State-dependent driving: A route to non-equilibrium stationary states

For small values of $f(\leq f_{c}=4.4$ $\rm{Hz})$, the coin remains stuck to the plate, while for larger values of $f$, it begins to slip with respect to the plate. Typical time traces of the velocity of the center of mass of the coin and a point fixed on the plate for $f=6$ Hz are shown in Fig. 2(a). In the phase in which the coin slips with respect to the plate, its center of mass drifts slowly in time. However, for small time intervals, its response $x_{\rm c}(t)$, namely, the location of its center of mass, is approximately sinusoidal, and has the same frequency of variation as that of the plate. The probability distribution of $|v|$, the magnitude of the velocity of the center of mass of the coin, is plotted in Fig. 2(c); the distribution is obtained by sampling the trajectories in the long-time limit over a time window. The variation of the time-averaged amplitude of displacement of the coin and its phase difference $\phi$ with respect to the plate in the long-time limit is plotted in Fig. 2, panels (d) and (e), respectively, as a function of the driving frequency.

Our main experimental observations may be summarized as follows:

1. 1.

   At long times, the velocity as well as the displacement of the coin is approximately sinusoidal in time (Fig. 2(a)).

2. 2.

   The velocity distribution of the coin has a tail that is exponential (Fig. 2(c)).

3. 3.

   The time-averaged amplitude of the displacement of the coin $x_{\rm c0}$ decreases with the drive frequency (Fig. 2(d)).

4. 4.

   The phase difference $\phi$ saturates to a large value with increase in the drive frequency (Fig. 2(e)). This is in contrast to what is seen in the case of a damped, driven harmonic oscillator with a real damping constant, which in turn precludes use of noisy harmonic oscillator to model our experimental scenario. The case of a complex damping constant is discussed in the appendix.
   

The most evident way to describe the system would be to use the Coulomb friction model   [18]. If the driving frequency is such that the amplitude of the sinusoidal forcing exerted by the plate is larger than the maximum static friction force, i.e., $|m(2\pi f)^{2}x_{\rm p0}|>\mu_{s}mg$, with $\mu_{s}$ being the coefficient of static friction between the coin and the plate and $m$ the mass of the coin, the coin will undergo both stick and slip motion periodically. In the stuck state, i.e., when $|m(2\pi f)^{2}x_{\rm p0}\cos(2\pi ft)|<\mu_{s}mg$, trajectory of the coin will follow the sinusoidal nature of the trajectory of the plate. However, as soon as the driving force exceeds $\mu_{s}mg$, the coin will start to slip with respect to the plate and it will then move under a constant acceleration due to the force of kinetic friction $\mu_{k}mg$. Here, $\mu_{k}$ is the coefficient of kinetic friction between the coin and the plate. Hence, the velocity of the coin will be partly sinusoidal and partly linear in a single period. This model gives rise to a deterministic velocity profile. This is in stark contrast to the noisy sinusoidal nature of the velocity observed in the experiments. In addition, the velocity distribution, the amplitude and the phase response of the coin as obtained from the Coulomb friction model of friction is very different from what is observed in the experiments. We have provided the details in the appendix.

The response of the system for $f>f_{c}$ and $f<f_{c}$ is qualitatively different. For $f<f_{c}$, the coin and the plate behave as a single rigid object, i.e., $x_{c}(t)=x_{p}(t)$, which is unlike the situation for $f>f_{c}$. The description below is limited to the latter case only.

Experiments suggest that the coin never gets stuck to the plate at the smallest time scale over which the motion of the coin and the plate is measured, i.e., $5.5$ $\rm{ms}$. Hence, to explain the experimental observations, we propose that the coin has access to (i) a ‘high-friction’ state (which in a limiting case becomes the stuck state) and (ii) a ‘low-friction’ state (or a slip state). Without the presence of ‘high-friction’ state, the coin would never experience the external driving applied to the plate and hence, the response of the coin would never reflect the waveform of the drive. However, over a large parameter regime, we find that the amplitude of the coin is smaller than that of the plate. Thus, it is reasonable to assume that the ‘high-friction’ state survives over a timescale that is smaller than the smallest time scale of measurement, and that during this time scale, $\tau_{\rm p}$, a fraction of the momentum of the plate gets transferred to the coin. Here, it becomes necessary to differentiate between a ‘stuck’ state (observed for $f<f_{c}$) and a ‘high-friction’ state. In contrast to a ‘high-friction’ state, in a ‘stuck’ state, the entire momentum and hence, the entire force would be transferred from the plate to the coin. We assume the transitions between the high-friction and the low-friction state to occur randomly in time. We thus propose a stochastic variant of the Coulomb friction model for the motion of the coin on the vibrated plate which captures the essential qualitative features of its dynamics observed in the experiments described above.

