# Particle dynamics and effective temperature of jammed granular
matter in a slowly sheared 3D Couette cell

Ping Wang  Chaoming Song  Christopher Briscoe  Hernán A. Makse  Levich
Institute and Physics Department, City College of New York, New York, NY
10031, US

###### Abstract

We report experimental measurements of particle dynamics on slowly sheared
granular matter in a three-dimensional (3D) Couette cell. A closely-packed
ensemble of transparent spherical beads is confined by an external pressure
and filled with fluid to match both the density and refractive index of the
beads. This allows us to track tracer particles embedded in the system and
obtain three-dimensional trajectories, $(r(t),\theta(t),z(t))$, as a function
of time. We study the PDF of the vertical and radial displacements, finding
Gaussian and exponential distributions, respectively. For slow shear rates,
the mean-square fluctuations in all three directions are found to be dependent
only on the angular displacement of the Couette cell, $\Delta\theta_{e}$,
$\langle\Delta z^{2}\rangle\sim\Delta\theta_{e}$, $\langle\Delta
r^{2}\rangle\sim\Delta{\theta_{e}}^{\alpha}$,
$\langle\Delta\theta^{2}\rangle\sim\Delta{\theta_{e}}^{\beta}$, where $\alpha$
and $\beta$ are constants. With $\Delta\theta_{e}$ proportional to the time
between measurements, the values of the constants, $\alpha$ and $\beta$, are
found to be sub-diffusive and super-diffusive, respectively. The linear
relation between $\langle\Delta z^{2}\rangle$ and angular displacement implies
a diffusive process, from which we can calculate an “effective temperature”,
$T_{\mbox{\scriptsize eff}}$, in the vertical direction, through a
Fluctuation-Dissipation relation. It is of interest to determine whether these
systems can be described by analogous equilibrium statistical mechanics
concepts such as “effective temperature” and “compactivity”. By studying the
dynamics of tracer particles, we find the effective temperature defined by the
Stokes-Einstein relation to be independent of the tracer particle
characteristic features, such as density and size, and dependent only on the
packing density of the system. For slow shear rate, both the diffusivity and
mobility of tracer particles are proportional to the shear rate, giving rise
to a constant effective temperature, characteristic of the jammed system. We
finally discuss the significance of the existence of $T_{\mbox{\scriptsize
eff}}$ for a statistical mechanics formulation of granular matter.

##  I Introduction

Fluctuation-Dissipation (FD) relations are commonly used in equilibrium
systems, derived from the notion that small perturbations and Brownian
fluctuations produce the same response in a given system landau . Mobility,
the constant of proportionality between a particles drift speed and a constant
external force, is extracted from velocity statistics of particles in a given
system. Diffusivity, calculated from fluctuation displacements of particles in
a system over time, represent the Brownian motion. The temperature of a system
in thermodynamic equilibrium can be extracted from a FD relation, defined as
the ratio of diffusivity and mobility, as is commonly used in the Einstein
relation. In equilibrium this temperature is taken to be the bath temperature.

As studies in granular matter have grown more important within the
environmental and industrial fields, the need to establish a scientific
framework that accurately predicts granular system responses on the continuum
level, beyond merely geometrical features, has also escalated. Granular
matter, when condensed to sufficiently high volume fractions, undergoes a
‘jamming’ transition to the jammed state. The jammed state is defined as the
condition when a many-body system is blocked in a configuration far from
equilibrium, such that relaxation cannot occur within a measurable time-scale.
For granular matter, the jammed state indicates a transition between a solid-
like behavior, and a liquid-like behavior. At high volume fractions, the
physical size of the constituent grains inhibits particle motion, thereby
rendering the system out of equilibrium, and the granular system behaves more
like a solid. Thermal motion does not govern the exploration of states in
jammed granular matter.

Theories proposed by Edwards and collaborators edwards  propose a statistical
mechanics for granular matter based on jamming the constituent grains at a
fixed total volume such that all microscopic jammed states are equally
probable and exhibit ergodicity. The exploration of reversible jammed states
is achieved via an external perturbation such as tapping or shear, not
Brownian motion as in thermal systems. There is an important difference
between reversible jammed states, and states that are only mechanically stable
within certain limits of perturbation magnitude. For example, pouring grains
into a container results in a pile at a particular angle of repose. This
mechanical equilibrium configuration is jammed regularly but not reversibly
jammed because in response to an external perturbation, the constituent
particles will irreversibly rearrange, approaching a truly jammed
configuration. Studying an ensemble of truly jammed, reversible states is
thereby suitable for a plausible application of statistical mechanics under
the present theory. These ensembles, inherently non-equilibrium systems, will
not be governed by the commonly used parameters of equilibrium statistical
mechanics, such as a bath temperature.

In recent studies theoretical mean-field models of glasses ckp  have
introduced the concept of an “effective temperature” as extracted from the FD
relations in non-equilibrium systems. While not equivalent to the equilibrium
bath temperature, the effective temperature reflects a change in the
relaxation time-scale of the system. These non-equilibrium systems extend
beyond glasses, and into granular media, where physical size of the
constituent grains inhibits motion, allowing for jammed systems far from
equilibrium. This concept has been furthered by computer simulations of
granular media and other non-equilibrium soft-matter systems liu3 ; bklm ;
sciortino ; barrat ; ono ; mk . It remains a question whether or not granular
media can be characterized by an effective temperature, thus revealing a
dynamic counterpart to the static “compactivity” as proposed by Edwards
edwards .

Figure 1: (Color online) Picture of experimental set-up. Transparent acrylic
grains and black tracers in a refractive index and density matched solution
are confined between the inner cylinder of radius $5.08$cm and the outer
cylinder of radius $6.67$cm.

Athermal systems require the input of energy by an external source to explore
the effective temperature review . One proposed method of calculating the
effective temperature of a jammed granular system is a slow shearing procedure
howell ; veje ; mueth ; mueth2 ; utter ; nedderman ; drake , leading to the
design of the experiment we present in swm . Slow shearing, at the quasi-
static limit, allows for extrapolation towards an effective temperature of
jammed, static, systems. The jammed system of interest is one of identical,
spherical grains, confined between the two cylindrical walls of a 3D Couette
cell. The grains are further confined by an external pressure in the vertical
direction. The inner cylinder of the Couette cell is slowly rotated to induce
shearing in the system. Tracer particles are inserted in the system, and their
trajectories recorded via multiple cameras surrounding the system. The Couette
cell is partially filled with a refractive index matched fluid to allow for
system transparency. The cylindrical walls are roughened by gluing grains,
identical to those of the bulk, such that crystallization is avoided.

Figure 2: (Color online) Top view of experimental set-up. The outer cylinder
is made of the same material as acrylic grains ($n\simeq 1.49$). Once the
refractive index is matched, light scattering from tracers will refract only
one times on the outer surface of the outer cylinder. A single particle is
captured by two cameras allowing the determination of the 3-dimensional
coordinates of the particle, $(r,\theta,z)$.

The cameras record tracer particle trajectories throughout the bulk, recording
data in cylindrical coordinates, $(r(t),\theta(t),z(t))$. Distributions of
tracer particle displacements are measured in each direction. As gravity is
the external force applicable to the mobility calculation in the current
formalism, only displacements in the z-direction are applied to the FD
relation. Additionally, average velocity profiles are calculated for each
direction, first with constant shear rate, $\dot{\gamma_{e}}$, and further
studied to determine shear rate dependence. Displacement measurements are
further limited to the “constant mobility and diffusivity” (CMD) region,
defined as the narrow range of radial coordinates such that the average
vertical velocity is roughly independent on radial distance. The PDF of
displacement distributions for each direction is presented. Further,
fluctuations in displacement are determined for each cylindrical direction,
and studied as a function of time. Radial displacement fluctuation is found to
be sub-diffusive, while angular displacement fluctuation is found to be super-
diffusive. Vertical displacement fluctuation is purely diffusive within the
time scales of the experiment, allowing for the validity of the FD relations
used herein. All displacement fluctuations are reduced to functions of angular
displacement, and the results are presented. Such relationships permit scaling
of the PDF curves with varying angular displacements due to changing shear
rates.

Utilizing the FD relations presented above, the diffusivity and mobility in
the z-direction are extracted from the tracer particle trajectories and the
effective temperature is realized. This effective temperature is found to be
independent of tracer particle properties, as shown in swm , and further
independent of the slow-shear rate. Moreover, the effective temperature may
then be considered a physical variable that characterizes the jammed granular
system, with respect to the generalization of the equilibrium statistical
mechanics of Boltzmann, as applied to non-equilibrium systems.

We further study the limits within which this effective temperature may be a
valid physical variable, as we determine mobility and diffusivity as a
function of shear rate. While diffusivity appears independent of shear rate,
even somewhat above the ‘slow’ regime, mobility shows a clear decrease in
magnitude as we explore shear rates above the slow regime, resulting in an
increase in the effective temperature as a function of shear rate.

In this paper, we will further report the experimental detail of particle
dynamics in swm . The outline is as follows:

##  II Experimental Method

###  II.1 Experimental Setup

Figure 3: (Color online) Sketch of experimental set-up. Note that the cylinder
is surrounded by 4 cameras, in the sketch we plot only two cameras. A single
particle is captured by two cameras allowing the determination of the
3-dimensional coordinates of the particle, $(r,\theta,z)$.

