Majorization and Dynamics of Continuous Distributions

In this work we show how the concept of majorization in continuous distributions can be employed to characterize mixing, diffusive, and quantum dynamics along with the H-Boltzmann theorem. The key point lies in that the definition of majorization allows choosing a wide range of convex functions ϕ for studying a given dynamics. By choosing appropriate convex functions, mixing dynamics, generalized Fokker-Planck equations, and quantum evolutions are characterized as majorized ordered chains along the time evolution, being the stationary states the infimum elements. Moreover, assuming a dynamics satisfying continuous majorization, the H-Boltzmann theorem is obtained as a special case for ϕ ( x ) = x ln x .