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 the decay of beer foam by measuring the foam heights and recording foam images as a function of time. It turns out that the foam decay does not follow a simple exponential law but a higher-order equation V(t)=a−bt−ct2.5, which can be explained as a superposition of two processes, that is, drainage and bubble rearrangement. The reorganisation of bubbles leads to the structure of an Apollonian gasket with a fractal dimension of D≈1.3058. Starting from foam images, we study the temporal development of bubble size distributions and give a model for the evolution towards the equilibrium state based upon the idea of Ernst Ruch to describe irreversible processes by lattices of Young diagrams. These lattices generally involve a partial order, but one can force a total order by mapping the diagrams onto the interval [0,1] using ordering functions such as the Shannon entropy. Several entropy-like and nonentropy-like mixing functions are discussed in comparison with the Young order, each of them giving a special prejudice for understanding the process of structure formation during beer foam decay.


Introduction
Beer foam is a fascinating subject not only for connoisseurs of selected cultivated beers but for scientists as well.Some years ago, in 2002, Leike [14] published his observations on the decay of beer froth.He found that the volume of beer froth decays exponentially with time.He looked for the height of the froth for different kinds of German beer as a function of time.His best fits represent the decay only in the beginning.However, Dale et al. have already published their fundamental results on the decay of beer foam in 1993 [8].They investigated the temporal change in the conductivity of the collapsing beer foam.They distinguished between three decay phases-the initial phase I, the consolidation phase C, and the residual R. For the first two phases I and C they estimated simple exponential laws for the decay of the foam mass.They concluded that at the first stage I of foam collapse the liquid beer drainage driven by gravity is the most important process.This process takes about 300 seconds.The consolidation stage C is characterised by an increase of the concentration of polypeptide material in the foam.The enrichment of surface active molecules in the bubble lamella-leading, for example, to bubble coalescence-is responsible for the collapse process.
Following these arguments, one may call in question the results presented by Leike [14].This leads to the problem, if it is possible to find a similar separation of different processes in the temporal development of the foam volume.Usually one describes foams by their bubble or cavity size distribution function.How does the bubble size distribution function develop during the decay of the foam?No answers, neither for beer foam, nor for other fast collapsing fluid/gas foams, are known to this question.The problem is that one cannot simply look into the three-dimensional body of the foam because of the refraction of light.
Especially in case of beer foam it is well known that the surface of the beer glass strongly influences the properties of the foam.Therefore, it seems to be very hard to conclude to the inner bubble size distribution function from looking at the outside of the glass.Nevertheless, there should be at least some correlations between the outer and the inner structure of the bubble arrangement, which could be figured out by looking at the temporal development of the foam as well as of the increase of the liquid beer flowing out of the foam.It is our aim to get information about the structure of the body of the foam by looking only from the outside.

Experimental setup
In order to investigate the decay of beer foam, at first one has to produce the foam.The traditional way of producing beer foam is to pour a glass of beer.Unfortunately, this method does not lead to reproducible results, since one cannot reproduce the initial bubble size distribution function.For this reason, we decided to foam up the unfoamed beer with ultrasound (Ultrasonik 28x; NEY) for 13 seconds.This unfoamed beer is allowed only to posses at most 5% of foam with respect to its fluid part.
It is well known that the shape of the glass and the amount of beer to be foamed will influence the decay of the beer foam.For this reason we investigated the decay of the foam in different measuring glasses (100 ml, 2.6 cm in diameter; 250 ml, 3.6 cm in diameter; and 500 ml, 5 cm in diameter) with different amounts of beer: 20 ml beer in the 100 ml measuring glass (abbreviation: 20/100), 50 ml beer in the 250 ml glass (50/250), and 100 ml beer in the 500 ml measuring glass (100/500).To investigate the temperature influence on the decay of foam, we used different temperatures in our beer foam experiments: Using these experimental conditions, the initial volume at time t 0 = 0 of the freshly generated foam after ultrasound treatment was about twice as much compared to the volume of the unfoamed beer.About 10 Vol.%-with respect to the foam volume-of the beer remained unfoamed.For example, in case of 20 ml beer we obtained 40 ml foam and 4 ml unfoamed beer after ultrasound foaming.
The volume of the foam and the liquid beer phase was measured in time intervals of 20 seconds by visual inspection.S. Sauerbrei et al. 3 In order to estimate the bubble size distribution function, we analysed photographs of collapsing beer foam by an optical measuring technique and computer-aided image processing [10].The measuring device consisted of a high resolution CCD-camera (JAI CV-A11), a telecentric lens, coaxial diffuse illumination, and a fast frame grabber card, leading to an accuracy up to 5 μm.The photographs were recorded every five seconds starting one minute after foaming up.In order to avoid optical distortions, the beer foam was generated in special square glass vessels (2.5 × 2.5 cm) manufactured in our laboratories.