The experiment is performed using a three-dimensional (3D) Couette cell, as
shown in Fig. 1, 2 and 3. The grains are confined between two cylinders of
height $19.0$cm. The inner cylinder is rotated via a motor, while the outer
cylinder remains fixed. The walls of the cylinders, in contact with the
grains, are roughened by means of a glued layer of identical granular
material, thereby minimizing wall slip. The walls of the inner and outer
cylinders are roughened by acrylic beads with diameter $3.97$ and $1.59$mm,
respectively. Testing the experiment with a rough inner wall and a smooth
outer wall resulted in packing crystallization. The grains are compacted by an
external pressure of a specific value (typically 386 Pa), introduced by a
moving piston at the top of the granular material, acting in the negative
z-direction.

Figure 4: (Color online) Trajectories of the $3.97$mm nylon tracers in Packing
1 showing the diffusion and response to the gravitational force when sheared
in the Couette cell.

Observation techniques are used to monitor the granular packing evolution as
it explores the available jammed configurations. The Couette cell is sheared
at the quasi-static limit, with slow frequencies $f=0.2\sim 4.2$ mHz defining
the external shear-rate $\dot{\gamma}_{e}=2\pi fR_{1}/(R_{2}-R_{1})=\omega
R_{1}/(R_{2}-R_{1})=0.004\sim 0.084$ $\texttt{s}^{-1}$, where $R_{1}=5.08$cm
and $R_{2}=6.67$cm are the radius of the inner and outer cylinders,
respectively, and $\omega$ is the angular velocity of the inner cylinder
(Notice that $R_{1}$ and $R_{2}$ are measured after the walls are roughened by
a glued layer of beads). The experiment is designed to measure the diffusivity
and mobility of tracer particles mk ; swm ; pch , as opposed to tracking the
motion of all constituent grains. The distance between the inner and outer
cylinder is less than 10 grain diameters to prevent bulk shear band formation
nedderman ; drake ; mueth ; veje ; utter  that may interfere with the
experimental measurements by altering the diffusivity.

A refractive index matching suspending solution is employed in order to create
a transparent sample. The suspending solution is also density matched to the
grains in order to eliminate pressure gradients derived from gravity in the
vertical direction, circumventing problems seen in previous experiments of
compactivity nowak  and other effects such as convection and size segregation
such as the Brazil nut effect inside the cell behringer . The solution used in
this experiment is approximately 74% weight fraction of cyclohexyl bromide and
26% decalin weeks . These steps avoids problems encountered in previous tests
of compactivity.

###  II.2 Packing Preparation

Figure 5: (Color online) A typical trajectory of the $3.97$mm nylon tracer for
3 hours in 3D plot. The dark gray and light gray cylinder indicate the outer
surface of the sheared inner cylinder and the inner surface of the static
outer cylinder respectively.

The granular system is a bidisperse, 1:1 by mass, mixture of spherical,
transparent Poly-methyl methacrylate (acrylic) particles, with density
$\rho=1.19$ and index of refraction $n\simeq 1.49$. The bidisperse mixture is
used in an effort to inhibit crystallization of the system. The respective
particle diameters are either $3.17$mm and $3.97$mm (Packing 1) or $3.97$mm
and $4.76$mm (Packing 2). The approximate same size ratio of each bidisperse
packing leads to approximately the same value of volume fraction for both,
being $0.62$ before shearing and $0.58$ during shearing.

Figure 6: (Color online) (a) Average angular velocity, $\omega_{\theta}(r)$,
(b) Average vertical velocity, $v_{z}(r)$, and (c) Average radial velocity,
$v_{r}(r)$, versus radial distance $r$ for various tracers and different
packings. Packing 1 and Packing 2 are run at
$\dot{\gamma}_{e}=0.048,0.024\mathrm{s}^{-1}$, respectively. In (a), solid
lines are exponential fitting. In (b), the positive velocity of nylon tracer
is due to the smaller density than acrylic’s. The negative velocity of delrin
tracer is due to the higher density than acrylic’s.

A negative consequence of utilizing a suspending solution includes possible
modification of the friction coefficient between the grains. While this cannot
be completely avoided, it is important to note that the liquid only partially
fills the cell (see Fig. 1), such that the pressure of the piston is
transmitted to the granular material exclusively, not to the fluid.
Additionally, hydrodynamic effects from partial cell filling are avoided by
the extremely slow rotational speeds applied to our system. The system remains
very closely packed, such that particles are not free to float in the fluid.
Therefore, the random motion of the particles is controlled by the ‘jamming’
forces exerted by the contacts between neighboring grains, not fluid
mechanics.

###  II.3 Implementation of Fluctuation-Dissipation Theory

Figure 7: (Color online) Average angular velocity of tracers,
$\omega_{\theta}(r)$, versus radial distance $r$ for various shear rate
$\dot{\gamma}_{e}$ in Packing 2. Black square, red circle, green triangle,
blue triangle-down, cyan diamond, magenta triangle-left, yellow triangle-
right, dark yellow hexagon are corresponding to
$\dot{\gamma}_{e}=0.008,0.016,0.024,0.032,0.041,0.048,0.060,0.084\mathrm{s}^{-1}$,
respectively. The inset plots the collapsing of average angular velocity
scaled by shear rate, $\omega_{\theta}(r)/\dot{\gamma}_{e}$, versus radial
distance $r$ for various shear rate $\dot{\gamma}_{e}$. The red solid curve is
the average result of the collapsing. The black solid curve is a exponential
fitting.

Cylindrical coordinates, $(r(t),\theta(t),z(t))$, of tracer particles are
obtained by analyzing images acquired by four digital cameras surrounding the
Couette cell. For systems in thermal equilibrium, a Fluctuation-Dissipation
(FD) relation may be utilized in an effort to calculate the bath temperature
of the system. This method may be extended to non-equilibrium systems, such as
jammed granular systems presented in this study. The FD relation is defined as
follows:

$\langle[x(t+\Delta t)-x(t)]^{2}\rangle\sim 2D\Delta t,$ (1)

$\langle[x(t+\Delta t)-x(t)\rangle\sim MF\Delta t,$ (2)

$\langle[x(t+\Delta t)-x(t)]^{2}\rangle=~{}2T_{\mbox{\scriptsize
eff}}~{}\frac{\langle x(t+\Delta t)-x(t)\rangle}{F}.$ (3)

The tracer particles must experience a constant force, $F$, in order to
calculate the mobility as defined above. The most convenient constant,
external, force, is gravity in the z-direction. If the effective temperature
is to be regarded as an intensive variable of the non-equilibrium system, it
requires independence from the tracer particles properties, and we present
data in favor of this result. However, we acknowledge that temperature
measures from multiple observables would be necessary to analyze the
underlying thermodynamic meaning of $T_{\mbox{\scriptsize eff}}$.

###  II.4 Properties of Tracer Particles

Tracer particles added to the bulk must have properties unique from the grains
comprising the bulk. However, tracer particles too small, or too large, with
respect to the acrylic grains described previously, would result in erroneous
measurements. Dynamics of tracer particles that were too small would be
dominated by “percolation effects” drahun , resulting in larger than expected
tracer particle displacements. Those too large would require shear rates above
the quasi-static limit we propose to study, or possibly have no dynamics at
all due to size limitations. With these notions in mind, two different types
of tracers, nylon ($\rho^{\prime}=1.12$) and delrin ($\rho^{\prime}=1.36$),
are employed, which result in different external forces,
$F=(\rho^{\prime}-\rho)Vg$, where $\rho$ and $\rho^{\prime}$ are the densities
of the acrylic particles and the tracers, respectively, $V$ is the volume of
the tracer particle and $g$ the gravitational acceleration. Variations in
tracer particle diameter and density allow us to study dynamical changes due
to a change in constant external force, while we remain within a range
appropriate to achieve results expected to be governed by the effective
temperature.

###  II.5 Particle Tracking Technique

Four digital cameras symmetrically surround the shear cell to track the tracer
particles with frame rate $\sim$ 5 frame/s, as shown in Fig. 3. The outer
cylinder is made of the same material as the grains (acrylic, $n\simeq 1.49$).
The refractive index is matched by the fluid such that the system can be
regarded as an optical whole, i.e., the light scattered from tracers refracts
only once at the outer surface of the outer cylinder, as shown in Fig. 2. The
determination of the 3D tracer position is achieved by a simple calculation
considering both system geometry and 2D projections captured by two adjacent
cameras.

Figure 8: (Color online) Average vertical velocity of tracers, $v_{z}(r)$,
versus radial distance $r$ for various shear rate $\dot{\gamma}_{e}$ in
Packing 2. Black square, red circle, green triangle, blue triangle-down, cyan
diamond, magenta triangle-left are corresponding to
$\dot{\gamma}_{e}=0.008,0.016,0.032,0.048,0.060,0.084\mathrm{s}^{-1}$,
respectively. The inset plots the collapsing of average vertical velocity
scaled by shear rate, $v_{z}(r)/\dot{\gamma}_{e}$, versus radial distance $r$
for various shear rate $\dot{\gamma}_{e}$. The red solid curve is the average
result of the collapsing.

