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					                                                                                                                          Final Report: 0604518

Final Report for Period: 07/2009 - 06/2010                                               Submitted on: 09/27/2010
Principal Investigator: Bonetto, Federico .                                              Award ID: 0604518
Organization: GA Tech Res Corp - GIT
Submitted By:
Bonetto, Federico - Principal Investigator
Perturbative Methods in Coupled Lattice Maps and Applications

                                                               Project Participants
Senior Personnel
           Name: Bonetto, Federico
           Worked for more than 160 Hours:           Yes
           Contribution to Project:


Graduate Student

Undergraduate Student

Technician, Programmer

Other Participant

Research Experience for Undergraduates

                                                               Organizational Partners

                                                         Other Collaborators or Contacts
Prof. Giovanni Gallavotti, Universita' di Roma 'La Sapienza'
Dr. Pierluigi Falco, Universita' di Roma 'Tor Vergata'
Prof. Guido Gentile, Universita di Roma 3
Prof. Joel Lebowitz, Rutgers University

                                                               Activities and Findings

Research and Education Activities:
My research activity, in the three years of the grant,developed mainly around the use of Dynamical System Theory for the study of problem
arising from Nonequilibrium Statistical Mechanics. For this reason I'm interested in Dynamical system with many degree of freedom and a
clear space-time structure.

The simplest version of such a class if Dynamical Systems is represented by Coupled Lattice Maps or Flow. In general, one considers a copy of
some simple dynamical system for each point of a subset the integer lattice in d dimension and then add a coupling between the individual
systems. The dynamics that arise in this way can be thought, for example, as schematization of a crystall. The dynamical systems considered
may be discrete or continuous in time and deterministic or stochastic. The main interest is in the property that characterize the global behavior
of the system when the considered subset become large, eventually the all lattice.

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                                                                                                                           Final Report: 0604518
My main methods of analysis is perturbation theory. This typically consist in the study of some rather particular class of system with an
emphasis on constructive results that can be compared with simulations and, possibly, real experiments.

In this line of research, four papers were published. They all deal with problems directly linked with the main subject of the proposal. One of
the results I was hoping to obtain is still incomplete. The proof of analiticity of the Lyapunov exponents for Coupled Anosov Map is complete
in the case one couple different maps thus avoiding degeneracy of the exponents. The more interesting case where degenerate exponents are
present is still incomplete but I think it will be completed soon.

The research was conducted with the collaboration of proff. J. Lebowitz from Rutgers University and G. Gallavotti from the University of
Rome 'La Sapienza'.

Results linked to the grant were presented in seminar held by me at several university among which University of California at Berkeley,
Priceton University and Universite' de Paris IX.

The funds of the grant were used to support my research during the summers. In these periods I visited the University of Rome, Paris
University and Rutgers University several times. These visit were partly supported with money from the grant. A. Amaricci and P. Falco both
visited me for some time at Georgia Tech and their visits were partly supported by the grant.

Finally, the grant was used to acquire the computational tools needed to investigate the numerically the systems we were studying analytically
and to present the result in seminars.


F. Bonetto, G. Gallavotti and G. Gentile,
A fluctuation Theorem in a Random environment

In this paper we extend the theory previously developed for perturbation of deterministic dynamical systems to the case where also a stochastic
perturbation is present. This is an interesting extension of the previous work and shows that our techniques can be used also in non
deterministic cases.

A. Amaricci, F. Bonetto and P. Falco,
Analyticity of the SRB measure for a class of simple Anosov flows

It deals with the geodetic flow on a surface of constant negative curvature. This is the simplest Anosov flow one can construct and can also be
realized as an Hamiltonian system with 2 degrees of freedom. We consider a smooth and small perturbation of this Hamiltonian system and
show that the relative SRB measure is smooth as a function of the perturbation. This is a first step toward the study of a system of coupled
Anosov flows. The work in this direction as already started and I think we will have some result in the coming year.

Bonetto, F; Lebowitz, JL; Lukkarinen, J; Olla, S
Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent Stochastic Reservoirs

The paper proves that the result obtained for the Harmonic chain can be partly extended to the case of a weak anharmonicity. Not all result can
be extended due to difficulties in controlling the large system limit. Moreover the paper show that the steady state can e characterized through a
minimum entropy production principle.

Bonetto, F.; Lebowitz, J.L.
Nonequilibrium Stationary Solution of Thermostated Boltzmann Equation in a Field.

In this paper we study the large N limit of a system of N particle interacting through a Gaussian thermostat and colliding with fixed obstacles.
The form of the interaction allow us to write a Boltzmann style equation for the limit. This equation can be exactly solved in 1 dimension.
Some of the properties found in the 1d exact solution can be extended to higher dimension via perturbative techniques.


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                                                                                                                            Final Report: 0604518
A preprint on the analyticity of the Lyapunov exponent for coupled Anosov maps is in preparation. In the case one couple different simple
Anosov systems (i.e. Arnold cat maps) the result is complete. In the case one couple identical systems, degeneration of the exponents appears
and one has to add hypotheses to ensure that the perturbation split the degeneration at some order in the perturbation. This part of the work is
still incomplete.

Training and Development:
Three young scientists partecipated in the project. All of them were graduate students when the project began.

A. Amaricci is now a Postdoctoral student at SiSSA-ISAS in Trieste.
P. Falco is a member (postdoctoral fellow) at the School of Mathematics of the Institute for Advanced Study in Princeton, NJ.
J. Lukkarinen is research fellow at the Department of Mathematics of Helsinki University.

All three of them learned research skills and ability while partecipating in the project and are going farward in their academic careers.

Outreach Activities:

                                                                Journal Publications

Amaricci, A; Bonetto, F; Falco, P, "Analyticity of the Sinai-Ruelle-Bowen measure for a class of simple Anosov flows", JOURNAL OF
MATHEMATICAL PHYSICS, p. , vol. 48, (2007). Published, 10.1063/1.274761

Bonetto F.;Gallavotti G.; Gentile G., "A fluctuation Theorem in a Random environmen", Ergodic Theory and Dynamical Systems, p. 21, vol.
28, (2008). Published, 10.1017/S0143385707000417

Bonetto, F; Lebowitz, JL; Lukkarinen, J; Olla, S, "Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent
Stochastic Reservoirs", JOURNAL OF STATISTICAL PHYSICS, p. 1097, vol. 134, (2009). Published, 10.1007/s10955-008-9657-

                                                    Books or Other One-time Publications

F. Bonetto, J.L. Lebowitz, "New Trends in Statistical Physics: Festschrift in Honor of Leopoldo Garcia-Colin's 80th Birthday", (2010). Article
in book., Published
Editor(s): Alfredo Macias, Leonardo Dagdug
Bibliography: ISBN: 981430753XWorld Scientific Publishing Company

                                                                Web/Internet Site

                                                             Other Specific Products

Contributions within Discipline:
This point was already explained in the Finding section.

Contributions to Other Disciplines:

Contributions to Human Resource Development:

Contributions to Resources for Research and Education:

Contributions Beyond Science and Engineering:

                                                                                                                                      Page 3 of 4
                                                                                       Final Report: 0604518

                                                             Conference Proceedings

                                           Categories for which nothing is reported:
Organizational Partners
Activities and Findings: Any Outreach Activities
Any Web/Internet Site
Any Product
Contributions: To Any Other Disciplines
Contributions: To Any Human Resource Development
Contributions: To Any Resources for Research and Education
Contributions: To Any Beyond Science and Engineering
Any Conference

                                                                                                Page 4 of 4

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