2026 May 1
- I've known box quantization since quasi forever for the explanation of the
density of states per energy in gases. The proof from a (3-d infinite square well) box
is an ideal limit and sources (like Google AI) say that you just take
the box to infinity to make the density states per energy generally valid.
However, I've always been unconvinced by that argument since
I keep wondering if there is a qualitative difference between
a box of any size containing homogenous medium
and the case where there is no box and the medium varies with position.
- After thinking about it a bit recently, I wonder if the following is
a valid perspective.
Consider a small enough region in a gas medium that it can be considered
embedded in a homogenous universe (e.g., the region being small compared variations
in density and pressure needed for hydrostatic equilibrium in atmospheres or stars),
but large compared to the mean free path of the particles considered classically.
Particles deep in the region sense an impenetrable wall in every direction
since in every direction a line intersects an other particle of finite size (thinking of the
other particles as classical).
Thus, there is an impenetrable wall in every direction, but distances to it vary
with direction and time.
The distances in general will be much larger than a mean free path.
The walls can't keep the deep particles trapped in the region
since clearly there would be a random walk for
the deep particles that leads to escape, but the random walk takes far longer
than a direct flight for the deep particles to escape to infinity.
But an any instant in time the deep particles are trapped
and have discrete eigenstates set by the impenetrable wall which
defines a complexly shaped walled region.
The density of states per energy in the walled region
might have a complex small energy scale behavior
as energy increases, but I'm guessing the large energy scale behavior
would be the same as a box of the same volume as the walled region since you can
distort walled region into the box without changing its volume.
There is distortion of the eigenstates, but you are not creating or destroying eigenstates.
At any instant in time the deep particles are in equivalent of a box.
As time passes, there is actually a flow in and out of the fluctuating walled region,
but thermodynamic equilibrium for the deep particles is set a on shorter time scale.
The upshot is that box quantization leads to the correct thermodynamic behavior
even though there is no actual box.
- Does this perspective make any sense or am I just being silly?
I'm being silly I think. Fact is how classical emerges from quantum
is still a mystery to me.