Camera calibration and determination of relative position is important as a
minimal asymmetry will result in a large calculation error of the tracer
particles coordinates, $(r(t),\theta(t),z(t))$. As opposed to directly
measuring relative positions of the cameras by physical devices, we utilize
computer programming. In order to simplify the calculation, we assume the
camera to be a pinhole, meaning all light coming into the camera coincides at
a single focal point.

Before each experiment, we record the images of a piece of grid paper attached
to the surface of the outer cylinder, acting as the 2D projections of the
outer cylinder for each camera. Next, we adjust the positions of four cameras
until each camera can give the approximately same 2D projections of the outer
cylinder. Then we use the computer program to generate a virtual cylinder,
along with four virtual cameras, according to the geometry of the shear cell,
In other words, we build a virtual space of the entire experimental setup and
the respective geometrical relations between its elements.

From the previous calibration procedure, we have the relative positions of the
four cameras to the shear cylinder with sufficient accuracy. In order to
further calibrate and know the exact position of cameras, we adjust the
relative position of cameras in our virtual space until the virtual 2D
projection of the cylinder to the cameras coincides exactly with the actual
projection, being the grid paper attached to the outer cylinder. When this
procedure is accomplished, the virtual space exactly coincides with the real
experimental setup space. Therefore the virtual relative position of cameras
are also the real positions.

Furthermore, in our virtual space, any point with 3D coordinates,
$(r,\theta,z)$, we can calculate its 2D coordinates in four virtual 2D
projections, $(x_{1},y_{1})\sim(x_{4},y_{4})$, by considering the geometry
relation to cameras. Oppositely, for any tracer particle, if we know its 2D
coordinates in four 2D projections, $(x_{1},y_{1})\sim(x_{4},y_{4})$, we can
exactly locate its 3D positions, $(r,\theta,z)$, since the virtual space is
equal to the actual one. The resulting vertical trajectories of the tracers
$z(t)$ are depicted in Fig. 4 showing that the nylon tracers not only diffuse,
but also move with a constant average velocity to the top of the cell. Fig. 5
shows a typical trajectory of tracer particle in 3D plotting.

##  III Results

###  III.1 Average Velocity Profiles

We first study the velocity profiles for a fixed shear rate,
$\dot{\gamma_{e}}$, followed by a study on the shear rate dependence in the
next section. The average velocity profiles in the angular direction,
$\omega_{\theta}(r)$, in the vertical direction, $v_{z}(r)$, and in the radial
direction, $v_{r}(r)$, are obtained by averaging the velocities of all tracer
particles over all times at each radius $r$, as shown in Fig. 6.

As observed in previous work mueth2 , we find that $\omega_{\theta}(r)$ can be
expressed in the exponential form demonstrated in Fig. 6a:

$\begin{split}\omega_{\theta}(r)&=\lambda_{1}\frac{\dot{\gamma}_{e}(R_{2}-R_{1})}{R_{1}}\exp(-\lambda_{2}\frac{r-R_{1}}{R_{2}-R_{1}})\\\
&=\lambda_{1}\omega\exp(-\lambda_{2}\frac{r-R_{1}}{R_{2}-R_{1}}),\end{split}$
(4)

where $\lambda_{1}$ and $\lambda_{2}$ are constants independent of shear rate,
tracer size and tracer type, depending only on the type of packings and
geometry of the shear cell. We find that $\lambda_{1}=0.77$ and $0.73$, and
$\lambda_{2}=2.15$ and $1.43$ for Packing 1 and Packing 2, respectively. When
$r=R_{1}$, $\omega_{\theta}(R_{1})=\lambda_{1}\omega$ being the angular
velocity of the first layer of grains closest to the sheared inner cylinder
with angular velocity $\omega$. Therefore $\lambda_{1}$ ($0<\lambda_{1}<1$)
can be taken as the efficiency of shearing, describing the amount of slip
between the inner rotating cylinder and the first layer of grains it contacts.
Packing 1 has a higher value of $\lambda_{1}$ than Packing 2, as the smaller
grains follow the rotating inner cylinder more easily. On the other hand, when
$r=R_{2}$, we find
$\omega_{\theta}(R_{2})=\lambda_{1}\omega\exp(-\lambda_{2})$, the velocity of
the last layer of grains closest to the static outer cylinder. This velocity
is non-zero, so that the shear band is located right at the outer cylinder,
avoiding the formation of shear bands in the bulk.

Figure 9: (Color online) Average radial velocity of tracers, $v_{r}(r)$,
versus radial distance $r$ for various shear rate $\dot{\gamma}_{e}$ in
Packing 2. Black square, red circle, green triangle, blue triangle-down, cyan
diamond, magenta triangle-left, yellow triangle-right, dark yellow hexagon are
corresponding to
$\dot{\gamma}_{e}=0.008,0.016,0.024,0.032,0.041,0.048,0.060,0.084\mathrm{s}^{-1}$,
respectively. The inset plots the collapsing of average radial velocity scaled
by shear rate, $v_{r}(r)/\dot{\gamma}_{e}$, versus radial distance $r$ for
various shear rate $\dot{\gamma}_{e}$. The red solid curve is the average
result of the collapsing.

In order to avoid the tracer particles sticking to the outer cylinder surface
and forcing its velocity to zero, we glue smaller size particles to roughen
the outer cylinder. This roughens the surface of the outer cylinder and avoids
crystallization. Further, this allows slipping of the bulk particles at the
outer cylinder, forcing the shear band to be located exactly at the outer
cylinder, not in the bulk. The glued particles are $1.59$mm, smaller than the
sheared granular material. The mean angular and vertical velocity,
$\omega_{\theta}$ and $v_{z}$, of tracer particles do not decay to zero even
if the tracers come close to outer cylinder surface. (See Fig. 7 and Fig. 8 at
$r=R_{2}$).

The exponential decay of $\omega_{\theta}(r)$ results in a local shear rate,
$\frac{d\omega_{\theta}(r)}{dr}$, dependent on radial distance. In the region
near the outer cylinder, $\omega_{\theta}(r)$ decays slowly with increasing
$r$ which leads to weak dependence of $\frac{d\omega_{\theta}(r)}{dr}$ on the
radial distance $r$. If we do Taylor expansion at $r=R_{2}$, the average
angular velocity of the tracers, $\omega_{\theta}(r)$, can be approximated to
a linear function of $r$, i.e.,
$\omega_{\theta}(r)\approx\lambda_{1}\omega\exp(-\lambda_{2})-r\frac{d\omega_{\theta}(r)}{dr}$
with constant local shear rate $\frac{d\omega_{\theta}(r)}{dr}=0.021$ s-1cm-1.
The diffusivity and mobility of the tracer particles strongly depend on the
local rearrangement of the grains. A constant shear rate results in homogenous
local rearrangement of the packings ensuring that the diffusivity and mobility
of tracers, dependent on local shear rate, remain approximately independent of
$r$. As shown in Fig. 6b, we find a plateau in the vertical velocity profile
which can be further seen in Fig. 8. Similar behavior is observed in the
vertical diffusivity profile, $D_{z}(r)$ as shown in Fig. 16, which we will
discuss in detail in Section III.4. We denote this the “constant mobility and
diffusivity region”, i.e., CMD region, $5.80$cm $<r<6.67$cm. Contrary to prior
work howell ; veje ; mueth ; mueth2 ; utter ; nedderman ; drake  on sheared
granular matter in the Couette cell, our experiment focuses only a narrow gap,
$15.9$mm, of Couette cell. The CMD region allows us to well define the
diffusivity and mobility of the tracer particles, such that we can calculate
the average vertical velocity, $v_{z}$, and the average vertical diffusivity,
$D_{z}$, by averaging the velocities of all tracers over all times in the CMD
region, significantly improving the statistics. In this study, the statistical
average and the measurements of tracer fluctuations will be confined only to
the CMD region.

Figure 10: (Color online) (a) PDF of the vertical displacements, $P(\Delta
z)$, of the $3.17$mm delrin tracers in Packing 1 for a given time interval
$\Delta t=50$s, and with $\dot{\gamma}_{e}=0.048\mathrm{s}^{-1}$. Tracer
trajectories are split into sub-trajectories confined in two regions, (i):
$5.08$cm $<r<5.80$cm, which is close to inner rotating cylinder, and (ii):
$5.80$cm $<r<6.67$cm, which is far away from inner rotation cylinder. We
compared the calculated $P(\Delta z)$ by using the sub-trajectories from the
regions of (i) and (ii) respectively, which are plotted as black triangle and
black circle. See more details in the main text. (b) PDF of the vertical
displacements, $P(\Delta z)$, of the $3.97$mm nylon tracers in Packing 1 with
$\dot{\gamma}_{e}=0.048\mathrm{s}^{-1}$, shifted by the average displacement
$\langle\Delta z\rangle$ and scaled by the root-mean-square deviation
$\langle\Delta z(t)^{2}\rangle^{1/2}$. The red solid curve is a Gaussian
distribution, $P(x)=0.4\text{e}^{-x^{2}/2}$.

