# Discrete Dynamics by Different  Concepts  of Majorization

Authors: S. Sauerbrei; P. J. Plath; M. Eiswirth
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

For the description of complex dynamics of open systems, an approach is given by different concepts of majorization (order structure). Discrete diffusion processes with both invariant object number and sink or source can be represented by the development of Young diagrams on lattices. As an experimental example, we investigated foam decay, dominated by sinks. The relevance of order structures for the
characterization of certain processes is discussed.


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## Body

## 1. Introduction

The  concept of majorization (order structure) goes
back to Hardy et al.
[1, 2] and wasinvestigated by Uhlmann and
Alberti [3–6], and Marshall and Olkin
[7] in detail. Majorization plays an important role in quantum
information theory; see Nielsen [8].This mathematical  concept offers, by various
extensions, many ways for the description of discrete process
dynamics. One of these extensions enables the inclusion of a sink (weak
submajorization [7]) or a source (weak supermajorization
[7]) for
objects during the process dynamics.A description of special processes by majorization was
developed by Zylka [9–12]. He defined an
attainability of states. By equalizing processes a set of attainable states,
which belongs to an initial state, forms a complicated geometrical structure in
state spaces. These structures are polyhedra which are in  general nonconvex.A discrete majorization can be represented by diagram
lattices (lattices of Youngdiagrams [13, 14]) which were introduced by Ruch [15]. The lattice
structures lead to a description of elementary transitions for both discrete
classical and discrete weak majorizations. A
rep-resentation of sets of attainability by lattices is also possible.Another variant of majorization can be formulated by
taking into account permutations. In [16], we investigated the
structures and lattices of Young diagrams and their permutations in a closed
system. Structures and their progression of weak majorization including
permutations are given in [17]. Here, we show further results of this approach.As an application of the above-mentioned mathematical
concepts, we use fast decaying liquid foams in the present work. There are
several methods to describe foams: measurement of foam volume, bubble
interaction like coalescence, gas exchange between bubbles, and pressure
differences between the inside and outside of a bubble (Young-Laplace law),
collapse of bubbles, and definition of the geometry of bubbles (Plateau laws
[18]). Some
methods were used in former papers [16, 17, 19].In this paper, we give a general approach to describe
the dynamics of complex systems, like a fast decaying foam, by order structures
which are based on the development of distributions.


## 2. The foam experiment

Foam structures were produced by frothing up with
ultrasound (Ultrasonik 28x; NEY) 20 mL of
nonfoamed beer (Haake Beck)
in a rectangular glass vessel (2.5cm×20cm×2.5cm) at a temperature of 24±1C°. Pictures were taken at five-second intervals with a CV-M10
CCD-camera with a telecentric lens (JENmetarTM1×/12LD). For illumination, a cold light source
(KL 2500 LCD) was used. The position of the camera was at the
22 mL mark of the
rectangular vessel, the image size was 6.4 mm by 5 mm
(Figure 1).Figure 1
A foam image at
190 seconds.Each bubble diameter was measured and ten bubble-size
intervals defined. Thus we obtained time series of bubble-size
distributions
with absolute values of the number of bubbles of one-size interval denoted byF={f(t)=(si(t))} with ∑i=110si(t)≥∑i=110si(t+1) and the corresponding normalized bubble-size
distributions, H={h(t)=(pi(t))} with ∑i=110pi(t)=1.
The set F can also be normalized to the first
bubble-size distribution with the maximum number of bubbles to
obtain(2.1)1=∑i=110si(t0)∑i=110si(t0),1≥∑i=110si(t)∑i=110si(t0)≥∑i=110si(t+1)∑i=110si(t0).The foam decay could not only be measured by the
decreasing foam volume but also by the decreasing number of bubbles in the foam
images. The number of bubbles was found to decrease monotonously over time
[17]; see Figure 2.Figure 2
The monotonously decreasing number of bubbles
of the foam pictures.In Table1, the bubble-size distribution of the image
in Figure 1 is given.Table 1
The number of intervalsi and their interval sizes to obtain bubble-size
distributions. As an example, the data of the bubble image in Figure
1 are
given: the absolute number of bubbles si and the relative frequencies pi.
Note that the last interval is open.
IntervalInterval sizeNo. of bubblesRelative frequencyi[μm]sipi10–86980.426286–172460.2003172–258290.1264258–344180.0785344–430110.0486430–516130.0577516–60240.0178602–68830.0139688–77440.01710774–40.017In a former paper [17], we used the same experimental conditions but the area imaged
by the camera and the diameter used for bubble-size distribution were larger by
a factor of 4 respectively 2, that is the image size was 1.3 cm by 1.0 cm
and the corresponding interval size was 173 μm, with images taken every 10 seconds.
Consequently, comparison to earlier results allows an assessment of the
influence of size and time scale.


