# Closed dynamical systems

C2 and C3 may be used in contexts that have nothing to do with induction. As we have just seen, this is the case with statistical mechanics; but the same is true for biology and economics. In these applications, exchangeability and invariance are considered in order to build up theories able to explain and foresee natural phenomena. Sometimes, with the clear intention to underestimate these applications, one speaks of probability models. In what follows we want to show how far this use of probability can be pursued. But in order to do this we must specify the scenario we have in mind.

We shall now consider closed dynamical systems. In order to explain what a closed dynamical system is, first of all we point out that, on account of what we have said about macroscopic quantities, when we speak of the state of a system we shall mean its occupation vector or (which is the same) its statistical distributions. A system is closed when its size does not change over time. A dynamical system is a system whose statistical distribution changes over time. Therefore, a closed dynamical system is a system whose size is constant notwithstanding changes in its statistical distribution. This means that, contrary to what we have done so far, we will consider systems that change not because new individuals enter the system but rather because the system's individuals change their attributes. From now on we shift our interest from induction and open evidence to theories and closed dynamical systems.

In considering closed dynamical systems, we focus on economic applications. Hence, to harmonize our terminology with the scenario we have in mind, we shall call agents the individuals and strategies the cells. As an abridgment, we shall say that "an agent is in j", not "an agent follows the strategy j." If the closed dynamical system is built up of *N* agents and *d* strategies, we will analyze the simplest transition the system may undergo. We call such a transition a unary move. Such a move occurs when an agent gives up a strategy (the starting one) and begins to follow another strategy (the resulting one). We shall also take more complex moves into account; but most of what we say concerns unary moves. The moves we are considering were first studied by physicists. When a particle undergoes a collision, for instance with the wall of the container, it changes its momentum. As a consequence, a (momentum) cell, say A, loses a particle and a different one, say B, gains a particle. In this sense a particle is destroyed in *A* and one is created in B. Physicists have adopted this more or less intuitive way of speaking. The terminology we shall use shows traces of this origin in that we shall speak of the "destruction" and "creation" of an agent in a strategy. More exactly, in order to say that an agent gives up the strategy j, we shall say that an agent is destroyed in j; conversely, we shall speak of creation in *j* if an agent chooses to follow the strategy j. A unary move can be seen as the juxtaposition of two steps: a destruction immediately followed by a creation. We stress that the starting strategy and the resulting strategy may be the same; obviously if this is the case, then no change occurs in the system. This is the way in which, from a probabilistic perspective, we shall analyze unary moves.