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