Transport properties in non-equilibrium and anomalous systems

Let us summarize, in a schematic fashion, the results achieved in this work: • Projection operations and entropy production: The effect of projections on entropy production is investigated. More specifically an equation with memory and colored noise is compared with the equivalent Markovian model. The latter leads to identify memory as a nonconservative force. These forces cease to contribute to entropy production only under the validity of the fluctuation dissipation relation of the second kind, which is equivalent to the detailed balance assumption. The projection from the Markovian to the non Markovian representation produces a loss of information detected from a decrease in mean entropy production. This change can be dramatic in the linear case, leading to the false conclusion that the system is in equilibrium. • Local velocity field in a granular gas: we designed a first granular dynamical theory describing non-equilibrium correlations and responses for a massive tracer. In the dilute regime, under the molecular chaos assumption, the tracer dynamics is Markovian and stationary, and the equation satisfies detailed balance, also if inelasticity is present. On the contrary, in a denser regime the dynamics of the tracer is non Markovian and memory effects are present. The equation with memory is introduced and gives a significant insight into the mechanisms of recollision and dynamical memory and their relation with the breakdown of equilibrium properties. It is remarkable that velocity correlations between the intruder and the surrounding velocity field, in the inelastic case, are responsible for both the violations of the equilibrium fluctuation dissipation relation and the appearance of a non-zero entropy production. • Einstein relations in subdiffusive models: we have considered two systems with subdiffusive behavior, showing that the proportionality between response function and correlation breaks down when “non equilibrium” conditions are introduced. In the case of a random walk on a comb model the response relation can be explicitly written, providing the out of equilibrium corrections to the Einstein relation. In the single file model an explicit formula is not available but we have shown that, by taking into account the coupling of the velocities of neighboring particles, it is possible to have a better prediction of the response. • The glassy ratchet as a non equilibrium thermometer: through numerical simulations in different models and different choices of the quench temperature, always chosen in the deep slowly relaxing regime, the “glassy ratchet” phenomenon is investigated. The drift velocity slowly decays in time and can be appreciably different from zero. The overall intensity of the drift, measured in terms of a “subvelocity”, is monotonically increasing with the distance from equilibrium, namely with the difference between the quench and effective temperatures. This observation supports the idea of regarding the ratchet drift as a “non equilibrium thermometer”: it can be used as a device capable to say how far a system is from equilibrium. In summary, the general message that one learns is that correlations among different degrees of freedom can work as a channel of energy exchange, being responsible of the breaking of the equilibrium fluctuation dissipation relation and acting as a primary source of entropy production. In a non-equilibrium context, a too strong projection operation and the consequent reduction of information leads to a reduced description with, in some cases, a vanishing entropy production. It must be noticed that, even if the scenario described appears very general, it has been tested in a quite controlled setup like the massive intruder in a granular gas model. The energy injection mechanism is homogeneous and the intruder follows a Langevin dynamics. One may wonder if this description is valid also for more complex situations. The Chapter 4 has been written as partial response to this general question. The subdiffusive models analyzed, even if with some differences, confirm the interpretations given for higher dimensions. A relevant ingredient in breaking the Einstein formula, for stationary regimes, is not the anomalous diffusion but the presence of currents driving the system out of equilibrium. The generalized response relations are a good tool to detect the main sources of non equilibrium present in the system. In this direction, in the single file model a non equilibrium correlation length can be defined and measured with simplicity, in analogy with the length coupled to a massive intruder in higher dimensions. However, this is only a partial result deserving a deeper investigation, since an equation for the tracer in the granular single file model is lacking and entropy production for this model is a challenging issue. A very promising direction for future researches is the study of fluctuating hydrodynamics, already faced in some recent works [136, 137]. The hydrodynamic equations are projections on slow modes and, as a consequence, they are associated to a loss of information. On the other hand, different models at a microscopic level can have the same hydrodynamic description, therefore the non-equilibrium currents “surviving” to a hydrodynamic projection are, in a sense, supposed to be more general with respect to the microscopic details of the model under investigation. Crucial, with respect to this “universality”, is the role of microscopic details in the specific form of non equilibrium fluctuations, which appear as noise in fluctuating hydrodynamics.