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Scientific Report 2007-2009
Theoretical physics
T16. On static and dynamic properties of complex systems in
statistical mechanics and quantum field theory
In the period under consideration, the research activities
have been mainly oriented toward the study of
the static and dynamic properties of complex systems,
with applications to the physics of elementary particles,
the physics of condensed matter, and biological systems.
The methods and techniques refer to statistical mechanics,
to the theory of stochastic processes, and dynamical
systems. The methods at the basis of our study of
spin glasses and neural nets let the physical intuition,
accumulated through the use of the replica trick and numerical
simulations, merge with the need for a rigorous
mathematical treatment. The essential ingredients are
given by powerful interpolation methods, and sum rules.
These methods led in the past years to the proof of relevant
results, in particular concerning the control of the
infinite volume limit, and the mechanism of the spontaneous
replica symmetry breaking.
We now give a concise review about the main results
obtained.
For the neural nets of Hopfield type, we have given
a characterization of the ergodic phase and the generalization
of the Ghirlanda-Guerra identities. Moreover,
a systematic interpolation method has been developed
which allows the characterization of the replica symmetric
approximation, and the possibility of introducing
functional order parameters for the description of
the replica symmetry breaking. Our method is based
on the transformation of the neural net into a bipartite
spin glass, where one of the party is given by usual Ising
spin variables, and the other party is given by Gaussian
variables. The quenched spin glass interaction is
assumed to be Gaussian. It is immediate to realize that
in general, for this kind of bipartite spin glass models,
universality does not hold in general, in contrast with
the Sherrington-Kirkpatrick model for a spin glass. The
variational principle arising in the expression of the free
energy in the infinite volume limit is of novel type, in
that it involves a mini-max procedure, in contrast with
the Sherrington-Kirkpatrick model for a spin glass. This
seems to be a general property of a very large class of
models. In particular, the mini-max variational principle
has been found to hold for general bipartite models,
of ferromagnetic and spin glass type. The replica
symmetric approximation is ruled by two order parameters,
connected with the values of the overlaps of the
Ising spin variables and the Gaussian variables, respectively,
connected by self-consistency relations. Obviously,
the replica symmetric approximation looses its
physical meaning at low temperatures, where the entropy
becomes negative. However, by our interpolation
methods, it is very simple to construct the full replica
broken scheme, by a deep generalization of the methods
developed for the spin glass case. The fully broken
scheme is believed to give the true solution of the model.
Diluted systems have been studied in the cases of ferromagnetic,
antiferromagnetic, and general interpolating
models. Also in these cases, interpolation techniques,
and the associated sum rules, have been found very useful.
The interest of the diluted model is given by the fact
that they give a kind of bridge between the mean field
models and the models with short range interaction.
The theory of self-oscillating mechanical systems has
been exploited for the study of speech formation, analysis
and synthesis, and musical instrument functioning.
It is possible to apply fully nonlinear schemes, by completely
avoiding any kind of exploitation of the Fourier
analysis. The role of the different peaks of the spectrum
in the Fourier analysis is played by the intervention
of successive Landau instability modes for the selfoscillating
system. Moreover, with the same methods,
we have studied tidal basins, and volcanic tremor of
Stromboli type, in the frame of a recent collaboration
with researchers at the Department of Physics at the
University of Salerno.
Finally, in recent times, we have developed the possibility
of giving simple models for the immunological
system, based on stochastic dynamical systems of statistical
mechanics far from equilibrium. The models are
simple enough to allow practical evaluations, in connection
with the known phenomenology, but they are very
rich in the possibility of introducing all basic feature of
the real system. This research is done in collaboration
with researchers at the Department of Physics of the
University of Parma.
Finally we would like to mention the study of the
quantum field theory formulation of the relativistic
Majorana equations, introduced in 1932 in a famous
paper on Nuovo Cimento, and the study of slowing
down, scattering and absorption of neutrons, by following
the original methods of Fermi, Wick, Bothe,
Heisenberg, with the purpose of a realistic assessment of
the validity of the approximations introduced by them,
in comparison with the modern methods of numerical
simulations in the nuclear reactor theory.
References
1. F. Guerra, Int. J. Mod. B23, 5505 (2009).
2. A. Barra et al., J. Math. Phys. 50, 053303 (2009).
3. A. Barra et al., J. Math. Phys. 49, 125217 (2008).
4. L. De Sanctis et al., J. Stat. Phys. 132, 759 (2008).
Authors
F. Guerra, A. Barra, G. Genovese
Sapienza Università di Roma 39 Dipartimento di Fisica