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Partitioning

Although the opposite of addition is usually considered to be subtraction, we can also consider partitioning to be the opposite of addition. "Partitioning involves splitting up a set to form separate or disjoint sets." Liebeck (1984, p.47) Is partitioning an example of divergent mathematics?

Example 1.
Within the natural numbers, the partitions of 5 are:
5=1+4, 5=4+1
5=1+1+3, 5=1+3+1, 5=3+1+1
5=1+1+1+2, 5=1+1+2+1, 5=1+2+1+1, 5=2+1+1+1
5=1+1+1+1+1
5=2+3, 5=3+2
5=2+2+1, 5=2+1+2, 5=1+2+2

Partitioning in statistical mechanics
Partitioning is used in physics in the branch of statistical mechanics. The practical applications are in the Maxwell-Boltzmann distribution in classical thermodynamics, and in the Fermi-Dirac and Bose-Einstein distributions in quantum mechanics. Partitioning is used to estimate the possible energy combinations of particles under investigation. The main difference between classical and quantum partitioning is that in the classical version particles can be labelled whereas in the quantum versions particles cannot.

Example 2. - an example of labelling:

Consider an experiment looking at the possible partitioning of 3 helium(4) atoms between three energy levels. If the experiment is in the classical domain, the individual atoms can be labelled (say, for example, as A, B, and C)

Therefore if there are three energy levels (1, 2, and 3). The atoms can be arranged in a number of ways

List notation
This makes use of HTML to display the partitions like an energy level diagram. The Ordered List tags in HTML mimic the layout of the diagram. However, they use a fractional amount of the memory as an inline image. View source (ALT V U on netscape) to see the html source code.

Example 2.1 - A possible partition of 3 atoms over 3 energy levels

  1. A
  2. BC

Example 2.2 - ...another possibility

  1. C
  2. AB
L-system notation
We can simplify the above descriptions by adopting a convention of notation.

Alphabet = A, B, C, /

The letters A, B, and C represent the 3 atoms.
The slash / represents a gap between energy levels

If we adopt the convention that the first energy levels is at the left, the above two examples can be described as follows:

Example 2.3 - L-system notation for example 2.1

A//BC

Example 2.4 - L-system notation for example 2.3

/C/AB

Exercise 1

Write the HTML to interpret the following examples of L-system notation as Ordered Lists.

  1. C//AB
  2. /BC/A
Returning to our Maxwell-Boltzmann distribution

The possible partitions in this case are:

ABC//,
AB/C/, AC/B/, BC/A/,
AB//C, AC//B, BC//A,
A/BC/, B/AC/, C/AB/,
A/B/C, A/C/B, B/A/C, B/C/A, C/A/B, C/B/A,
A//BC, B//AC, C//AB,
/AB/C, /AC/B, /BC/A,
/A/BC, /B/AC, /C/AB,
//ABC

(Oops: additional exercise - find my mistake, the missing partition!!)

Exercise 2

Find the grammatic rules for an L-system that generates all the possible partitions of a 3 atom, 3 energy level, Maxwell-Boltzmann distribution. Start with the axiom ABC// and generate all true statements.

(I'm not sure if this is possible. I've created a qbasic program that generates these partitions. It shows the possible transitions as a graph - partrans.bas)

Exercise 3 For those who have completed Exercise 2

Consider the partition for 5 atoms between 7 energy levels.
Starting from your answer to Exercise 2, change the starting axiom to ABCDE//////

Does your L-system accurately predict the correct partitions?


The quantum case

In the quantum case the atoms cannot be labelled and the possible partitions are listed below. As Helium(4) atoms are Bosons, more than one particle can exist in each energy level.

111//,
11/1/,
11//1,
1/11/,
1/1/1,
1//11,
/11/1,
/1/11,
//111,

If the particles in this example were fermions, such as electrons, only one particle could exist in each energy level. Therefore the only possible distribution is:

1/1/1

These differences in partitioning describe some of the ways in which the quantum world is different from the classical world.


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Created 16/1/00
Last modified 17/1/00