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Register a new userThe classical granular temperature and slightly beyond
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One goal of this paper is to discuss the classical definition of granular temperature as an extension of its thermodynamic equivalent and a useful concept which provides an important characterization of fluidized granular matter. Following a review of some basic concepts and techniques (with emphasis on fundamental issues) we present new results for a system that can exhibit strong violations of equipartition, yet is amenable to description by classical granular hydrodynamics, namely a binary granular gas mixture. A second goal of this article is to present a result that pertains to dense granular and molecular solids alike, namely the existence of a correction to the elastic energy which is related to the heat flux in the equations of continuum mechanics. The latter is of the same (second) order in the strain as the elastic energy. Although recent definitions of temperatures for granular matter, glasses and other disordered many-body systems are not within the scope of this article we do make several general comments on this subject in the closing section.
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The condition of thermal equilibrium simplifies the theoretical treatment of fluctuations as found in the celebrated Einsteins relation between mobility and diffusivity for Brownian motion. Several recent theories relax the hypothesis of thermal equilibrium resulting in at least two main scenarios. With well separated timescales, as in aging glassy systems, equilibrium Fluctuation-Dissipation Theorem applies at each scale with its own effective temperature. With mixed timescales, as for example in active or granular fluids or in turbulence, temperature is no more well-defined, the dynamical nature of fluctuations fully emerges and a Generalized Fluctuation-Dissipation Theorem (GFDT) applies. Here, we study experimentally the mixed timescale regime by studying fluctuations and linear response in the Brownian motion of a rotating intruder immersed in a vibro-fluidized granular medium. Increasing the packing fraction, the system is moved from a dilute single-timescale regime toward a denser multiple-timescale stage. Einsteins relation holds in the former and is violated in the latter. The violation cannot be explained in terms of effective temperatures, while the GFDT is able to impute it to the emergence of a strong coupling between the intruder and the surrounding fluid. Direct experimental measurements confirm the development of spatial correlations in the system when the density is increased.
Pyrochlore magnets are candidates for spin-ice behavior. We present theoretical simulations of relevance for the pyrochlore family R2Ti2O7 (R= rare earth) supported by magnetothermal measurements on selected systems. By considering long ranged dipole-dipole as well as short-ranged superexchange interactions we get three distinct behaviours: (i) an ordered doubly degenerate state, (ii) a highly disordered state with a broad transition to paramagnetism, (iii) a partially ordered state with a sharp transition to paramagnetism. Thus these competing interactions can induce behaviour very different from conventional ``spin ice. Closely corresponding behaviour is seen in the real compounds---in particular Ho2Ti2O7 corresponds to case (iii) which has not been discussed before, rather than (ii) as suggested earlier.
It is demonstrated, by numerical simulations of a 2D assembly of polydisperse disks, that there exists a range (plateau) of coarse graining scales for which the stress tensor field in a granular solid is nearly resolution independent, thereby enabling an `objective definition of this field. Expectedly, it is not the mere size of the the system but the (related) magnitudes of the gradients that determine the widths of the plateaus. Ensemble averaging (even over `small ensembles) extends the widths of the plateaus to sub-particle scales. The fluctuations within the ensemble are studied as well. Both the response to homogeneous forcing and to an external compressive localized load (and gravity) are studied. Implications to small solid systems and constitutive relations are briefly discussed.
We review generalized Fluctuation-Dissipation Relations which are valid under general conditions even in ``non-standard systems, e.g. out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperperturbed dynamics. In these relations, typically one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in non-standard cases, including driven granular media, systems with a multiscale structure, active matter and systems showing anomalous diffusion.
We experimentally investigate the response of a sheared granular medium in a Couette geometry. The apparatus exhibits the expected stick-slip motion and we probe it in the very intermittent regime resulting from low driving. Statistical analysis of the dynamic fluctuations reveals notable regularities. We observe a possible stability property for the torque distribution, reminiscent of the stability of Gaussian independent variables. In this case, however, the variables are correlated and the distribution is skewed. Moreover, the whole dynamical intermittent regime can be described with a simple stochastic model, finding good quantitative agreement with the experimental data. Interestingly, a similar model has been previously introduced in the study of magnetic domain wall motion, a source of Barkhausen noise. Our study suggests interesting connections between different complex phenomena and reveals some unexpected features that remain to be explained.
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