## 3. Concepts of majorization

In the
following, we briefly introduce the concepts of classical
(Section3.1) and weakmajorizations
(Section 3.2). These concepts form
the basis of our investigations. In Section 3.3, the discrete
majorization is derived from number theory and lattice theory and is extended
by a special case of majorization that is based on permutations. A restricted
dynamics that underlies majorization will be presented in Section 3.4.


## 3.1. Classical majorization

Classical majorization belongs to order theory and is
a partial order [7]. Comparing two distribution vectors by means of classical
majorization, one can distinguish both in terms of statistics. Let h(t)=(pi(t)) and h(t+1)=(pi(t+1)) be two distribution vectors which are
normalized to unity, ∑i=1npi(t)=∑i=1npi(t+1)=1.
To define the relative statistical order between both distribution vectors, one
has to sort the vector entries (relative frequencies) in decreasing order and
calculate the partial sum vectors h˜(t) and h˜(t+1).
By a greater relation, the partial sum vectors can be compared
componentwise as follows:(3.1)h(t)=(0.80.10.1),h˜(t)=(0.80.8+0.10.8+0.1+0.1)=(0.80.91),(3.2)h(t+1)=(0.60.20.2),h˜(t+1)=(0.60.81)<<=(0.80.91)=h˜(t).In (3.2), one sees  the partial sum vector comparison by the greater
relation in the last term. In this case, the greater relation is preserved and
distribution vector h(t) majorizes distribution vector h(t+1),
denoted by h(t)≻h(t+1),
or in other words the transition from h(t) to h(t+1) characterizes a process of decreasing
statistical order.In general, classical majorization can be expressed as
follows [7]: let h(pi(t)) and h(pi(t+1))∈ℝn be two vectors the entries of which are sorted
in decreasing order, p1≥p2≥⋯≥pn.
Then(3.3)h(t)≻h(t+1)  if(3.4)∑1kpi(t)≥∑1kpi(t+1),k=1,…,n−1,(3.5)∑i=1npi(t)=∑i=1npi(t+1).Considering positive semidefinite distribution vectors
withn≥3,
incomparableness can occur which ischaracteristic for a partial order.
Incomparableness mathematically means that the greater relation does not
persist for increasing k;
see (3.6). Such relations are denoted by h(t)⊁h(t+1) and h(t)⊀h(t+1),
abbreviated by h(t)×h(t+1).
This phenomenon of incomparable distributions and their significance for
process dynamics will be discussed in Section 4,(3.6)h(t)=(0.60.20.2),h(t+1)=(0.50.50),h˜(t)=(0.60.81)><=(0.511)=h˜(t+1).


## 3.2. Weak majorization

Classical majorization is applicable to closed
systems; see (3.5). A process which is characterized by a loss or a gain of
objects can be described by weak majorization [7].First, we consider a loss of objects, theweak submajorization. Then for the
distribution vectors of an open system with sink we have(3.7)∑i=1nsi(t)≥∑i=1nsi(t+1).The statistical evaluation by
weak submajorization works the same as in classical majorization. One sorts the
entries in decreasing order and compares the partial sum vectors componentwise.In general, weak submajorization can be expressed as
follows [7]: let f(t)=(si(t)) and f(t+1)=(si(t+1))∈ℝn be two vectors the entries of which are sorted
in decreasing order, s1≥s2≥⋯≥sn.
Then f(t) weakly submajorizes f(t+1),
denoted by(3.8)f(t)≻wf(t+1)if(3.9)∑1ksi(t)≥∑1ksi(t+1),k=1,…,n−1,and the relation (3.7) holds.Both classical majorization and weak submajorization
allow incomparableness. Note that because of (3.7) the classical majorization is
contained in the weak submajorization, that is, if h(t) majorizes h(t+1), h(t)≻h(t+1),
then h(t) also weakly submajorizes h(t+1), h(t)≻wh(t+1).
But we like to distinguish both cases because of (3.7) and (3.5) by using the
different notations ≻ and ≻w.
In (3.10) and (3.11), two distribution vectors f(t) and f(t+1) are given. The sums of their entries are
different, ∑i=1nsi(t)=1 and ∑i=1nsi(t+1)=0.9;
see (3.7). We call the entries of distribution vectors with ∑i=1nsi<1weak
frequencies. The derivation of such distributions is shown in Section 2.
As an example in (3.11), the greater relation is preserved and therefore vector f(t) weakly submajorizes f(t+1), f(t)≻wf(t+1).
Such a transition can be characterized by diffusion and sink, but both are
weighted in quite different ways:(3.10)f(t)=(0.60.20.2),f(t+1)=(0.50.20.2),(3.11)f˜(t)=(0.60.81)>>>(0.50.70.9)=f˜(t+1).An example of incomparableness of the weak
submajorization is given in (3.12) and (3.13). The greater relation is not
preserved and so neither f(t) weakly submajorizes f(t+1) nor f(t) is weakly submajorized by f(t+1),
denoted by f(t)⊁wf(t+1) and f(t)⊀wf(t+1),
abbreviated by f(t)×f(t+1),(3.12)f(t)=(0.60.20.2),f(t+1)=(0.70.10.1),(3.13)f˜(t)=(0.60.81)<>>(0.70.80.9)=f˜(t+1).Theweak
supermajorization is defined by the componentwise comparison of partial sum
vector entries the original entries of which are rearranged in increasing
order, s1≤s2⋯≤sn.
Then f(t) weakly supermajorizes f(t+1),(3.14)f(t)≻wf(t+1)if(3.15)∑1ksi(t)≤∑1ksi(t+1),k=1,…,n.


