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The authors have declared that no competing interests exist.

Conceived and designed the experiments: AG AP AS. Performed the experiments: AG AP. Analyzed the data: AG AP AS AV. Contributed reagents/materials/analysis tools: AG AP AS AV. Wrote the paper: AG AP AS AV.

The condition of thermal equilibrium simplifies the theoretical treatment of fluctuations as found in the celebrated Einstein’s 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. Einstein’s 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.

Several fundamental results of statistical mechanics are obtained under the crucial assumption of thermal equilibrium. A celebrated example of the power of the equilibrium hypothesis is given by the theoretical treatment of Brownian motion developed by Einstein at the beginning of the 20th century

Recently, it has been shown that even in out-of-equilibrium systems a relation between response and spontaneous fluctuations still exists

Often in nonequilibrium systems the different timescales are not clearly separated and the picture in terms of effective temperature does not hold. Instances of this entanglement of scales appear in climate and turbulence

A paradigmatic case in which Eq. (3) can be tested is that of strongly fluidized granular media

A sketch of the setup illustrates the essential components. A wheel rotating around a fixed axis is suspended in a cylindrical cell containing steel spheres. The cell is shaken in order to fluidize the material and obtain a granular gas. The wheel performs a Brownian-like dynamics, randomly excited by collisions with the spheres. A small motor is coupled to the wheel axis, in order to apply an external impulsive perturbation. An angular encoder reads the angular velocity of the wheel. Statistical properties of the velocities of the spheres are collected through a fast camera, placed above the system. A detailed description is presented in Methods section.

The measurements of interest in our experiment are the response of the angular velocity

Response function

For higher values of the gas density, panels (b) and (c) of

The GFDT discussed above, Eq. (3), accounts for all the observations of

The static properties of the system are fully described by the joint probability density function (PDF)

PDF of the rotator’s angular velocity rescaled by

Assuming a factorization among

that used with (5) for the GFDT gives

The emergence of the relevance of coupling between wheel and fluid, going from the dilute case to the dense one, already appears in the study of autocorrelations functions. At low packing fraction, the shape of

Absolute value of autocorrelations in log-log scale (symbols denote positive values) for different densities.

Both these features imply the existence of more than one time-scale. In a molecular fluid at equilibrium, however, even when

In order to find an explicit form for the correlation functions appearing in the GFDT, Eq. (4), it is necessary to understand the role of the relevant degrees of freedom coupled with

Correlation between the angular velocity of the probe

The granular medium, made of

At the currently used maximum acceleration (

We would like to thank MD. Deen for technical support and A. Petri for useful comments on the manuscript. The authors acknowledge the support of the Italian MIUR under the grants: FIRB-IDEAS n. RBID08Z9JE. AP and AV acknowledge the support of the Italian MIUR under the grant PRIN n. 2009PYYZM5.