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 kinetic energy, we can expect to see
some exchange of internal energy.
Pause to consider some (external) quantities whose units are action:
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
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
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
momentum probably involving spin; what's its analogue of temperature ?). Each
superposition is characterized by a
temperature: it partakes of the nature of
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).
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 it gets into any more 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 R to obtained from S by integrating this
distribution as a function of s, with
in so far as specifying our density.
sound-bite, 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 will this be T ?