To be consistent with the experiment, we choose to describe the motion of the center of mass of the coin in an inertial frame, i.e., in the laboratory frame. With respect to such a frame, the plate on which the coin is placed is being vibrated sinusoidally in time, so that its velocity at time $t$ reads $v_{\rm p}(t)=v_{\rm p0}\sin(2\pi ft)$. When the coin is in the ‘high-friction’ state, the equation of motion of its center of mass is the usual Newton’s equation of motion:

where $v$ is the velocity of the coin, while $F(t)$ is the force experienced by the coin due to transfer of momentum from the plate to the coin. Here, the quantity $m_{\rm p}$ is the mass of the plate and $\tau_{\rm p}$ as defined earlier is the timescale over which the ‘high-friction’ state survives and during which the momentum of the vibrating plate is transferred to the coin. On the other hand, when the coin is in the ‘low-friction’ (slip) state, it is detached from the plate, and consequently, it moves in presence of an effective damping force that dissipates in time the initial momentum of the coin at the instant of detachment from the plate. Hence, in the ‘low-friction’ state, the equation of motion of center of mass is given by $m\partial_{t}v=-\gamma(v-v_{p})$, where $\gamma>0$ is a phenomenological dissipation constant. The coin toggles randomly in time between the ‘high-friction’ and the ‘low-friction’ state. Introducing a random variable $\xi$ taking on values 1 or 0 corresponding to the ‘high-friction’ and the ‘low-friction’ state, respectively, one may on the basis of the foregoing write down the equation of motion of the center of mass as a stochastic differential equation, a Langevin-like equation, of the form

Equation (9) has to be supplemented by another equation describing the time evolution of the instantaneous $\xi$, namely, $\xi(t)$. While derivation of an exact form of the latter would invariably involve a detailed modeling of the dynamics of the contact region via which the friction force is transmitted, we here offer a phenomenological description for the evolution of $\xi$ in terms of a stochastic Markov process, namely, between times $t$ and $t+{\rm dt}$, the variable $\xi(t)$ is updated to read $\xi(t+{\rm d}t)=1$ with probability $\beta{\rm d}t$, while $\xi(t+{\rm d}t)=0$ with the complementary probability $1-\beta{\rm d}t$. Here, $\beta>0$ is a dynamical parameter. It then follows that the random time $\tau$ between two successive occurrences of the value unity for $\xi$ is distributed as an exponential: $p(\tau)=\beta e^{-\beta\tau};\leavevmode\nobreak\ \tau\in[0,\infty)$, and that the average $\tau$ is given by $\langle\tau\rangle=1/\beta$. As a function of time, $\xi(t)$ appears as a set of impulses distributed randomly in time, as shown in Fig. 1(c). For a given drive, the parameter $\beta$ is a phenomenological parameter. To fit our experimental data we require $\gamma\propto 1/(f-f_{c})$. This suggests strain rate induced weakening of the contact zone which could possibly arise from the reduction of contact area with increasing strain rate   [37]. A reduction in $\gamma$ results in re-scaling of the relaxation time $\tau_{p}\sim m/\gamma\propto(f-f_{c})$, which thereby influences the momentum transfer process via $F(t)=m_{p}v_{p}/\tau_{p}$. For the coin to slip, the contact zone must reach a critical size   [38]. Further, given $\tau_{p}$ describes the dynamics at the contact it also likely to influence the time scale associated with the growth of the slip region of the contact zone. We thus assume $\beta^{-1}$ to be a function of $\tau_{p}$. This forms the rationale for assuming a linear dependence between $1/\beta$ and $(f-f_{c})$ which is used to fit the data. The specific forms of $\gamma$ and $\beta$ as a function of $f$ are given in the appendix.

