| Summary: | 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 <i>H</i>-Boltzmann theorem. The key point lies in that the definition of majorization allows choosing a wide range of convex functions <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula> 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 <i>H</i>-Boltzmann theorem is obtained as a special case for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ϕ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>x</mi> <mo form="prefix">ln</mo> <mi>x</mi> </mrow> </semantics> </math> </inline-formula>.
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