Dynamical evolution in non-commutative discrete phase space and the derivation of classical kinetic equations

By considering a lattice model of extended phase space, and using techniques of non-commutative differential geometry, we are led to: (a) the concept of vector fields as generators of motion and transition probability distributions on the lattice; (b) the emergence of the time direction on the basis of the encoding of probabilities in the lattice structure; (c) the general prescription for the evolution of the observables in analogy with classical dynamics. We show that, in the limit of a continuous description, these results lead to the time evolution of observables in terms of (the adjoint of) generalized Fokker-Planck equations having: (1) a diffusion coefficient given by the limit of the correlation matrix of the lattice coordinates with respect to the probability distribution associated with the generator of motion; (2) a drift term given by the microscopic average of the dynamical equations in the present context. These results are applied to one- and two-dimensional problems. Specifically, we derive: (I) the equations of diffusion, Smoluchowski and Fokker-Planck in velocity space, thus indicating the way random-walk models are incorporated in the present context; (II) Kramers' equation, by further assuming that, motion is deterministic in coordinate space.