Nisan comments on Nisan’s Shortform

[Nisan]

We can derive Newton’s law of cooling from first principles.

Consider an ergodic discrete-time dynamical system and group the microstates into macrostates according to some observable variable $X$. ($X$ might be the temperature of a subsystem.)

Let’s assume that if $X=x$, then in the next timestep $X$ can be one of the values $x-dx$, $x$, or $x+dx$.

Let’s make the further assumption that the transition probabilities for these three possibilities have the same ratio as the number of microstates.

Then it turns out that the rate of change over time $\frac{dX}{dt}$ is proportional to $\frac{dH}{dX}$, where $H$ is the entropy, which is the logarithm of the number of microstates.

Now suppose our system consists of two interacting subsystems with energies $E_1$ and $E_2$. Total energy is conserved. How fast will energy flow from one system to the other? By the above lemma, $\frac{d{E}_1}{dt}$ is proportional to $\frac{dH}{d{E}_1}=\frac{d{H}_1}{d{E}_1}-\frac{d{H}_2}{d{E}_2}=C_1-C_2=\frac{1}{T_1}-\frac{1}{T_2}$.

Here $C_1$ and $C_2$ are the coldnesses of the subsystems. Coldness is the inverse of temperature, and is more fundamental than temperature.

Note that Newton’s law of cooling says that the rate of heat transfer is proportional to $T_2-T_1$. For a narrow temperature range this will approximate our result.

[Nisan]

I’d love if anyone can point me to anywhere this cooling law (proportional to the difference of coldnesses) has been written up.

Also my assumptions about the dynamical system are kinda ad hoc. I’d like to know assumptions I ought to be using.