Thermodynamic Scattering

We may describe a molecule in terms of a state (Hilbert) vector which consists of a thermodynamic superposition of internal states; we may consider two such molecules, each with its own proper spectrum and temperature, approaching one another on a collision course; we are going to be interested in statistical truths about the states of the two molecules after the collision.

Separate from the internal state of each molecule, it has an overall momentum with which is associated a kinetic energy. These are normally decoupled from the internal state but, in a collision, the kinetic energy clearly has some scope for getting mixed up with the internal energy. In so far as the collision involves an exchange of momentum or kinetic energy, we can expect there to be some scope for an exchange of internal energy.

Pause to consider some (external) quantities whose units are action:

• amount of angular momentum exchanged between the two bodies
• amount of kinetic energy exchanged, times duration of collision
• in the frame of reference of their centre of mass, their (equal but opposite) momentum times the distance between their two extrapolated trajectories at the moment of closest approach, whether projecting their prior trajectories forward, projecting their subsequent trajectories backwards or taking the difference between the two actions these imply
• amount of momentum exchanged, times … what distance ? One possibility is to derive one length from the lengths of the trajectories of the two particles between when they make contact and when they break off contact.

One may use the kinetic energy transferred as a yard-stick in what follows; or one may combine the above (e.g. add them up) to obtain an action, divide this by the duration of the collision and use the resulting energy as yard-stick. What matters (to the internal thermodynamics), is that the collision's kinetic aspects yield a characteristic energy: combining this with the temperature of each molecule should give us means of assessing how the molecules' individual thermodynamic distributions get disturbed.

Each molecule's state is understood as a superposition of what I'll describe as sound-bite states, i.e. ones we have some easy way to describe, among the attributes of which we find energy as a common theme (we probably also have total angular momentum probably involving spin; what's its analogue of temperature ?). Each superposition is characterized by a temperature: it partakes of the nature of each sound-bite to a degree which is controlled by the ratio between the latter's energy and the superposition's temperature.

Chemical equilibria are likewise linked to ratios between temperature and characteristic energies of the reactions which might turn some constituents of the mixture into others.

I am drawn to ask what perturbation each molecule's thermodynamic superposition induces on that of the other. Suppose (not necessarily reliably) the molecules' states to be unentangled after the collision (and if that's gibberish to you, be at peace and ignore it). When considering the collision's effect on either molecule, it seems sensible to suppose that the change in externally-observed energy and momentum of any given constituent is wont to be of similar order to (that constituent's share of) the total transfer of energy and momentum. This would suggest that the (kinetic) energy transferred (or some kindred energy, as discussed above) would make a good characteristic energy in terms of which to try to build an analogy between theories of chemical equilibrium and probabilities of transitions within the molecules. So I'll look at the total change in internal energy of each molecule (and when the collision is elastic, i.e. the external energies add up the same after as before, the internal energy change of one molecule is exactly opposite to that of the other – but collisions involving exchange of internal energy probably aren't elastic).

So suppose the other molecule's lost a total (internal) energy E and we're looking at the molecule that's gained this. It starts out in a thermodynamic-equilibrium superposition S with temperature T, gains energy E and ends up in a superposition R, putatively with temperature U, though R isn't initially guaranteed to be in thermodynamic equilibrium; but I do expect the molecule's internal dynamics to bring it into equilibrium before long, albeit not necessarily before it gets into further collisions.

In so far as S partakes in the nature of some sound-bite, s, we can look at what distribution on the available sound-bites to expect from putting energy E into s; overall, we should expect to obtain R from S by integrating this distribution as a function of s, with in so far as specifying our density.

Now, a sound-bite state, s, can contribute to S even though its energy may be somewhat off the (overall) energy of S (provided, of course, that S partakes in the natures of some others whose contributions are wrong in the other direction); but the further off it is, the less S partakes in its nature (that's a characteristic of the thermodynamic distribution; T provides a yard-stick for the energy-error). Likewise, the distribution for s plus energy E might hope to have a characteristic temperature; but how will it be related to T and E ?

Written by Eddy.