It is interesting and pertinent that we draw a parallel between Eq. (9) and the usual form of the Langevin equation that one encounters in describing say the paradigmatic Brownian motion. Besides the nature of the stochastic noise, which in the latter is a Gaussian, white noise and in Eq. (9) is a dichotomous noise, one very important difference is the following. In the case of the Brownian motion, the strength of the noise term is a constant that does not depend on the value of the dynamical variable in question (more precisely, the constant is by virtue of the fluctuation-dissipation theorem related to the equilibrium temperature of the ambient medium). By contrast, in Eq. (9), the strength of the noise term is explicitly dependent on the value of the dynamical variable $v$ being studied. Another point worth mentioning is that the noise in the case of Brownian motion is considered stationary, while in our case, the noise is non-stationary.

Another interesting parallel that one may draw is between the dynamics (9) and the dynamics of a driven, damped harmonic oscillator evolving in presence of noise. In such an approach, one may consider the noise to be multiplicative, in the sense that it appears as a coefficient of the damping term in the dynamics, i.e., $m\partial_{t}v=-\gamma\xi(t)(v-v_{p})+F(t)$. The problem with this approach is that even when the coin is slipping, there will be the unphysical dynamics of momentum transfer between the coin and the plate. In contrast, the model presented here is nevertheless founded on physical reasoning, as presented in the paragraph preceding Eq. (9). Note that in the dynamics described by Eq. (9), the force experienced by the coin is being continually reset in time between the pure sinusoidal drive and the entirely dissipative form as the random variable $\xi$ toggles in time between its two-possible values. Equation (9) may be considered representative of a class of stochastic dynamical systems in which the force resets stochastically in time between a set of possible values, the two-state process considered here being the simplest one may conceive.

The dynamics described by Eq. (9) involves two time scales $1/\beta$, setting the average time between two impulses, and $1/f$. We work in the regime in which $1/\beta\ll 1/f$. Consequently, in the long-time limit, we expect the velocity $v$ to vary sinusoidally in time with occasional jump in values as an effect of toggling acting on a time scale that is faster compared to the sinusoidal variation. As far as the parameter $\gamma$ is concerned, its magnitude would set the cut-off scale of the velocity in the slip state.

Figure 2, panels (b)-(e) compare the results obtained from numerical simulation of the dynamics (9) with the experimental findings. In panel (b), we see that consistent with our expectations mentioned above, the velocity does represent an almost sinusoidal variation in time and correspondingly, the velocity distribution in (c) exhibits a peak at a value around the amplitude value of the sinusoidal variation in (b). The toggling phenomenon manifests itself as deviations from a pure sinusoidal variation in (b) and contributes to a width around the peak in the distribution in (c). It is remarkable that our phenomenological model (9) is able to reproduce in a rather striking manner the experimental features of the coin dynamics in (a) and the velocity distribution in (c). Moreover, the distribution has a tail that is exponential, as seen in panel (c). In simulations, this is a result of dissipation set by $\gamma$ in the dynamics (9). Moreover, the amplitude and the phase response as obtained from the simulations are in good agreement with the experimental data (see Fig. 2 (d) and (e)). As for the coin-trick problem that was proposed in the introduction, we expect the coin to move with a constant velocity. The magnitude of the velocity would depend on the coefficient of friction and the acceleration of the card being pulled out.

Effectively, this is a two-dimensional version of the previous problem, wherein the particle-substrate interaction generates the randomness in the motion. Individually, both the $x$- and the $y$-components of the motion are approximately sinusoidal in nature. The abrupt alteration in the trajectory marked as $P_{1}$ in Fig. 4 (b) is an example of such scattering. Occasional collisions with the boundary leads to additional randomness in the trajectories. An instance of this kind of scattering is shown as $P_{2}$ in Fig. 4 (b). These scattering events introduce a random phase difference between the two components which results in the randomization of the trajectories. The probability distribution of the magnitude of the $x$-component of the velocity, $|v_{x}|$, of a rolling sphere for representative value of $f$ is plotted in Fig. 3(c). A similar distribution is seen for $|v_{y}|$ also. In the tails, the velocity distribution varies as $P(|\mathbf{v}|)\sim\exp(-|\mathbf{v}|^{\alpha})$, with $\alpha\approx 1$. This is very different from the well-known Maxwell distribution, in which case $\alpha=2$.

