The Apollonian decay of beer foam bubble size distribution and the lattices of young diagrams and their correlated mixing functions

We present different methods to characterise thedecay of beer foam by measuring the foam heights and recordingfoam images as a function of time. It turns out that the foamdecay does not follow a simple exponential law but a higher-orderequation V ( t ) = a − b t − c t 2.5 , which can be explained as asuperposition of two processes, that is, drainage and bubblerearrangement. The reorganisation of bubbles leads to thestructure of an Apollonian gasket with a fractaldimension of D ≈ 1.3058 . Starting from foam images, westudy the temporal development of bubble size distributions andgive a model for the evolution towards the equilibrium statebased upon the idea of Ernst Ruch to describe irreversibleprocesses by lattices of Young diagrams . These latticesgenerally involve a partial order, but one can force a total orderby mapping the diagrams onto the interval [ 0 , 1 ] using ordering functions such as the Shannon entropy . Several entropy-like and nonentropy-like mixingfunctions are discussed in comparison with the Youngorder , each of them giving a special prejudice for understandingthe process of structure formation during beer foam decay.