\def\CP{\hbox{{$\cal P$}}} 
\def\ltwid{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\documentstyle[aps,multicol,epsf]{revtex}
\begin{document}
\title{Bimodal Diffusion in Power-Law Shear Flows}
\author{E.~Ben-Naim and S.~Redner}
\address{Center for Polymer Studies and Department of Physics, 
Boston University, Boston, MA 02215}
\def\CP{\hbox{{$\cal P$}}} 
\author{D. ben-Avraham}
\address{Department of Physics, Clarkson University, Potsdam, NY 13699}
\maketitle
\begin{abstract}
The motion of dynamically-neutral Brownian particles which are
influenced by a unidirectional velocity field of the form $\vec
v(x,y)=v_0 \vert y\vert{}^{\beta }\, {\rm sgn}(y)\hat x$, with $\beta
\ge 0$, is studied.  Analytic expressions for the two-dimensional
probability distribution are obtained for the special cases $\beta =0$\
and $\beta =1$.  As a function of $\beta$, the longitudinal probability
distribution of displacements exhibits bimodality for $\beta < \beta
_c\, $\ and unimodality otherwise.  A simple effective velocity
approximation is introduced, which provides an integral form for the
longitudinal probability distribution for general $\beta$, and which
predicts the existence of this transition.  A numerical exact
enumeration of the probability distribution yields $\beta_c = 3/4$.  The
power-law model parallels the behavior found for tracer motion in a
class of non-Newtonian fluids, where a unimodal to bimodal transition is
also found to occur.
\end{abstract}
P.A.C.S. Numbers: 05.40.+j, 02.50.+s, 47.50.+d, 47.15.-x
\begin{multicols}{2}
\section{Introduction}
The phenomenon in which a Brownian particle moves faster than
diffusively arises in a number of diverse situations.  In the context
of fluid mechanics [1-3], for example, a dynamically neutral tracer
particle in a linear shear flow, with fluid velocity $\vec
v(x,y)\propto y\hat x$, exhibits a mean-square longitudinal
displacement which grows in time as $\langle x(t)^2\rangle\sim t^3$.
It is also well-known that the probability distribution of
longitudinal displacements for the tracer particles is Gaussian,
$P_L(x,t)\equiv\int P(x,y,t)\, dy\sim e^{{\rm const.}\times
x^2/t^3}$.  In the context of random media, enhanced diffusion can
arise when there is a steady unidirectional convection field whose
magnitude depends stochastically on the transverse coordinate [4-5].
One such example is a system where the magnitude of the longitudinal
velocity is itself determined by a random walk [6].  In a typical
realization of this "random walk" shear flow, the longitudinal
velocity at transverse co-ordinate $y$ increases as $y^{1/2}$, a
feature which leads to faster than ballistic motion of a tracer
particle.  The competition between transverse mixing and the large
effective longitudinal steps imposed by the flow field also leads to a
{\it bimodal\/} form for $P_L(x,t)$ in a single random shear flow
configuration.  This intriguing aspect of the system motivates our
present work.

In this paper, we investigate the motion of a dynamically-neutral
tracer particle which moves in a deterministic velocity field, $\vec
v(x,y)= v_0 |y|^{\beta}\, {\rm sgn}(y)\hat x$ (Fig.~1).  This can be
viewed as the average over many configurations of the random walk
shear flow problem.  We find, rather strikingly, that the probability
distribution of longitudinal displacements exhibits a transition from
unimodal to bimodal behavior as $\beta$ is varied.  One of our primary
goals is to elucidate this phenomenon.  A potential application of
this result is that power-law shear flow qualitatively mimics the flow
field of a non-Newtonian fluid under fixed external shear.  In this
latter system, we have also found that there is a transition from a
unimodal to bimodal probability distribution when the flow condition
of the fluid is varied.  Thus the shape of $P_L(x,t)$ may provide
useful information about the flow field of the ambient fluid.

In section II, we discuss some basic scaling properties of $P_L(x,t)$
in power-law shear flow.  The exact solutions for the full
two-dimensional probability distribution in the special cases of
$\beta=0$ and $\beta=1$ are then presented in the next two sections.
In the case of ``split-flow'', $\beta=0$, our solution represents the
extension of the classical arcsine law [7], which is equivalent to the
distribution of longitudinal displacements, to the full distribution
in the plane.  We then describe a new way to obtain the probability
distribution for the case of linear shear flow, $\beta=1$.  While the
exact solution in the continuum limit is well-known [1-3], our
simple approach for the discrete case clearly exhibits the connection
between Brownian motion in linear shear flow, and a temporally
inhomogeneous diffusion process, where the diffusion coefficient grows
as $t^2$.  In section V, we present an effective velocity
approximation for writing $P_L(x,t)$ for general $\beta$, which
predicts a unimodal to bimodal transition at $\beta=8/9$.  In section
VI, numerical results from exact enumeration are given, which suggest
that the critical value of $\beta$ is 3/4.  In section VII, we present
exact enumerations for the qualitatively similar problem of tracer
motion in the flow field of a non-Newtonian fluid under a constant
external shear stress.  A unimodal to bimodal transition is again
found as the flow conditions of the fluid is varied.

