Energy distribution and effective temperatures in a driven dissipative model

We investigate nonequilibrium behavior of driven dissipative systems, using a model we recently presented [Phys. Rev. Lett., 93, 240601 (2004)]. We solve the non-Boltzmann steady state energy distribution and the temporal evolution to it, and find its high energy tail to behave exponentially. We demonstrate that various measures of effective temperatures generally differ. We discuss infinite hierarchies of effective temperatures defined from moments of the nonexponential energy distribution, and relate them to the “configurational temperature,” measured directly from instantaneous particle locations without any kinetic information. We calculate the “granular temperature,” characterizing the average energy in the system, two different “fluctuation temperatures,” scaling fluctuation-dissipation relations, and the “entropic temperature,” defined from differentiating the entropy with respect to energy.

Figure 1

The possible interactions in our model: dissipative two particle interaction (left) and conservative system-bath interaction (right).

Figure 2

Contours of the ratio $⟨e⟩∕T_B$ between the average energy per particle in the system and the bath temperature vs the restitution coefficient $\alpha$ and the coupling strength $f$ to the bath, as given by Eq. (4).

Figure 3

Steady state energy distribution for restitution coefficient $\alpha =0.5$ and coupling strength $f=0.5$ (resulting in $T_G=T_B∕2$) obtained in a numerical simulation of the model (solid line). Exponential distributions at temperatures $T_B$ and $T_G$ are given for reference (dashed lines).

Figure 4

(Color online) The hierarchies of effective temperatures $T_R^{\left(n\right)}$ and $T_M^{\left(n\right)}$ for several model parameters, obtained by substituting Eq. (7) in Eqs. (9).

Figure 5

(Color online) Temporal evolution to the steady state energy distribution (dashed line) for restitution coefficient $\alpha =0.5$ and coupling strength $f=0.5$ obtained in a numerical simulation of the model, starting with all particles having energy $e_0=0$ (ascending lines after $N$ and $2N$ interactions in the system), or $e_0=5T_B$ (descending lines after $N$, $2N$, $3N$, $4N$, and $5N$ interactions).

Figure 6

The interactions between the system $\left\{e_i\right\}$ and the probe $\left\{x_i\right\}$, introduced in addition to those given in Fig. 1.

Figure 7

Contours of the ratio $T_F∕T_G$ between the fluctuation temperatures and the granular temperature vs the restitution coefficient $\alpha$ and the coupling strength $f$ in the limit of weak coupling between the probe and the system $(h\to 0)$, as given in [40]. In this limit, $T_F^{\left(N\right)}=T_F^{\left(1\right)}$.

Figure 8

Contours of the ratio $T_F^{\left(N\right)}∕T_F^{\left(1\right)}$ between the many-particle and single-particle fluctuation temperatures vs the coupling strengths $f$ and $h$ in the maximal dissipation limit $(\alpha =0)$, as given in [41].

Figure 9

(Color online) The deviation of the entropy $s$ from its equilibrium value $s^{\mathit{eq}}$ vs the average energy $⟨e⟩$ for several values of the restitution coefficient $\alpha$ and for coupling strength $f=0.5$.

Figure 10

(Color online) The ratio $T_S∕T_S^{\mathit{eq}}$ of the entropic temperature to its equilibrium value vs the average energy $⟨e⟩$ for several values of the restitution coefficient $\alpha$ and for coupling strength $f=0.5$.