Undergrad Modeling Thermal Equilibrium in Interacting Einstein Solids: A Python Approach

[Isaac0427]

TL;DR

How can we model the interaction between two Einstein solids within short periods of time?

When I learned about Einstein solids in thermal physics, we assumed the fundamental assumption of statistical mechanics. For two interacting Einstein solids, I completely understand why this is valid after a considerable amount of time has passed. But, how can we model these solids as they get to thermal equilibrium?

I’m thinking of this in terms of python code (I thought doing a model like this would be a fun idea for my semester project in coding), so I’m thinking about discrete time steps as opposed to continuous time.

Here are my thoughts so far:
For each time step, every packet of energy has a probability of switching solids. Within a solid, though, the fundamental assumption is still assumed. The probability of switching from solid a with $N_a$ oscillators to solid b with $N_b$ oscillators would need to be less than or equal to $\frac{N_b}{N_a+N_b}$. Similarly, the probability of switching from solid b to solid a would need to be less than or equal to $\frac{N_a}{N_a+N_b}$.

Thus far, how does it sound? To me, assuming the fundamental assumption within the solid seems reasonable even considering time, but is it? I know there is a time it takes heat to travel across an object, but can I assume that this is negligible compared to the time it takes heat to travel within objects?

If my model is ok thus far, my next thought is about what these “switching” probabilities depend on. The easiest thing is that there is some constant $0\leq c\leq 1$, which is the same for both solids, where the probability of switching from a to b is $c*\frac{N_b}{N_a+N_b}$, and the probability of switching from b to a is $c*\frac{N_a}{N_a+N_b}$.

Do you think it is reasonable to assume that c is the same for both solids? I also have argued with myself about temperature dependence (i.e. a given energy packet is more likely to escape from a solid with a lot of energy packets than from a solid with few), but I figured that an individual energy packet does not know or care about how many energy packets are nearby (and not repelling, as with, say, charge). By nature of the way I set this up, there would be a higher chance of SOME energy packet escaping a higher temperature solid, as there are more energy packets compared to oscillators, but the probability for any individual packet is independent of temperature.

I would really appreciate any feedback you can give!

 

[DrClaude]

Do you think it is reasonable to assume that c is the same for both solids?

It has to be, as it is a property of the two objects together.

You'll have to check that your model reproduces Newton's law: https://en.wikipedia.org/wiki/Newton's_law_of_cooling

 

[Isaac0427]

It has to be, as it is a property of the two objects together.

You'll have to check that your model reproduces Newton's law: https://en.wikipedia.org/wiki/Newton's_law_of_cooling

It definitely checks out in the high temperature limit ($T=\frac{\epsilon q}{Nk}$) with two identical solids. I can't seem to figure out it it works with solids of different sizes.

EDIT: I figured out that in the high temperature limit, it agrees with Newton’s law of cooling for any sized solids. Thank you!