**Speaker:** Giovanni Gallavotti (INFN and Professor Emeritus, University of Rome La Sapienza, Italy) -

**Title:** "Entropy, Irreversiblity and Probability in chaotic systems"

**Abstract:**

PDEs and Fractals • Geometry with its applications has been at the heart of

the development of partial differential equations and boundary value problems since the very

beginning. In physics, biology, economics, and other applied fields, a variety of new problems are

now emerging that display unusual geometrical, analytical and scaling features, possibly of fractal

type. The objective of these lectures is to acquire the view of outstanding mathematicians on the

subject of differential equations and fractals, and their developments and applications, in a broad

perspective encompassing both classical highlights and contemporary trends.

Irreversibility became central in thermodynamics with the

problem of determining the maximum efficiency of a thermic

machine and led to the II law of Thermodynamics which can be

seen as the beginning of the theory of irreversibility dominated

by the new physical quantity named Entropy. The immediately

attempted interpretations of the II law as a consequence of the

existence of atoms (still an hypothesis at the time) expose the

apparently irreconcilable microscopic reversibility and

macroscopic irreversibility: after Maxwell’s derivation of the

macroscopic dissipation equations from the molecular

(reversible) motion and Boltzmann’s H-theorem the basic ideas

were slowly assimilated and accepted into a consistent

thermodynamic theory of equilibrium states. • The closely

related problem of a thermodynamic theory of stationary states

out of equilibrium (like a steady flow of a fluid under constant

forcing), in which dissipation plays a key role, did not go beyond

the first order corrections to equilibrium thermodynamics, with

the major development of Onsager’s reciprocity of transport

coefficients resting on microscopic reversibility. In the ‘60s

chaotic notions (already known to play a key role in celestial

mechanics since Poincaré) became essential and, therefore,

objects of intense research, and in the ‘80s the simulations of

microscopic evolutions gave new impulse to the theory and the

problems of nonequilibrium thermodynamics and its statistical

mechanics interpretation. • After a brief review of the early

developments, the recent works on the irreversible chaotic

evolutions will be described and, using as example the Navier-

Stokes fluid equations, a close analogy between the statistical

theory of equilibrium and a statistical theory of nonequilibrium

will be discussed with particular attention to the mechanical

interpretation of viscosity (or more generally friction) and its

relation with chaotic evolutions.

Sponsored by WPI and hosted by the Department of Mathematical Sciences

Refreshments available before the lecture • Participation of faculty and students is most welcome