Temporal behaviour of the foam volume.
We investigated the volume dependence V (t) of beer foam decay measuring the height of the foam and the level of liquid beer as a function of time.In every case we averaged arithmetically five independent measurements under the same conditions.The data obtained were approximated by various curve fitting calculations.Out of the tested fitting functions we always found the following fitting formula V (t) with highest priority: fulfilling the constraints Using (3.1), we estimated the coefficients for many kinds of beer and different temperatures in different glasses.Taking Beck's Pils from Bremen, for instance, we obtained in case of the (20/100/24 • C) experiment equation (3.3) (see Figure 3.1): This is not an ordinary (simple) exponential function, but it can be interpreted as a product function decomposable into two separate functions, which describe different processes of the foam decay at different time intervals: for 0 s ≤ t ≤ 160s, and The coefficient a ≈ a 1 represents the foam volume V (0) ≡ V 0 ; V 1 (0) = exp(a 1 ) ≈ exp(a) = V 0 immediately after the ultrasound foaming up of the beer.b 1 is the coefficient of the exponential decay at the beginning, which is mainly caused by the drainage of the liquid beer out of the foam (see Figure 3.2).The essential new factor is the term −c 2 t 2.5 or exp(−c 2 t 2.5 ), respectively.The corresponding function V 2 (t) = exp(a 2 )exp(−c 2 t 2.5 )which has never been described before-represents the rearrangement of the bubbles in the second time interval C (see Figure 3.2).The coefficient a 2 represents the fictive volume V 2 (0) = exp(a 2 ) of this second process at t 0 = 0.The character of this process will be discussed later.
The temperature dependence is shown in Table 3.1 for Beck's Pils in the (20/100) experiments.The lower the temperature is the smaller the amount of initial foam volume V 0 = exp(a) in lnV (t) after ultrasound treatment of beer is and the faster the drainage is.This can be seen from the coefficients b and b 1 , respectively.The coefficients of the rearrangement process in the phase C decrease with lowering the temperature, which indicates a slowing down of the underlying process of bubble rearrangement.Table 3.2 shows an example of the influence of the size of the beer glass on the decay of beer foam.
Taking wider glasses, we needed more beer for inspection of the decay of foam.The wider the glasses are and the more beer has to be foamed up, the slower the drainage is and the rearrangement processes are in tendency.For wider glasses it is much more difficult to separate the two processes (see Figure 3.3).
Comparing different kinds of beer at the standard experimental conditions (20/100/ 24), Table 3.3, we obtained the coefficients shown in Table 3.3.As can be seen, in case of   Diebel's Alt one gets more foam and the lowest decay by drainage and during rearrangement.In other words, under these conditions the foam of Diebel's Alt is more stable than that of Beck's or Vitamals.It seems to be possible to enhance such kinds of measurements in order to compare the foam quality of different kinds of beer.
6 The Apollonian decay of beer foam 3. An example for the decay of beer foam which is not separable under these conditions.Dale et al. suggested that the exponential shrinking of the volume according to (3.4) is mainly caused by the drainage [8].A first inspection seems to support this idea, since the fitting curve V 1 (t) = exp(a 1 )exp(−b 1 t) for the data of the initial phase I (0s ≤ t ≤ 160s) is an exponential one.However, this drainage process is at the same time overlaid by the beginning of the rearrangement.Possibly, the separation into two parts of the overall function (3.3) based on the experimental data enables studying the underlying processes in more detail.

Drainage process.
In the region between 200 s and 380 s the function of the temporal dependence of the volume has been determined to be V 2 (t) = exp(a 2 )exp(−c 2 t 2.5 ) (3.5), where V 2,0 ≡ V 2 (0) is the fictitious initial volume for this process of bubble rearrangement.If we calculate the difference between the experimental data in the first time interval I and the function V 2 (t), we can estimate the function of the pure drainage without the contribution of the rearrangement process by for 0 s ≤ t ≤ 160s.
The best fit for these data is the following parabola (see Table 3.4 and Figure 3.4): Even simpler parabola will give reasonable results like, for example, or  however, all these fitting functions do not fulfil the (artificial) constraint lim t→∞ V d (t) = 0, which means that the experimental data are embedded in the function V 2 (t) for the interval 200 ≤ t ≤ ∞, unless a third process starts to govern the decay of the foam at time t ≥ 330s.An exponential approach of the type exhibits systematic deviations from the experimental data.So, the above-mentioned parabola (3.7) is the best description for the experimentally observed temporal decay of the foam volume, if one considers drainage as its only reason.For the moment we are far from understanding this process.Therefore, we have to restrict ourselves presently only on experimental data and first numerical results of the very complex behaviour of the foam decay by drainage.