Figure 11: (Color online) PDF of the radial displacements, $P(\Delta r)$, of
the $3.97$mm nylon tracers in Packing 1 with
$\dot{\gamma}_{e}=0.048\mathrm{s}^{-1}$ for different time intervals. A
symmetric distribution around zero displacement indicates that there is no net
flow in the radial direction. The solid lines are exponential fitting,
$P(\Delta r)\sim\text{exp}(-\frac{|\Delta r|}{r_{o}})$, where
$r_{o}=0.17,0.25,0.32$ for $\Delta t=100,300,500$s, respectively. The inset
shows the rms fluctuations, which gives the value of $\alpha=0.67$.

We find $v_{r}(r)$ to be flat for different types of tracer particles and for
different packings except when the tracers are close to the inner and outer
cylinder, i.e., $r=R_{1}$ and $r=R_{2}$, as shown in Fig. 6c. $v_{r}(R_{1})$
is negative and $v_{r}(R_{2})$ is positive, indicating the inner and outer
cylinder walls can slightly attract the tracers. It should be noted that the
statistics presented in this study does not incorporate data from the regions
close to the inner and outer cylinder to avoid these boundary effects.

Figure 12: (Color online) PDF of the angular displacements, $P(\Delta\theta)$,
of the $3.97$mm nylon tracers in Packing 1 with
$\dot{\gamma}_{e}=0.048\mathrm{s}^{-1}$ for different time intervals. Due to
Taylor dispersion effects the distribution shows an asymmetric shape. The rms
fluctuations shown in the inset reveal a faster than diffusion process.

###  III.2 Shear Rate Dependent Average Velocity Profiles

Next, we study the dependence of the particle velocity on the external shear
rate. According to Eq. (4), the velocity profile in the angular direction,
$\omega_{\theta}(r)$, is proportional to the external shear rate
$\dot{\gamma}_{e}$. We can collapse $\omega_{\theta}(r)$ by scaling the shear
rate. The results are shown in the inset of Fig. 7 for Packing 2. The
collapsing of $\omega_{\theta}(r)/\dot{\gamma}_{e}$ shows a periodic shape
superimposed to exponential decay with a very small amplitude, also found in
the velocity profile of $v_{r}(r)$ (see Fig. 9). The periodic length is
roughly equal to the grain particle size and reflects the different layers of
grains in the radial direction. This periodicity is weaker in Packing 1 than
Packing 2, since the particle size of Packing 1 is smaller than that of
Packing 2.

The collapsing method can be further applied to $v_{z}(r)$, as seen in Fig. 8.
After scaling by the shear rate, $v_{z}(r)/\dot{\gamma}_{e}$ also shows a flat
plateau indicating the CMD region.

###  III.3 Probability Distribution of Displacements

Fig. 10a shows the results of the probability distribution of the
displacements $\Delta z$ in the vertical direction for a given time interval
$\Delta t$. The data corresponds to the $3.17$mm delrin tracers in Packing 1.
Usually, 20 tracers are used for calculations. Tracer trajectories are split
into sub-trajectories confined in two regions, (i): $5.08$cm $<r<5.80$cm,
close to the inner rotating cylinder, and (ii): $5.80$cm $<r<6.67$cm, i.e.,
CMD region, close to the outer cylinder. We compare the calculated $P(\Delta
z)$ by using the sub-trajectories from the regions of (i) and (ii)
respectively, which are plotted as black triangle and black circle in the Fig.
10a. The data in the inner region (i) clearly display an asymmetric tail for
$\Delta z<0$. This extra spreading is similar to the phenomena of the Taylor
dispersion taylor .

Figure 13: (Color online) PDF of the vertical displacements, $P(\Delta z)$, of
the $4.76$mm nylon tracers in Packing 2 for various effective angular
displacement $\Delta{\theta_{e}}$ and effective shear rate $\gamma_{e}$. The
PDFs are scaled by $\Delta{\theta_{e}}^{1/2}$ and shifted by the mean
displacement $\langle\Delta z\rangle$. The red dashed line is the Gaussian
fitting, $P(x)\sim\text{exp}[-(\frac{x}{0.147})^{2}]$. The collapsing of PDFs
indicates that the RMS fluctuations of the vertical displacements follow the
relation, $\langle\Delta z^{2}\rangle\sim\Delta{\theta_{e}}$.

Taylor dispersion appears when diffusion couples with the gradient of flow
giving rise to a larger dispersion along the flowing direction (see for
instance utter  for a study of Taylor dispersion in granular materials). In
the present experiment, the shear rate of granular flow in the angular
direction exhibits exponential decay, as shown in Fig. 6b. The larger shear
rate in the inner region (i) results in larger packing rearrangement, which
gives rise to a larger dispersion in the vertical direction. In this case it
is not possible to extract the bare diffusion constant. On the contrary, for
the region (ii), i.e., CMD region, as we mentioned, the gradient of the flow,
i.e., the shear rate is approximately constant, giving rise to a Gaussian
diffusion, as shown in Fig. 10a. By measuring the width and the mean value of
this Gaussian distribution of

$\begin{split}P(\Delta z)&\sim\exp[{-\frac{(\Delta z-\langle\Delta
z\rangle)^{2}}{2\langle\Delta z^{2}\rangle}}]\\\ &\sim\exp[{-\frac{(\Delta
z-M_{z}F\Delta t)^{2}}{4D_{z}\Delta t}}],\end{split}$ (5)

we can define the diffusivity and mobility, $D_{z}$ and $M_{z}$, which lead to
the effective temperature of the granular packing discussed in the following
section. In the Fig. 10b, we define a new scaled variable $x=\frac{\Delta
z-\langle\Delta z\rangle}{{\langle\Delta z^{2}\rangle}^{1/2}}$ and plot $P(x)$
for different $\Delta t$, all the curves are found to collapse into a single
curve

$P(x)\sim\text{e}^{-x^{2}/2}.$ (6)

In this experiment, we will focus our measurements in the region away from the
inner boundary (region (ii), i.e., CMD region), where the mobility is a
constant (as shown a plateau in the inset of Fig. 8) and Taylor dispersion
effects are absent.

Figure 14: (Color online) PDF of the radial displacements, $P(\Delta r)$, of
the $4.76$mm nylon tracers in Packing 2 for various effective angular
displacement $\Delta{\theta_{e}}$ and effective shear rate $\gamma_{e}$. The
PDFs are scaled by $\Delta{\theta_{e}}^{\alpha/2}$, where $\alpha=0.67$. The
red dashed line is exponential fitting,
$P(x)\sim\text{exp}(-\frac{|x|}{0.089})$. The collapsing of PDFs indicates
that the RMS fluctuations of the radial displacements follow the relation,
$\langle\Delta r^{2}\rangle\sim\Delta{\theta_{e}}^{\alpha}$.

We find exponential fluctuations for the probability distributions of the
tracer particles in the radial direction as shown in Fig. 11,

$P(\Delta r)\sim\text{e}^{-\frac{|\Delta r|}{r_{o}}},$ (7)

where $r_{o}$ is a function of $\Delta t$. The symmetric shape for $P(\Delta
r)$ indicates the absence of a shear induced segregation, as observed with
multiple sizes of grains, as there is no net flow of the tracer particles
towards either cylindrical wall within the time-scales of the experiment. We
also observe no average motion of the tracer particles towards the center of
the Couette cell except within a small range of radial distance, around
$0.12$cm, close to both walls where particles experience a slight attraction
to boundaries. These features are shown in Fig. 6c and Fig. 9.

The analysis of the radial displacement fluctuation reveals a power law, sub-
diffusive, process:

$\langle\Delta r^{2}\rangle\sim{\Delta t}^{\alpha}$ (8)

as shown in Figure 8, where $\alpha=0.67$ for both the delrin and nylon
tracers.

The data taken for the angular displacement is in the direction of the flow,
and affected by Taylor dispersion as shown in the non-Gaussian tail of the
displacement distribution $\Delta\theta(t)$ in Fig. 12. This leads to a power
law, super-diffusive process, illustrated by

$\langle\Delta\theta^{2}\rangle\sim{\Delta t}^{\beta}$ (9)

as seen in the analysis of the fluctuations of $\Delta\theta$ shown in Fig.
12, where $\beta=1.67$ and $1.30$ for Packing 1 and Packing 2 respectively.

We further study how shear rate affects the displacement probability
distribution. We find that for small shear rate, the probability distributions
of displacements in the three cylindrical coordinates are independent of the
shear rate, depending only on the sheared displacement, i.e., the external
rotating displacement, defined as

$\begin{split}\Delta\theta_{e}&=\dot{\gamma}_{e}\Delta t\\\ &=\omega\Delta
tR_{1}/(R_{2}-R_{1})\\\ &=\Delta\theta_{i}R_{1}/(R_{2}-R_{1})\end{split}$ (10)

where $\Delta\theta_{i}$ is the rotating displacement of the inner cylinder.
This result is expected. Since we shear the Couette cell very slowly, the
diffusion of the tracers depends only the number of granular packing
configurations sampled by the Couette cell, which depends only on the sheared
displacement.

