# Statistical mechanics - why is temperature not a mechanical variable

1. Jun 24, 2008

### knulp

statistical mechanics -- why is temperature not a mechanical variable

Hi, I have heard that temperature is not a mechanical variable. That is, that even if you knew the positions and momenta of all the particles in some system, you still couldn't calculate the temperature, because temperature (and entropy, and free energy, etc) are ensemble variables.

Why is that?

By the way, one implication of this statement is that temperature is not really the average kinetic energy of a system, at least in some cases. Say you had a dilute (better yet, ideal) system of independent gas (argon) atoms and you knew the mass of any particle (they all have the same mass) and its velocity. You could then calculate kinetic energy (0.5 * m*v*v, right?) and average kinetic energy, therefore kinetic energy (and average kinetic energy) is a mechanical variable. But temperature is not. So temperature is not really average kinetic energy.

So, what is temperature?

Thanks!

2. Jun 25, 2008

### CompuChip

Re: statistical mechanics -- why is temperature not a mechanical variable

There is a nice definition of temperature, as the quantity which is the same between two systems in thermal equilibrium. So if we allow energy, but not particles or volume, to flow between two systems, at some point the energy flow will stop. At that time the systems are in thermal equilibrium and there is some quantity which is the same for both of them, called temperature.

If you want to have this thoroughly treated, you should look into a book, like the one by Kittel and Kroemer (Introduction to Thermal Physics).

3. Jun 26, 2008

### Andy Resnick

Re: statistical mechanics -- why is temperature not a mechanical variable

I have to state up-front that I have a heretical view of thermodynamics. For example, I am reading Truesdell's "Rational Thermodynamics" which is really opening my eyes.

In my view, 'Temperature' is some property of a system that is independent of it's constitution. It does not have to be linked with 'equilibrium states' and is instead intimately linked with entropy.

Temperature should also de-linked from statistical-mechanical treatments. It's not helpful and is one of the reasons extending thermodynamics to nonequilibrium processes is so difficult.

As I said, my view is at variance with, say, Kittel. Or Reif. The standard "just-so" pseudo-mathematical treatments. For example, why is thermo*dynamics* presented in a time-independent form?