ANALISE DA DINAMICA DE PARTICULAS BROWNIANAS INTERAGENTES A PARTIR DE REDES DE MAPAS ACOPLADOS - Szmoski, Romeu Miquéias (2009)
The Brownian motion is one important topic of the non-equilibrium statistical mechanics and it is related to many natural phenomena. The first observations and theories on this motion were essential for understand the microscopic behavior of the nature and its influence on macroscopics observables. In this dissertation, we studied the dynamics of a system composed of several interacting Brownian particle from the point of view of coupled maps lattices. We use a map with a direct correlation to the abovementioned motion and we employ four different kinds of couplings in order to represent several ways of interaction among the particles. Using nonlinear dynamics tools, we observe the situations in which the particles velocities synchronize or show a tendency to the synchronized state. We also obtain algebrics expressions for the Lyapunov spectra of lattices with regular couplings whose interactions decays with distance as a power-law and we raise two hypotheses about Lyapunov exponents of a lattice with the coupling probability decreasing with the distance, as follows: the exponents of this lattice converge to the exponents of the lattice whose interactions decay with the distance in agreement to a power-law when the number of particles is very large; and the Lyapunov exponents of this lattice are given by the sum of the probabilities products of the each coupling matrix by eigenvalues of these matrixes. The values obtained for the Lyapunov exponents by means of the expressions deducted are in agreement with those obtained by numerical approximations techniques. Regarding distributions of the velocities of the particles, we observed that occur an aproximation to a Gaussian distribuition when the intensity of the coupling tends to its maximum.