Dynamics of a tracer granular particle as a nonequilibrium Markov process

The dynamics of a tracer particle in a stationary driven granular gas is investigated. We show how to transform the linear Boltzmann equation, describing the dynamics of the tracer into a master equation for a continuous Markov process. The transition rates depend on the stationary velocity distribution of the gas. When the gas has a Gaussian velocity probability distribution function (PDF), the stationary velocity PDF of the tracer is Gaussian with a lower temperature and satisfies detailed balance for any value of the restitution coefficient $\alpha$. As soon as the velocity PDF of the gas departs from the Gaussian form, detailed balance is violated. This nonequilibrium state can be characterized in terms of a Lebowitz-Spohn action functional $W\left(\tau \right)$ defined over trajectories of time duration $\tau$. We discuss the properties of this functional and of a similar functional $\overline{W}\left(\tau \right)$, which differs from the first for a term that is nonextensive in time. On the one hand, we show that in numerical experiments (i.e., at finite times $\tau$), the two functionals have different fluctuations and $\overline{W}$ always satisfies an Evans-Searles-like symmetry. On the other hand, we cannot observe the verification of the Lebowitz-Spohn-Gallavotti-Cohen (LS-GC) relation, which is expected for $W\left(\tau \right)$ at very large times $\tau$. We give an argument for the possible failure of the LS-GC relation in this situation. We also suggest practical recipes for measuring $W\left(\tau \right)$ and $\overline{W}\left(\tau \right)$ in experiments.

Figure 1

Measure of transition rates in Monte Carlo simulations. Sections of the function $D\tilde{Y}(u_{\sigma },u_{\sigma }^{\prime })∕t\mathrm{d}{u}_{\sigma }\mathrm{d}{u}_{\sigma }^{\prime }$ for the values $u_{\sigma }=-0.1$, $u_{\sigma }=0.9$, and $u_{\sigma }=3.5$ for three different cases. Top: the gas has Gaussian velocity PDF with temperature $T=1$ and $\alpha =1$ (thermodynamic equilibrium). Middle: the gas has a Gaussian velocity PDF and $\alpha =0.5$ (detailed balance in absence of thermodynamic equilibrium, $T_{*}=0.6\equiv T^{\prime }<T$). Bottom: the gas has a non-Gaussian velocity PDF characterized by a first Sonine correction with $a_2=0.1$ (nonequilibrium stationary state, $T_{*}=0.62\approx T^{\prime }<T$ and $a_2^{*}=0.057⪡a_2$). The dashed lines mark the theoretical expressions given in Eqs. (37, 38, 39).

Figure 2

(Color online) Verification of the fluctuation relations for $W\left(\tau \right)$ (squares) and $\overline{W}\left(\tau \right)$ (circles). Each line is composed of two graphs and shows the results for a particular choice of $\alpha$ and $a_2$: the left graph is at small times and the right graph at large times. In each frame, the inset contains the PDFs of $W$ and $\overline{W}$, while the main plot shows $a{s}_{\tau }\left(W\right)$ vs $W$ as well as $a{s}_{\tau }\left(\overline{W}\right)$ vs $\overline{W}$. The red (solid) line in the insets is a Gaussian fit for the PDF of $\overline{W}$. The times $\tau$ are rescaled with the tracer mean free time ${\tau }_c$ (which varies with the parameters). For each choice of parameters, inside the right graph, we show the value of $⟨W\left(\tau \right)⟩$ and of $⟨w⟩\equiv ⟨W\left(\tau \right)⟩∕\tau \equiv ⟨\overline{W}\left(\tau \right)⟩∕\tau \equiv {\lim }_{\tau \to \infty }W\left(\tau \right)∕\tau \equiv {\lim }_{\tau \to \infty }\overline{W}\left(\tau \right)∕\tau$, which marks the distance from equilibrium. The dashed line has slope 1.

Figure 3

Cumulants of the fluctuations of $W\left(\tau \right)$ (empty symbols) and $\overline{W}\left(\tau \right)$ (solid symbols). (a) Average values of $W\left(\tau \right)$ and $\overline{W}\left(\tau \right)$ as a function of $\tau$ and for different choices of the parameters $\alpha$ and $a_2$. (b) Second cumulants of $W\left(\tau \right)$ and $\overline{W}\left(\tau \right)$ as a function of $\tau$. (c) Ratio between the second cumulant of $\overline{W}\left(\tau \right)$ and that of $W\left(\tau \right)$ as a function of $\tau$. (d) Ratio between $5\sqrt{{⟨W{\left(\tau \right)}^2⟩}_c}$ and $⟨W\left(\tau \right)⟩$. When this ratio is much smaller than 1, the probability of observing negative events in the fluctuations of $W\left(\tau \right)$ becomes extremely low. All the times $\tau$ are rescaled by the tracer mean free time.