"Heat", in classical thermodynamics, is a derived abstraction. It's defined as the sum over a bunch of classical energies distributed among particles that behave deterministically. So this is just the measurement trick: we can't measure it therefore it's behavior is "random" from our perspective. But that's a practical limit and nothing to do with "determinism" in a mathematical sense.

When we consider things at the macro scale, we may find properties that don't appear at the micro scale, this is called emergence, and that's what thermodynamics is about.

Particle motion may be deterministic, and importantly, time-reversible, when we have too many particle to individually consider, the rules change, and that's when you are talking about entropy and temperature. To consider an extreme example, our brain is made of subatomic particles, and yet, psychology is not at all like particle physics. The same can be said of finance, where global economics have laws that don't apply to individual transactions and vice-versa.

It is like shuffling a deck of card is considered random, though it is not at all the case, in fact, a skilled magician can control the shuffle and pretty much order the deck in any way he likes. But for the purpose of playing cards, it is considered random, and theory is built on that.

The question is slightly inaccurate, it should be "classical mechanics". Classical thermodynamics is very much stochastic.

Stochastic does not mean undeteministc. It means that tiny perturbations to initial conditions lead to huge differences in time evolution.

See this gif for example https://gereshes.com/2019/02/18/chaos-and-the-double-pendulu...

Thermal phenomena like heat transfer can arise in systems that are deterministic at the microscopic level. Indeed, at the time of Boltzmann (pre quantum) one of the major questions is how to go from deterministic but complex & unpredictable classical particle dynamics to continuum models like the heat equation. Kinetic theory is one piece of that bridge between scales.

In more recent times these questions are still studied, e.g., within mathematical physics / ergodic theory circles. Look up "Lorentz gas", "Fourier law", etc. Usually to get anything interesting one needs to hook these systems up to "reservoirs", which are usually stochastic. In principle one could replace the reservoirs by another large, chaotic classical system but that makes the mathematical questions too hard, and having some randomness in a small corner of the system and studying how its influence spreads is still very challenging but more tractable.

In classical thermodynamics heat transfer is modelled using non-stochastic and fully deterministic PDEs.

The examples in this article all seem to involve infinite forces and speeds... I don't think they're as interesting as they are made out to be.