Time evolution of entropy, in various scenarios

While undergraduate texts describe the change, ΔS in entropies, from time t = 0 to $t=\infty$, they do not speak about exactly how, $\Delta S\left(t\right)\equiv S\left(t\right)-S\left(0\right)$, the change in entropy up to time t, looks. Here, a 'derivation' of the continuous version of Shannon entropy (for position), $-{\int }_{-\infty }^{\infty }p\left(x\right)ln\left(p\right(x\left)\right)dx$, where p(x) is the probability distribution for the particle, from the Boltzmann equation, is sketched and employed for the theoretical study of the time evolution of Shannon entropies for diffusion from a point source, classical free expansion (under circumstances where the mean free path is much larger than the length of the box, and vice versa), and 'versions' of quantum evolution of a particle initially localised in a Gaussian, and quantum free expansion. For the quantum versions, striking similarities to the classical counterparts are highlighted; furthermore, for quantum free expansion, the Poincare period is determined and discussed at some length. The quantum versions studied here work with pure states and zero temperature. (paper)