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