The Apollonian rearrangement.
The temporal behaviour of the bubble rearrangement in the consolidation stage C (200s < t < 380s) is described in a very good agreement by the equation V 2 (t) = exp(a 2 )exp(−c 2 t 2.5 ) (3.5).Dale et al. assumed that the increase of the concentration of polypeptide materials in the foam lamella is responsible for this consolidation [8].We observed a dramatic change in the bubble size distribution function, which starts from a very sharp distribution function of very small bubbles just after ultrasound foaming up.Simple visual inspections of the changes in the foam during this time interval indicate that the bubble rearrangement is the essential process.Within time, large bubbles survive surrounded by small and very small bubbles, which are filling the remaining spacing between the larger bubbles (Figure 3.5).These structures are strongly suggestive for the structure of an Apollonian gasket (Figure 3.8).S. Sauerbrei et al. 9  Apollonios of Perge (262 to 192 BC [28]) put two circles (of different sizes) inside a large one, both touching each other and the large one at three different points of contact (see Figure 3.6).In both the two remaining circular triangles he inscribed the largest possible circles (called Apollonian circles, represented with magenta colour in Figure 3.6), each of them having contact with the original circles at three points.He ended up with five circles and six spacing circular triangles [17].
G. W. Leibniz (1646-1716) [17] repeated this procedure ad infinitum (compare Figure 3.7): Imagine a circle; inscribe within three other circles congruent to each other, and of maximum radius; proceed similarly with each of these circles and within each interval between them, and imagine that the process continues to infinity ... (quoted from Mandelbrot, The Fractal Geometry of Nature, page 170) [15,16].His construction is known today as the Apollonian packing or Leibniz packing [16].Following the notation of Mandelbrot, a finite set of Apollonian circles constructed this way is called Apollonian gasket (Figure 3.8).
The Apollonian cascade is not self-affine.Nevertheless, one can use the Hausdorff-Besicovitch definition to define a measure.Boyd [3,4] has estimated numerically this fractal dimension D to be in the range of 1.30197 < D < 1.314534 [4], or D ≈ 1.3058 [3].(3.11)In order to confirm the assumption that the bubble rearrangement during the consolidation phase C is just the formation of an Apollonian gasket, we estimated the Hausdorff-Besicovitch dimension D for the photographs, which have been taken from the bubbles on the walls inside the beer glasses.For this purpose we used a special square glass vessel in order to get plane surfaces.The grey valued pictures of the bubbles have to be transformed firstly into binary pictures (black and white pictures).Using cellular automata we estimated the borderlines of the black disks getting a set of circles (see Figure 3.9).
From these pictures of the borderlines we calculated the fractal dimensions D of the arrangement of circles at different times.The dimension D of the set of circles converges to the dimension of the Apollonian gasket (Figure 3.10).Apollonian spheres (this reproduction has been taken from [18]; the publishing house does not exist any more).
In order to quantify the statement of the temporal behaviour of the fractal dimension D, we estimated the spline functions (3.12) (see Figure 3.10): for the decay of the foam volume at 8 • C and 24 • C in case of Beck's Alkoholfrei beer (see Figure 3.10 and Table 3.5 for the temperature dependence of the coefficients).
It is an interesting result that one can correlate the fractal dimension D of the pictures of the foam surfaces with the state of the decomposition of the foam.As a consequence, one can state that the fractal Hausdorff-Besicovitch dimension D of these pictures is a measure for the geometrical rearrangement process of the foam bubbles.
One may argue that one cannot conclude from the inspection of the outer surface of the foam to its inner part.Therefore it should not be allowed to extrapolate the Apollonian rearrangement observed on the foam surface to the processes in the interior of foam.On the other hand, there is an early work on Apollonian packages of spheres [7,18], which has already been used to describe dense mixtures of particles of very different sizes (Figure 3.11).Moreover, Japanese mathematicians of the Wasan school discussed in 1822 a similar problem on wooden Sangaku Tables in order to put Kissing Spheres inside a sphere [20,22].
Sometimes, one can even recognise bubbles belonging somehow to the inner part of the foam and one can detect a structure, which resembles strongly an Apollonian gasket of spheres.This is by no means a proof, but it is a strong indication that the rearrangement  of foam bubbles as an Apollonian package of spheres occurs in the interior of foam in a similar way as compared to the interface area of the foam and the glass wall.

The liquid content of the foam.
In brewery techniques it is important to know the amount of liquid phase in the foam at time t.Obviously, one can record the increasing volume of fluid beer under the foam (see Figure 3.12) getting also the remaining portion of the liquid phase in the foam [2].
The time-dependence of the liquid volume fraction in the foam is given by where α V and β V are the coefficients of the corresponding regression curve (β V ≈ 1).On the other hand, from the recorded photographs (Figure 3.13) one can estimate the size of the bubble area A B (t) which covers the foam picture at time t.Referring to the total image area A I , one can calculate the remaining non-bubble area A L (t) which represents the liquid part of the two-dimensional foam image (3.14), (3.14) Figure 3.14 shows the temporal development of the two-dimensional liquid area of the images and Figure 3.15 a snapshot of the bubbles after 165 seconds.Now, one can correlate the liquid volume V (t) (see Figure 3.12) of the three dimensional foam and the two-dimensional liquid portion A L (t) (see Figure 3.14).For this purpose one has to normalise the amplitudes of both functions: V (t)/V (0) and A L (t)/A I (Figure 3.16).
This representation shows a clear functional correlation between the classically obtained time dependence of liquid foam volume fraction and the optically determined time dependence of the two-dimensional liquid portion of the foam.

