbalized their solution, and the experimenter recorded solution accuracy. Participants were then prompted to press a button to indicate whether they had solved the problem insightfully (i.e., resulting from an "aha" moment in which the solution suddenly intrudes on ongoing thought) or analytically (i.e., in which the solution resulted from deliberate, conscious manipulation of the elements of the problem, as in hypothesis testing; Bowden et al., 2005). If participants were unable to come to a conclusion as to how the problem was solved, they refrained from pressing anything, and the program continued after 4 s.
EEG Recording and Data Processing
Eighty-four channel electroencephalographic data were recorded with tin electrodes embedded in a nylon cap (Electro-Cap International, Eaton, OH, United States) using the MANSCAN EEG recording system (SAM Technology, Inc., San Francisco, CA, United States) and extended 10–20 system locations referenced to digitally linked mastoid electrodes. Data were preprocessed using the EEGLAB toolbox in Matlab 7.14 (Mathworks, Inc., Natick, MA, United States). Bad channels were removed by visual inspection. Data were segmented and filtered using a 1-Hz high-pass and 55-Hz low-pass FIR filter. Movement artifacts were removed using an amplitude threshold ranging from -300 to 300 μV (Hoffman and Falkenstein, 2008). ICA weights were calculated using EEGLAB's FASTICA algorithm and submitted to the ADJUST artifacting tool (Mognon et al., 2011). Previously removed channels were replaced by interpolation. Analyses were conducted in SPM 12's EEG toolbox (Litvak et al., 2011). Fast Fourier transforms (FFT) were calculated from 2 to 55 Hz in frequency steps of 2 Hz (Hamming windowed), robust averaged, and log transformed within session, then transformed into 3D Scalp × Frequency images. Tests were performed with a p < 0.001 cluster-correction threshold.
Behavioral Data Analysis
Growth-curve analysis (Mirman, 2014) was used to analyze change over time in the relative accumulation of solutions over the course of the 72 CRA problems presented during each session. GCA offered information both about the influence of priming on solution type and the time course of this influence. All analyses were undertaken with R version 3.1.1 using the lme4 package (version 1.1-7).
Solution Difference (Insight – Analytical Solutions)
The time-course of changes in solving style (insight versus analysis) was modeled with second-order orthogonal polynomials using fixed effects of priming on all time terms (in all analyses in this report, this refers to the intercept, linear, and quadratic terms) and with participant and participant-by-condition (near versus far priming) random effects on all time terms. In this analysis, the intercept term refers to the average solution difference score, the linear term refers to the change in the solution difference score over time, and the quadratic term captures the curvature of the data—specifically, the increase and then subsequent decrease of the solution difference score over time, or vice versa. The far-priming condition was treated as baseline with parameters being estimated for the near-priming condition. Parameter-specific p-values were estimated using the normal approximation.
Solution Accumulation
The overall time-course for each condition (near versus far priming) was modeled with second-order orthogonal polynomials using fixed effects of solution type on all time terms and with participant and participant-by-solution type (analytical versus insight solution) random effects on all time terms. In this analysis, the intercept term refers to the average number of each solution type, the linear term captures the solution accumulation rate, and the quadratic term reflects the change the rate of solution accumulation over the course of the experiment. Insight solutions were treated as baseline with parameters being estimated for analytical solutions. Parameter specific p-values were estimated using the normal approximation.
Mean Performance
In the far-future priming condition, participants reported an average of 11.30 (SD = 6.03) correct insight solutions and 12.26 (SD = 7.34) correct analytical solutions. They solved 8.44 (SD = 11.41) problems incorrectly, and timed out in 39.44 (SD = 11.61) trials. In the near-future priming condition, participants reported an average of 10.78 (SD = 5.44) correct insight solutions and 10.81 (SD = 5.76) analytical solutions. They solved an average of 10.30 (SD = 14.31) problems incorrectly, and timed out in 38.85 (SD = 14.34) trials. Neither the positive affect (t = -1.00, p = 0.329) nor the negative affect (t = 0.486, p = 0.632) PANAS scores significantly differed between conditions (see Table 1).
TABLE 1. Positive and negative affect scores by condition.
There were no significant differences between priming conditions in terms of total correct solutions (p = 0.106), total incorrect solutions (p = 0.429), and total timeouts (p = 0.897). Data for all of the following models can be found in Table 2.
TABLE 2. Model fit results for each analysis.