## 3.3. Discrete majorization

To define
discrete majorization, we have to combine number theory with order theory. One
important topic in number theory is integer partitions which were first studied
by Euler [20, 21]. For many years, one of the most
intriguing and difficult questions about them was how to determine the
asymptotic properties of the number of partitions of an integer p(n) as n gets large. This question was finally answered
by Hardy et al. [22, 23].A partition of an integern is a representation of this integer as a sum
of natural numbers; examples for partitions of n=6 are given as
follows:(3.16)6,5+1,4+2,4+1+1,3+3,…,1+1+1+1+1+1.Each partition of n can be represented by a Young diagram
[13, 14]: there are n boxes and each term of the partition can be
assigned to a row of the diagram; see Figure
3. The set of the partitions of n can be represented as vectors which are
embedded in the n-dimensional vector space by extending the
partition terms with zeros, shown as
follows:(3.17)(600000),(510000),(420000),(411000),(330000)…(111111).All diagrams (or partitions or
partition vectors) are comparable by majorization and form lattices. These
diagram lattices and the idea to use them for process description go back to
Ruch [15]. In
Figure 4, the totally ordered n=5 diagram lattice and the partially ordered n=6 diagram lattice are given. Incomparableness of
the discrete classical majorization occurs for lattices with n≥6.
One pair of incomparable diagrams of the n=6 lattice is labeled as a and b on the left. Weak majorization can be combined
with diagram lattices. In contrast to the arrows which show the direction from
the least upper bound to the greatest lower bound and represent the classical
majorization (“→” means ≻), the dashed arrows between the lattices on
the left correspond to the weak submajorization (“⤏” means ≻w); see Figure 4. An example for
incomparableness in terms of weak submajorization is given by diagrams c and d.
The dashed arrows between the lattices on the right correspond to the weak
supermajorization, (“⤏” means ≻w). Additionally, by application of the
different concepts of weak majorization the order of the diagrams changes; see
Figure 4.Figure 3
The Young diagrams of the partitions in (3.16).Figure 4
(Left) Two Ruch lattices to show
the order of discrete majorization. The arrows show the direction from the
least upper bound to the greatest lower bound and characterize the discrete
classical majorization. The dashed arrows between both lattices describe the
weak submajorization [17]. The diagrams a and b characterize incomparableness in terms of
classical majorization and the diagrams c and d incomparableness of the weak submajorization.
(Right) The same Ruch lattices, but the dashed arrows represent the weak
supermajorization.The order of weak majorization is connected by dual
diagrams (dual partitions). A dual diagram labeled with an asterisk is the
diagram obtained by reflecting the Young diagram along the main diagonal—an
exchange of rows and columns; see Figure5. It holds for the corresponding
dual diagrams that if x weakly submajorizes y, x≻wy,
then the dual diagram of y weakly supermajorizes the dual diagram of x, y⋆≻wx⋆.
Additionally, there is an algorithm to describe the progression of the diagrams
from the least upper bound to the greatest lower bound. Note that the rules for
classical majorization were given by Ruch [15],(1)
a
diagram withn boxes containing only one box in the lowest occupied
row weaklysubmajorizes a diagram with n−1 boxes by eliminating this one box [17]. Due to the dual
diagrams, the weak supermajorization obeys the rule,(2)
a diagram withn boxes, always containing boxes in the first
row, weakly supermajorizes an n+1 diagram by gaining one box in the first row. Note that these
structures are transitive. It holds that if(3.18)x≻y,y≻wz,thenx≻wz,and if(3.19)x≻wy,y≻z,thenx≻wz.In an analogous manner, (3.18) and
(3.19) hold for weak supermajorization.Figure 5
The Young diagramx and its dual diagram x⋆.Additionally, there exists a special case of
majorization,perm(utation)-majorization, which is based
on classical and weak majorizations but the rearrangement of the vector entries
is omitted and the set of permutations is taken into account
[17]. For weak perm-supermajorization, the permutations
can be compared by changing the entry sequence, s1→sn, s2→sn−1,…,sn→sn−(n−1)=s1.
Then the partial sum comparison follows. The progression of the structures
which is constructed by perm-majorization (we called these structures partition-permutation-structures)
without changing the number of boxes is described in [16],(3)
from any rowi containing at least one box, one box shifts to
the next row i+1.Ifx weakly perm-submajorizes y, xperm≻wy, then(4)
only boxes of the last possible row can be
eliminated. Note that this
expression is not identical with the algorithm for weak submajorization. If x weakly perm-supermajorizes y, xperm≻wy, then(5)
only the first row can gain boxes, also if
this row contains no box. A part of the n=4 pp-structure combined with weak
perm-submajorization and weak perm-supermajorization is given in
Figure 6. The
diagram in Figure 6 top right is an example for the rules (4) and (5) above.Figure 6
An example for the algorithms 4 and 5.


