Statistical mechanics of a nonlinear discrete system

Statistical mechanics of the discrete nonlinear Schr"odinger equation is studied by means of analytical and numerical techniques. The lower bound of the Hamiltonian permits the construction of standard Gibbsian equilibrium measures for positive temperatures. Beyond the line of $T=\mathrm{\infty }$, we identify a phase transition, through a discontinuity in the partition function. The phase transition is demonstrated to manifest itself in the creation of breather-like localized excitations. Interrelation between the statistical mechanics and the nonlinear dynamics of the system is explored numerically in both regimes.