Solution Difference-Scores
The effect of priming significantly improved model fit on the quadratic term, χ2 = 12.75, p < 0.001, indicating that a curvilinear model best fits the data. Solution pattern and consistency differed significantly between conditions over the course of the experiment, as reflected by differences in the steepness of the quadratic curvature between the near- and far-future priming conditions. Specifically, participants in the far-future priming condition produced consistently more analytical solutions in the initial stages of the experiment, Estimate = 2.80, SE = 1.12, p = 0.013. Conversely, significance in the opposite direction in the near-future priming condition indicates that participants utilized more insightful solving immediately after priming, Estimate = -5.54, SE = 1.51, p < 0.001. This difference grew smaller as the experiment progressed (Figure 2).
FIGURE 2. Solution difference score by priming type. Model fit of the solution difference score (insight solutions – analytical solutions) by priming type (near-future versus far-future) over the series of CRA problems.
Near-Future Priming Condition
The effect of solution type significantly improved model fit on the quadratic term, χ2 = 7.14, p = 0.008, indicating a curvilinear model as the best fit of the data. Solution type did not significantly affect the intercept or the linear terms, p = 0.599, indicating that there was no significant difference in solution type in the near-priming condition; overall, participants tended to apply analytical and insightful methods about equally often. However, the effect of solution type on the quadratic term reflects differences in the steepness of quadratic curvature between the two conditions. This can be related to solution-type accumulation over time. Specifically, with near-future priming, insights initially accumulated somewhat more rapidly than analytical solutions, Estimate = -2.47, SE = 0.73, p = 0.001. However, the curvature of analytical solutions was also significant, but in the opposite direction, Estimate = 2.75, SE = 0.99, p = 0.006, which suggests that they were mildly suppressed by near-future priming.
Although participants applied roughly equal numbers of insightful and analytical solving methods over the course of the experiment, the rate of accumulation of each solution type differed (Figure 3). Insightful solutions accumulated slightly more rapidly than analytical solutions in the initial portion of the experiment.
FIGURE 3. Near-future priming solution accumulation. Model fit of the accumulated solutions (insight versus analytical) over the series of CRA problems in the near-future priming condition. Although participants used roughly the same number of each solution type, insightful solutions initially accumulated somewhat more rapidly than analytical solutions.
Far-Future Priming Condition
The effect of solution type significantly improved model fit on the quadratic term, χ2 = 8.10, p = 0.004, indicating a curvilinear model as the best fit of the data (Figure 4). As in the near-priming condition, there was no significant difference in solution type, p = 0.306, but, rather, there was a significant difference in the rate of solution accumulation over time, as indicated by the steepness of the curvature in the analytical condition, Estimate = -2.80, SE = 0.95, p = 0.003. Specifically, analytical solutions initially accumulated more rapidly than insights.
FIGURE 4. Far-future priming solution accumulation. Model fit of the accumulated solutions (insight versus analytical) over the series of CRA problems in the far-future priming condition. Although participants used similar numbers of each solution type, analytical solutions initially accumulated more rapidly than insightful solutions.
Similar to the near-future priming condition, participants utilized relatively equal numbers of insightful and analytical solutions over the course of the experiment. However, the rate of accumulation differed. In this condition, analytical solutions accumulated more rapidly than insightful solutions in the initial portion of the experiment.
Resting-State EEG Data
The resting-state EEGs were subjected to frequency-domain analyses. To test for priming differences across all frequency bands (2–50 Hz), a flexible factorial model was created with the factors order (of priming condition) and priming-condition (near- versus far-future conditions). The first contrast tested the main effect of priming-condition in an F-test. No clusters survived at a cluster-forming threshold of p < 0.001. Because in-preparation analysis of other resting state data that we have collected shows that differences in resting-state beta-band oscillations are the strongest predictor of subsequent problem-solving strategy, we performed a focused analysis of priming condition constrained to the beta band (13–30 Hz). Again, no clusters survived at a cluster-forming threshold of p < 0.001. In sum, these analyses revealed no significant brain-activity differences between the near-future and far-future priming conditions after 5 min of priming (Figure 5). Means of the logged beta EEG power values for selected representative electrodes are shown in Table 3.
FIGURE 5. Mean EEG beta-frequency power values in log (μV2) for selected electrode sites by priming condition. All bar charts are scaled 0 to -10 in units of log(μV2), with negative plotted up and error bars reflecting the standard error.