Similar to previous experiment, we postulate that the noise in the trajectory arises from occasional slipping of the ball with respect to the plate. These partial slips have a concomitant toggling between the sliding and the rolling state of the system, and hence, similar to the previous example, the resulting noise can also be considered dichotomous. Thus, the Langevin-like equation of motion of the velocity of the center of mass of the ball has the form

with $\sigma=\{x,y\}$. Here, $\xi(t)=0$ and $1$ correspond to sliding and rolling states respectively. It is to be noted that contrary to the coin on the plate problem, here momentum transfer from the plate to the sphere happens in the rolling (‘low-friction’) state. This is due to the fact in a pure rolling motion (without slipping), the contact point of the sphere is temporarily at rest with respect to the plate. On the other hand, in the sliding state which in this case is a relatively ‘high-friction’ state, the sphere moves in the presence of the damping force.

However, when there are more than one particle in the system, the inter-particle collisions also contribute to the over all noise in the trajectories. In the following subsection, we consider the case of many spheres on an orbital shaker.

The trajectories shown in Fig. 4 (b) and (c) are obtained by averaging a sequence of images, each taken with a time interval of $100$ ms. The micrographs in Fig. 4 (c) to (d) highlight the influence exerted by one particle on the other. When two spheres collide tangentially, two events can occur - (i) the spheres alter their direction and continue swirling, or, (ii) they form a bound dyad-like pair; This pair moves together mostly in a reciprocating manner, rolling back and forth in the direction that is perpendicular to the line joining the centers of the two balls. In the direction that is along the line joining the two centers, the pair does a sliding motion. Figure 4(c) shows two trajectories $\pi_{1}$ and $\pi_{2}$ that correspond to a collision and formation of a transiently stable dyad structure. In its dyad form, the two trajectories move parallel to each other. This structure is short lived and often spontaneously break apart. When it does so, the trajectories $\pi_{1}$ and $\pi_{2}$ depart from each other. The destabilization of the dyads can be brought about either by a particle-substrate or a particle-particle scattering event. In the case of the trajectory shown in Fig. 4(c), it is the particle-substrate scattering that destabilizes the dyad.

Occasionally, a third ball bumps into the two-particle pair. This can either destabilize the dyad or can form a compact three-particle triad. For spheres in contact, rolling in the same direction causes shearing of the contact region   [3]. Thus, a triad cannot roll and, hence, it either becomes static or does small sliding motion. Figure 4(d) shows a sequence of micrographs that captures the event corresponding to the formation of the triad. These micrographs are obtained by using exposure time of $10$ ms. This long exposure produces a motion-induced dilation effect of the objects. Thus, moving objects in these micrographs will appear extended, and its shape tells us about the nature of motion, e.g., a roller’s motion will appear as a curved line, and dyads will appear as parallel straight lines. For static objects, motion-induced elongation is absent, and the objects appear as dots. The leftmost frame shows a dyad and a nearby rolling isolated sphere. The other two particles in the frame are present to provide a reference. After collision, the three spheres form a static triad. Absence of substantial motion for this triad can be inferred from the lack of motion- induced dilation (see the second frame of the micrograph in Fig. 4 (d) and first frame in Fig. 4 (e)). For any higher-order structure involving more than 3 spheres that can form, the only available mode of movement is sliding. So the transitions in these structures are limited to stuck to sliding states. For the values of $f$ reported in this paper, the clusters (dyads, triads and higher order structures) are only transiently stable. Structures involving more than 3 particles are destabilized by particle collisions. An instance of this destabilization can be seen in Fig. 4(e). Figure  4(f) shows the corresponding sequence of formation and dissolution of particle clusters for a system of 11 particles that is being driven. The system begins with no clustering, i.e., $n=1$ and then quickly shows clustering. We find that dyads $n=2$ are the most frequently formed clusters and that these can be stable for few seconds. Occasionally, we do observe triads $n=3$, but these structures compared to the dyads are relatively less stable.

The very fact that two moving particles can come to a halt after a collision points us to instances where the momentum is not conserved. Moreover, the transitions between the rolling, siding and the stuck states are hysteretic in nature, i.e., the frequency at which a moving particle makes a transition to the stuck state is lower than the corresponding transition from a stuck to a moving state. Figure. 4 (a) shows this hysteresis between stuck and rolling states for a single sphere.

With each hysteretic transition, a certain amount of kinetic energy is lost. It would thus be natural to expect higher particle density to result in higher frictional dissipation and hence, lower velocities. However, contrary to this, we observe in Fig. 5 (a), (b) and (c) the tail of the velocity distribution $P(|\mathbf{v}|)\sim\exp(-|\mathbf{v}|^{\alpha})$, with $\alpha\approx 1$, to increase with the density.