Bubble size distribution.
Usually, foam is characterised by its bubble size distribution function.Starting with a very narrow distribution function (Figure 3.17) immediately after the ultrasound treatment of the beer sample, we observed a fast broadening during the initial state I of drainage due to ordinary diffusion (Figure 3.18(a)).But in the consolidation stage C we observed a very surprising behaviour.Instead of continuing the normal diffusion process, the distribution function splits into many separated peaks (Figure 3.18(b)).Only very few and quite different bubble sizes survived.This very strange behaviour encouraged us to look for another method of description for the development of the temporal size distribution function.Starting from the famous articles of Ruch on diagram lattices as structural principle [23] on the principles of increasing mixing [25] and on information extent and information distance [24] we suggest a model for bubble size distribution described in the following section.16 The Apollonian decay of beer foam

Diagram lattices and mixing functions as a model for bubble size distributions
Originally Ruch used lattices constructed from Young diagrams [26] in order to describe the development of a discrete distribution function.Together with Schönhofer [26] he introduced a greater relation for Young diagrams [31,32], in order to answer questions which appear in connection with the theory of chirality functions.This greater relation enabled him to construct a lattice.He could show that this lattice introduces a partial order of the Young diagrams in contrast to the total order which has been introduced by Young [31].For describing the time development of irreversible processes in closed systems he mapped the diagrams which are partitions of natural numbers onto the Shannon entropy [23,27].Going back to Jaynes [12,13], a comparable idea has been developed at the same time by Uhlmann [29] on the Shannon entropy and related functionals on convex sets.Some years later Uhlmann and Alberti described dissipative motions in state space by use of stochastic transformations of convex sets like probability vectors on algebras over real numbers and star algebras like C and W [1]. Surprisingly, this powerful idea of lattices based on Young diagrams has later been used to rank biodiversity indices for the comparison of the water quality of lakes [6].For example, if the diagram consists only of one row (Figure 4.2), then all bubbles belong to the same bubble size interval.This is a realistic assumption for the foam in status nascendi immediately after ultrasound treatment (cf. Figure 3.17).During the temporal development, larger bubbles are created, which means in our model that the number of the small initial bubbles diminishes for the benefit of larger bubbles, that is, the diagram develops (see Figure 4.3).Following the definition of Ruch [23], A diagram γ is called greater than a diagram γ , denoted by γ ⊃ γ , if γ can be constructed from γ by moving boxes exclusively upward, that is, from shorter rows into larger or equal ones.From this definition Ruch transformed a diagram to a smaller neighboured one by moving one box from a larger row into a smaller one.
It is also known [23] that the lengths of the rows of a diagram represent a partition of the integer n, that is, the decomposition of n into a sum of integers.In this case, the definition for the greater relation given by Ruch [23] is the following: A partition is called greater than another, if the transition from a smaller to the greater one can be made in steps, whereby partition numbers increase at the expense of others which are not larger (see Figure 4.4).For further details the reader is kindly requested to look at the original literature [23,26].Following Ruch's rule for the construction of his partition lattice or diagram lattice, the most striking result of the greater relation is the fact that one gets a total order on S. Sauerbrei et al. 19 The totally ordered sets are given in grey coloured boxes, whereas the partially ordered set for n = 6 is represented with red and pink coloured boxes.The pink coloured diagrams of the same level are incomparable elements of this lattice.The numbers p with 1 ≤ p ≤ m at the right side of the diagrams demonstrate their sequence with respect to the total order given by Young, which is identical with the order of the sums S(O γ ) of the partial sums (4.4) [26] in case of the lattice for n = 6.
the set of all possible diagrams for n boxes only if n ≤ 5, but a partial order, if n > 5 (see Figure 4.5).
The greater relation is originally defined [26] by the vector of the partial sums o i of the ith row of the partition diagram γ, each vector having the length n and filled up with zeros if necessary, for example, 20 The Apollonian decay of beer foam A partition γ is greater than a partition γ , denoted by γ ⊂ γ, if and only if where n is given by the total number of all boxes of the diagram.We demonstrate this statement by a comparison of the above-mentioned diagram γ with two other incomparable diagrams (see Figure 4
But if we compare both the diagrams γ and γ (Figure 4.6) with each other, we cannot say that γ is smaller than γ nor that the opposite is true since the components of the vectors of the partial sums do not fulfil the condition o i ≤ o i for all i = 1,...,n or its negation, respectively.For this reason the diagrams γ and γ are not comparable.
Some additional statements should be mentioned.One can define a distance d i (γ,γ ) between two neighboured diagrams γ and γ as the number of rows which a box has to be shifted in order to step from the upper row of the initial diagram γ to the lower row of the destination diagram γ .The sum of all distances on a way from the greatest diagram in the diagram lattice consisting of only one row with n boxes to the smallest diagram which contains always one box in each of the n rows is given by the expression (4.3) It is remarkable that this sum of distances does not depend on the way from top to bottom, that is, the chosen prejudice of how to step through the incomparable diagrams of the partially ordered lattice does not affect this sum.
The second remark concerns the sum S(O γ ) of the partial sums of elements in the vector O γ of the partial sums.The set of these sums reflects the structure of the partially ordered set of diagrams.In the case of diagrams with n < 7, incomparable elements are characterised by the same number S(O γ ) = S(O γ ).Then, we can define that a diagram γ is called greater than a diagram γ , if the inequality  The row n − 7 can be calculated from the sums of n = 7.
(4.5) between the sum of partial sums (partitionings) holds: This definition defines a total order onto the set of all diagrams.So, we can enumerate the diagrams in a natural way by integers 1 ≤ p ≤ m; p,m ∈ ℵ, where m is the number of diagrams or partitions which can be constructed from n boxes or from the integer n (see Figure 4.5).But there is no polynomial expression to calculate the value m for a given number n, since this problem is a nonpolynomial NP-complete problem [9].However, one can find iteration equations [30] which give the value m for each natural number n of boxes (see Table 4.1 for n = 1,..., 16).A solution of this problem was presented by Rademacher using a series expansion [21].

