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Non-Equilibrium Steady States

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Transport Properties in Non-Equilibrium and Anomalous Systems

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Abstract

In this chapter a brief collection of the results present in literature and used in this work is described. We starts with a derivation of the Langevin equation in a way that makes clear the assumptions on the basis of equilibrium dynamics. Then, generalized response relations are presented and the role of entropy production is discussed. Since a large part of this work regards the study of granular gases, the second part of the chapter is entirely devoted to them, paying attention to the still open problems in dense regimes. This is not a chapter of a review article, and for this reason it could appear incomplete. However, it must be seen as an occasion to present the common ground where there are the basis of our research, and it proposes some questions which are developed and, at least partially, solved in the rest of the work.

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Notes

  1. 1.

    In this thesis we always measure the temperature in scales of energy, namely we set the Boltzmann constant \(k_{B}\) equal to one.

  2. 2.

    We will omit the expression “of the first kind” When the kind is not specified we always refer to this relation.

  3. 3.

    We consider a numerable set of trajectories for simplicity of notation.

  4. 4.

    We put \(\rho \equiv \rho _{inv}\) for simplicity.

  5. 5.

    \(\Theta \) is the Heavyside step function.

  6. 6.

    A similar definition is often introduced also in the frequency domain conjugated to the variable \(t\) [27].

  7. 7.

    The index \(i\) has been removed for simplicity.

  8. 8.

    The second order term trivially vanishes for functions of the form \(g(x)=\ln [f(x)/f(-x)]\).

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  1. LPTMS, Université Paris Sud and CNRS, Orsay Cedex, France

    Dario Villamaina

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Villamaina, D. (2014). Non-Equilibrium Steady States. In: Transport Properties in Non-Equilibrium and Anomalous Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-01772-3_2

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