Angelo Vulpiani - Universit&agrave; di Roma <i>La Sapienza</i> #
The role of chaos for the foundation of statistical mechanics #
A basic question of the Statistical Mechanics, starting from the
Boltzmann's ergodic hypothesis, is the connection between the
dynamics and the statistical properties. This is a rather 
difficult
task and, in spite of the mathematical progress, for long time
basically ergodic theory had a marginal relevance in the 
development
of the statistical mechanics (at least in the physics 
community).
Partially this was due to a widely spreaded opinion (based also 
on the
belief of influential scientists as Landau) on the key role of 
the
many degrees of freedom and the practical irrelevance of
ergodicity. This point of view found a mathematical support on 
some
results by Khinchin.
<br/>
On the other hand the discovery of the deterministic chaos 
beyond its
e undoubted relevance for many natural phenomena, showed how the
similar statistical features observed in systems with many 
degrees of
freedom, can be generated also by the presence of deterministic 
chaos
in simple systems.
<br/>
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.
<br/>
We discusse the role of ergodicity and chaos for the validity of
statistical laws. Detailed studies show in a clear way
that  chaos is not a fundamental ingredient for the validity of
the equilibrium statistical mechanics.
Therefore the point of view that good statistical
properties need 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 by the equilibrium
statistical mechanics.
<br/>
Basically one can say that thermodynamics can be seen as an
emergent property of large systems.