DISORDERED-SYSTEMS AND BURGERS TURBULENCE

In order to celebrate the memory of my friend Giovanni, I could not im agine a better topic than one which lies at the interface between the studies of turbulence and the statistical mechanics of disordered syst ems, iris two favorite fields in physics. I have recently written a pa per on this subject, in the Proceedings of the International Conferenc e of Mathematical Physics 1997, and what I have written here is basica lly the content of this recent paper. The basic properties of scaling and intermittency in fully developped turbulence can be studied in a l ot of details in the case of stirred Burgers turbulence, using the map ping between Burgers' equation and the problem of a directed polymer i n a random medium. In the limit of large dimensions, a replica symmetr y breaking solution of the polymer problem provides the full probabili ty distribution of the velocity difference u(r) between points separat ed by a distance r much smaller than the correlation length of the for cing. This exhibits a very strong intermittency which is related to re gions of shock waves, in the fluid, and to the existence of metastable states in the directed polymer problem.