## 3.4. Attainability

In [9–12], Zylka described a simple model system in
which he considered n bodies (objects), the only property of which
is their temperature. Then the system is completely described by a temperature
distribution. The process dynamics of these temperature distributions is
characterized by a two-body heat
exchange. That means only two bodies are connected for a period of time and
an amount of heat can be transported from the hotter body to the cooler one.
Then two other bodies are connected and so on. By this process dynamics, Zylka
derived the set of distributions which are attainable from a fixed initial
distribution.We apply this restricted mixing process dynamics to
discrete dynamics. Let a Young diagram be characterized by its number of boxesn which are distributed over three rows, then
the corresponding diagram vector is x=(xi)∈ℕ3 with x1≥x2≥x3 and its trace is defined by tr(x)=∑i=13xi=n.
Then the set of attainable diagrams K(x) can be generated by picking two entries xi, xj from x with i<j.
The entry with smaller index i loses boxes, and the other entry j gains boxes as long as xi≥xj.
Such a transition is denoted by an arrow with (i,j), (i,k),
or (j,k) for i=1, j=2,
and k=3.
An example for the set of attainable diagrams of the initial diagram (610) is given in Figure
7.Figure 7
The initial Young diagram (610) with its set of attainable diagrams. The
arrows characterize the individual mixing transitions with the corresponding
indices of the diagram vector, i=1, j=2,
and k=3.The setK(x) can be ordered by perm-majorization and is
represented by a diagram lattice in Figure
8. Let x and x′ be two elements of K(x), then the join in the lattice is defined by x∪x′=max(x˜j,x˜j′) and the meet is defined by x∩x′=min(x˜j,x˜j′), where x˜ and x˜′ are the partial sum vectors of x and x′.Figure 8
The setK(x) of Figure 7 is ordered by perm-majorization
and represented by a diagram lattice.