\section{Scaling Properties}

For a fluid in which the flow field is $\vec v(x,y)=v_0\,|y|^\beta{\rm
sgn}\,(y)\hat x$, the probability distribution of an immersed
dynamically-neutral Brownian particle obeys the convection-diffusion
equation
\begin{equation}  
{\partial P(x,y,t)\over \partial t}+{\rm sgn}(y)|y|^\beta v_0
{\partial P(x,y,t)\over \partial x}=D{\partial^2 P(x,y,t)\over\partial y^2},
\end{equation}
where the subdominant contribution of diffusion in the longitudinal
$(x)$ direction has been neglected at the outset.  Our primary goal is
to determine the distribution of longitudinal displacements,
$P_L(x,t)\equiv\int P(x,y,t)\,dy$, with the initial conditions $P(\vec
r,t=0)=\delta(0)$.  The transverse displacement behaves in a purely
diffusive manner, $y\sim\sqrt{Dt}$, as can immediately be confirmed by
integration of Eq.~(1) over $x$.  On the other hand, the root-mean
square longitudinal displacement, $x_{\rm rms}$ may be roughly estimated by
noting that the longitudinal velocity at time $t$ is $v_x(y(t))\sim
v_0(Dt)^{\beta/2}$.  Consequently
\begin{equation}
x_{\rm rms}\sim (v_0t)\,(Dt)^{\beta /2},
\end{equation} 
with the correlation length exponent $\nu=1+\beta/2$.  The increase in
longitudinal velocity with time scale is the underlying mechanism which
leads to $x_{\rm rms}$ growing faster than linear in time.  It is interesting
to notice that for $\beta =0$, $x$\ is independent of the diffusion
coefficient $D$.

For describing the probability distribution, it will be convenient
to introduce the scaled longitudinal and transverse displacements,
\begin{equation}
\xi=x/(v_0t)\,(Dt)^{\beta/2}\qquad
\hbox{ and}\qquad \eta=y/\sqrt{Dt},
\end{equation}
respectively.  In terms of these scaled variables, we may also write the
probability distribution in the following convenient scaling form,
\begin{equation}
f(\xi,\eta)\equiv (v_0t)\,(Dt)^{(1+\beta)/2}\,P(x,y,t).
\end{equation}
>From this, we also define the scaled form of the longitudinal
probability distribution as $f_L(\xi)\equiv\int f(\xi,\eta)\,d\eta$.
We expect that this function has the asymptotic behaviors
$f_L(\xi)\to{\rm const.}$ as $\xi\to 0$, and $f_L(\xi)\sim
e^{-\xi^\delta}$, as $\xi\to\infty$.  We can immediately deduce the
value of the large-distance shape exponent, $\delta$, by constructing
a rough estimate for the probability of finding the extreme walks that
contribute to the tail of the distribution [7-8].  For power-law shear
flow, the longitudinally stretched walks must have each transverse
step in the same direction, in order that the walk has the largest
possible velocity at each time step.  This implies that the
probability of finding a stretched walk decays as a pure exponential
in $t$, $e^{-\alpha t}$.  On the other hand, a stretched walk has a
longitudinal displacement which scales as $x_{\rm max}(t)\sim\int^t
y^\beta\,dy\sim t^{1+\beta}$.  This maximal displacement corresponds
to a scaled displacement $\xi\sim t^{\beta/2}$, and correspondingly,
$f_L(\xi)\sim e^{-t^{\delta\beta/2}}$.  Since we have argued that
this probability decays as a pure exponential in $t$, we conclude that
$\delta=2/\beta$.  Using $\nu=1+\beta/2$, the expression for $\delta$
can be written as $\delta=|1-\nu|^{-1}$.  This is of the same form as
the classical Fisher relation [9] between the shape and size
exponents, $\delta=(1-\nu)^{-1}$ for the usual situation where
$\nu<1$.