As emphasized in the previous text, the statistical average and the
measurements of the tracer fluctuations is confined to the CMD region, such as
the $D_{z}$ shown in Fig. 17a. We calculate the $D_{z}$ by measuring the width
and the mean value of the Gaussian distribution of $P(\Delta z)$, and obtain
the $P(\Delta z)$ by averaging the displacement fluctuations of all tracers
over all time in the CMD region. Next, we apply a different method to reveal
how $D_{z}(r)$ depends on the radial distance $r$, as shown in the Fig. 16. We
first obtain $P(\Delta z,r)$ for a certain radial distance $r$, then we
calculate $D_{z}(r)$ by measuring the width and the mean value of the Gaussian
distribution of $P(\Delta z,r)$. In Fig. 16, we see that the tracer particles
have higher diffusivity close to the inner cylinder than the outer. Since
$D_{z}\sim\dot{\gamma}_{e}$, we can collapse all the $D_{z}(r)$ for various
shear rates, as shown in the inset of Fig. 16. The collapse of
$D_{z}(r)/\dot{\gamma}_{e}$ shows a plateau close to the outer cylinder,
consistent with our previous discussion of the CMD region.

Figure 15: (Color online) PDF of the angular displacements, $P(\Delta\theta)$,
of the $4.76$mm nylon tracers in Packing 2 for various effective angular
displacement $\Delta{\theta_{e}}$ and effective shear rate $\gamma_{e}$. The
PDFs are shifted by the mean displacement $\langle\Delta\theta\rangle$ and
scaled by $\Delta{\theta_{e}}^{\beta/2}$, where $\beta=1.30$. The red dashed
line are Gaussian and exponential fittings for $x<0$ and $x>0$, respectively.
The collapsing of PDFs indicates that the rms fluctuations of the angular
displacements follow the relation,
$\langle\Delta\theta^{2}\rangle\sim\Delta{\theta_{e}}^{\beta}$.

$\displaystyle\langle[z(t+\Delta t)-z(t)]^{2}\rangle$
$\displaystyle\sim\Delta\theta_{e}$ (11a) $\displaystyle\langle[r(t+\Delta
t)-r(t)]^{2}\rangle$ $\displaystyle\sim{\Delta\theta_{e}}^{\alpha}$ (11b)
$\displaystyle\langle[\theta(t+\Delta t)-\theta(t)]^{2}\rangle$
$\displaystyle\sim{\Delta\theta_{e}}^{\beta}$ (11c)

Eq. (11) implies that one can collapse the probability distribution of the
displacements, $P(\Delta z)$, $P(\Delta r)$ and $P(\Delta\theta)$ for
different shear rates and time intervals by scaling $\Delta z$, $\Delta r$ and
$\Delta\theta$ respectively to $\Delta z/{\Delta\theta_{e}}^{1/2}$, $\Delta
r/{\Delta\theta_{e}}^{\alpha/2}$ and
$\Delta\theta/{\Delta\theta_{e}}^{\beta/2}$. The results are presented in Fig.
13, 14 and 15.

Figure 16: (Color online) Diffusivity $D_{z}$ versus radial distance $r$ for
various shear rate $\dot{\gamma}_{e}$ in Packing 2. Black square, red circle,
green triangle, blue triangle-down, cyan diamond, magenta triangle-left are
corresponding to
$\dot{\gamma}_{e}=0.008,0.016,0.032,0.048,0.060,0.084\mathrm{s}^{-1}$,
respectively. The inset plots the collapsing of diffusivity scaled by shear
rate, $D_{z}/\dot{\gamma}_{e}$, versus radial distance $r$ for various shear
rate $\dot{\gamma}_{e}$. The red solid curve is the average result of the
collapsing.

###  III.4 Effective Temperature

Packing 1, 1:1 mass mixture of $3.17$mm & $3.97$mm acrylic beads,
$\dot{\gamma}_{e}=0.048\mathrm{s}^{-1}$
---
tracer |  $d$ (mm) |  $\rho$ (g$\cdot$cm-3) |  $D_{z}$ ($10^{-8}$ m2$\cdot$s-1) |  $M_{z}$ ($10^{-2}$ s$\cdot$kg-1) |  $T_{\mathrm{eff}}$ ($10^{-7}$ J)
acrylic | $3.17$ | $1.19$ | $2.5\pm 0.3$ |  |
delrin | $3.17$ | $1.36$ | $2.4\pm 0.3$ | $24\pm 3$ | $1.0\pm 0.2$
delrin | $3.97$ | $1.36$ | $1.2\pm 0.1$ | $9.3\pm 0.9$ | $1.3\pm 0.2$
nylon | $3.97$ | $1.12$ | $1.1\pm 0.1$ | $9.5\pm 0.9$ | $1.2\pm 0.2$
ceramic | $3.97$ | $3.28$ |  | $2.2\pm 0.2$ |
brass | $3.97$ | $8.4$ |  | $1.7\pm 0.1$ |
Packing 2, 1:1 mass mixture of $3.97$mm & $4.76$mm acrylic beads,
$\dot{\gamma}_{e}=0.024\mathrm{s}^{-1}$
tracer |  $d$ (mm) |  $\rho$ (g$\cdot$cm-3) |  $D_{z}$ ($10^{-8}$ m2$\cdot$s-1) |  $M_{z}$ ($10^{-2}$ s$\cdot$kg-1) |  $T_{\mathrm{eff}}$ ($10^{-7}$ J)
nylon | $3.97$ | $1.12$ | $1.8\pm 0.1$ | $19.0\pm 0.9$ | $0.95\pm 0.07$
nylon | $4.76$ | $1.12$ | $1.6\pm 0.1$ | $15.7\pm 0.4$ | $1.0\pm 0.1$
Table 1: Diffusivity and mobility for the different types of tracer and
packings.

We present results for the diffusivity in the $z$ direction, the only
direction where the effective temperature can be calculated due to the
vertically acting external force. The Gaussian distribution in $P(\Delta z)$
allows us to apply the FD relation to the particle displacements, as the
diffusivity is proportional to the variance of a Gaussian distribution in
displacements. Exponential fluctuations do not possess this same property, but
it is important to note that the radial direction has no constant applied
external force. It remains a possibility that a well-defined effective
temperature for displacements in the radial direction could exist. To test
whether the effective temperature is isotropic, as done in danna , may be of
great interest in future studies.

A common method of performing a time average to measure transport coefficients
is employed (see Chapter 5.3 in rapaport ) by dividing the trajectory of a
single tracer particle into a series of trajectories, having evenly spaced
start times, separated by time interval $\Delta t$. The diffusion constant is
obtained by averaging over the aggregate of tracers and over the initial time
intervals, allowing for the use of merely 20 tracer particles in this
particular system. Correlations between measurements are ensured to have
decayed almost to zero, rendering time-translational invariance valid in this
system, without any measurable“aging”, since under shearing, system reaches
the “stationary state” coniglio . Furthermore, doubling the number of tracer
particles leaves $D_{z}$ unchanged, indicating independence of the diffusion
constant from the number of tracers that explore the jammed configurations of
this non-equilibrium system.

Analysis of the vertical particle displacements in the CMD region reveals a
Gaussian distribution, broadening over time, as seen in Fig. 10 and Fig. 13.
For sufficiently long times period, the mean square fluctuations grow linearly
(see Fig. 17a):

$\langle[z(t+\Delta t)-z(t)]^{2}\rangle\sim 2D_{z}\Delta t,$ (12)

where $D_{z}$ is the self-diffusion constant in the vertical direction. For
the both nylon and delrin $3.97$mm tracers in Packing 1 we obtain $D_{z\
\mbox{\scriptsize 3.97mm}}\approx(1.15\pm 0.1)\times 10^{-8}$ m2/s.

Figure 17b shows mean value tracer particle positions, extracted from the peak
of the Gaussian distribution, as a function of time. The mobility in the
vertical direction, $M_{z}$, is defined as

$\langle z(t+\Delta t)-z(t)\rangle\sim M_{z}F\Delta t.$ (13)

The applied force on the tracers, $F=(\rho-\rho^{\prime})Vg$, is the
gravitational force due to density mismatch where $\rho$ and $\rho^{\prime}$
are the densities of the acrylic particles and the tracers, respectively, $V$
is the volume of the tracer particle and $g$ the gravitational acceleration.
The value of the mobility for the both nylon and delrin $3.97$mm tracers in
Packing 1 is $M_{z\ \mbox{\scriptsize 3.97mm}}\approx(9.4\pm 0.9)\times
10^{-2}$ s/kg.

Fig. 17a further reveals a downward curvature of the mean-square fluctuations,
for sufficiently long times period. Additionally, an apparent cut-off time for
the tracer particles fluctuation measurements is shown. These effects are due
to the finite size effect imposed upon the tracers by the finite trajectories
and should be inversely proportional to the tracer particles velocities.
Tracer particles with larger mobility will have larger mean velocities and
take a shorter time to complete its trajectory in the cell. The cut-off
discussed in reference to Fig. 17a is prominently displayed in the $3.17$mm
delrin tracers of Packing 1, having the largest mobility, hence increased mean
velocities, as shown in Fig. 17b. The larger mobility results in the shortest
cut-off time for the diffusivity. Conversely, $3.97$mm delrin tracers of
Packing 1 have a smaller mobility, hence a longer cut-off time for the
diffusivity. It is important to note that for all tracer particles studied
here, the cut-off is observed for distances larger than a few particles
diameters, ensuring that the study examines the structural motion of the
grains and not internal motion inside of “cages”.

