For a two‐degree‐of‐freedom dynamical system in which the dynamics depends on a parameter &mgr;, the effect of an externally imposed random variation (e.g., collisions) of &mgr; is considered. In particular, mapping equations in conjunction with particle conservation are used to directly derive transport coefficients. The analytic method and nature of the results depend on a stochasticity parameterKwhich describes the nonlinearity of the collision‐free dynamics and a collisionality parameter &sgr;. For largeKor &sgr;, Rechester, Rosenbluth, and White’s method of diagrams in Fourier space is used to calculate stochastic diffusion. For smallKand moderate &sgr;(≳K3/2), an expansion in powers ofKis used to solve the map equations and thus obtain plateau transport coefficients. For smallKand small &sgr;, an expansion in powers of &sgr; leads to banana transport coefficients. The results are applied to the calculation of radial resonant transport coefficients for tandem mirrors; the plateau and banana particle and energy fluxes are equivalent to those previously obtained from drift‐kinetic theory, and the stochastic fluxes, not directly calculable in the drift‐kinetic approach, are given explicitly.