\section{Split Flow}
   
In this section we present an exact solution for the two-dimensional
probability distribution of displacements in the case of ``split-flow'',
where the velocity field is $\vec v(x,y)=v_0\,{\rm sgn}(y)\,\hat x$
(Fig.~2).  Within a discrete version of the problem, the transverse
behavior is simply a symmetric random walk, while the longitudinal
displacement is proportional to the difference, $n_+\!-n_-$, where $n_+$
and $n_-$ are the number of steps that the transverse random walk spends in the
upper- and lower-half planes, respectively.  Since $n_++n_-$ equals the
total number of steps $n$, the probability distribution in the plane can
be found once we determine $\CP_n(y,n_+)$, the probability that an
$n$-step random walk is at transverse position $y$ and has spent
$n_+$\ steps in the upper half-plane.

Consider first, the special case of $\CP_n(y=0,n_+)$, the probability
that an $n$-step walk, which starts and ends at $y=0$, spends $n_+$
steps in the upper half-plane.  Rather surprisingly, this probability is
{\it independent\/} of $n_+$ [10].
\begin{equation}
\CP_n(y\!=\! 0,n_+)\sim \sqrt{1/ 2\pi n^3}.
\end{equation}
To obtain the corresponding probability for arbitrary $y$, it is helpful
to visualize the individual random paths that contribute to this
probability as the sequence of points $\{i,y_i\}$ in a space-time
representation (Fig.~3).  For a walk which starts at $\{0,0\}$ and ends
at $\{n,y_n\}$ there will be a ``break point'', which is the last time
that the walk passes through the axis $y=0$.  Without loss of
generality, we need only consider the case $y_n> 0$.  We define the
break point to occur at time step $n-k$, and divide the trajectory from
$\{0,0\}$ to $\{n,y_n\}$ into a ``return'' segment, from $\{0,0\}$ to
$\{n-k,0\}$, and a ``first passage'' segment, from $\{n-k,0\}$ to
$\{n,y_n\}$.  The probability for the full trajectory can then be
computed as the convolution of the probabilities of these two segments.

The initial return segment takes $n-k$ steps, and by construction,
$n_+-k$ of these lie in the upper half plane.  Consequently, the
probability for this segment is $\CP_{n-k}(y\!=\!0,n_+\!-\!k)$.  By
construction, the subsequent $k$-step segment touches $y=0$ only once.
Hence the transverse co-ordinate of the time-reversed segment is exactly a
$k$-step first-passage trajectory from $y=y_n$ to $y=0$, and the
probability for this event is [10], 
\begin{equation}
f_k(y)\sim y\sqrt{2/\pi k^3}\,e^{-y^2/2k}.
\end{equation}
The convolution of these two probabilities yields
\begin{eqnarray}
\CP_n(y,n_+) &=& \sum _{k=y}^{n_+} \CP_{n-k}(0,{n_+}\! -\! k)\, f_k(y)
\nonumber\\ 
&\sim & \int\limits_{k=y}^{n_+}dk{y \over {\pi\sqrt{k(n-k)}^3}}\,e^{-y^2/2k}
\end{eqnarray} 
The lower limit of $k$\ is $y$, since the walker must spend at least
$y$\ steps in the first-passage segment of the trajectory in order to
reach transverse position $y$.  Changing variables to $u=y^2/2k$,
defining $x=v_0(n_+\!-\!n_-)$, and replacing $n$ by $t$, the integral
can be performed.  From this result we deduce that
\begin{eqnarray}
f(\xi,\eta )&\sim&
{1\! -\! \eta ^2\over \sqrt{2\pi}}\,{\rm erfc}\left(\eta\sqrt{{1-\xi\over 
2(1+\xi)}}\right)\, 
e^{-\eta^2 /2}+\, \nonumber\\
&&{\eta\over \pi} \sqrt{{1+\xi\over1-\xi}}\
e^{-\eta^2 /1+\xi}\, 
\end{eqnarray} 
where $\xi$\ and $\eta $ are the scaled longitudinal and transverse
coordinates defined in Eq.~(3) (note that $|\xi|\! <\! 1$).  

It is instructive to examine the asymptotic behaviors of this
probability distribution.  Consider first the behavior of longitudinal
slices, {\it i.e.}, fix $\eta$ and vary $\xi$.  For $\eta\ll 1$ and
$-1+2\eta^2\!<\!\xi\!<\!1$,  Eq.~(8) reduces to
\begin{equation}
f(\xi,\eta\to 0)\sim {1\over\sqrt{2\pi}}+{2\eta
\xi\over\pi\sqrt{1-\xi^2}}.
\end{equation} 
Indeed, this function is independent of $\xi$ when $\eta\!=\!0$.  For
constant positive $\eta$, the profiles of $f(\xi,\eta)$ are sharply
peaked at $\xi=1$, as determined by the leading behavior of $\simeq
1/\sqrt{1-\xi}$.  For $\xi\to -1$, however, the probability decays
exponentially as $e^{-\eta^2/1+\xi}$ (see Fig.~4(a)).  Note also that
the area of the constant-$\eta$ profiles is proportional to $\exp
(-\eta^2/2)$, as expected.  For fixed non-zero $\xi$, the profiles of
$f(\xi,\eta)$ are peaked at a non zero value of $\eta$ which depends on
$\xi$ (Fig.~4(b)).  More surprisingly, note that the area of the
profiles for fixed $\xi$ increases as $\xi$ increases, as required by
the arcsine law.