Figure 17: (Color online) (a) Autocorrelation function of tracers. (b)
Response function of tracers. (c) Log-log plot of effective temperatures for
various tracers and different packings as obtained from a parametric plot of
their autocorrelation function versus response function. (d) Same as (c) but
in a linear-linear plot. The slopes for different tracer diffusivity vs.
mobility curves return the same average value of $T_{\mbox{\scriptsize
eff}}\approx(1.1\pm 0.2)\times 10^{-7}$J as given by Eq. (3).

According to a Fluctuation-Dissipation relation, we calculate
$T_{\mbox{\scriptsize eff}}$:

$T_{\mbox{\scriptsize eff}}=\frac{F\langle[z(t+\Delta
t)-z(t)]^{2}\rangle}{2\langle z(t+\Delta t)-z(t)\rangle}.$ (14)

Fig. 17c shows a parametric plot of fluctuations and responses, with $\Delta
t$, as the parameter, as extracted from Fig. 17a and Fig. 17b. A linear
relationship exists between diffusivity and mobility, with a slope of
$T_{\mbox{\scriptsize eff}}$. We obtain for the both nylon and delrin $3.97$mm
tracers in Packing 1, $T_{\mbox{\scriptsize eff}}\approx(1.25\pm 0.2)\times
10^{-7}\text{J}$.

If the effective temperature is to be regarded as an intensive thermodynamic
quantity, changing the tracer particle size should give rise to a different
diffusion and mobility yet result in the same measurement of effective
temperature. The above calculation is repeated for delrin tracers of $3.17$mm
in Packing 1. We find that while the mobility and diffusivity change
dramatically with respect to tracers of $3.97$mm, ($D_{z\ \mbox{\scriptsize
3.17mm}}=(2.4\pm 0.3)\times 10^{-8}$ m2/s and $M_{z\ \mbox{\scriptsize
3.17mm}}=(2.4\pm 0.3)\times 10^{-1}$ s/kg) as shown in Table 1, due to the
change in tracer size, their ratio remains unchanged. In all cases $D_{z}$ and
$M_{z}$ are inversely proportional to the size of the tracers, but the
effective temperature remains approximately the same, as seen in Fig. 17c,
with an average value over all tracers of

$T_{\mbox{\scriptsize eff}}\approx(1.1\pm 0.2)\times 10^{-7}\text{J}.$ (15)

Though this effective temperature is high with respect to the bath
temperature, we note that a plausible scale for the system energy review , is
$(\rho-\rho^{\prime})gd$, the gravitational potential energy to move a nylon
tracer particle one particle diameter, $d$. A corresponding temperature would
arise from the conversion of this energy into a temperature via the Boltzmann
constant, $k_{B}$, is $T_{\mbox{\scriptsize eff}}=2.7\times 10^{13}k_{B}T$ at
room temperature (T = 300K). This specific value serves as a coarse-grained
estimate, since the tracer size and density clearly shift its value, and we
focus on the order of magnitude. This large value is expected review , and
agrees with computer simulation estimates for an athermal granular system mk .
Therefore, our calculated value for $T_{\mbox{\scriptsize eff}}$ in a sheared
granular system appears reasonable within the boundaries of the present
theory.

###  III.5 Linear Response Regime

Figure 18: (Color online) Mobility, $M_{z}$, versus the external force, $F$,
for $3.97$mm tracers in Packing 1. The black squares are experimental data
which is coming from different types of tracer, they are (from left to right)
nylon, delrin, ceramic and brass. The red dashed guide line is a function of
$8.0\exp{[-(x/0.007)^{4}]}+1.7$.

In an effort to further test the concept of the effective temperature as an
intensive quantity, a linear response regime in the system is of great
interest. Such a regime would imply that mobility and diffusivity are
independent of the external gravitational force as $F\to 0$. The external
force is varied by changing the density of the tracers of the same size. This
is realized experimentally in Packing 1 by the introduction of delrin
($\rho^{\prime}=1.36$) tracers of $3.97$mm diameter, the density of which is
higher than that of nylon ($\rho^{\prime}=1.12$).

Analysis of the trajectories reveals that the mobility is approximately the
same for both the delrin and nylon tracers with the same diameter and is
thereby independent of the external force, as shown in Fig. 17b and Fig. 18.
Further, the external force should have no effect on the diffusivity. By
calculating the diffusivity of the non-tracer particles via dying acrylic
tracers and analyzing their trajectories, as shown in Fig. 17a, the diffusion
of the acrylic tracers of size $3.17$mm (for which no external force is
applied) is the same as the diffusion of the delrin tracers of the same size
(for which the gravitational force is applied). A further example of this
property would be using two different tracers with different sizes, but having
the same external force applied. One would calculate two different values of
the diffusivity, due to the variation in tracer size, without having any
variation in external force.

Nonlinear effects appear for tracers heavier than delrin, implying that
mobility depends on the external force for large enough forces. We find that
for a $3.97$mm ceramic tracer $(\rho^{\prime}=3.28)$ in Packing 1 the mobility
is $M_{z\ \mbox{\scriptsize ceramic}}=(2.2\pm 0.2)\times 10^{-2}$ s/kg and for
a brass tracer $(\rho^{\prime}=8.4)$, $M_{z\ \mbox{\scriptsize brass}}=(1.7\pm
0.1)\times 10^{-2}$ s/kg as shown in Table 1, smaller than the mobility of the
nylon and delrin tracers of the same size. This behavior is expected since if
a linear regime exists in the system, it will be valid only within certain
limits, i.e., $M_{z}$ remains a constant for small value of external force,
$F$, as shown in Fig. 18. It is here that our experiments approach the
boundaries presented above for estimated of energy scales for the sheared
granular system. Our effective temperature measurements are therefore limited
to those tracer particles for which we experience a linear regime with respect
to both mobility and diffusivity.

Lastly, the experiment is again repeated for a different packing of spherical
particles, noted earlier as Packing 2. Having nearly the same volume fraction
of particles being both packings of spherical particles, one would expect
$T_{\mbox{\scriptsize eff}}$ to remain unchanged, as it is a measure of how
dense the particulate packing is (i.e. a large $T_{\mbox{\scriptsize eff}}$
implies a loose configuration, e.g. random loose packing, while a reduced
$T_{\mbox{\scriptsize eff}}$ implies a more compact structure, e.g. random
close packing). It is found that although differences exist between the two
packings with respect to mobility and diffusivity, as shown in Fig. 17a,b, the
effective temperature remains approximately the same, as shown in Fig. 17c. It
should be noted that both packings are composed of spherical particles and the
statement regarding effective temperature as a measure of particulate packing
density would not be true if the packings are composed of particles of, for
instance, different shapes, even if they have the same volume fraction.

Figure 19: The dependence of (a) diffusivity $D_{z}$, (b) mobility $M_{z}$ and
(c) effective temperature $T_{\mbox{\scriptsize eff}}$ on the shear rate
$\dot{\gamma}_{e}$ for the $4.76$mm nylon tracers in Packing 2. The solid
lines in (a) and (b) are linear fitting [note that the line in (b) is a
fitting only for the first 6 data points at the small value of shear rate
$\dot{\gamma}_{e}$]. We find that $D_{z}\sim\dot{\gamma}_{e}$ and
$M_{z}\sim\dot{\gamma}_{e}$, while $T_{\mbox{\scriptsize eff}}=D_{z}/M_{z}$ is
approximately constant for sufficiently small $\dot{\gamma}_{e}$. This quasi-
static regime coincides with the appearance of a rate-independent stress in
experiments tardos , that $T_{\mbox{\scriptsize eff}}$ is interpreted as the
temperature of the jammed states. The height of flat solid line in (c) is
calculated from the slope of lines in (a) and (b), which indicates a constant
effective temperature $T_{\mbox{\scriptsize eff}}=(1.2\pm 0.2)\times 10^{-7}$J
at the small value of shear rate $\dot{\gamma}_{e}$.

###  III.6 Shear Rate Dependence

We further explore the effective temperature as an intensive quantity by
analyzing diffusivity and mobility as a function of the shear rate. We show in
Fig. 19 that the effective temperature seems to become approximately constant,
as long as the particulate motion is slow enough such that the system is very
close to jamming. We find that

$D_{z}\sim\dot{\gamma}_{e},\ \ \ \ \dot{\gamma}_{e}\lesssim
0.06\texttt{s}^{-1}$ (16) $M_{z}\sim\dot{\gamma}_{e},\ \ \ \
\dot{\gamma}_{e}\lesssim 0.04\texttt{s}^{-1}$ (17)

while $T_{\mbox{\scriptsize eff}}=D_{z}/M_{z}$ remains approximately constant
for sufficiently small $\dot{\gamma}_{e}$.