Mapping Young diagrams on scalar functions.
If one normalises the number of boxes ν γ,i in the ith row of a given Young diagram γ = (v γ,1 ,ν γ,2 ,...,ν γ,i ,...,ν γ,n ) of n boxes (or the corresponding partition) with respect to the overall number of boxes n according to (4.6), one can map all diagrams onto a scalar function, for example, the Shannon entropy I(γ) [23,27]: By this way one introduces a total order on the set of all Young diagrams belonging to a given number n of boxes.
On the other hand, one can also obtain a total order γ ≥ γ on the set of diagrams or partitions for any given number n by the definition [26]: γ > γ , if moving from top to bottom in the diagram lattice by comparison of corresponding rows the first row is reached, where the relation ν i > ν i holds.This definition by Ruch and Schönhofer [26] of a total order on the set of diagrams corresponds in the case of n < 7 to the total order which is given by the sum S(O γ ) (4.4) of partial sums.For the further discussion let us mention that the sum S of all normalised row values p γ,i (4.6) of a partition vector γ is equal to one (simplex condition), that is, the trace of the vector is one, By the simplex condition (4.8) one obtains a normalised and bounded domain-the simplex S n -on which the Shannon entropy is defined.Also, the function over this domain of definition itself, that is, the Shannon entropy I(γ) = F 1 (γ) (4.7) can be normalised for any number n of boxes by (4.9) (see Figure 4.8 for n = 3): (4.9) This normalised Shannon entropy F 2 (γ) maps the Young diagrams onto real numbers of the closed interval [0,1]: It is now very important to realise that the function F 2 (γ) is a concave function [19] (also called convex from the above [5,11]) over the simplex S n which in addition fulfils the restrictions The Shannon information entropy functions F 1 (γ) or F 2 (γ), respectively, can be considered as a measure for the gain of information if one executes an experiment where the probabilities of its events are distributed under the restriction of the simplex constraints functions-comparable with the normalised Shannon entropy F 2 -which map the Young diagrams onto the closed interval [0,1] and which fulfil condition (4.11), respectively, over n-dimensional simplices S n as domains are (i) the normalised general tent function F 3 (γ) (see Figure 4.9 for n = 3), (ii) the normalised general logistic function F 4 (γ), (iii) the normalised general biquadratic function F 5 (γ) (stretched to the simplex boundaries, see Figure 4.10 for n = 3), (iv) the normalised general k-quadratic function F 6 (γ) (stretched to the simplex boundaries), which becomes the logistic function for k = 1 and the biquadratic function for k = 2, respectively, There are further functions fulfilling condition (4.11) over n-dimensional simplices S n , such as (i) the normalised general biquadratic function F 7 (γ) (delta-like compressed to the centre of simplex), Both functions are not concave over the whole simplex, on which they are defined (see Figure 4.11), although they fulfil condition (4.11), that is, they are symmetric to the centre of equally distributed values p γ,i = 1/n with function value one, and they take the function value zero if one p γ,i gets 1 and all other p γ, j vanish; but their mapping interval exceeds the value 1 in general.
A nonpolynomial type of concave functions fulfilling condition (4.11) which maps the Young diagrams onto the closed interval [0,1] over n-dimensional simplices S n as well is given by the normalised general product of trigonometric functions F 9 (γ): The representations of functions F 2 (γ), F 3 (γ), F 5 (γ), and F 8 (γ) in Figure 4.8 to Figure 4.11 on the S 2 simplex over triangular coordinates were chosen to give an intuitive insight into these types of functions on higher simplices S i with i > 1. Recall that a Young diagram γ of n boxes corresponds always to a simplex S n−1 with n vertices defining the domain of the functions F i (γ).In fact, our application leads to functions over much higher simplices which cannot be drawn easily in a three-dimensional graphical representation.The functions F 1 (γ) to F 6 (γ) as well as F 9 (γ) are concave functions on the simplex S i , that is, the following equation [19]: holds for any variable x i with i = 1,2 (x i should be identified with p γ,i in our context).On the other hand, the functions F 7 and F 8 are neither concave nor convex over the whole simplex (see Figure 4.11), but it is remarkable that both functions exhibit several maxima although they are normalised to 1 in the centre of the simplex where all variables have equally distributed values x i , for example, x 1 = x 2 = x 3 = 1/3 in case of the simplex S 2 .This kind of functions could be of interest with respect to structure formations in processes far away from equilibrium and will be discussed in greater detail in Section 4.5.