The longitudinal probability
distribution is obtained by integrating Eq.~(8) over $\eta$ and gives
the arcsine law [10]:
\begin{equation} 
f_L (\xi)\sim {1 \over {\pi \sqrt{1-\xi^2}}} 
\end{equation} 
In addition to being bimodal, the distribution is singular at the
extrema $|\xi|=1$. Thus the split flow velocity field, which corresponds
to the case $\beta=0$ in power-law shear flow, gives rise to a
probability distribution with an extreme degree of
bimodality(Fig. 4(c)).

\section{Linear Shear Flow}

A well-known, and exactly-soluble [1-3] limit of power-law shear is
the case of linear shear flow.  Here we present an alternative method of
solution for the discrete version of diffusion in linear shear which
provides useful insights into the nature of the resulting longitudinal
motion.

We define the discrete limit of linear shear flow as a random walk which
moves from $(x_k,y_k)$ to $(x_k+ v_x(y_k+\epsilon_k),y_k+\epsilon_k)$ at
the $k^{\rm th}$ step.  Here $\epsilon_k$ is a random variable that describes
the individual transverse hop at the $k^{\rm th}$ step and is governed by the
time independent probability distribution $p_y(\epsilon_k)$.  Since
$v_x(y_k)=v_0y_k$ in linear shear flow, the two-dimensional displacement
after $n$ steps can be written as
\begin{eqnarray}
(x_n/v_0,y_n)&=&(y_1+y_2+\cdots+y_n\>,\>\epsilon_1+\epsilon_2
+\cdots+\epsilon_n),\nonumber\\
&=&(\epsilon_1+(\epsilon_1+\epsilon_2)+\cdots
,\epsilon_1\>+\>\epsilon_2+\cdots+\epsilon_n),
\nonumber\\ 
&=&\epsilon_1(n,1)+\epsilon_2(n\! -\!1,1)+\cdots+\epsilon_n(1,1).
\end{eqnarray}
To analyze this sum, it is helpful to reverse the order of the
$\epsilon_k$'s.  Since their distribution is independent of time, this
redefinition does not affect the resulting two-dimensional distribution.
Consequently, we may write the displacement after $n$ steps as
\begin{equation}
(x_n,y_n)=\epsilon_1(v_0,1)+\epsilon_2(2v_0,1)+\cdots
+\epsilon_n(nv_0,1).
\end{equation} 
According to this expression, the longitudinal displacement in linear
shear flow is equivalent to a one-dimensional random walk in which the
magnitude of the $k^{\rm th}$ step is simply proportional to $k$.  This
fundamental relation is the key that allows us to obtain the full
two-dimensional probability distribution in linear shear flow by
elementary methods.
  
For this purpose, we introduce the structure function [11]
\begin{equation}
\Gamma_n (\vec\theta\>) =\sum_{\vec r}
P_n({\vec r}\> ) e^{ i{\vec r\cdot\vec\theta}\> } .
\end{equation}
where $P_n ({\vec r}\,)$\ is the probability that an $n$-step random
walk is at $\vec r$.  According to the convolution theorem,
\begin{equation}
\Gamma_n({\vec \theta}\>)
=\prod_{k=1}^n
\lambda_k(\vec\theta\>) ,
\end{equation}
where $\lambda_k(\vec\theta\,)$ is the single-step structure function,
\begin{equation}
\lambda_k
({\vec\theta}\,) =
\sum_{\vec r} p_k ({\vec r}\,)e^{ i{\vec
r\cdot\vec\theta}\, },
\end{equation}
and $p_k({\vec r}\,)$ is the single step probability distribution at the
$k^{\rm th}$ time step.  Our discussion has thus far been applicable to any
distribution of the $\epsilon_k$'s, and we now specialize to the special
case of a symmetric random walk in the $y$ direction.  Thus at the $k^{\rm th}$
step, the random walk moves by an amount $\pm(kv_0,1)$, each with
probability 1/2 (Fig.~5).  Therefore, the corresponding single-step
structure function is,
$\lambda_k(\theta_x,\theta_y)=\cos(k\theta_x\!+\!\theta_y)$.  Evaluating
the product in 1 gives the structure function
\begin{equation}
\Gamma_n(\vec\theta\>
 )=\cos(v_0\theta_x +\theta_y )
\cos(2v_0\theta_x+\theta_y)\cdots
\cos(nv_0\theta_x+\theta_y).
\end{equation}
Moments of the probability distribution can be found by
appropriate differentiation of the structure function,
1.  This calculation gives
\begin{equation}
\langle{x_n^j y_n^k}\rangle = i^{j+k}\,{\partial^{j+k}\,\Gamma_{\!
n}(\vec\theta\>)\over
\partial\theta_x^j\partial\theta_y^k}\,\biggl\vert_{\theta=0}.
\end{equation}
From this, we find, for example
\begin{equation}
\langle x_n^2 \rangle\sim v_0^2n^3/3,\qquad\qquad\qquad 
\langle x_n y_n\rangle\sim v_0n^2/2.  
\end{equation}
Thus the mean-square longitudinal displacement in linear shear grows as
$t^3$, and perhaps less well-appreciated, the longitudinal and
transverse displacements are coupled, so that cross-correlations are
non-zero.