It is within this quasi-static range where the effective temperature could be
identified with exploration of the jammed configurations. As it remains an
important assumption of this study that the system is being continuously
jammed, shear rates high enough to impact the effective temperature
measurement imply systems that are not continuously exploring jammed
configurations. As we study the nature of the jammed granular packings, it is
logical to presume that quasi-static shearing will provide systems of
interest. The limit of $T_{\mbox{\scriptsize eff}}$ as $\dot{\gamma}_{e}\to 0$
may result in an effective temperature for the static jammed configuration.
The quasi-static shear rate regime observed could be analogous to the shear-
rate independent regime observed in the behavior of shear stress in slowly
sheared granular materials savage ; tardos . This solid friction-like behavior
has been previously studied savage ; tardos  and occurs when frictional forces
and enduring contacts dominate the dynamics. This regime has been also
observed in recent computer simulations of the effective temperature of
sheared granular materials ono ; NingXu . Our calculations of
$T_{\mbox{\scriptsize eff}}$ for systems close to jamming exclude the systems
outside of the quasi-static range, in accordance with prior studies.

##  IV Outlook: Significance Of $T_{\mbox{\scriptsize eff}}$ for a Statistical
Mechanics of Grains

In contrast to measurements of slow mode temperatures, exemplified by
$T_{\mbox{\scriptsize eff}}$, we also measure the temperature of the fast
modes as given by the root mean square (RMS) fluctuations of the velocity of
the particles. It should be noted that these velocities are not instantaneous,
as the time necessary to obtain an instantaneous velocity is much smaller than
the time between measurements. Nevertheless, we can obtain an estimate of the
kinetic granular temperature, $T_{k}$, from $T_{k}=\frac{2}{3}E_{k}$, where
$E_{k}=\frac{1}{2}m\overline{v^{2}}$ with $\overline{v^{2}}$ the average
kinetic energy of the grains. We obtain $T_{k}=9.17\times 10^{10}k_{B}T$, or
$3.77\times 10^{-10}J$ and $T_{k}=1.54\times 10^{11}k_{B}T$, or $6.34\times
10^{-10}J$, for $3.17$ mm and $3.97$ mm delrin tracers in packing 1,
respectively. Here, $T={298.15K}$, the room temperature, and
$k_{B}=1.3806504\times 10^{-23}JK^{-1}$. This kinetic granular temperature is
smaller than $T_{\mbox{\scriptsize eff}}$ and differs for each type of tracer
indicating that it is not governed by the same statistics. Similar results
have been obtained in experiments of vibrated granular gases menon . The
significance of this result is that fast modes of relaxation are governed by a
different temperature. This result is analogous to what is found in models of
glasses and computer simulations of molecular glasses (see for instance Refs.
ckp ; liu1 ; mehta ; coniglio ; wolf ). In the glassy phase of these models,
the bath temperature is found to control the fast modes of relaxation and a
different, larger, effective temperature is found to control the slow modes of
relaxation. Similarly, we find a granular bath temperature for the fast modes
and a larger effective temperature for slow modes of relaxation.

It is possible to identify $T_{\mbox{\scriptsize eff}}$ as the property of the
system governing the exploration of jammed configurations. As this particular
non-equilibrium system remains athermal, the ’bath’ temperature in which the
grains exist is immaterial, as shown above. Particle diffusion is of the order
of several particle diameters over the time scale of the experiment (see Fig.
4 and 17a) implying that exploration of the available jammed configurations
occurs via rearrangements of the particles outside their “cages”, suggesting
that the trajectory of the system can be mapped onto successive jammed
configurations explored by the system.

Incorporating certain experimental conditions of reversibility, and
ergodicity, a statistical mechanics formulation may well describe a jammed
granular system edwards ; mehta2 . Under the primary assumption that different
jammed configurations are taken to have equal statistical weight, observables
can be calculated by “flat” averages over the jammed configurational space
edwards ; kurchan ; bklm ; coniglio2 ; mk ; luck ; mehta3 . This assumption,
advocated by Edwards and collaborators, has been thoroughly debated in the
literature (see for instance coniglio ; cavendish ). Existing work suggests
the effective temperature obtained by applying a fluctuation-dissipation
theory to non-equilibrium systems is analogous to performing a “flat” average
over the jammed configurational space, at least for frictionless systems mk .
Additionally, the effective temperature can be identified with the
compactivity introduced in edwards , resulting from entropic calculations of
the granular packing kurchan ; bklm ; mk . Experimentally testing these ideas
is difficult as the entropy of the jammed configurations is not easily
measured, and it is not possible to obtain the compactivity from entropic
considerations in the present study.

The exploration of reversible jammed states in granular matter bears
similarity to that of inherent structures in glasses. Inherent structures form
a network of attractive basins within an energy landscape, and the system
explores these basins as governed by their stability over the slow-relaxation
time of the glass. It should be noted, however, that there exists a crucial
difference between glasses and grains. In liquids energy remains conserved,
while energy is dissipated in granular systems through frictional contact and
path dependent forces between grains. Thus, a driven granular system will
quickly come to a mechanically stable, or jammed, state after the removal of
the driving forces. By its nature, energy is not conserved in a granular
system. As energy conservation is the crucial property used to define an
energy ensemble in statistical mechanics, the use of energy to characterize
granular systems is questionable. Thus, while $T_{\mbox{\scriptsize eff}}$
seems to imply the exploration of reversible jammed states within an energy
ensemble, with

$P(E)\sim\text{e}^{-\frac{E}{T_{\mbox{\scriptsize eff}}}}$ (18)

describing the nature of the exploration, the validity of the energy ensemble
to describe granular matter in the absence of energy conservation remains an
open question. Here, we set the analogous Boltzmann constant for grains equal
to unity for simplicity.

Noting the drastic difference between the bath temperature and the effective
temperature in a granular system, we are inspired towards a more careful
analysis of the energy ensemble in slowly driven granular systems. The work of
Edwards has promoted the concept of a volume ensemble, where the free volume
per grain in a static granular system replaces the energy as the conserved
quantity of the non-equilibrium system, at a particular volume fraction
edwards ; mehta2 . The basis for using the volume ensemble stems from the
ability to conserve volume in a given packing and additivity of volume per
grain. Further, it is possible to explore the configuration of states at a
fixed volume, via experiment or simulation. The statistical mechanics is then
derived using methods similar to Boltzmann statistics for equilibrium systems.
From these methods, one can obtain the compactivity, $X$, as a derivative of
the entropy with respect to the volume, enabling the calculation of an
equation of state in the volume ensemble as follows.

$X^{-1}=\frac{\partial{S}}{\partial{V}}$ (19)

The compactivity, $X$, is thereby assumed to be an equilibrium measure of a
system within the framework of the volume ensemble, much like the bath
temperature of the energy ensemble. This assumption can be realized by
performing an ABC experiment and testing a zero-th law of thermodynamics for
volumes cavendish . According to the zero-th law of thermodynamics, if system
A and C are in thermal equilibrium with system B respectively, then A and C
are in thermal equilibrium with equal temperature. In granular system, such an
experiment would require two granular systems with distinct volumes, $V_{1}$
and $V_{2}$, with the same $X$. Bringing these two systems together should
result in a granular system of volume $V=V_{1}+V_{2}$, at the same $X$, if the
assumption is valid. This experiment is feasible due to the fact that it is
always possible to prepare a system at a given volume fraction and will be the
subject of future study and experiment, facilitated by recent theoretical
findings chaoming_nature .

Similar to the conservation of volume, boundary stress may also be a conserved
quantity in jammed granular systems, and Edwards statistical mechanics for
volume distributions could be applied analogously to the distribution of
boundary stresses, $\Pi$, or forces, referred to as the force ensemble
Edwards_new ; Henkes . The angoricity, $A$, is calculated as the derivative of
the entropy with respect to the boundary stress, and an additional equation of
state is thereby achieved as follows:

$A^{-1}=\frac{\partial{S}}{\partial{\Pi}}$ (20)

This result can be combined with that of the volume ensemble in an effort to
accurately define the statistical mechanics of static jammed granular matter.
Such an approach remains a topic of ongoing research.

However, slowly driven granular systems introduce yet another ensemble, the
energy ensemble, from which the above defined $T_{\mbox{\scriptsize eff}}$ is
derived. While the above results reveal that $T_{\mbox{\scriptsize eff}}$ does
not tend to zero as the magnitude of the driving force decreases, indicating
extrapolation to a non-zero static quantity, it remains unclear how the
effective temperature may relate to the compactivity and angoricity as defined
by Edwards statistics. Are we defining a new static quantity by determining
the static limit of $T_{\mbox{\scriptsize eff}}$, or are we expanding the
statistical mechanics of jammed granular matter to include dynamic systems by
relating $T_{\mbox{\scriptsize eff}}$, $X$ and $A$? $T_{\mbox{\scriptsize
eff}}$ is obtained in the quasi-static limit $\dot{\gamma_{e}}\rightarrow
0^{+}$, while the volume and force ensembles correspond to
$\dot{\gamma_{e}}=0$, exactly. Is it possible that a relation between
$T_{\mbox{\scriptsize eff}}$, $X$ and $A$ can be expected?

