We review the general aspects of the concept of temperature in equilibrium and non-equilibrium statistical mechanics. Although temperature is an old and well-established notion, it still presents controversial facets. After a short historical survey of the key role of temperature in thermodynamics and statistical mechanics, we tackle a series of issues which have been recently reconsidered. In particular, we discuss different definitions and their relevance for energy fluctuations. The interest in such a topic has been triggered by the recent observation of negative temperatures in condensed matter experiments. Moreover, the ability to manipulate systems at the micro and nano-scale urges to understand and clarify some aspects related to the statistical properties of small systems (as the issue of temperature's "fluctuations"). We also discuss the notion of temperature in a dynamical context, within the theory of linear response for Hamiltonian systems at equilibrium and stochastic models with detailed balance, and the generalised fluctuation-response relations, which provide a hint for an extension of the definition of temperature in far-from-equilibrium systems. To conclude we consider non-Hamiltonian systems, such as granular materials, turbulence and active matter, where a general theoretical framework is still lacking.

A unified derivation of the off equilibrium fluctuation dissipation relations (FDR) is given for Ising and continous spins to arbitrary order, within the framework of Markovian stochastic dynamics. Knowledge of the FDR allows to develop zero field algorithms for the efficient numerical computation of the response functions. Two applications are presented. In the first one, the problem of probing for the existence of a growing cooperative length scale is considered in those cases, like in glassy systems, where the linear FDR is of no use. The effectiveness of an appropriate second order FDR is illustrated in the test case of the Edwards-Anderson spin glass in one and two dimensions. In the second one, the important problem of the definition of an off equilibrium effective temperature through the nonlinear FDR is considered. It is shown that, in the case of coarsening systems, the effective temperature derived from the second order FDR is consistent with the one obtained from the linear FDR.

We study the behavior of the stationary velocity of a driven particle in an environment of mobile hard-core obstacles. Based on a lattice gas model, we demonstrate analytically that the drift velocity can exhibit a nonmonotonic dependence on the applied force, and show quantitatively that such negative differential mobility (NDM), observed in various physical contexts, is controlled by both the density and diffusion time scale of obstacles. Our study unifies recent numerical and analytical results obtained in specific regimes, and makes it possible to determine analytically the region of the full parameter space where NDM occurs. These results suggest that NDM could be a generic feature of biased (or active) transport in crowded environments.PACS numbers: 05.40.Fb,83.10.Pp Introduction.-Quantifying the response of a complex system to an external force is one of the cornerstone problems of statistical mechanics. In the linear response regime, a fundamental result is the fluctuationdissipation theorem, which relates system response and spontaneous fluctuations. Within the last years a great effort has been devoted to generalizations of this theorem to nonequilibrium situations [1][2][3][4], when the time reversal symmetry is broken, and also to elucidating the effects of the higher order contributions in the external perturbation [5][6][7][8][9][10][11]. From experimental perspective, theoretical understanding of the latter issues is of an utmost importance in several fields, such as active microrheology [12][13][14] and dynamics of nonequilibrium fluids [15,16].A striking example of anomalous behavior beyond the linear regime is the negative response of a particle's velocity to an applied force, observed in diverse situations in which a particle subject to an external force F travels through a medium. The terminal drift velocity V (F ) attained by the driven particle is then a nonmonotonic function of the force: upon a gradual increase of F , the terminal drift velocity first grows as expected from linear response, reaches a peak value and eventually decreases. This means that the differential mobility of the driven particle becomes negative for F exceeding a certain threshold value. Such a counter-intuitive "getting more from pushing less" [17] behavior of the differential mobility (or of the differential conductivity) has been observed for a variety of physical systems and processes, e.g. for electron transfer in semiconductors at low temperatures [18][19][20][21], hopping processes in disordered media [22], transport of electrons in mixtures of atomic gases with reactive collisions [23], far-from-equilibrium quantum spin chains [24], some models of Brownian motors [25,26], soft matter colloidal particles [27], different nonequilibrium systems [17], and also for the kinetically constrained models of glass formers [28][29][30].Apart of these examples, negative differential mobility (NDM) has been observed in the minimal model of a driven lattice gas, which captures many essential features of the behavior in r...

We study the mobility and the diffusion coefficient of an inertial tracer advected by a twodimensional incompressible laminar flow, in the presence of thermal noise and under the action of an external force. We show, with extensive numerical simulations, that the force-velocity relation for the tracer, in the nonlinear regime, displays complex and rich behaviors, including negative differential and absolute mobility. These effects rely upon a subtle coupling between inertia and applied force which induce the tracer to persist in particular regions of phase space with a velocity opposite to the force. The relevance of this coupling is revisited in the framework of non-equilibrium response theory, applying a generalized Einstein relation to our system. The possibility of experimental observation of these results is also discussed.

The rectification of unbiased fluctuations, also known as the ratchet effect, is normally obtained under statistical nonequilibrium conditions. Here we propose a new ratchet mechanism where a thermal bath solicits the random rotation of an asymmetric wheel, which is also subject to Coulomb friction due to solid-on-solid contacts. Numerical simulations and analytical calculations demonstrate a net drift induced by friction. If the thermal bath is replaced by a granular gas, the well-known granular ratchet effect also intervenes, becoming dominant at high collision rates. For our chosen wheel shape the granular effect acts in the opposite direction with respect to the friction-induced torque, resulting in the inversion of the ratchet direction as the collision rate increases. We have realized a new granular ratchet experiment where both these ratchet effects are observed, as well as the predicted inversion at their crossover. Our discovery paves the way to the realization of micro and submicrometer Brownian motors in an equilibrium fluid, based purely upon nanofriction.

A Generalized Langevin Equation with exponential memory is proposed for the dynamics of a massive intruder in a dense granular fluid. The model reproduces numerical correlation and response functions, violating the equilibrium Fluctuation Dissipation relations. The source of memory is identified in the coupling of the tracer velocity V with a spontaneous local velocity field U in the surrounding fluid: fluctuations of this field introduce a new timescale with its associated lengthscale. Such identification allows us to measure the intruder's fluctuating entropy production as a function of V and U , obtaining a neat verification of the Fluctuation Relation.

Static and dynamical structure factors for shear and longitudinal modes of the velocity and density fields are computed for a granular system fluidized by a stochastic bath with friction. Analytical expressions are obtained through fluctuating hydrodynamics and are successfully compared with numerical simulations up to a volume fraction ∼ 50%. Hydrodynamic noise is the sum of external noise due to the bath and internal one due to collisions. Only the latter is assumed to satisfy the fluctuation-dissipation relation with the average granular temperature.Static velocity structure factors S ⊥ (k) and S (k) display a general non-constant behavior with two plateaux at large and small k, representing the granular temperature T g and the bath temperature T b > T g respectively. From this behavior, two different velocity correlation lengths are measured, both increasing as the packing fraction is raised. This growth of spatial order is in agreement with the behaviour of dynamical structure factors, the decay of which becomes slower and slower at increasing density.

Abstract. We study the stochastic motion of an intruder in a dilute driven granular gas. All particles are coupled to a thermostat, representing the external energy source, which is the sum of random forces and a viscous drag. The dynamics of the intruder, in the large mass limit, is well described by a linear Langevin equation, combining the effects of the external bath and of the "granular bath". The drag and diffusion coefficients are calculated under few assumptions, whose validity is well verified in numerical simulations. We also discuss the non-equilibrium properties of the intruder dynamics, as well as the corrections due to finite packing fraction or finite intruder mass. Granular Brownian motion2