## 4. Majorization as process dynamics

## 4.1. General

A simple experiment of foam decay exemplifies process dynamics in a statistical sense. The application of majorization to
bubble-size distributions is possible if one understands the order of
majorization as a process dynamics. Hence, let us summarize majorization in
words of process dynamics. Now, diagram boxes are objects and rows are states.Classical majorization
Let us consider an
ordinary statistical diffusion process with invariant object number. The
attractor of this process is the equal distribution. The problem of
incomparableness of classical majorization has not yet been discussed for
physicochemical systems. In general, a transition of incomparableness consists
at least of two elementary transitions and one of these is inverse. For
instance, the transition from diagrama to diagram b (Figure 4) consists of the transitions from
diagram a to c and from c to b, a≻c and c≺b.
The second transition is an inverse one. In this case, the inverse transition
describes an increasing statistical order. Process dynamics of increasing
statistical order leads one to suppose that the process is based on structure
formation.Weak majorization
This concept is
subdivided into weak submajorization and weak supermajorization, and contains
classical majorization. This relation describes a process of statistical
diffusion with noninvariant number of objects. The order of weak
submajorization characterizes a diffusion with sink. The sink obeys rule (1),
see Section3.3. The attractor of weak submajorization is the death of the system. Incomparableness is
given if the statistical order increases and a sink occurs; see the transition
from diagram c to diagram d in Figure 4. But also a process of
predominating sinks and small sources causes incomparableness. It can be seen
that there are more boxes (objects) in the first row of diagram d than in the first row of diagram c.
Weak supermajorization leads to a diverging number of objects which are
distributed over all states. A diffusion process with source (rule (2)) is described. An increasing statistical
order and a source leads to incomparableness. Note that in the presence of
sinks and sources, two order structures (weak sub- and supermajorization,
resp.) can be assigned to the system. These two order structures
(Figure 4)
show that incomparable diagrams in terms of weak submajorization are comparable
in terms of weak supermajorization and vice versa. In addition, alternating
transitions of weak sub- and supermajorization can lead to incomparableness in
the classical sense. A simple example is given by(4.1)(411)≻w(410)≻w(510)≻(420)≻(330),(411)⊁(330),(411)⊀(330).The lattices in Figure 4 show
this fact clearly.Permutation-majorization
This concept contains classical and weak
majorizations and offers the possibility of multimodal distributions. With a
constant object number, the process is defined by rule (3),
that is, every object stepwise passes through all states whereby the
statistical order can decrease or increase and multimodal distributions are
possible. By taking permutations into account, the large increase in the overall number of
possible distributions leads to the result that many more states are
incomparable; see Figure6. The process ends if all objects are in the last
possible state (lowest row). If the object number decreases, weak
perm-submajorization obeys rule (4),
that is, an object has to pass through all states (classical perm-majorization)
before it leaves the system. The weak perm-supermajorization obeys the process
of classical perm-majorization with a source in the first state.