Finally, the probability distribution $P_n(\vec r\,)$\ can be easily
extracted from 2 by the standard procedure of expanding
this expression for small $\theta_x$ and $\theta_y$, re-exponentiating,
and then performing an inverse Fourier transform of the result.  This
gives the long time limit of the probability distribution as
\begin{equation}
P_n(x,y)\sim {\sqrt{3}\over \pi\, v_0\, n^2} 
e^{-y^2/2n}\,e^{-6(x-\langle{v}\rangle n)^2/v_0^2n^3},
\end{equation}
where $\langle{v}\rangle=v_0y/2$ is the average velocity of the transverse
coordinate in the range $[0,y]$.  For fixed $y$ the probability profile
is a Gaussian function which is peaked at $x=\langle{v}\rangle n$ 
and whose width grows as $t^{3/2}$. 

An advantageous consequence of our method of solution is that it exposes
the basic equivalence between convection-diffusion in linear shear flow
and a temporally inhomogeneous random walk process in which the step
length grows linearly in time.  In the continuum limit, this leads to a
diffusive process with a time dependent diffusion coefficient $D(t)=
\Delta x(t)^2/\Delta t=D(v_0t)^2$.  Consequently, the longitudinal
probability distribution obeys,
\begin{equation}
{\partial P_L(x,t) \over \partial t}=D(v_0t)^2\,{\partial^2 P_L(x,t)\over
\partial x^2}.
\end{equation}
The solution to this equation is the Gaussian $f_L(\xi)\sim
\sqrt{3/2\pi}e^{-3\xi^2/2}$ (following the notations of Eqs.~(3) and
(4)), a result which also follows directly from 1.  Since the
distribution is a pure Gaussian, the large distance tail of this
distribution which is predicted by the argument given in Sec.\ II, is
actually valid over the entire range of $\xi$.

\section{Effective Velocity Approximation}

Since the exact expression for $P_L(x,t)$ is bimodal for $\beta=0$ and
unimodal for $\beta=1$, the distribution must change between these two
shapes as $\beta$ varies between 0 and 1.  We now wish to determine
the location and the nature of this shape transition.  To derive the
functional form of $P_L(x,t)$ for general $\beta$, we introduce the
following effective velocity approximation [6]. We hypothesize that at
a fixed value of $y=y_0$, the probability distribution $P(x,y_0,t)$ is
given by the Gaussian form of 1 , but with the average
longitudinal velocity, $\langle{v}\rangle$, now proportional to
$y^\beta$.  This may be realized by allowing the parameter $v_0$ to
acquire a dependence on $y$ which is determined by
$\langle{v(y)}\rangle\propto v_0(y)y=y^\beta$, or,
\begin{equation}
v_0(y)\propto y^{\beta-1}.
\end{equation}
With this hypothesis, 3 predicts that at fixed $y=y_0$, the
longitudinal dispersion, $\delta x(y_0)$, is proportional to $y_0^{\beta
-1}$, a result which we have also confirmed numerically.  Thus for
$0\leq\beta<1$, the relative longitudinal width decreases with $y_0$,
although the velocity increases in $y_0$.

