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My computer is blowing heat, and all it's supposed to be doing is sending pulses of voltage through tiny circuits to flip and check the condition of FETs.

Ignoring the source of the electricity, I know that all of the conductive paths have some non-zero resistance, and so they generate heat as they carry current. Through engineering and material enhancement the resistance of conductors can be reduced. In fact, in the limit we can actually put the conductors into a state of superconductivity in which they literally have no resistance and so would generate no heat.

I understand that (at least in labs) the same has been shown for FETs.

So in the limit we can produce a computer with literally zero electrical resistance. Is this correct?

And if this is correct, then is it true that calculation and data storage and retrieval are in principle thermodynamically neutral? Or have I missed some "leak" in the process that requires an increase of entropy?

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There are reversable and irreversible operations. Most operations we see in modern computers are irreversible, which means they are not thermodynamically neutral, even when perfectly implemented, because they are "logically irreversible". Logically irreversible means the operation fundamentally loses information, even if the physical implementation was as lossless as you can get. The layman's phrasing (which works really well) is to say that if you "erase" information, it is not reversable.

As an example, consider two registers, A and B, each containing a number. If I calculate A XOR B and store the result in B, I did not "erase" anything because if I take this new B (call it B'), and do A XOR B', I get B back. No information was lost. On the other hand, if I calculate A OR B, and store the result in B, I do erase information. If any bit in the resulting B' is a 1, I don't know whether that 1 came from A, B, or both.

There are ways to be logically irreversible and be complete. The Toffoli gate is a logic gate capable of doing any Boolean logic reversibly. This gate has been implemented using quantum mechanics, demonstrating the potential for a logically reversible and theoretically physically reversible process.

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  • $\begingroup$ Helpful detail (pointed out by @lemon): If a computer performs irreversible operations (as do all conventional computers to date) then it does increase entropy and literally generates heat – the lower bound on which is shown via Landauer's principle. $\endgroup$ – feetwet Jun 22 '17 at 17:15
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    $\begingroup$ The OP did ask about theoretical limits in principle, so this answer is basically what was requested. However, it would also be helpful to point out that the basic thermodynamic limits on computation are many orders of magnitude away from being relevant in real-world computers. $\endgroup$ – Ben Crowell Jun 22 '17 at 17:54

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