There exists the further requirement of energy conservation for the validity
of a Boltzmann approach that would guarantee:

$T_{\mbox{\scriptsize eff}}^{-1}=\frac{\partial{S}}{\partial{E}}$ (21)

As discussed above, energy is constantly dissipated in a driven granular
system, through Coulomb friction and path-dependent tangential forces between
grains. However, the input of energy by the external driving force brings the
system to a steady state where the average energy is constant over the time-
scale of the experiment. This steady state energy could be likened to the
conserved variable in a statistical formalism depicted in Eq. 21, thereby
introducing a thermodynamic meaning for $T_{\mbox{\scriptsize eff}}$.

In a compressed emulsion system the absence of Coulomb friction and inter-
particle tangential forces greatly simplifies the formalism brujic . Jamming
occurs due to osmotic pressure, and the system remains athermal as a result of
the large particle size. A well defined potential energy exists due to the
absence of tangential forces, corresponding to the deformation of the
particles at the inter-particle contact points. Therefore, a restriction to
use the energy ensemble in an effort to describe a jammed system is lifted, as
frictional tangential forces no longer hinder energy conservation.

Computer simulations of frictionless emulsion droplets cavendish  incorporate
a simulated annealing method employing an auxiliary temperature to sample the
available jammed configurations. The simulated annealing method assumes a
Boltzmann distribution, or a flat average assumption, over the jammed states
of the emulsion. The $T_{\mbox{\scriptsize eff}}$ obtained by Eq. 21 with
simulated annealing methods mk  is very close in value to the
$T_{\mbox{\scriptsize eff}}$ obtained via the FDT calculations as in the
present work. Such a result could indicate that ergodicity holds in this
frictionless system, a further justification of the methods presented herein.
Therefore, a firmer basis for the validity of using the $T_{\mbox{\scriptsize
eff}}$ obtained in the quasi-static limit to describe the statistical
mechanics of the same system at the static limit is achieved.

Further, $T_{\mbox{\scriptsize eff}}$ remains approximately constant with
varying tracer particle size, implying a zero-th law of thermodynamics for
slowly sheared jammed granular systems. These statements further provoke the
necessity for an ABC experiment, to test the zero-th law for the effective
temperature, as well as similar experiments for the compacitivity and
angoricity. Such experiments may enlighten us to understand under what
conditions $P(E)\sim\text{e}^{-\frac{E}{T_{\mbox{\scriptsize eff}}}}$,
$P(V)\sim\text{e}^{-\frac{V}{X}}$ and $P(\Pi)\sim\text{e}^{-\frac{\Pi}{A}}$
may be valid in describing the statistics of the jammed and nearly jammed
granular systems.

At this point, we believe that the most prominent direction is the exploration
of the volume and pressure ensembles. Our understanding is that these
ensembles may be sufficient to characterize the jammed state of granular
matter, while the energy ensemble may be necessary for slowly moving granular
systems. These are open questions at the present time. Recent papers in the
theory and simulation front suggest that the compactivity characterizes the
system into a phase diagram at the isostatic point, while the angoricity will
be necessary to describe the pressure ensemble of compressible granular matter
chaoming_nature .

##  V Summary

In summary, this study focuses on the dynamics of slowly sheared granular
matter in a 3D Couette cell. A mixture of spherical, transparent and bi-
disperse grains are confined between two cylinders, having walls roughened by
glued identical grains, with the inner cylinder rotated via motor. We compact
the grains by means of an external pressure in the negative z-direction. Fluid
matching the density and refractive index of the grains partially fills the
cell, allowing tracking of tracer particle trajectories as a function of time.
Tracers of varying density and size are used. Multiple cameras track the
tracer particle positions relative to the cylinders.

We find that the angular velocity of the tracer particles,
$\omega_{\theta}(r)$, follows an exponential relation with $r$, defined by the
type of packing and geometry of the Couette cell. The velocity of the last
layer of grains is non-zero, such that the shear band is located at the outer
cylinder and ensures no formation of shear bands in the bulk. Near the outer
cylinder $\omega_{\theta}(r)$ decays slowly with increasing $r$, such that
$\omega_{\theta}(r)$ can be approximated linearly with a constant local shear
rate. The constant local shear rate ensures that the mobility and diffusivity
of tracers, dependent on local shear rate, remain approximately independent of
$r$. We define this region the “constant mobility and diffusivity region”, or
the CMD region.

An “effective temperature”, $T_{\mbox{\scriptsize eff}}$, is realized by a
fluctuation-dissipation relation generalized to granular materials.
Statistical measurements are confined exclusively to the CMD region. The
mobility in the vertical direction, $M_{z}$, is found to be proportional to
the shear rate, $\dot{\gamma}_{e}$, for small enough values of
$\dot{\gamma}_{e}$. As $D_{z}$ is also found proportional to shear rate,
collapsing all the $D_{z}(r)$ for various shear rates shows a plateau in the
CMD region. An approximately constant effective temperature is obtained from
measurements of the mobility and diffusivity, under a constant external
applied force, and with sufficiently small shear rates. This effective
temperature is calculated by an analogous equation used in equilibrium
statistical mechanics. We find this effective temperature to be independent of
the tracer particle properties, and dependent only on the packing density of
the system. While this result describes an intensive property of the system,
it remains an important future study to test the effective temperature against
the laws of thermodynamics. More specifically, a test of the zeroth-law of
thermodynamics with respect to these non-equilibrium jammed systems could
expand the scope of $T_{\mbox{\scriptsize eff}}$ beyond that of an intensive
quantity of a particular system. A well defined effective temperature in the
radial direction may exist, though its existence would require a constant
external force applied in the radial direction.

The probability distribution of the displacements in the radial direction,
$P(\Delta r)$, reveals exponential fluctuations. The analysis of the
fluctuations reveals a power law, sub-diffusive, process, $\langle\Delta
r^{2}\rangle\sim{\Delta t}^{\alpha}$, with $\alpha$ less than unity. A similar
analysis for fluctuations in the angular direction reveal a super-diffusive
process, $\langle\Delta\theta^{2}\rangle\sim{\Delta t}^{\beta}$, with $\beta$
greater than unity. Lastly, the probability distribution of the displacements
in the vertical direction are found have a Gaussian distribution such that
$\langle\Delta z^{2}\rangle\sim{\Delta t}$. It is this linearity that defines
vertical displacement as a diffusive process, and allows for the use of the
Fluctuation-Dissipation relation to calculate the diffusivity in the vertical
direction. We further discover a linear relationship between angular
displacement and the time between measurements $\Delta{\theta_{e}}\sim{\Delta
t}$, such that all mean square fluctuations can be defined in terms of
$\Delta{\theta_{e}}$ for the small shear rates of our experiments.

In the CMD region, the linear approximation of $\omega_{\theta}(r)$
proportional to approximately constant external shear rate,
$\dot{\gamma}_{e}$, allows for the collapsing of all tracer particle velocity
curves via dividing $\omega_{\theta}(r)$ by the shear rate. This collapse
reveals a periodic shape with a small amplitude and periodic length roughly
equal to the grain size. The effect is shown to be weaker in packings with
smaller size grains. We further apply this remarkable scaling feature
$v_{z}(r)$ and $v_{r}(r)$, achieving similar results.

It is important to note that the effective temperature, defined in this study
for small shear rates, does not remain constant as the shear rate increases.
While previous studies have discovered an an increasing effective temperature
via simulations, we have measured diffusivity and mobility separately in an
effort to calculate $T_{\mbox{\scriptsize eff}}$ through a fluctuation-
dissipation relation. We find the diffusivity in the z-direction remains
approximately constant throughout the range of shear rates used in this
experiment, while the mobility in the z-direction approaches a plateau.
exclusively increasing $T_{\mbox{\scriptsize eff}}$. Such an effect in the
radial direction would be of great interest for future studies in sheared
granular dynamics.

The nearly constant value of $T_{\mbox{\scriptsize eff}}$ with respect to
varying tracer particle size indicates that a zero-th law of thermodynamics
for slowly sheared jammed granular systems could be valid and prompts one to
perform an ABC experiment, fully testing the zero-th law for the effective
temperature.

As we work towards a more complete description of the statistical mechanics of
jammed granular matter, we strive to incorporate the varied statistical
ensembles into one fundamental picture. These ensembles include the energy
ensemble, as described herein, along with the volume and force ensembles, as
proposed by Edwards. Such an incorporation may link static quantities of
compacitivity and angoricity, describing volume and force ensembles,
respectively, to the dynamic effective temperature presented in this study,
derived from the energy ensemble. The exact nature of the relation between
such quantities remains an open topic. Ultimately, these quantities will help
to develop a thorough statistical description for jammed granular matter and
reveal an equation of state. A deeper topic of concern is the formation of a
clear definition of energy in jammed granular matter. Energy is not conserved
in frictional systems and it remains open to debate as to how one would
incorporate energy into the statistical mechanics.

One possible approach to describe the energy of jammed systems is to consider
the similarities between the inherent structure formalism of glasses and the
exploration of jammed states in granular matter, at least for the case of
frictionless granular systems. Inherent structures probe a network of
potential energy basins within an energy landscape. Such an approach toward
the jammed states of granular matter may assist in understanding exactly what
is meant by energy within the framework of a non-equilibrium system.

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We are deeply grateful to M. Shattuck for help in the design of the Couette
cell and J. Kurchan and J. Brujić for discussions. We acknowledge financial
support from the DOE, and NSF.