In our effective velocity approximation we use the two-dimensional Gaussian
distribution, 2, together with the velocity given in
, and integrate over the transverse coordinate to obtain an
approximate form for $P_L(x,t)$.  In performing these steps, it is
convenient to re-express the integrand in terms of the scaled
coordinates $\xi$ and $\eta$ and use symmetry of the integrand in $\eta$
to write the integral over positive $\eta$ only.  This computation gives
(dropping all irrelevant numerical factors),
\begin{equation}
f_L(\xi)\sim \int\limits_{\eta =0}^\infty d\eta\,
\eta^{1-\beta}\,e^{-2\eta^2}
e^{-6\xi^2\eta^{2-2\beta}}\cosh(6\xi\eta ^{2-\beta}).
\end{equation}
For large $\xi$, the asymptotic behavior of the integral is determined by
the maximum of the function in the exponential at $\eta_0 =\xi^{1/\beta}$.
Performing the integral by the Laplace method yields the large distance
behavior
\begin{equation}
f_L(\xi)\sim e^{-\xi^{2/\beta}},
\end{equation}
which gives the same shape exponent as that predicted by the naive
argument of Sec.\ II.

The transition point between bimodality and unimodality 
can be determined by evaluating the second derivative of
1 at $\xi\!=\!0$.  Omitting irrelevant factors, this
calculation yields
\begin{eqnarray}
{\partial^2\over \partial \xi^2}f_L(\xi)\biggl |_{\xi=0}\,
&\propto&
\int\limits_{\eta =0}^{\infty} d\eta e^{-2\eta^2} 
[3\eta^{5-3\beta}-\eta^{3-3\beta}]\nonumber\\
&\propto& 
\Gamma \left({4-3\beta\over
2}\right)\left[{8\over 9}-\beta\right],
\end{eqnarray}
where the last expression is valid only for $\beta<4/3$.  For
$\beta>4/3$, the second derivative of $f_L(\xi)$ at $\xi=0$ diverges,
indicative of a cusp in the distribution function at the origin.  The
effective velocity approximation therefore predicts that the transition
point between unimodality and bimodality occurs at $\beta_c=8/9$,
compared to our numerical estimate $\beta_c=3/4$ (see below).

Similar qualitative results are obtained if we use an effective velocity
approximation with the two-dimensional distribution of split-flow as the
basis for writing the general $\beta$ distribution.  In this case, the
amplitude of the velocity $v_0$ must now acquire the $y$ dependence
$v_0(y)=y^\beta$, in order to obtain the appropriate magnitude of the
longitudinal velocity at $y=y_0$.  Using this result, and following step
by step the reasoning that led to 2, we obtain an
expression for $f_L(\xi)$ which is characterized by a shape exponent
$\delta=2/\beta$ and the longitudinal dispersion $\delta x(y_0)\sim
y_0^{\beta-1}$, in agreement with the results obtained by using the
distribution of linear shear as the basis for the effective velocity
approximation. The analytic expression that determines the critical
value of $\beta$ is complicated and numerically we estimate that
$\beta_c\approx 0.5$.  Thus both effective velocity approximations
predict qualitatively similar properties for the longitudinal
distribution.

\section{Exact Enumeration of the Probability Distribution}

To test our predictions about the nature of the longitudinal probability
distribution, we now present the results of a numerical exact
enumeration [12] in power-law shear flow.  At the start
of the enumeration, there is a unit probability at the origin, and we
evolve the probability distribution according to the following recursion
relation
\begin{equation}
P_{n+1}(x,y)=\sum_{y'=y\pm1} \,{1\over 2}\,P_n(x-v_x(y'),y').
\end{equation}
This evolution process takes into account both transverse diffusion and
longitudinal convection.  In the case of a non-integer value for the
velocity, this recursion formula would lead to probability being
propagated to positions between lattice sites.  When this occurs, the
interstitial probability element is split longitudinally between the two
nearest-neighbor sites, with the relative weight of each component
ensuring that the correct average displacement occurs.

Over the temporal range of our enumeration (up to 128 time steps), the
longitudinal distribution is a scaling function of $\xi$ to a high
degree of accuracy.  This verification of scaling thus excludes the
possibility that the longitudinal distribution could undergo a unimodal
to bimodal transition as a function of time.  Consequently, the
numerical determination of $\beta_c$ is relatively unambiguous.  In
Fig.~6(a) we plot the scaled probabilities {\it v.s.}\ the scaled coordinate
for various values of $\beta$.  At $\beta=1$, the distribution is a
Gaussian, and as $\beta$ is decreased the distribution develops broader
shoulders.  As $\beta\to\beta_c$, the distribution becomes flat at the
maximum, indicative of the vanishing of the second derivative at
$\xi=0$.  Numerically, we find the critical value of $\beta$ to be
$0.75\pm 0.01$.  As $\beta$ is decreased beyond this point, the
bimodality of the distribution becomes more pronounced, and ultimately a
singularity develops at $|\xi|=1$ when $\beta=0$.  Qualitatively, these
features mirror our theoretical predictions.

