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Scientific Report 2007-2009
Theoretical physics
different contexts. The first contributions is related to statistical mechanics and quantum field
theory, and is connected to the rigorous investigation of the static and dynamic properties of
complex systems [T16]. A large number of applications (to the physics of elementary particles, to
the physics of condensed matter and to biological systems) have been investigated. The methods
used for understanding spin glasses and neural networks let physical intuition merge with a rigorous
mathematical treatment. The essential ingredients are given by powerful interpolation methods
and sum rules. For neural networks of Hopfield type a characterization of the ergodic phase and
a generalization of the Ghirlanda-Guerra identities have been given. Diluted systems have been
studied in the cases of ferromagnetic, anti-ferromagnetic and general interpolating models. The
theory of self-oscillating mechanical systems has been used for the study of speech formation.
Simple models for the immunological system based on stochastic dynamical systems of statistical
mechanics far from equilibrium have been studied.
The “Macroscopic fluctuation theory of irreversible processes” has been studied in [T17]. A
macroscopic theory for a class of thermodynamic systems out of equilibrium has been proposed,
funded on the analysis of a large family of stochastic microscopic models. The theory that has
been proposed has many features of substantial improvement with respect to the theory developed
long ago by Onsager and then by Onsager-Machlup which applies to states close to the equilibrium
and does not really include the effect of non trivial boundary reservoirs. This treatment is based
on an approach developed in the analysis of fluctuations in stochastic lattice gases. Developments
in this field are of paramount importance, and the research of our group is giving an important
contribution.
“Equilibrium statistical mechanics for one dimensional long range systems” has been analyzed
in [T18]. The project is based on studying a one dimensional system of particles interacting via a
long range attractive potential, and techniques developed for spins on a lattice are exploited. The
one dimensional nature of the system allows to control the hard core contribution. “Markov chains
in a graph” are analyzed in [T19]. Aldous conjecture about finite graphs claims that “The random
walk and the interchange process on a finite connected simple graph have the same spectral gap”.
This conjecture has been proven for complete multipartite graphs by researchers of our group,
using a technique based on the representation theory of the symmetric group. Also a further
result has been obtained, with a similar proof valid for a different Markov chain called initial
reversals.
A number of very interesting nonlinear problems complete the list of the ideas discussed in
this report. We can identify four main issues: (1) “Optical solitons in resonant interactions
of three waves” in [T20]; (2) “Propagation and breaking of weakly nonlinear and quasi one
dimensional waves in Nature” in [T21]; (3) “Towards a theory of chaos explained as travel on
Riemann surfaces” in [T22]; (4) “Discrete integrable dynamical systems and Diophantine relations
associated with certain polynomial classes” in [T23]. As far as the optical solitons of point one are
concerned our group has discovered a new multi-parametric class of soliton solutions of the model
of the resonant interaction of three waves, that describe a triplet made of two short pulses and
a background. As far as the quasi one dimensional waves of point two are concerned an inverse
spectral transform has been developed for families of multidimensional vector fields, and it has
been used to construct the formal solution of the Cauchy problem for non linear PDE’s, and to
give an analytic description of the breaking of multidimensional waves in Nature. For point three
a new dynamical system has been introduced, interpretable as a 3-body problem in the complex
plane, to improve the understanding of the role of movable branch points in the onset of chaotic
motions in a deterministic context. At last (issue four) new Diophantine properties related to the
integrable hierarchy of nonlinear PDE’s associated with the KdV equation have been obtained,
and new discrete integrable systems have been identified.
Enzo Marinari
Sapienza Università di Roma 23 Dipartimento di Fisica