## 4.2. Foam dynamics

Foam decay can be subdivided into two processes,
drainage and rearrangement [16, 17, 19, 24]. Sometimes a third process is mentioned which describes the drying of foam
[25]. During ultrasound degassing,
only very small bubbles are formed. The corresponding distribution shows a
sharp peak. During drainage, the liquid flows out of the foam and the
distribution broadens. The rearrangement phase is characterized by processes
like coalescence and bursting of bubbles or growing of larger bubbles on cost of
smaller ones. During this phase, certain bubble packings can be formed
[19, 26].The bubble-size distributions of our measurements (see
Section2) are ordered by the different concepts of majorization which have
been introduced in Section 3. Tables 2 and 3 show these relations, which
are denoted by h(k)Rh(k+1) for distributions which are normalized to
unity and f(k)Rf(k+1) if a sink is taken into account. R is the relation and k=t[s]/5s describes the time steps. Additionally, we
distinguish between five-second intervals, h(k)Rh(k+1) or f(k)Rf(k+1),
and ten-second intervals, h(2k)Rh(2k+2) or f(2k)Rf(2k+2),
in order to compare the results of [17]. Transitions of incomparableness are denoted by × for all relations.Table 2
The order of the different concepts of majorization
concerning bubble-size distributions is given. One sees the time steps in the
first column. The second and third columns show the relationR of classical majorization with different time steps. The transitions
of the weak submajorization can be examined in the fourth and fifth
columns differing in the time steps. Incomparable distributions are denoted by ×.
k=t[s]/5sh(k)Rh(k+1)h(2k)Rh(2k+2)f(k)Rf(k+1)f(2k)Rf(2k+2)0≻≻≻w≻w1≻≻w2≻≻≻w≻w3≻≻w4≻≻≻w≻w5×≻w6××≻w≻w7×≻w8×≻×≻w9≻≻w10≻≻≻w≻w11×≻w12××≻w≻w13≻≻w14×××≻w15×≻w16××××17××18××≻w≻w19××20××≻w≻w21×≻w22≻×≻w≻w23××24×≻≻w≻w25≻≻w26××≻w≻w27×≻w28≻×≻w≻w29××30≻≻≻w≻w31×≻w32×≻≻w≻w33×≻w34×××≻w35×≻w36×≻≻w≻w37×≻w38××≻w≻w39≻≻w40×≻≻w≻w41≻≻w42×××≻w43×≻w44××≻w≻w45×≻w46××≻w≻w47×≻w48≻≻≻w≻w49×≻w50××Table 3
The evaluation of the different concepts of perm-majorization.
In the first column the time steps are given. One sees the relationR of the perm-majorization in the second column and the weak
perm-submajorization in the third column. Incomparableness are
denoted by ×.
k=t[s]/5sh(k)Rh(k+1)h(2k)Rh(2k+2)f(k)Rf(k+1)f(2k)Rf(2k+2)0perm≻perm≻perm≻wperm≻w1perm≻perm≻w2perm≻perm≻perm≻wperm≻w3×perm≻w4×perm≻perm≻wperm≻w5××6××perm≻wperm≻w7×perm≻w8×perm≻×perm≻w9×perm≻w10×perm≻perm≻wperm≻w11×perm≻w12××perm≻wperm≻w13×perm≻w14×××perm≻w15×perm≻w16××××17××18××perm≻wperm≻w19××20××perm≻wperm≻w21×perm≻w22××perm≻wperm≻w23××24×perm≻perm≻wperm≻w25×perm≻w26××perm≻wperm≻w27×perm≻w28××perm≻wperm≻w29××30××perm≻wperm≻w31×perm≻w32×perm≻perm≻wperm≻w33×perm≻w34×××perm≻w35×perm≻w36××perm≻wperm≻w37×perm≻w38××perm≻wperm≻w39×perm≻w40×perm≻perm≻wperm≻w41×perm≻w42×××perm≻w43×perm≻w44×××perm≻w45×perm≻w46××perm≻wperm≻w47×perm≻w48××perm≻wperm≻w49×perm≻w50perm≻×In the second and third columns of Table2, the
relation of the classical majorization is given. The fourth and fifth columns
contain the relation of the weak submajorization. In Table 3, the comparison
of the bubble-size distributions by perm-majorization (second and third
columns) and weak perm-submajorization (third and fourth columns) is given. We
also applied the weak supermajorization and weak perm-supermajorization,
running the process backwards.Classical majorization and its perm-variant
By classical majorization we obtain a partially
ordered set of bubble-size distributions in the second and third columns of
Table2. During drainage, transitions of classical majorization predominate,
that is, the drainage obeys an ordinary diffusion process. During rearrangement
phase, there are a lot of incomparable distributions and inverse transitions of
classical majo-rization, respectively. The incomparableness increases by
reducing the time steps. Such transitions of incomparableness characterize an
increasing statistical order which can be caused by structure formation. The
foam is attracted by a structure (a certain packing of different sized bubbles)
which cannot be attained because a loss of bubbles predominates. This supports
the results of the former paper [17]. Incomparableness can also be the result of sources and
sinks. The two incomparable distributions in (3.12) give an example. That means
the loss of bubbles predominates, see Figure
2, but there can still be sources
of small bubbles even in the rearrangement phase either by formation of small
bubbles out of the liquid or by the collapse of larger bubbles.
It is very interesting that the order structure of our
measurement is comparable to the measurements of [17], but the behavior of the Shannon entropy
[27],(4.2)I(h)=1log2n∑i=1npi,shows no process separation; see
Figure 9. Our current measurement shows an approximately increasing Shannon
entropy. The measurement of [17] was described by an increasing Shannon entropy during
drainage and a decreasing and increasing (zig-zag behavior) entropy
during rearrangement. This supports the assumption of an inhibited structure
formation. It is possible that the reduction of the image size leads to an
increasing statistical influence of small bubbles. The large number of small bubbles
predominates the statistics. The maximum value of the Shannon entropy is
approximately 0.28.
In [17], the maximum value is 0.64.
The high frequency of the first bubble size class of our current measurement
inhibits an increase of the entropy measure.
Application of classical perm-majorization (Table3,
second and third columns) leads to an increase of incomparableness in