4.3.
Scalar functions on the simplex S 1 .Figure 4.12 shows a comparison of the scalar functions y 1 ≡ F 1 (γ) to y 9 ≡ F 9 (γ) defined in the previous section for the special case of the well known simplex S 1 , that is, the closed interval [0,1].The choice of the S 1 simplex means that we consider only two variables or a Young diagram of only n = 2 boxes, respectively, where the simplex condition of the domain of definition is given by Under these assumptions the Shannon entropy functions F 1 (γ) (4.7) and F 2 (γ) (4.9) take the form The tent function F 3 (γ) (4.12) transforms to y 3 according to (4.22).Note that y 3 is an affine function [19] since it is concave (convex from the above) and convex (convex from the below) simultaneously.
The logistic function F 4 (γ) (4.13) gets the well-known shape respectively.For k → ∞, the function y 6 tends to the quadratic pulse-function H of width one given by (4.25): 1, for 0 < x i < 1, 0, otherwise. (4.25) The product function of trigonometric functions F 9 (γ) (4.18) which is also concave but of nonpolynomial type becomes The general biquadratic function F 7 (γ) (4.16) and the k-quadratic function F 8 (γ) (4.17) both of which are delta-like compressed to the centre of simplex take now the simple forms k , e.g., y 8 = 2 20 x 10 (1 − x) 10 for k = 10, (4.27) respectively.Both these functions y 7 and y 8 reflect the fact that they are neither concave nor convex over the whole closed interval [0,1].But they have a maximum at x 1 = x 2 = 1/2 in contrast to their corresponding functions F 7 (γ) and F 8 (γ) defined on higher dimensional simplices which in general show a minimum at the centre of the simplex and various maxima symmetrically distributed around this point.All these functions can be used as measures characterising the disorder or mixing of the system under inspection.We call the functions F 1 (γ) to F 6 (γ) and  The partially ordered lattice of the diagrams describes the system without prejudice for choosing a way from top to bottom in the lattice, that is, from order to disorder.The classical Shannon entropy is by no means a unique choice to measure disorder or order, respectively.The choice of a special entropy-like mixing function is nothing else than the introduction of a prejudice concerning a special mapping by a mixing function of the diagrams or the corresponding partitions onto the interval [0,1].The projection may change the sequence of steps from top to bottom in the lattice, that is, the sequence of states on the way from order to disorder.For example, if one takes the biquadratic F 5 (γ) (4.14) of the diagrams with n = 6 boxes as a function of the correlated Shannon entropy F 2 (γ), the incomparable diagrams of the pairs γ 4 and γ 5 as well as γ 7 and γ 8 , respectively, permute with each other in their sequence (see also Figures 4.5    As can be seen from Figure 4.16 there is no unique system to select a total order which is preferred with respect to all other choices which can be made.The Young order is comparable to all the other possibilities forming a total order on the partially ordered lattice of diagrams.But from Figure 4.16 one can see clearly that the set of all entropy-like mixing functions forms a family of functions which tends to reflect the Young order and that the k-quadratic functions F 6 (γ) (4.15) which are stretched to the simplex boundaries correspond strictly to the Young order for the limiting case with k → ∞; and even the biquadratic function F 6 (γ) (4.14) reproduces the Young order exactly up to n = 13.

