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Scientific Report 2007-2009<br />
Condensed matter physics and biophysics<br />
C16. Fluctuation-dissipation relations in non equilibrium statistical<br />
mechanics and chaotic systems<br />
One of the most important and general results concerning<br />
statistical mechanics is the existence of a relation<br />
between the spontaneous fluctuations and the response<br />
to external fields of physical observables (FDR).<br />
This result has applications both in equilibrium statistical<br />
mechanics, where it is used to relate the correlation<br />
functions to macroscopically measurable quantities such<br />
as specific heats, susceptibilities and compressibilities,<br />
and in nonequilibrium systems, where it offers the possibility<br />
of studying the response to time-dependent external<br />
fields, by analyzing time-dependent correlations.<br />
The idea of relating the amplitude of the dissipation to<br />
that of the fluctuations dates back to Einsteins work on<br />
Brownian motion.<br />
The FDR result represents a fundamental tool in<br />
nonequilibrium statistical mechanics since it allows one<br />
to predict the average response to external perturbations,<br />
without applying any perturbation. In fact, via<br />
an equilibrium molecular dynamics simulation one can<br />
compute correlation functions at equilibrium and then,<br />
using the GreenKubo formula, obtain the transport coefficients<br />
of model liquids without resorting to approximation<br />
schemes.<br />
Although the FDR theory was originally applied to<br />
Hamiltonian systems near thermodynamic equilibrium,<br />
it has been realized that a generalized FDR holds for a<br />
vast class of systems with chaotic dynamics of special interest<br />
in the study of natural systems, such as geophysics<br />
and climate. A renewed interest toward the FDR has<br />
been motivated by the study of the entropy production<br />
rate in systems arbitrarily far from equilibrium.<br />
Recent developments in nonequilibrium statistical<br />
physics give evidence that the fluctuation-dissipation relations,<br />
which hold in systems described by statistical<br />
mechanics, have an important role even beyond the traditional<br />
applications of statistical mechanics, e.g. in a<br />
wide range of disciplines ranging from the study of small<br />
biological systems to turbulence, from climate studies to<br />
granular media, etc.<br />
It is well known that in systems with aging and glassy<br />
behaviours there are non trivial relations among response<br />
functions and correlation funcion. Such a feature<br />
holds even for many systems which are ergodic and have<br />
an invariant phase space distribution, which is reached<br />
in physically relevant time scales. In particular in all the<br />
cases where there are strong correlations among different<br />
degrees of freedom.<br />
In our research we study a generalized FDR which<br />
holds under rather general conditions, even in non-<br />
Hamiltonian systems, and in nonequilibrium situations.<br />
In addition, we discuss the connection between this FDR<br />
and the foundations of statistical mechanics, fluid dynamics<br />
climate, and granular materials.<br />
As example we can cite the study of a multi-variate<br />
linear Langevin model, including dynamics with memory,<br />
which is used as a treatable example to show how<br />
the usual relations are recovered only in particular<br />
cases. This study brings to the fore the ambiguities<br />
of a check of the FDR done without knowing the<br />
significant degrees of freedom and their coupling. An<br />
analogous scenario emerges in the dynamics of diluted<br />
shaken granular media. There, the correlation between<br />
position and velocity of particles, due to spatial inhomogeneities,<br />
induces violation of usual FDRs. The<br />
search for the appropriate correlation function which<br />
could restore the FDR, can be more insightful than<br />
a definition of non-equilibrium or effective temperatures.<br />
References<br />
1. A. Puglisi. et al., J. Stat. Mech.-Theory Exp. P08016<br />
(2007)<br />
2. U. Marini Bettolo, et al. Phys. Rep. 461, 111 (2008)<br />
3. D. Villamaina, et al., J. Stat. Mech.-Theory Exp. L10001<br />
(2008)<br />
4. D. Villamaina et al., J. Stat. Mech.-Theory Exp. P07024<br />
(2009)<br />
Authors<br />
A. Vulpiani, D. Villamaina, A. Baldassarri 3 , A. Puglisi 3<br />
http://tnt.phys.uniroma1.it/twiki/bin/view/TNTgroup/<br />
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<strong>Sapienza</strong> Università di Roma 69 Dipartimento di Fisica