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[OngBonga]

So this "incredibly low probability"... am I to assume that while I sit by my fire, the vast majority of the heat is moving from a warm place to a cool place, but not quite all of it? I assume the heat moving from the cooler area to the warmer area is utterly overwhelmed by the heat moving in the opposite direction, rather like trying to piss up a waterfall.

What causes that "incredibly low probabiltiy"? Is it environmental, or completely random? Or do we not know?

[MadMojoMonkey]

So this "incredibly low probability"... am I to assume that while I sit by my fire, the vast majority of the heat is moving from a warm place to a cool place, but not quite all of it? I assume the heat moving from the cooler area to the warmer area is utterly overwhelmed by the heat moving in the opposite direction, rather like trying to piss up a waterfall.

What causes that "incredibly low probabiltiy"? Is it environmental, or completely random? Or do we not know?

Heat transport is a function of relative temperature between 2 sources, and some coefficient of heat diffusion, which characterizes the specific material through which the heat is flowing.

Newton's Law of Cooling:
dT/dt = -k(T(t) - T_inf)

The time rate of change of the temperature of an object is proportional to the difference in temperature between that body and its "ambient" environment.

Solving this 1st order differential equation yields an equation of the form
T(t) = T_inf + (T_init - T_inf)*e^(-kt)

That e^(-kt) part is what we're looking at now.

I'm running out of time.
I'm trying to get to a point where I state that the transport of heat from low temp to high temp is exponentially less than the transport from high temp to low temp the further apart the high and low temps are. When the high and low temps approach the same value due to this exchange process, the amount of transport approaches equilibrium. I.e. when things are the same temperature, the amount of heat flowing between them is the same.

(I need to add a lot of caveats to pin that down to a specific case, but I hope you can do some of that yourself. Feel free to ask for clarity where I trailed off. I'll come back to it later.)

[OngBonga]

I'm also very much glad this isn't a "random" function, because to be quite honest, whenever science talks of "random", what I actually read is "unknown dynamic".