TABLE 3. EEG beta power values for selected electrodes log(μV2).
Research by Förster et al. (2004) indicates that thinking about the distant future promotes both creative processes (such as insight) and creative outputs and suppresses analytical reasoning. Our data contradict this. Distant-future thought primed analytical problem solving while near-future thinking primed insightful solving. Moreover, the shapes of the fitted curves illustrate a deterioration of these priming effects over approximately 30 min (the time course of the stimulus presentation procedure). The priming effect was more pronounced in the far-future priming condition than in the near-future condition. This was not unexpected because the near future is similar to the present. Far-future thought would plausibly induce a greater change in mind-set and a more pronounced priming effect because the far future is comparatively dissimilar to the present.
One possible explanation for the difference between our findings and Förster et al.'s (2004) is that future-thought priming effects may be highly dependent on the specific content of the priming scenarios. For example, our scenarios may have prompted more concrete construals, regardless of priming condition, than those used in the Förster et al. (2004) study. Thinking about detail-oriented tasks such as finding a place to live or finding a job may produce an inherently more low-level construal than thinking about life in general. However, if this were the case, then we might expect predominantly analytical solutions in both priming conditions. This did not occur – near-future priming gave a small temporary boost to insightful solving. Another hypothesis is that the priming scenarios could have induced mood changes strong enough to override temporal-construal priming (Subramaniam et al., 2009). However, the absence of any significant priming effects on the PANAS mood questionnaire results weighs against this hypothesis. Finally, it is possible that the tasks that Förster et al. (2004) used did not tap creativity or insight and that their participants were using analytic thought to accomplish them. Because participants may solve so-called classic insight problems by using analytical methods (Danek et al., 2016), the present study used a method that revealed on a trial-by-trial basis the type of processing that each participant used to solve each CRA problem.
One potential explanation for our findings is that high-level construal, such as thinking about the distant future, may engage executive processes involved in working memory maintenance and inhibition of prepotent long-term memory representations more than low-level construal. Indeed, several studies indicate that imagining a future event draws heavily on working memory and other executive processes required for analytical problem solving (e.g., D'Argembeau et al., 2010; Zavagnin et al., 2016). It is expected that far-future priming would draw more heavily upon these processes than near-future priming because an event in the near future is very similar to an event in the present. Specifically, imagining that you are looking for a job next week is not significantly different than imagining that you are currently looking for a job. The only details that must be retained in working memory are the few slight deviations from one's current situation; namely, that one has to find a job. In contrast, one is likely to assume that things will be quite different 10 years from the present. One may assume that they are married, possibly with children, and may have other family responsibilities or interests. They will likely expect to have different career options than they presently have. Thus, when imagining the distant future, one has to maintain in working memory all of these new features, while inhibiting some features of the present that conflict with those being imagined. In essence, imagining the distant future is likely a more computationally complex simulation than imagining the near-future. These findings, together with research indicating that enhanced analytical problem solving depends on working memory capacity more than insightful problem solving (Fleck, 2008; Wiley and Jarosz, 2012; DeCaro, 2016), lend credence to the idea that thinking about the distant future primes analytical thinking by activating these executive processes.
Interestingly, the temporary facilitation of analytical problem-solving in the distant-future condition did not produce a significant change in the total number of analytical solutions compared to insights. This suggests that not only does the priming's facilitating effect deteriorate, but analytical solving may actually be suppressed for a short time, as in a rebound effect. Because analytical problem-solving requires deliberate, focused attention (Kahneman, 2011) and because executive processes are susceptible to resource depletion (e.g., van der Linden et al., 2003; Persson et al., 2007), it is plausible that a rebound effect may occur due to cognitive fatigue from sustained analytical thought. This rebound effect is not as robust in the near-future priming condition, which may be in keeping with the idea that insightful problem-solving is largely unconscious (Fleck, 2008), and would plausibly induce less cognitive fatigue. However, it may also be less robust because the effect of near-priming is weaker in general.
Regarding the temporary nature of the priming effect, there are two important implications. The relative brevity of such effects may be responsible for some previous failures to replicate social priming effects if the test phases of those experiments were either too long or too delayed after a weak priming phase. Indeed, had we examined behavioral priming effects averaged over the session rather than analyzing the time-courses of these priming effects, we could have missed them altogether. This is consistent with other recent research that suggests that priming effects of future thought may not be as robust as previously suggested (Stins et al., 2016). Thus, the dynamic properties of priming should be taken into consideration in future studies.
