Dynamics and Statistical Mechanics of Systems with Long Range Forces
In collaboration with
A. Rapisarda (Univ. di Catania) and
S. Ruffo (Univ. di Firenze) we have studied relaxation to the equilibrium,
chaotic properties and anomalous diffusion in the HMF (Hamiltonian Mean Field) model.
The model describes a system of N fully-coupled rotors and exhibits a second order phase transition
from a clustered phase to a homogeneous one as a function of energy.
We find strong chaos in correspondence to the phase transition:
the largest Lyapunov exponent and the Kolmogorov-Sinai entropy show a
peak at the critical energy. This result is valid in the thermodynamic
limit and provides a new bridge between Hamiltonian chaos in
systems with many degrees of freedom
and equilibrium statistical mechanics
[PRL98]
[PhysD99].
Anomalous diffusion is observed in a transient out-of-equilibrium
regime (quasi-stationary states) and for a small range of energy, below the critical one.
The superdiffusive behavior of our system is due to Levy walks
of single particles and becomes normal at equilibrium,
after a relaxation time which diverges with N
[PRL99].
The quasi-stationary state is characterized by non-Gaussian velocity distributions
[PRE01].
Kolmogorov-Sinai Entropy-Rate vs. Thermodynamic Entropy
in Chaotic Conservative Systems
This is a project I started in collaboration with
M. Baranger
(Center for Theoretical Physics, MIT) when I was postdoctorate fellow
at MIT. We studied the connections between two
quantities, both called entropies and used in two different fields:
the entropy of a thermodynamic system S (the entropy of Boltzmann and
Clausius), and K, the Kolmogorov-Sinai entropy, defined by the
mathematicians to describe the dyamical instabilities of a trajectory
in the phase space and to measure chaos. We considered different
chaotic conservative systems (for example a gas, or various
conservative chaotic maps) started in far-from-equilibrium initial
conditions, and we studied the relaxation to equilibrium. The
distribution function, initially localized in a particular region of
phase space changes of shape: it stretches in some directions and
contracts in others, and it bends on itself many times; soon it
diffuses all over the phase space becoming a complicate fractal. The
occupied volume stays constant because of the Liouville's theorem,
though the shape of the distribution has turned into a filamentous
structure. Because of this fractalization process the coarse grained
entropy S(t) increases linearly in time until the system reaches
equilibrium. In
[PRL99] we showed, by means of
numerical simulations, that the rate of entropy increase dS/dt is equal to
K.
The Edge of Chaos
This is a collaboration started in 1999
with
C. Tsallis (Santa Fe Institute and CBPF, Rio de Janeiro),
A. Rapisarda (Univ. di Catania) and
M. Baranger
(Center for Theoretical Physics, MIT) about
the special phenomena happening at the so-called edge of
chaos , i.e. at the transition point between a non chaotic and a
chaotic phase. This issue has been the subject of much speculation in
connection with the self-organization of complex systems and even with
Tsallis' non-extensive thermodynamics. In
[PLA00]
(see also our letter to the Editor of Science
[Science03] )
we illustrate with the simplest dissipative system, the logistic map,
how to generalize for a system at the edge of chaos
important concepts, such as the entropy and
the sensitivity to initial conditions.
My old love: Nuclear Multifragmentation
Recent experiments in proton-nucleus and nucleus-nucleus collisions
at energies around the Fermi energy have revealed the production
of nuclear fragments with size distributions exhibiting power laws.
These observations have raised the question whether critical phenomena
can be explored in the context of heavy-ion collisions.
Many are the open problems: nuclei are finite systems with
only few hundreds of constituents,
while phase transitions are defined in the thermodynamic
limit; moreover a nuclear collision is not an equilibrium process.
Crical behavior and multifragmentation in nuclear systems has been
my main research interest during my Ph.D.
Under the supervision of
M. Di Toro (Univ. di Catania) and
A. Bonasera (LNS Catania),
we have developed statistical as well as dynamical models to describe
nuclear fragmentation from a microscopical point of view
[PRL94] ,
[PRC95] ,
[PRL95] , and we
have closely collaborated with different experimental groups
[PRL96] .
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