\section{An Application to Non-Newtonian Fluids}

A potential application of the abstract power-law shear flow problem is
to Brownian motion of a dynamically-neutral tracer particle in a class
non-Newtonian fluids, where the shear (assumed to be along
the $y$ direction) and the force along $x$, $\tau_{xy}$, are related by,
\begin{equation}
\tau_{xy}\propto \left({dv_x \over dy}\right)^n. 
\end{equation}
Thus the exponent $n$ quantifies the non-linearity of the
fluid [13];
for $n=1$, one recovers a Newtonian fluid,
while if $n\ne 1$, one has a ``power law'' fluid.  Under the application
of a shear, a power-law fluid develops a flow field in which the
longitudinal velocity has a functional dependence on the transverse
coordinate which qualitatively resembles that of power-law shear flow.
Hence, we might expect that tracer motion in a non-Newtonian fluid will
exhibit the same rich spectrum of behavior as in power-law shear.  In
particular, by varying the exponent $n$, or equivalently, by varying
external parameters of the flow itself, it should be possible to obtain
both bimodal and unimodal distributions of longitudinal displacements
for tracer particles.

For example, consider the case of plane Couette flow.  For two parallel
planes at $y=-L$, and $y=L$, moving at velocities $-v_0$\ and
$+v_0$, respectively, along the $x$ direction, the steady state
velocity profile is 
\begin{equation}
v_x(y)=v_0\,{\rm sgn}(y)\,
{{\lambda^m-(\lambda-|y|)^m}\over{\lambda^m-(\lambda-L)^m}},
\end{equation}
where $m=1+1/n$, and $\lambda$ is a constant with the dimension of a
length whose value depends on details such as the external pressure
gradient, velocity at the boundary, {\it etc.} Qualitatively, this velocity
profile is very similar to that in power-law shear flow (Fig.~7), and
there is a similar correspondence between the probability distributions
(Fig.~6(b)).  The analogy with power-law shear flow should hold as long
as $\sqrt{Dt}<L$, as the influence of the transverse boundaries of the
system are irrelevant within this temporal range.  Our enumeration for
the case $\lambda =L=32$\ after 32 time steps exhibits a transition from
bimodality at small $n$ to unimodality when $n_c\approx 1/7$.  This
transition can also be produced by fixing the value of $n$ and varying
$\lambda$.  For example, the situation where $n=1/7$ and $\lambda\gg L$
corresponds to linear shear, where the probability distribution is
unimodal.  Conversely, the case where $n=1/7$ and $\lambda<32$ should
lead to a bimodal distribution.  Thus the transition can be obtained
either by tuning $n$, the non-linearity exponent of the fluid, or by
tuning $\lambda$, which can be achieved by experimentally-controllable
parameters contained in $\lambda$.  Therefore, the unimodal to bimodal
transition is an aspect of transport in non-Newtonian fluids which
should be amenable to experimental observation.

\section{Conclusions}

We have elucidated some unusual features of the motion of
dynamically-neutral tracer particles which are carried by a power-law
shear flow, where $\vec v(x,y)=v_0|y|^\beta{\rm sgn}(y)\hat x$.  Our main
result is that the probability distribution of longitudinal
displacements, $P_L(x,t)$, is bimodal for small $\beta$ and unimodal for
larger $\beta$.  From a practical viewpoint, this result may apply
to tracer motion in a non-Newtonian fluid.  The existence of this
transition follows from consideration of the special cases $\beta=0$
(split flow), where the distribution is bimodal, and $\beta=1$ (linear
shear flow), where the distribution is unimodal.  For these cases, we
also obtained the exact expressions for the two-dimensional probability
distributions.  

These results which serve as a starting point for writing an effective
velocity approximation for $P_L(x,t)$ for general $\beta$, an approach
which appears to capture the essential mechanism underlying the
unimodal to bimodal transition.  This mechanism can be appreciated by
viewing the full longitudinal distribution, $P_L(x,t)$, as a
superposition of longitudinal distributions for each layer with fixed
$y=y_0$.  Each component layer distribution is characterized by its
average position, $\langle x(y)\rangle$, its width, $\delta x(y)$, and
its weight, $e^{-y^2/t}$.  The competition between these three
factors, as the component distributions for different values of $y$
are superposed, governs the shape of $P_L(x,t)$.

For the case of small $\beta$, $\langle{x(y)}\rangle$ rapidly increases as a
function of $y$, while $\delta x(y)$ is decreasing in $y$.  These
features lead to a bimodal form of $P_L(x,t)$ when the superposition
of layer distributions is performed.  On the other hand, for
$\beta\ltwid 1$, $\langle x(y)\rangle$ increases nearly linearly in
$y$, while $\delta x(y)$ is nearly independent of $y$.  Consequently,
the superposition of distributions for each layer should lead to a
unimodal shape.  The effective velocity approximation involves taking
either the two-dimensional distribution of split flow or linear shear
flow, and incorporating a physically-motivated functional form for
$\langle x(y)\rangle$ which concomitantly determines $\delta x(y)$.
The competition between these two quantities when the superposition
over all layers is performed provides a simple and reasonably accurate
description of the unimodal to bimodal transition.


