*Pascal’s triangle as he drew it in his 1654 book Traité du triangle arithmétique.*

I always thought Pascal’s triangle was invented with its origin at the top like this Δ and all the rows ranged below. But when Pascal drew it, he tipped the base of the triangle over so that the other two sides ranged horizontal (*Rangs paralleles*) and vertical (*Rangs perpendiculaires*), and numbered the rows and columns as shown. Each number in the array is thus specified by a row-and-column coordinate pair. This turns out to have thermodynamic significance, as we shall see.

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**Think small**

Thermodynamics is a big subject but it can equally well be applied to very small systems consisting of just a few atoms. Such systems play by different rules – namely quantum rules – but that’s ok, the rules are known. So let’s imagine that our thermodynamic system is an idealized solid consisting of three atoms, each distinguishable from the others by its unique position in space, and each able to perform simple harmonic oscillations independently of the others.

Harmonic motion is quantized, such that if the energy of the ground state is taken as zero and the energy of the first excited state as ε, then 2ε is the energy of the second excited state, 3ε is the energy of the third excited state, and so on. Suppose that from its thermal surroundings our 3-atom system absorbs one unit of energy ε, sufficient to set one of the atoms oscillating. Clearly, one unit of energy can be distributed among three atoms in three different ways – 100, 010, 001 – or in more compact notation [100|3].

Now let’s consider 2ε of absorbed energy. Our system can do this in two ways, either by promoting one oscillator to its second excited state, or two oscillators to their first excited state. Each of these energy distributions can be achieved in three ways, which we can write [200|3], [110|3]. For 3ε of absorbed energy, there are three distributions: [300|3], [210|6], [111|1].

The distribution of n units of ε (n = 0,1,2,3) among three oscillators (N=3) can be summarized as

n=0:[000|1] = 1 way

n=1:[100|3] = 3 ways

n=2:[200|3],[110|3] = 6 ways

n=3:[300|3],[210|6],[111|1] = 10 ways

Compare this with the distribution among four oscillators (N = 4)

n=0:[0000|1] = 1 way

n=1:[1000|4] = 4 ways

n=2:[2000|4],[1100|6] = 10 ways

n=3:[3000|4],[2100|12],[1110|4] = 20 ways

There is a formula for computing the total number of ways n units of energy can be distributed among N atoms, or to put it another way, the total number of microstates W available to a system of N oscillators with n units of energy

In every case the number is a binomial coefficient, and the numbers generated can be matched to Pascal’s upended triangle by assigning N (1,2,3 …) to the rows and n (0,1,2 …) to the columns as shown below

Here is the connexion between thermodynamics and Pascal’s triangle, which neatly tabulates the total number of microstates available to an idealized solid comprising N atoms with n units of energy, each atom able to perform simple harmonic oscillations independently of the others.

The reason why the first row consists solely of the number 1 is that one atom (N=1) can have only one microstate regardless of the number of energy units it absorbs. It is also to be noted that the rows read the same as the columns due to the property of the binomial coefficient

and that the series of numbers in rows 2, 3, 4, 5 etc are the natural numbers, triangular numbers, tetrahedral numbers, pentatope numbers etc.

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P Mander October 2020

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