## Saturday 3 October 2015

### Perturbation Theory

I have often mentioned that what we have in a set of wargame rules is a set of models. These cover such things as movement, ranged combat, close combat and morale. Each of these models can be well founded, based on empirical data and so on. I am sure I have commented on what exactly that might mean in the past, and how, in fact, fudge factors have to be introduced because, for example, men under fire take longer to spread out into a line, or do not, themselves, fire as accurately (or even, at all) as those who are not, or who are in parade ground situations.

Perhaps the interesting thing is how the models interact, however. The situation being modelled is not that of a specific model but of a set thereof. A model of movement, for example, can be modified by a model of firing. If the soldiers moving come under fire, then we expect that, in some sense, that will affect how the movement is carried out, potentially at least.

In practice, of course, we have modifiers and reaction tests. If a unit is moving and come under fire, we might apply a reaction test to it. We sum up the various external and internal factors such as surprise, cohesion, and training, cover and so on, roll dice and reach some sort of conclusion. The firing model has interacted with the movement one. As another example, many rule sets state that some troops can move half a full move and fire. Again, one model has changed another. Similarly, we might assess the damage from combat as being more than one man per figure (I think that was a Tercio thing) and make a forced morale check on that basis. In that case the combat model and the morale model have interacted and modified each other. It can work the other way around, as well. A morale failure can mean that a unit recoils and hence, if reengaged by their opponents, then fights at a disadvantage.

I think that there is a little bit of a problem for the unwary rule writer here, however. Our models can conflict, and hence we can arrive at odd situations. I am sure we have all seen them on the table. A rather shaky D class regiment of new recruits sees off two regiments of superb cavalry in short order. We shake our heads and mutter ‘these things happen’ and get on with winning the battle by other means. What has happened is that a combination of a morale model and a combat model has given some low probability event; somehow the models have conspired against the normal course of events.

In a sense this is not a problem. I dare say we could find examples all over history where unreliable troops have performed well above expectations. But it is as well to be aware of what is happening here. The models are throwing up low probability events. It is a bit like the interminable arguments over breaking squares in Napoleonic wargames. It did happen, but not very often, and our models probably make it too probable.

One way of looking at how models interact is using perturbation theory. This is a quintessential physics viewpoint. It arises when we want to model something a bit more complex than the normal. We can solve, for example, the quantum mechanical equations for a particle in a box, or a hydrogen atom in a vacuum. That we, we can write down a mathematical equation for the electron in a potential well produced by a proton, and solve it. The solution gives us a set of waves, of states in which the electron can exist.

We would like to solve more complex problems, however; for example, a hydrogen atom in an electric field. Unfortunately, we cannot write down the Schrodinger equation for this and solve it. The mathematics does not work out easily. So what we can do is treat the electric field as a small correction to the already known solutions for the atom in a vacuum. This is fine, so long as the electric field is fairly weak.

I won’t bore you with all the mathematical bits of how this is solved, but the upshot of it is that the extra bit, the perturbing field, mixes the solutions we found before. That is, instead of being in a pure state, say state 1, as a given solution, the electron in the hydrogen atom is in a mixed state, a bit of state 1 plus a bit of state 2. Given that normalisation is upheld in this (that is, the electron has to be somewhere) we can see that the application of the electric field could force the electron to shift from state 1 to state 2; the probability of this is given by the ratio of the ‘bits’ of each state the perturbed electron has access to.

Thus the electron can jump from one state to another by the application of an electric field. Classically, we would say that it gained energy from the field (or lost it, it can go the other way), but quantum mechanically, I hope you can see that it is a bit more complex than that. Nevertheless, its application to wargame rules is, I hope, fairly clear. The models perturb each other, and we need to be specific as to how that happens, and clear as to which models can modify which other models.

The thing is, I think, that if we do not do this modification clearly, we can end up with situations where one model tells us something (for example the combat model says ‘run away’) while another model tells us something else (for example ‘stand firm’). We need to be clear as to which model takes precedence. In Polemos: SPQR it is the combat model. In older rule sets, as I recall, it was the morale model.

I am not saying that one is better or worse than another, but we do need to be clear as to which model is the perturber, and which the basis. And of course, what we have here is weak interaction of the models. This leads to a slow evolution of the states of our units. Another approach is via catastrophe, but that is another story.