We thank F. Leyvraz for numerous useful discussions during the course of
this work.  We also gratefully acknowledge financial support through
grants from the ARO \& NSF (SR), and NSF \& PRF (DbA).

\begin{thebibliography}{99}


\bibitem{} G. K.~Batchelor, {\sl J. Fluid Mech.} {\bf 95}, 369 (1979).
\bibitem{} R. T.~Foister and T. G. M. Van De Ven, {\sl J. Fluid Mech.} 
{\bf 96}, 105 (1980). 
\bibitem{} R. Mauri and S. Haber, {\sl SIAM J. Appl. Math.}, 
{\bf 46}, 49 (1986).
\bibitem{} G.~Matheron and G.~de Marsily, 
{\sl Water Resources Res.} {\bf 16}, 901 (1980).
\bibitem{} J.~P.~Bouchaud, A.~Georges, J.~Koplik, A.~Provata, and
S.~Redner, {\sl Phys. Rev. Lett.} {\bf 64}, 2503 (1990).
\bibitem{} D. ben-Avraham, F. Leyvraz, and S. Redner, {\sl Phys. Rev. A} 
{\bf xx} ,xxxx (1992) .
\bibitem{} W.~Feller, {\sl An introduction to Probability Theory and its 
Applications}, Vol 1, (Wiley, New York, 1968).
\bibitem{} G.H.~Weiss and R.J.~Rubin, {\sl Adv. Chem. Phys.} {\bf 52}, 
363 (1983).
\bibitem{} P.~Pincus, {\sl Macromol.} {\bf 9}, 386 (1976).
\bibitem{} P.~G.~deGennes, {\sl Scaling Concepts in Polymer Physics} 
(Cornell University Press, 1979).
\bibitem{} M. E.~Fisher, {\sl J.\ Chem.\ Phys.} {\bf 44}, 616 (1966) .
\bibitem{} S.~Havlin and D.~ben-Avraham, {\sl Adv. Phys.} {\bf 36}, 695 (1987).
\bibitem{}  R.B.~Bird, R.C.~Armstrong and O.~Hassager
{\sl Dynamics of Polymeric Liquids}, Vol 1, Chap.~5 (Wiley, New York, 1977).
\end{thebibliography} 
\section*{Figure Captions}

Fig 1 The velocity profile of power-law shear flow and a
schematic illustration of the trajectory of a dynamically neutral
Brownian particle in this flow field.

Fig 2 The velocity profile of split flow.

Fig 3 Space-time representation of a random walk path which contributes
to $\CP_n(y,n_+)$.  This trajectory is decomposed into a ``return''
portion (- - - -) and a first-passage portion (------), and are
separated by the break point at time step $n-k$.  

Fig 4 Profiles of the two-dimensional probability distribution in split
flow for: (a) fixed $\eta$, {\it i.e.}, $f(\xi,\eta={\rm fixed})$, with the
cases $\eta=0.1$ (solid) and $\eta=1.0$ (dotted) shown; (b) fixed $\xi$,
{\it i.e.}, $f(\xi={\rm fixed},\eta)$, with the cases $\xi=0$ (solid) and 
$\xi=0.8$ (dotted) shown; (c) the arcsine law which gives the
longitudinal probability distribution, $f_L(\xi)$.

Fig 4 The sequence of single step distributions in linear shear flow.

Fig 5 (a) Exact enumeration results for the scaling function $f_L(\xi)$ 
{\it v.s.} $\xi$ in power-law shear flow at 64 time steps for (i) $\beta=0.25$
(dashed), (ii) $\beta=0.75$ (dotted), and (iii) $\beta=1.0$ (solid).
(b) Exact enumeration results at 32 time steps for $P_L(x,t)$ {\it v.s.} $x$ in
the flow field given by Eq.~(26) for a power-law fluid for the cases
of (i) $n=1/14$ (dashed), (ii), $n=1/7$ (dotted), and (iii) $n=1/4$
(solid).

Fig 6 Comparison of the velocity profiles of power-law shear flow 
with $v_0\!=\!1$ and $\beta=0.45$ (dashed), and of a power-law fluid in
Couette flow, (Eq.~(26)), with $v_0\!=\!L\!=\!1$, $\lambda=4$, and
$n=1/9$ (solid).


\end{multicols}
\end{document}