Furthermore, though we observed temporary priming effects on behavior after participants received two 5-min priming sessions, the first 5-min priming phase was insufficient to cause any detectable changes in resting-state brain activity, the likely mediator of priming effects (Kounios et al., 2008). This is likely because the priming duration was too brief. Although behavioral effects could be observed after 10 min of priming, these were short-lived. This indicates that the effect of priming on problem-solving is relatively weak. Some prior priming studies have used short periods of even less-immersive priming, thus decreasing the likelihood of obtaining even subtle effects.
To summarize, growth-curve analysis showed that high-level construals engaged by distant-future thought transiently primed analytical solving while low-level construals engaged by near-future thought transiently primed insightful solving. Further research should investigate whether the direction, duration, and intensity of such priming effects are determined by specific features of the priming scenarios and whether other types of priming are similarly fleeting.
JK devised the conceptual idea for the study. All authors contributed to study design. JA and MK collected pilot data. MT-H and BE collected the data and performed the data processing and analysis. MT-H wrote the manuscript with support from JK.
This work was supported by the National Science Foundation grant 1144976.
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Keywords: creativity, problem solving, temporal construal, growth curve analysis, insight
Citation: Truelove-Hill M, Erickson BA, Anderson J, Kossoyan M and Kounios J (2018) A Growth-Curve Analysis of the Effects of Future-Thought Priming on Insight and Analytical Problem-Solving. Front. Psychol. 9:1311. doi: 10.3389/fpsyg.2018.01311
Received: 16 January 2018; Accepted: 09 July 2018;
Published: 30 July 2018.
Amory H. Danek, Universität Heidelberg, Germany
Saskia Jaarsveld, Technische Universität Kaiserslautern, Germany
Mark Beeman, Northwestern University, United States
Kristin Grunewald, Northwestern University, United States, in collaboration with reviewer MB.
Copyright © 2018 Truelove-Hill, Erickson, Anderson, Kossoyan and Kounios. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Monica Truelove-Hill, mlh349@drexel.edu
This article is part of the Research Topic
Novel Approaches for Studying Creativity in Problem-Solving and Artistic Performance
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redpajama_set_name: RedPajamaCommonCrawl


Last week was slow because I was on a cruise. This left just two days to go shopping. Still, I managed to earn $228 in fees and $160 in reimbursements in those two days.
10 gas stations and one fast food restaurant.
A busy day. I shopped three fast food restaurants, 3 gas stations, 4 mailing centers and a truck stop. This was a reasonable number of shops but they were spread out so I did a lot of driving. I've mentioned before that the last few days of the month can be lucrative because companies are trying to meet deadlines and are willing to offer nice bonuses.


Didn't exercise this morning or last night or in the past 5 years? No problem! You still need to eat. You still deserve to eat.
Is your fitness equipment collecting dust or doubling as an extra place to hang your clothes? It's probably not the right fitness equipment for you, so why not get rid of it this year and have some extra money (by selling it) to do a physical activity you actually enjoy?
Hanukkah has passed and it's almost Christmas which means you've most likely already encountered holiday cookies. If you're lucky, you've been somewhere there's an assortment of cookies. You may be wondering which cookies are "best" to eat.
I was in a yoga class the other day and the instructor wished everyone a happy upcoming Thanksgiving and she cautioned us all to "be careful" about our food choices. Um…. what?! Be careful? Be careful about what? We could assume she meant be careful you don't eat "too much" or eat "bad foods", but I like this new list of the only things you need to be careful about when eating on Thanksgiving. Be careful your food isn't held at improper temperatures, be careful the food you're eating is free of your allergens, and be careful you're not eating according to diet rules.
As American Thanksgiving approaches, you're probably seeing articles telling you which foods to avoid and which are lower calorie and therefore the "better" choice. These traditional "eat this", "not that" posts completely negate a person's personal preferences, hunger, and satisfaction levels.


Today, in NY NY a very significant event in the rug world occurred but few in that rug world knew about it.
An as top of the line as you can get furnishing carpet with some minor distinguished provenance was sold for a bit more than $2,000,000.oo with premium, which makes it, we believe, the most expensive rug yet sold.
We might be wrong about this, as we are not very interested in furnishing carpets and don't follow their prices at auction.
