Consider the following thought experiment with $2$ conducting objects $A$ and $B$,

initial setup

where object $A$ is initially at a much higher temperature than object $B$.

The objects are now placed in perfect thermal contact,


Assume that once the objects are in contact all sides are covered with a perfectly insulating material so that no heat can escape (an isolated system).

From Fouriers' Law of heat conduction since there is a temperature gradient between $A$ and $B$ heat will flow from $A$ to $B$. Then after some (long) time thermodynamic equilibrium will be reached, the temperature gradient will be zero and thus no more heat will flow between $A$ and $B$.

My question is concerning how the thermodynamic equilibrium was reached:


Since heat has been transferred from $A$ to $B$, unless I'm mistaken this will place $B$ momentarily at a higher temperature than $A$. So the direction of heat flow will reverse. Subsequently object $A$ will be at a higher temperature than $B$ so as before the direction of heat flow will reverse, and so on...

Will this oscillating process ever stop (I guess it would have to; otherwise thermodynamic equilibrium will never be reached)? Or will the oscillations decay in time?

I know the heat is transfered due to particles in the hotter object vibrating and colliding with neighboring particles and as a result heat is transferred. But once these collisions reach the end of the connected cooler object, that object in turn will be at a higher temperature (unless I'm missing something) and the heat will flow back to the previous object. Am I completely wrong about this situation and the heat flow direction does not oscillate whatsoever?

  • 2
    $\begingroup$ Your beating Newton's balls here, what I mean is you're treating temperature as a linear momentum, that's elastically transferred upon interaction. In reality, the energy is spread among degrees of freedom of say a molecule. Now the heat diffusion equation has proper modes as solutions, so in some situations there may be a bit of oscillation, but I would expect it to be over damped. The only situation I can think of off hand is two plasmas with a collinear anisotropic temperature. Oh! And flux pumping a superconductor with thermal waves (very cool!) $\endgroup$
    – R. Rankin
    Jul 17 '16 at 10:54
  • $\begingroup$ If I'm reading yout question correctly, you are starting with a wrong assuption: There is no oscillation. You may be treating heat as if it has momentum, but it does not. $\endgroup$ Jul 17 '16 at 10:56
  • $\begingroup$ @Terry I know what you are saying and the above comment also raises the same point. I know that heat does not have momentum. What I am trying to say is what happens once the far end of the cool object becomes hot? Will the heat direction of flow reverse? $\endgroup$
    – BLAZE
    Jul 17 '16 at 11:00
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    $\begingroup$ Ah, I think I now see what you are trying to ask: The far end gets warmer, then presumably starts cooling down again; an oscillation. But it only starts to cool down again if the entire system is placed in contact with some larger, external heat sink. If all of this was taking place within some perfect insulating container (an impossibility of course), then all the points you show will experience monotonically increasing temperature increases until equilibrium -- until a single system-wide uniform temperature -- is reached. No oscillations, just monotonistic temperature increases. $\endgroup$ Jul 17 '16 at 11:10
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    $\begingroup$ The problem with the oscillation assumption is that it considers your two conductors as atomic, monolithic objects. Instead, subdivide each block in smaller pieces (representing the molecules) and the actual process becomes much clearer. When the hot surface touches the cold surface, heat is transferred to the top layer of molecules, which then transfers to the second layer, etc... From the macro perspective there is no oscillation, but there IS transfer back and forth between each layers of molecules. $\endgroup$ Jul 17 '16 at 20:21

You are mistaken. You seem to be assuming that there is some kind of inertia in the process of heat transfer, as in water sloshing about in a tank. There is no such inertia here, so there is no oscillation.

You write:

Since heat has been transferred from A to B, unless I'm mistaken this will place B momentarily at a higher temperature than A.

Yes, you are mistaken about this. The Fourier Model is continuous: a finite excess of heat energy is not transferred which makes B hotter than A. In the model, the amounts transferred are infinitesimal, and cause infinitesimal changes in temperature.

You are also missing that there is a reverse process going on at the same time. As soon as the temperature of B rises even a little, the rate of heat flow back towards A also increases. There is no time delay in waiting for the 'heat wave' to be reflected from the far end of B.

In the Fourier Model, the process of heat flow out of each element of material is not directional. It is a diffusion process which happens at random equally in all directions regardless of the temperature of adjacent elements. But the amount of out-flow increases with temperature, with the result that there is a net flow of thermal energy away from regions with higher temperature and towards those with lower temperature.

When B reaches the same temperature as A there is a dynamic equilibrium between the flows from A to B and B to A. Heat does not continue flowing preferentially from A to B because of inertia. When you apply the mathematical model the result is not an oscillation, but an exponential decay in the temperature difference between A and B.

Outside of the ideal model, because heat flow is a random process, in reality there are small random fluctuations in temperature between two bodies which are in thermal equilibrium. However, these fluctuations are insignificant unless the bodies are microscopic (or your thermometer is highly accurate), and they are not oscillations.

  • $\begingroup$ There's even a mechanical analogy with no oscillation: you could consider the heat flow to be heavily damped. $\endgroup$ Jul 17 '16 at 17:35

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