A purely hardware based computer could exist, where every program is in the hardware and not 'executing via the CPU / registers' and it could be powered by fire (heat -> energy -> power source).
Video cards / GPUs are 'similar' to this at 'some' level, but they still have memory and registers, although they do execute a predefined 'program' in hardware.
This differs greatly from just heating silicon to a level where anomolies occur (stray energy causing hardware to perform in a unexpected way).
If it where possible, without memory (of any kind), chances are it would be in the 60 - 120 degress Celcius range using specialist designed hardware that can only execute predefined functions (although that doesn't mean an entire OS plus application suite couldn't be stored on silicon, it would require a large amount of silicon, or other building block for the 'computer' though... then again early operating systems could fit in the cache of current CPUs, but it would still be classed as 'memory' at some level - as it would be more similar to executing a ROM without shadowing it within the same die that contains a execution unit).... but introducing random elements into a system designed to only execute things a certain way would lead to all sorts of instability (unless the heat is turned into a power source, at a nanostructure, or likely much larger level and not interfering with the 'computer' in dangerous ways).
In summary: Directing any form of energy through a circuit / process, in that there is a power source (fire, mechanical, electricity, etc) a meaningful input (controlled heat source creating energy -> in a certain way so that it is meaningful) a process, and a desired or meaningful output, then yes it could be classed as a computer. Some of the earliest computers where mechanical (not electronic), and just really rather large (half) automatic abacus. So yes, you could construct a rather large mechanical prototype and have it perform some calculations without 'storing' those calculations per say. Then the design could be scaled down to an electronic version, I doubt it would be efficient or powerful though.
Think about how a transistor works, and valves, etc, and before that, and so on, perhaps a mechanical transistor powered by fire or steam did once exist.
If you just had a giant maze / grid of transistors, etc, and 'damaged' some from heat, and some where powered by the heat (raises complexity but not impossible), then if done in a highly controlled way, statistically speaking one could create a kind of computer by destroying part of the 'grid' so that other parts 'route' in a particular way, to produce something with a power source, a input, a process and a output.
The point though ? - It would be totally pointless line of research as we have lithography concepts (similar to the above) already in use for manufacturing far more advanced (and far more useful / more powerful designs) integrated circuits / microprocessors. (Which have standards like registers and memory as it improves the usefulness and efficiency of the end product)..... so there is absolutely no point 'researching' this since we've surpassed the concept several generations already. (to the point where people have forgotten it, and think it is 'crazy')
You are a strange one, but who knows, 100 (or so) years ago you may have been onto something.
That's just my 2c
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