The modified repeat rate described within a thermodynamic framework

Abstract

Derek J. de Solla Price viewed science as a complex system and anticipated that the science of science can be developed via an analogy to thermodynamics. The main point of this article is to show a direct equivalence between a thermodynamic framework and the classical theory of evenness. It illustrates how thermodynamically inspired terms can lead to the measures used to quantify diversity (or lack thereof), balance, evenness, consistency, or concentration. A real-world example based on intersectional inequalities in science is used as an illustration.

Appendix

Construction of different distributions with the same modified repeat rate

There exists an interesting way of constructing multisets with the same modified repeat rate. We illustrate this in the case of four numbers (N = 4). The method explained here is due to Sridhar Ramesh (no date available).

Consider a three-dimensional cube with vertices a, b, c, and d on the upper facet and e, f, g, and h on the lower facet. Vertices are named counterclockwise and vertex e is situated below vertex a (see Fig. 3).

Fig. 3

Cube used to perform Ramesh’s algorithm

In this theory vertices a, c, f, and h are called the white vertices, while vertices b, d, e, and g are called the black vertices. A cube has six facets, here bounded by a,b,c,d; a,b,f,e; b,c,g,f; c,d,h,b; a,d,h,e and finally by e,f,g, and h. Now assign to each facet a non-negative number and finally, assign to each vertex the sum of the numbers of the three facets adjacent to it. This number is called a V-number. The multiset of the four V-numbers of the white vertices has the same sum and the same sum of squares as the multiset of the four V-numbers of the black vertices, and hence they have the same modified repeat rate. A simple example is given in Table 2.

Table 2 An example of Ramesh’s algorithm

The multisets {12,11,11,8} and {13,10,10,9} have the same sum, namely 42, and the same sum of squares, namely 450. Proof of this property can be given based on symmetry properties of the assigned numbers, but we provide a very elementary proof by just doing the calculations for an abstract assignment of numbers to facets, as shown in the third column of Table 2a and of 2b.

We see that the sum of the white V-numbers is: (u + v + y) + (u + w + x) + (v + w + z) + (x + y + z) = 2(u + v + w + x + y + z ), which is equal to the sum of the black numbers: (u + v + w) + (u + x + y) + (v + y + z) + (w + x + z). Similarly, one can easily check that (u + v + y)² + (u + w + x)² + (v + w + z)² + (x + y + z)² = (u + v + w)² + (u + x + y)² + (v + y + z)² + (w + x + z)².

The previous construction deals with the case N = 4. Performing this construction two, three, or more times yields the cases N = 8, 12, etc. (take the union of the obtained multisets). Taking two sets with 4 numbers and appending the same number to both yields an example with N = 5; appending two or three numbers to both multisets (the same numbers) yields examples for the cases N = 6 and N = 7. The construction performed on a cube and also be done on a square yielding the case N = 2, such as {3,7} and {5,5}.

As, for N = 4, the sum of the white V-numbers is equal to the sum of the black V-numbers, and is twice the sum of the freely chosen numbers u,v,w,x,y,z one can form as many couples of multisets with the same modified repeat rate as there are numbers with the same sum. This reasoning also applies to other values of N.