  We study approximations of non-Gaussian stationary processes having long
range correlations with microcanonical models. These models are conditioned by
the empirical value of an energy vector, evaluated on a single realization.
Asymptotic properties of maximum entropy microcanonical and macrocanonical
processes and their convergence to Gibbs measures are reviewed. We show that
the Jacobian of the energy vector controls the entropy rate of microcanonical
processes.
  Sampling maximum entropy processes through MCMC algorithms require too many
operations when the number of constraints is large. We define microcanonical
gradient descent processes by transporting a maximum entropy measure with a
gradient descent algorithm which enforces the energy conditions. Convergence
and symmetries are analyzed. Approximations of non-Gaussian processes with long
range interactions are defined with multiscale energy vectors computed with
wavelet and scattering transforms. Sparsity properties are captured with $\bf
l^1$ norms. Approximations of Gaussian, Ising and point processes are studied,
as well as image and audio texture synthesis.
