Before studying this chapter, the reader is advised to (re-)read Section 6.1.4. In fact, the present chapter is devoted to a generalization of that example where random ‘destructions’ are followed by ‘creations’ whose probability is no longer uniform over all the categories, but follows a rule due to L. Brillouin and directly related to the Pólya distribution discussed in Chapter 5.

After reading this chapter, you should be able to:

1. • use random destructions (à la Ehrenfest) and Pólya distributed creations (à la Brillouin) to study the kinematics of a system of n elements moving within g categories;

2. • define unary, binary, …, m-ary moves;

3. • write the appropriate transition probabilities for these moves;

4. • use detailed balance to find the invariant distribution for the Ehrenfest–Brillouin Markov chain;

5. • discuss some applications to economics, finance and physics illustrating the generality of the Ehrenfest–Brillouin approach.

Merging Ehrenfest-like destructions and Brillouin-like creations

Statistical physics studies the macroscopic properties of physical systems at the human scale in terms of the properties of microscopic constituents. In 1996, Aoki explicitly used such a point of view in economics, in order to describe macro variables in terms of large collections of interacting microeconomic entities (agents, firms, and so on). These entities are supposed to change their state unceasingly, ruled by a Markov-chain probabilistic dynamics.