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Criterion for Equilibrium

If our two systems are initially prepared separately with energies  U₁  and  U₂  other than the most probable, what will happen when we bring them into contact so that  U  can flow between them? The correct answer is, of course, "Everything that possibly can happen." But there is a bigger variety of possibilities for certain gross distributions of energy than for others, and this makes those gross distributions more likely than others. The overall entropy is thus a measure of this likelihood. It seems inevitable that one will eventually feel compelled to anthropomorphize this behaviour and express it as follows:^15.13

All random systems "like" variety and will "seek" arrangements that maximize it.

In any case, the tendency of energy to flow from one system to the other will not be governed by equalization of either energy or entropy themselves, but by equalization of the rate of change of entropy with energy,   ${\partial \sigma \over \partial U}$.  To see why, suppose (for now) that more energy always gives more entropy. Then suppose that the entropy  $\sigma_1$  of system   ${\cal S}_1$  depends only weakly on its energy  U₁, while the entropy  $\sigma_2$  of system   ${\cal S}_2$  depends strongly on its energy  U₂. In mathematical terms, this reads

$$\hbox{\rm Suppose} \qquad {\partial \sigma_1 \over \partial U_1} \; < \; {\partial \sigma_2 \over \partial U_2}$$

Then removal of a small amount of energy  dU  from   ${\cal S}_1$  will decrease its entropy  $\sigma_1$,  but not by as much as the addition of that same energy  dU  to   ${\cal S}_2$  will increase its entropy  $\sigma_2$.  Thus the net entropy   $\sigma_1 + \sigma_2$  will be increased by the transfer of  dU  from   ${\cal S}_1$  to   ${\cal S}_2$. This argument is as convoluted as it sounds, but it contains the irreducible essence of the definition of temperature, so don't let it slip by!

The converse also holds, so we can combine this idea with our previous statements about the system's "preference" for higher entropy and make the following claim:

Energy U will flow spontaneously from a system with smaller ${\displaystyle {\partial \sigma \over \partial U} }$ to a system with larger ${\displaystyle {\partial \sigma \over \partial U} }$.

If the rate of increase of entropy with energy   $\left(\partial \sigma \over \partial U \right)$  is the same for   ${\cal S}_1$  and   ${\cal S}_2$,  then the combined system will be "happy," the energy will stay where it is (on average) and a state of "thermal equilibrium" will prevail.