# Discrete systems

### Simplest system

The space of states representing a coin - heads or tails - is the simplest interesting system. To describe the rules for updating (laws of motion), we assume that the motion is stroboscopic, and that for each update, we must define what happens to each element in the space of states.

Figure 1.1 - The simplest system

In figure 1.1, we describe possible rules of motion for the system (a), which we can write as follows:

(b) $\left\{ \begin{matrix} H \to H \\ T \to T \end{matrix} \right.$

(c) $\left\{ \begin{matrix} H \to T \\ T \to H \end{matrix} \right.$

(d) $\left\{ \begin{matrix} H \to H \\ T \to H \end{matrix} \right.$

(e) $\left\{ \begin{matrix} H \to T \\ T \to T \end{matrix} \right.$

All of these rules are deterministic, but they are not all valid rules for updating. That is, they are not all laws of physics.

The suspect rules are those that cannot be written as permutations of the state space, namely (d) and (e). They fail because they involve a loss in information along the flow. Although they are deterministic, the information lost is about previous states. In (e), if you are at $T$, then you don't know if you were previously at $H$ or $T$.

So, the system needs to be both deterministic and reversible. In terms of the diagrams, this means that for each node in the system, there has to be exactly one arrow arriving at the node and one arrow departing.

In fact, (b) and (c) are the only two possible laws of nature for heads and tails.

### A die - six states

Figure 1.2 - States of a die

Going to a slightly more complicated system - a die with six states - we find that there are many valid rules for updating. There are in fact $6! = 720$ rules, which is the number of permutations of a die.

Figure 1.2 shows three possible such rules, each of which can be written as products of cycles, which form permutations.

The simplest one is the first, which is just a single 6-cycle: $(123456)$.

The second is pair of 3-cycles: $(135)(246)$.

The third is a mixture of cycles: $(12)(345)(6)$.

The space of states, along with the rules for updating is called the phase space. The last two examples show that the phase space can break up into self-contained regions, which don't get mixed up with other parts of the space.

This is the basis for conservation laws, and in this particular case, conservation of information.