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Inverse Temperature

What do we expect to happen if the systems are out of equilibrium? For instance, suppose system   ${\cal S}_1$  has an energy   $U_1 < \hat{U}_1$.  What will random chance "do" to the two systems? Well, a while later it would be more likely to find system   ${\cal S}_1$  with the energy  $\hat{U}_1$  again. That is, energy would tend to "spontaneously flow" from system   ${\cal S}_2$  into system   ${\cal S}_1$  to maximize the total entropy.^15.16 Now stop and think: is there any familiar, everyday property of physical objects that governs whether or not internal energy ( HEAT) will spontaneously flow from one to another? Of course! Every child who has touched a hot stove knows that heat flows spontaneously from a hotter object [like a stove] to a cooler object [like a finger]. We even have a name for the quantitative measure of "hotness" -- we call it TEMPERATURE.

Going back to Eq. (9), we have a mathematical expression for the criterion for THERMAL EQUILIBRIUM, whose familiar everyday-life equivalent is to say that the two systems have the same temperature. Therefore we have a compelling motivation to associate the quantity   ${\partial \sigma \over \partial U}$  for a given system with the TEMPERATURE of that system; then the equation reads the same as our intuition. The only problem is that we expect heat to flow from a system at high temperature to a system at low temperature; let's check to see what is predicted by the mathematics.^15.17 Let's suppose that for some initial value of   $U_1 < \hat{U}_1$  we have

$${\partial \sigma_1 \over \partial U_1} > {\partial \sigma_2 \over \partial U_2} .$$

Then adding a little extra energy  dU  to   ${\cal S}_1$  will increase  $\sigma_1$  by more than we decrease  $\sigma_2$  by subtracting the same  dU  from   ${\cal S}_2$  [which we must do, because the total energy is conserved]. So the total entropy will increase if we move a little energy from the system with a smaller   ${\partial \sigma \over \partial U}$  to the system with a larger   ${\partial \sigma \over \partial U}$.  The region of smaller   ${\partial \sigma \over \partial U}$  must therefore be hotter and the region of larger   ${\partial \sigma \over \partial U}$  must be cooler. This is the opposite of what we expect of TEMPERATURE, so we do the obvious: we define   ${\partial \sigma \over \partial U}$  to be the INVERSE TEMPERATURE of a system:

$${\partial \sigma \over \partial U} \; \equiv \; {1 \over \tau}$$ (15.10)

where (at last)  $\tau$  is the TEMPERATURE of the system in question. We can now express Eq. (9) in the form that agrees with our intuition:

$$\hbox{\rm Condition of {\sc thermal equilibrium}:}$$

$$\tau_1 = \tau_2$$ (15.11)

-- i.e. if the temperatures of the two systems are the same, then they will be in thermal equilibrium and everything will be most likely to stay pretty much as it is.

As you can see, TEMPERATURE is not quite such a simple or obvious concept as we may have been led to believe! But now we have a universal, rigorous and valid definition of temperature. Let's see what use we can make of it.