Nonentropy-like mixing functions.
In contrast to the concave entropy-like mixing functions discussed above, the functions y 7 and y 8 or-more generally-F 7 (γ) and F 8 (γ), respectively, which are not concave over the whole simplex of their definition, show the following remarkable properties.
(1) In case of S 1 or the closed interval [0,1] as domain of definition, y 7 and y 8 are concave around the centre of S 1 at x = 0.5 and convex (from the below) towards the boundaries at x = 0 and x = 1, respectively (see Figure 4.12).In other words, they have two crossing points with the affine tent function y 3 symmetrically arranged with respect to the centre of S 1 both of them moving to x = 0.5 for k → ∞.
As a consequence, only in the region around x = 0.5, that is, in case of highly disordered systems (close to equilibrium or almost equally distributed), y 7 and y 8 behave as entropy-like mixing functions, yielding a high increase of information if one executes an experiment.But beyond the crossing points with the affine tent function y 3 -that is, on the convex sides-they rapidly tend to zero.If one now takes into account that only at the vertices of the simplex the uncertainty of an experiment is zero or-equivalently-the knowledge is maximal (complete), one can conclude that in case of these non-overall concave functions even at comparably less knowledge the additional gain in information is very small.(2) As can be seen exemplarily from Figure 4.11 the functions F 7 (γ) and F 8 (γ) defined on higher simplices S i with i > 1 show in general a minimum at the centre of the simplex and various maxima symmetrically distributed around this point; hence they exhibit function values greater than one (see Figure 4.16, red polygon) if they are normalised to 1 for an equilibrated or equally distributed system corresponding to the Young diagram with the highest number m of the correlated Young order.As a consequence, they have convex (from the below) regions not only over the vicinities of the simplex vertices but also a convex area over the centre of the simplex and concave parts symmetrically arranged with respect to the simplex centre.If one interprets this behaviour of functions F 7 (γ) and F 8 (γ) in terms of statements about information content, an experiment in the immediate neighbourhood the equilibrated or equally distributed system does not lead to the maximum gain of information, although there is maximum uncertainty and-as a consequence-the least known knowledge.On the other hand, somewhat farther away from the equilibrium point there are several (equivalent) situations with maximum increase of information, but already some knowledge.The consequences from the function behaviour in convex regions on the simplex corners are the same as described above for y 7 and y 8 .Considering these interesting properties we suggest-with the necessary cautionthat the functions F 7 (γ) and F 8 (γ) may be of interest for the description of dynamically fluctuating systems and the explanation of structure formations in processes.

Conclusion
It was our aim to characterise the decay of beer foam by quite different methods like measuring the temporal behaviour of the foam volume, the fractal dimension of the twodimensional photographs, of the bubble arrangements, the estimation of liquid content in the foam phase, and the bubble size distribution functions.It is our finding that the foam volume does not shrink simple exponentially but in terms of higher order according to the equation ln V (t) = a − bt − ct 2.5 (3.1).
The term ct 2.5 describes the reorganisation of the bubble arrangements which will lead to an Apollonian gasket of bubbles.This process could be described by the fractal dimension of the correlated bubble pictures and tends to the fractal dimension D ≈ 1.3058 [3] of the Apollonian packages of circles.Both methods have a great potential of industrial and environmental application, since they allow to follow the decay of foams easily.This could be demonstrated via the estimation of the actual fluid part remaining in the foam during its decay.
The bubble size distribution function of led to the question how to compare different bubble size distribution diagrams.Following the idea of Ruch [23], we described the bubble size diagrams by Young diagrams forming a partially ordered lattice.The temporal S. Sauerbrei  sequence of Young diagrams reflects the movement of a closed system towards its equilibrium attractor.Mapping the Young diagrams on scalar functions, for example, the Shannon entropy, one can follow the temporal development of the system by its correlated Shannon entropy or information content, respectively (Figure 5.1).It was our expectation to find a smooth, monotonously growing information function in dependence on time.Surprisingly we got an oscillating behaviour of this information function for 0 • C, 8 • C, and 24 • C, respectively.This can be understood qualitatively if we take into consideration that our system is by no means a closed system and, as a consequence, one should be able to observe structure formation processes during time intervals of its development.On the other hand, we started with a very narrow bubble size distribution function immediately after the ultrasound degassing of the beer, where one should expect classically a broadening of this distribution function and an increase of the information entropy, but not the occurrence of a multimodal distribution function oscillatory swinging up and down.
To get an insight into all these problems we investigated the behaviour of different mixing functions mapped on the simplex S n which is correlated to a lattice of Young diagrams with n boxes.There is an infinite number of concave functions which can be used to characterise the mixing of states or bubble sizes, respectively, which is described by one Young diagram.But any choice of a special function out of this set of concave functions corresponds to the introduction of a prejudice with respect to the definition of a total order on the partial order of the lattice of Young diagrams with n boxes.Therefore, we concluded that not only one function, but the total set of all concave functions which are entropy-like mixing functions is an appropriate, unprejudiced measure for the mixing of states or bubbles, respectively.In addition, we suggested that the nonentropy-like mixing functions may be an adequate description for structure formations in processes.
Obviously, the decay of beer foam is finished before reaching the equilibrium-which of course can never be reached.Before vanishing totally, the foam system remains in a state which can roughly be described by distribution functions belonging to states anywhere in the middle of a partially ordered lattice of Young diagrams of many boxes.It will be the task of our next work to describe the oscillating movement of the foam system into this region based on an analysis of the doubly stochastic matrices which transform the vectors of the bubble size distribution functions during their temporal development into each other.

Figure 3 . 1 .
Figure 3.1.The volume of the beer foam as a function of time.The fitting function is given by (3.3).The arrow approximately indicates the point, where drainage switches to bubble rearrangement as dominating process.

