# In the context of statistical mechanics can anyone define temperature?

I was told that defining temperature as the "average kinetic energy of the particles in a system" is not accurate enough.

Well, we have come to have a more general sense of temperature in statistical physics. It has to do with total energy instead of kinetic energy. We define temperature as

$$\frac{1}{T} = \left( \frac{\partial S(E)}{\partial E}\right)_{V, N}$$

Where ##S(E)## is the entropy the system when it has total energy E and the ##\left. \right)_{V, N}## means hold the volume and number of particles constant while taking the derivative.
The entropy (of an ergotic system in equilibrium) is in turn proportional to the log of the number of states with that energy, ##\Omega(E)##
$$S(E) = k_B \ln \Omega(E)$$
This is also can be expressed in terms of a more general formula
$$S(E) = -k_B \sum_i p_i \ln p_i$$
Where the sum is over micro-states and ##p_i## is the probability for microstate i.

So, one important difference from "average kinetic energy" is that this definition involves the total energy, not just kinetic. Potential energy from interactions gets a role in determining temperature.

So for example we can consider the "atoms" in a magnet. Depending on the temperature the "atoms" that make up the magnet have a differing tendency to align with an external magnetic field. An important part of this temperature has nothing to do with motion. Instead it has to do with the different possible alignment configurations. So, the statistical physics framework allows us to define a temperature for practically any system that has a total energy, even those that don't involve motion.

In systems where the total energy is the sum of kinetic plus a potential energy that is a function of the positions, there tends to be a relationship between the kinetic energy and the total energy. This is called the virial theorem. For example average kinetic energy can be proportional to the total energy. So for these types of systems, when we say something about the average kinetic energy, we are also saying something about the total energy as well.

But it's really from the total energy that the statistical physics sense of temperature arises.

Last edited:
• Monsterboy