Some might groan at RK's characterization but in reality, that's all this late in the Period, cotton warp and wefted cartoon carpet is – a luxurious technical achievement that has no soul.
OK you might now groan again what gives a carpet soul and why doesn't this one have one?
Let's start with the second half of that question. The weavers of this grand 27 by 11 foot carpet slavishly followed a cartoon and a small one at that.
RK has always found the typical flip-flopping re-use of a cartoon, as this rug and the designs of many other "Classical Carpets" demonstrate, disconcerting to say the least. And although one can admire the perfect math at work here, the big "so what" is all we see and think.
There is no message, no connection to anything other than the designer's skill at tendering a "design" and a weaver's to reproduce it.
Being dated to the late 16th century in the catalog is something we don't buy and would rather propose mid-17th.
We did not go to see it in person, though we know from experience how beautiful these carpets can be, especially when in excellent condition – undamaged pile and color – as supposedly the case here.
But beauty without substance wears quickly thin and rugs of this ilk are, we imagine, great to walk on but, and this we know, nothing to meditate with.
For all those who don't already know, RK only likes rugs one can meditate with, but, let's go on here.
The 2 million price paid today is ten times that paid, in the same auction room by the way, in 1980 when this rug last changed hands.
The lurid tale of how it, and the rest of this 'estate', end up at sothebys is quite a tale but we don't have the energy to gossip about it here.
But we do have the strength to wonder out loud why sotheby did not hype this carpet to the max.
They sure have blown their horns for lesser goods but this, really wonderful and rare furnishing rug, was basically ignored.
We all know the brightest bulbs aren't burning in sothebys carpet dept. but this one should have been a natural for even them.
Plus hali, their often time advert paid mouthpiece, didn't give it a mention, well at least one we saw.
RK suspects the lack of promo wasn't an oversight but part of an agenda that might be almost as lurid as Monsieur Safra's demise, only no-one died here!
In closing let us state two million dollars for a rug is a big deal….but……not really, when many a painting, with probably similar commercialistic parentages (be it Titian, Picasso or Warhol) regularly sells for ten times as much as it made.
Like paintings, there are good rugs, great rugs and then there are masterpieces. Was this rug a masterpiece? We say no but neither was it something to be have been closeted away in an "estate" sale with barely a mention.
And in the scheme of things the "Safra" rug is worth far in excess of the price it brought, as is almost always the case with masterpiece rugs at auction.
Why? Stay tuned we might one day address that fact but for now we appreciate it too much to discuss in public.
Reading the hali "review" of the sale of this carpet entitled "Safra's a Winner for Sotheby's" made RK feel like he'd landed on Mars without his Martian map. Yikes, do they think just by writing from their myopic advertising-driven viewpoint they can change history?
'Scuze us but we'd like to inquire of hali's erstwhile "new" editor ben evans, who probably penned this electronic screed, "where's the goes from"? If it's the Rothschild piece sold in 1999, we'd have to say that market is moving as slowly as frozen maple syrup, not even worth the mention in our opinion. And since the Rothschild piece outsold Safra's such a comparision is a bit short on tooth.
Now why is that ben, aka jimmy olsen, evans? Don't trade buyers use phone bids? Come on benny-boy either level with your readers that you know who bought it or forget offering dumb crystal ball gazing as fact.
By the way, ben, that , "exceptionally decorative, dense, small-scale all-over design" is nothing more than a rather elaborate, and rocco version in our estimation, interpretation of the old mina-khani pattern. And surely not one for the record books, though we do grant this rug is very, very decorative.
Vis-à-vis the consignor, we heard an interesting story about how when Mrs. Safra first saw the auction preview she almost hit the roof because of the sloppy job sotheby's had done arranging her treasures.
Supposedly she started squawking at the staff she'd stop the sale unless something was done immediately. After one of the sotheby's head honchos was called in to appease her, we heard, she then started right in on him.
Well, after the staff spent some hours re-arranging the preview to her direction and satisfaction we guess she was satisfied, the sale did go on.
Did her Michaelian Carpet sell as well as her other masterpieces? We say no, not really, in any true comparative sense.
Carpets are still extremely undervalued and sotheby's mediocre effort, even though it produced a price of 2 million, proved the point in spades. If their advertising campaign failed to bring the carpet's sale to the attention of important rug collectors, who knows how many other interested buyers also remained in the dark about its sale?