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Scientific Report 2007-2009
Condensed matter physics and biophysics
C17. Chaos, complexity and statistical mechanics
The main aim of statistical mechanics is to describe
the equilibrium state of systems with many degrees of
freedom; while dynamical systems theory can explain the
irregular evolution of systems with few degrees of freedom.
Also macroscopic systems are dynamical systems
with a very large number of degrees of freedom. However
while the low dymensional systems have been largely investigated,
and their basic features are well understood;
the study of systems with many degrees of freedom and
many characteristic times (e.g. climate and turbulence)
is a difficult task. The reason of that is mainly due to the
fact that, in these cases, the usual indicators (Lyapunov
exponents and Kolmogorov-Sinai entropy) are not very
relevant.
The Kolmogorov-Sinai entropy and the Lyapunov exponents
quantify rather well the degree of time ”complexity”
in systems with few degrees of freedom. A complexity
characterization is more difficult when many degrees
of freedom and many time scales are present. For
instance in developed turbulence this can be achieved using
the ϵ-entropy, which measures the information content
at different scale resolution. The climatic systems,
where the fluctuations at different scales are comparable,
are much more complicated. Other interesting situations
arise in non chaotic systems (i.e. with zero Lyapunov
exponent) but with irregular behaviour and in
discrete-states systems, with regard to their continuum
limit. The latter topic is tied up with the semiclassical
limit and decoherence in quantum mechanics, and with
deterministic algorithms to produce random numbers.
Perhaps in physics the most relevant example of high
dimensional systems is the dynamics of macroscropic
bodies studied in statistical mechanics. From the very
beginning, starting from the Boltzmanns ergodic hypothesis,
a basic question was the connection between
the dynamics and the statistical properties.
The discovery of the deterministic chaos (from the anticipating
work of Poincaré to the contributions, in the
second half of the XX-th century, by Chirikov, Hénon,
Lorenz and Ruelle, to cite just the most famous) beyond
its undoubted relevance for many natural phenomena,
showed how the typical statistical features observed in
systems with many degrees of freedom, can be generated
also by the presence of deterministic chaos in simple
systems. For example low dimensional models can emulate
spatially extended dynamics modelling transport
and conduction processes.
Surely the rediscovery of deterministic chaos has revitalized
investigations on the foundation of Statistical
Mechanics forcing the scientists to reconsider the connection
between statistical properties and dynamics. However,
even after many years, there is not a consensus
on the basic conditions which should ensure the validity
of the statistical mechanics. Roughly speaking the
two extreme positions are the traditional one, for which
the main ingredient is the presence of many degrees of
freedom and the innovative one which considers chaos a
crucial requirement to develop a statistical approach.
One aim of our research has been to show how, for understanding
the conceptual aspects of the statistical mechanics,
one has to combine concepts and techniques developed
in the context of the dynamical systems with statistical
approaches able to describe systems with many
degrees of freedom. In particular we discussed the relevance
of non asymptotic quantities, e.g ϵ-entropy, and
the role of pseudochaotic systems, i.e. non chaotic systems
with a non trivial behaviour.
Vivid examples of such a feature is shown by numerical
studies which evidenciate in a clear way that for high
dimensional Hamiltonian systems chaos is not a fundamental
ingredient for the validity of the equilibrium
statistical mechanics. This happens for instance for
the transport properties of systems with many degrees
of freedom, e.g. diffusion coefficient, which are not
sensitive to the presence of chaos. Such results support
the point of view that to have good statistical properties
chaos is unnecessarily demanding: even in the absence
of chaos, one can have (according to Khichin ideas)
a good agreement between the time averages and the
predictions of the equilibrium statistical mechanics.
References
1. F. Cecconi et al. J. Stat. Mech.-Theory Exp. P12001
(2007)
2. M. Falcioni et al. Physica A 385, 170 (2007)
3. P. Castiglione et al Chaos and Coarse Graining in
Statistical Mechanics (Cambridge University Press, 2008)
4. M. Cencini et al Chaos: From Simple Models to Complex
Systems (World Scientific, 2009)
Authors
M. Falcioni, A. Vulpiani, F. Cecconi 3 , M. Cencini 3 , L.
Palatella 3
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