Internal temperature and scale separation

The internal noise observed in the study of the longitudinal and transversal structure factor can be characterized by means of an average local granular temperature $T_{\sigma }$:

$$T_{\sigma}=\langle \vert\mathbf{v}- \langle \mathbf{v} \rangle_\sigma \vert^2 \rangle_\sigma$$ (5.71)

where $\langle ... \rangle_\sigma$ means an average on a region of linear size $\sigma$.

If we call $L(t)$ a characteristic correlation length of the system, since when $\sigma \gg L(t)$ the local average tends to the global (zero) momentum, then

$$\lim_{\sigma \to \infty} T_{\sigma} = E$$ (5.72)

For $\sigma < L(t)$, instead, $T_{\sigma} < E$. The behavior of $T_\sigma$ in the uncorrelated (Haff) regime and in the correlated (asymptotic) regime for two different values of $r$ is presented in Fig. fig_tsigma.

A very important observation is the following: for quasi elastic systems $T_\sigma$ exhibits a plateau for $1\ll \sigma\ll L(t)$ that identifies the strength of the internal noise and individuates the mesoscopic scale needed by a hydrodynamics description. The local temperature ceases to be well defined for smaller $r$: this clearly suggests the absence of scale separation between microscopic and macroscopic fluctuations in the strongly inelastic regime [93].

Figure: The scale dependent temperature, $T_{\sigma }$, defined in the text, as function of the coarse graining size $\sigma$ for $\tau =24$ (incoherent regime) and $\tau =860$ (correlated regime for both choices of $r$). In the early incoherent regime the total energy per particle and $T_{\sigma }$ remain nearly indistinguishable for a wide window of values of $\sigma$ (length scales): for $\sigma <L(\tau )$ the thermal energy becomes much smaller than the kinetic energy, a clear indication of the onset of macroscopic spatial order. This becomes evident in the correlated regime. Furthermore, in the correlated regime the more elastic case presents a short wavelength plateau, indicating a well defined mesoscopic temperature, and therefore a clear separation between the microscopic scales (the first slope) and the macroscopic scales (the second slope).