comparison to classical majorization. Additionally, the incomparableness
increases from 10-second time steps to 5-second time steps. An increase of
incomparable distributions by changing from classical majorization to classical
perm-majorization is also given by [17]. In contrast to the current measurement, the total order
during drainage is preserved in both classical majorization and classical
perm-majorization referring to [17].
The increase of incomparableness in terms of classical
perm-majorization depends on the fact that the omission of the rearrangement of
the vector entries leads to an influence of certain bubble-size classes, that
is, not only the relative frequencies independent of bubble-size class
influence the partial sum comparison, but also both of the relative frequency and
the corresponding bubble-size class. By classical perm-majorization, weighting
for bubble-size classes is introduced.Figure 9
The Shannon entropy development of the
bubble-size distribution setH.Weak submajorization and its perm-variant
Both weak
submajorization and weak perm-submajorization show only few incomparable
distributions. The incomparableness increases by reducing the time steps; see
the fourth column in Tables2 and 3. The order structure (practically a
total order) of the fifth column in Table 2 and Table 3 is also given by the
measurement of [17]. In terms of weak
submajorization, both measurements obey a diffusion process with a sink which
occurs if a bubble belonging to the next vicinal-size class has formed. This
dynamics would allow for bubbles of all sizes but the smallest one to be
eliminated. In contrast, weak perm-submajorization only allows a bubble of the
largest possible size to disappear, which may, therefore, be the realistic
concept for sinks in systems where a critical size is required for elimination.
Additionally, foam decay obeys the complicated diffusion process defined by
rule (3).
Since the number of bubbles is monotonously
decreasing, see Section2, the fewtransitions of incomparableness (fourth
column in Tables 2 and  3) could be related to a source of a certain
bubble-size class or an increasing statistical order. Probably, within ten
seconds the sink predominates in such a manner that diffusion processes or a
small source cannot be taken into account by weak submajorization and weak
perm-submajorization. In summary, the order structures in terms of weak
submajorization and weak perm-submajorization seem to be time-step-variant, but
image size-invariant.Weak supermajorization and its perm-variant
Applying weak
supermajorization and weak perm-supermajorization, the bubble-size
distributions of all measurements are incomparable with very few exceptions.
These results are easy to understand. If the weak
submajorization and the weak perm-submajorization show an approximately total
order, the inversion of these orders can lead to incomparableness in terms of
weak supermajorization and weak perm-supermajorization. In other words, the
inversion of a statistical diffusion (decreasing statistical order) with sink
is an increasing statistical order with source, respectively incomparableness.
This inversion seems also to be valid for the weak perm-submajorization and
weak perm-supermajorization. Note that the source occurs in the first size
class. Therefore, only small bubbles are formed by running the process
backwards. These order structures seem to be independent of time and image-size
scale.Attainability
By the concept
of attainability, a two-body dynamics is described which is based on equalizing
processes. Zylka described a two-body heat exchange [9–12], but instead of
bodies and their temperature we consider bubble-size classes and their relative
frequencies.
In [16, 24], we showed the approximate oscillating behavior of individual bubble
sizes. In the beginning, the first size class decreases and the vicinal-size
class increases. Both size classes obey an equalizing process until the next
size class appears which grows on cost of one or of both of the other classes.
This behavior exactly corresponds to the process dynamics of Zylka. Indeed, the
set of the bubble-size distributions is in the set of attainable distributions
starting from the first bubble-size distribution.
The fact that the set of our bubble-size distributionsH is attainable starting from the first
distribution means that the statistical dynamics of the bubble sizes obeys an
exchange of two size classes concerning the relative frequency. Now it would be
interesting to investigate an exchange between three or more size classes.
For the set of bubble-size distributionsF there does not exist a concept of
attainability, a kind of weak
attainability taking sources or sinks into account. To introduce a source
or sink can depend on the original set of attainability K(x) or on the elements of that set. Let us
consider the first possibility for the introduction of a sink by the concept of
weak perm-submajorization. One generates the set of K(x) and picks out of this set the diagram of the
highest order x′ that can obey algorithm (4); see Section 3.3. After eliminating a box of the lowest row, x′perm≻wy,
the set of attainability K1(y) can be generated. Both sets, K(x) and K(y),
would represent the set of weak attainability.
The other possibility is that one picks out of setK(x) all diagrams which can obey algorithm (4). The
resulting diagrams would represent the set K2(y).
The number of diagrams of the set K1(x) does not equal K2(x).
In an analogous manner, a source can be introduced. Still both concepts of weak
attainability have to be investigated in detail in terms of consistency and
application to process dynamics in future work.


