We study the interaction between gas and particles in a protoplanetary disk, using both analytical and numerical approaches. We first present analytical expressions for the trajectories of individual particles undergoing gas drag in the disk, in the asymptotic cases of very small particles (Epstein regime) and very large particles (Stokes regime). Using a Boltzmann averaging method, we obtain an analytic expression for the evolution of the average density, velocity, and dispersion of the particles as a function of distance above the midplane of the disk. Using successive moments of the Boltzmann equation, we derive the equivalent fluid equations for the average motion of the particles; these are intrinsically different in the Epstein and Stokes regimes. A simple closure of the moment equations is proposed in both regimes. These fluid equations provide much better prospects for the study of more complex problems related to protoplanetary accretion disks, since for general initial size and phase-space distributions the evolution of the average behavior of the particles can be evaluated numerically with much less computational time than that required for the numerical integration of the orbits of all individual particles. In a companion paper, for instance, we use them for the analysis of a shearing instability induced by the sedimentation of the particles. In the present work we test the adequacy of the fluid formulation against a set of idealized numerical experiments. In the Epstein regime, we study an idealized uniform initial distribution of small particles. We obtain a set of analytic solutions for the fluid equations, which are found to be in good agreement with those obtained from numerical integration of the orbits of many particles. We also verify that any initial velocity dispersion is quickly damped out by the surrounding gas on the short stopping timescale, which provides closure and justifies the description of the particles as a fluid with a linear drag force and negligible pressure. In the Stokes regime, as the large particles oscillate across the midplane with declining amplitude, their velocity dispersion remains comparable to their average speed. Their sedimentation is analogous to the cooling of a pressure-supported fluid. We propose an empirical closure scheme for the moment equations of the Stokes particles fluid and test it against idealized numerical experiments. In both cases, this method can eventually be applied to study the evolution of particle distributions in protostellar disks after additional effects such as collision, sublimation, and condensation are included.