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Solving the sourceless heat equation with dimensionless variables $u_{xx} = u_t$ on a disk with $u(x,0)=f(x)$, one gets under suitable assumptions: $u(x,t) = \sum C_n e^{-n^2t}e^{inx}$, $C_n$ being the $n$th Fourier coefficient of $f$. This is the only possible solution.

This contradicts with my understanding of irreversible processes. While $(x,t) \mapsto u(x,-t)$ is not a solution, we can go back in time: given a heat distribution I can infer what the heat was previously simply by computing $u(x,t)$ with $t<0$. Am I right in being bothered by this fact?

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  • $\begingroup$ In principle, you could go backwards, but even the tiniest errors in high-n fourier components would be magnified. $\endgroup$ Commented Jul 6, 2018 at 11:00

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Doing a mathematical model of a transient process like transient heat conduction is similar to creating a movie of the process. You can run the movie in reverse, and it appears that the process has reversed but this doesn't mean that the reverse process actually took place physically (the bomb did not unexplore and reassemble itself spontaneously).

If you study the transient heat conduction equation mathematically, you can precisely calculate the rate of entropy generation within the system as a function of time (as time increases). You will find that, at every location within the system, the rate of entropy generation is positive, and the change in entropy from the initial state to the final state of the combined system and surroundings is positive. This is a sure confirmation that the process is irreversible.

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  • $\begingroup$ I know, I'm not saying that I can make the process go back in time, but rather that I myself can see back in time. I find it weird that one can do that with an irreversible process. $\endgroup$
    – user115153
    Commented Jul 4, 2018 at 20:01
  • $\begingroup$ I guess, for whatever reason, it doesn't seem weird to me. The first law of thermodynamics, which the energy balance for the transient heat conduction situation represents, does not impose the constraints required by the 2nd law of thermodynamics. $\endgroup$ Commented Jul 4, 2018 at 21:38
  • $\begingroup$ Yet Fourier's law of heat flow $j_{th} = -\lambda\nabla T$ does, or so I was told, and is necessary to derive the heat equation. $\endgroup$
    – user115153
    Commented Jul 4, 2018 at 23:20
  • $\begingroup$ True. Good point. But, for transient heat conduction situations, the transient entropy balance equation (derivable directly from the transient heat conduction equation) indicates that this conductive flux results in entropy generation only if time is moving forward; if time is moving backwards, the rate of entropy production from this flux comes out negative. $\endgroup$ Commented Jul 5, 2018 at 0:47
  • $\begingroup$ I don't deny that heat diffusion is irreversible. I'm merely wondering how to reconcile the irreversibility with the ability to look in the past in the manner I have described, or indeed if there is anything to reconcile at all. My question is: 'don't you find this weird?' $\endgroup$
    – user115153
    Commented Jul 5, 2018 at 11:58
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I think you will find that propagating a temperature distribution backward in time will result in distributions with negative temperatures in some places. Some references.

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