分野別セミナー

After normalization, an evolving real non-negative function may be viewed as a probability density. From this we may derive the corresponding evolution law for Shannon entropy. Parabolic equations, hyperbolic equations and fourth-order “diffusion” equations evolve information in quite different ways. Entropy and irreversibility can be introduced in a self-consistent manner and at an elementary level by reference to some simple evolution equations such as the linear heat equation. It is easily seen that the 2nd law of thermodynamics is equivalent to loss of Shannon information when temperature obeys a general nonlinear 2nd order diffusion equation. With the constraint of prescribed variance, this leads to the central limit theorem.


With fourth order diffusion terms, new problems arise. We know from applications such as thin film flow and surface diffusion, that fourth order diffusion terms may generate ripples and they do not satisfy the Second Law. Despite this, we can identify the class of fourth order quasi-linear diffusion equations that increase the Shannon entropy.


This will be a general-interest talk, with most technical details omitted.