Common circuits use electricity to transmit power and signals. I know we transfer heat power using things like HVAC (heating, ventilation, and air conditioning).
I don't know anything that encodes/modulates information and transmits it via heat. There are systems with heat as a feedback signal, but they aren't modulating some completely independent information (say the stock market prices) onto the heat signal. I imagine entropy makes it harder to use heat as information, but there should still be some ways to? Even if not industrial there should be some scientific paper about it.
So I don't know of any existing heat circuits, in industry or in academia. Why is that? Ex: a calculator which does calculations via the processing of heat signals (not an electrical calculator that is merely powered by heat).