Figure 3 . 2 .
Figure 3.2.The foam volume as a function of time.The three different functions ln V , lnV 1 , and lnV 2 are put together in this figure for comparison.

Figure 3 . 5 .
Figure 3.5.Apollonian foam at the inner walls of a plane beer glass.

Figure 3 . 9 .Figure 3 . 10 .
Figure 3.9.Cellular automata are used to estimate the borderlines of the black disks representing the bubbles in the foam: (a) transformation rules of the cellular automaton, (b) black disks and surrounding borderlines (pink coloured), exemplarily drawn for two bubbles.

Figure 3 . 12 .
Figure 3.12.Time dependence of the volume of the liquid beer phase flowing out of the foam (dark blue), the portion of fluid beer in the foam (blue) and fitted curve (red).

Figure 3 . 13 .
Figure 3.13.Determination of the liquid fraction of foam from a foam image taken as snapshot of shrinking beer foam with trapped bubbles areas.

Figure 3 . 14 .
Figure 3.14.Temporal development of the two-dimensional liquid area of the foam pictures according to (3.14).For the marked point at 165 s, see Figure 3.15.

Figure 3 . 1 AFigure 3 . 16 .
Figure 3.15.Snapshot of the two-dimensional arrangement of the bubbles at the inner walls of the glass after 165 s.

4. 1 .
Histogram representation of bubble size distribution functions and Young diagrams.Now, let us assume that a row with n boxes in a Young diagram is a representation of the number, the frequency, or even the probability of bubbles, the size of which belongs to the same interval (see Figures 4.1and 4.2).

Figure 4 . 3 .Figure 4 . 4 .
Figure 4.3.Development of a Young diagram by shifting one box from the upper row to the next lower row.

Figure 4 . 5 .
Figure 4.5.Diagram lattices of n boxes and m diagrams within each lattice.The totally ordered sets are given in grey coloured boxes, whereas the partially ordered set for n = 6 is represented with red and pink coloured boxes.The pink coloured diagrams of the same level are incomparable elements of this lattice.The numbers p with 1 ≤ p ≤ m at the right side of the diagrams demonstrate their sequence with respect to the total order given by Young, which is identical with the order of the sums S(O γ ) of the partial sums (4.4)[26] in case of the lattice for n = 6.

Figure 4 . 6 .
Figure 4.6.Two incomparable diagrams (see Figure 4.5 for n = 6) are shown together with their correlated partition vectors γ and γ and the vectors O γ and O γ of the corresponding partial sums.

Figure 4 .
7 shows (a) the Shannon entropies of the diagrams as a function of the consecutive numbers p of the totally ordered set of entropy values, and (b) the corresponding partial sums of Shannon entropies up to p.Both functions are increasing monotonous and discrete functions of the ordering numbers p of the diagrams which correlate with the progression of the diffusion process.

Figure 4 . 7 .
Figure 4.7.Progression of diffusion process: (a) Shannon entropies of the Young diagrams as a function of the ordering number p according to (4.4),(b) partial sums of Shannon entropies up to p as a function of p.

Figure 4 . 13 .
Figure 4.13.Comparison of the normalised concave entropy-like mixing functions y 1 ,..., y 6 and y 9 on the simplex S 1 , that is, the closed interval [0,1].The tent function y 3 is an affine function, since it is concave and convex simultaneously.

y 1 4 . 4 .
to y 6 and y 9 , respectively, entropy-like mixing functions (see Figure4.13), since they form a family of concave functions on the interval [0,1] the maxima of which are 1 in case of x i = 1/n for all indices i. Ordering of mixing functions in higher dimensional diagram lattices.However, depending on our choice of the entropy-like mixing function the sequences of the diagrams change, which can easily be seen from the occurrence of extreme values if one compares the entropy like mixing functions with each other (Figure4.14).
and 4.14).The sequences of those diagrams which are comparable do not switch their sequence when choosing another partition function than the Shannon entropy (see Figure4.15).30The Apollonian decay of beer foam 0 entropy F 2 (γ) Concave partition functions F i (γ) in dependence on the normalised Shannon entropy n = 6

Figure 4 . 14 . 5 F 2
Figure 4.14.Dependence of concave partition functions F i (γ) on the normalised Shannon entropy of the Young diagrams in case of n = 6.

Figure 4 . 16 .
Figure 4.16.Dependence of partition functions F i (γ) on the Young order of partial sums for the Young diagrams in case of n = 10.

Table 3 .
2. Influence of the size of the beer glass on the decay of the beer foam (Beck's Alkoholfrei) at 24 • C.

Table 3 .
3. Comparison of beer foam decay for different kinds of beer under the same constraints of the 20/100/24 • C experiments.

Table 4 .
1. Iterative calculations for the number m of Young diagrams for a given number n of boxes.
et al. 33 Alkoholfrei at 24 • C Figure 5.1.The Shannon information entropy of the bubble size distribution of Beck's Alkoholfrei beer at 24 • C as a function of time.