## 5. Discussion

The change of the time steps from 10 to 5 seconds
particularly influences the order structures of the normalized bubble-size
distributions by an increasing number of incomparable bubble-size
distributions. In contrast, for the order structures of the variants of weak
majorization including permutations, the reduction of the time steps leads to a
marginal change since only a few incomparable distributions occur additionally.
Also the reduction of the image size is only of significance for the normalized
bubble-size distributions. The corresponding order structures of classical
majorization and classical perm-majorization are preserved but the time
development of the Shannon entropy (4.2), see Figure
9, changes conspicuously.
Instead of a process separation [17], an approximately monotonous function behavior is given which
is caused by the strong influence of the smallest bubble-size class. Note that
although this function behavior is given, the distributions are predominantly
incomparable. The other order structures are not influenced by changing the
image size with the exception of the order structure of weak supermajorization
which shows a slight increase of incomparableness by the size reduction.For the set of normalized bubble-size distributions,
further investigations concerning the dependency of order structures on the
time scale (time steps) are important. The question is whether a further
reduction of the time steps leads to an increase of incomparableness or shows a
loss of incomparable distributions but a gain of transitions of decreasingh(t)≻h(t+1) or increasing h(t)≺h(t+1) statistical order.Provided that the reduction of the time interval leads
to incomparableness of normalized bubble-size distributions and does not change
the comparable distributions in terms of weak submajorization, then one has to
define transitions for which it holds thatf(t)≻wf(t+1) and the same distributions are incomparable in
terms of classical majorization:(5.1)(411000)≻w(320000)(5.2)but1/6(411000)⊁1/5(320000),(5.3)1/6(411000)⊀1/5(320000).But a further transition of
decreasing order, (320000)≻(311000),
leads to two comparable pairs of distributions in terms of both classical and
weak submajorizations:(5.4)(411000)≻w(311000),(5.5)1/6(411000)≻1/5(311000).It is also possible that there
is a transition of weak submajorization and the transition of the same
normalized distributions is of increasing order:(5.6)(411000)≻w(410000),(5.7)1/6(411000)≺1/5(410000).Let transitions consist of a
part of sink and a part of entropy
distance, which characterizes the relative difference of statistical
entropy between the compared distributions. Since the sink of the examples
above is constant, the statistical order decreases from (410000) to (320000) to (311000):(5.8)(410000)≻(320000)≻(311000),and hence the entropy distance
to the initial distribution (411000) increases; one may say that for the transition
(5.1) the sink and the entropy distance are relatively
balanced, but for (5.4) the entropy distance is greater and for
(5.6) is less.If the incomparableness increases by diminishing the
time intervals, the decaying foam would be a regular diffusion process with a
sink. In case that the temporal reduction leads to transitions of decreasingh(t)≻h(t+1) or increasing h(t)≺h(t+1) statistical order in terms of classical
majorization and the total order of weak submajorization is preserved, the
examples of (5.4)–(5.7) describe the experimentally observed development of the
bubble-size distributions. Then the foam decay process would be characterized
by phases of predominating diffusion or sink. For our foam statistics, it is of
high importance to understand how the sink and the diffusion are connected.The evaluation of the order structures of decaying
foam by normalized bubble-size distributions shows interesting results like
process separation [17] and sets of attainability, but this concept is problematic
and, respectively, it is dependent on certain conditions like time and
image-size scales. Additionally, one has to take into account that this system
is open and that normalization emphasizes certain bubble-size classes according
to the image size. Therefore, normalization may misrepresent the statistics,
since the number of bubbles decreases.


## 6. Outlook

Characterization of processes by order
structures can in general lead to an understanding of the transitions allowed
in a system and the states which are attainable from initial conditions without
specifying any kinetics, assigning transition probabilities, or explicitly
running sim-ulations. While classical majorization is appropriate for closed
systems, the occurrence of sinks or sources can be described by weak sub- or
supermajorization, respectively. The inclusion of permutations makes sure that
sinks can only occur in the last row (weak perm-submajorization). Sources can
only enter in the first row whether permutations are included or not. This is
not only realistic for the example described (foam dynamics), but for any
system in which small nuclei form (1st row), then grow (while moving downward),
and finally leave the system when they become too large. This is the case, for
example, for precipitation reactions or water droplets forming in the
atmosphere, or more generally for any process which can meaningfully be divided
into discrete steps and has clearly defined sources and sinks (say raw material
and finished product).The advantage of order structures is therefore that
they can be used to classify a large class of systems. However, it is only
possible to determine which states are attainable, not which will actually be
reached. Since there is no explicit time dependence, it can in the presence of
sinks and sources not even be determined whether an attractor exists (let alone
what it would look like). To this end, a specific kinetics has to be assumed
(resp., transition probabilities assigned for stochastic models). Obviously a
large variety of different kinetics can be defined on a given order structure
depending on the system(s) under consideration. In most cases, one would expect
to reach a stationary state, but also oscillatory behavior is possible (e.g.,
if larger particles grow faster and the reaction velocities depend on (at
least) one additional variable, such as monomer concentration or degree of
supersaturation) [28, 29]. Despite these limitations, we
believe that order structures may be a valuable (additional) tool for the
abstract characterization of processes also in open systems. It could be
possible in the future to determine basic properties of a given system by
assigning it to certain types of order structures.



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*Source: 474782-2008-07-22.xml*