Why isn't temperature definable for a single, classical particle?

Question

I read that the relation between temperature and kinetic energy of an ideal gas is only applicable to a large number of particles so that the mean value for kinetic energy has to be considered, so the temperature concept makes sense only as a statistical quantity. But why is that?

Using the aforementioned relation for a single, classical, non-rotating particle, it should always be possible to apply some work on it such that its velocity with respect to the ideal box that contains the particle is zero, hence making the kinetic energy of the particle zero too. But doing so, the temperature obtained is zero kelvin.

This leads to the fact that the temperature to kinetic energy relation only applies to a large number of particles. However, it sounds odd to me. Albeit being highly improbable, such a zero kinetic energy configuration should be possible for a large collection of particles too. This means that temperature is somehow related to entropy, but again, if I'm not wrong, the entropy of such a system is zero.

In conclusion, why is that the temperature for a single classical particle can't be defined? Does it exist a different temperature definition that makes it possible?

Answer 1

Thermal energy is the energy contained in the internal degrees of freedom of a system, so there are two problems with saying that the KE of a single particle gives its temperature.

First, the system’s overall momentum, angular momentum, and so forth are by definition external degrees of freedom. Since they are external then by definition they are not considered in defining thermal energy. For a single classical point particle there are no internal degrees of freedom, hence thermal energy is completely undefined.

Second, it is important to be aware that thermal energy is distributed among all of the internal degrees of freedom. After studying the ideal gas law it is easy to be fooled into thinking that thermal energy is internal kinetic energy, but that is only the case for an ideal mono atomic gas where there are no other internal degrees of freedom. Other gasses have rotational and bending degrees of freedom, and solids have all sorts of complicated internal interactions. Thermal is not generally just microscopic kinetic energy.

Answer 2

Let us consider a situation where we put one "blue" molecule in a gas of "red" molecules. We allow it to equilibriate with a heat bath of temperature T.

In this context the "blue" molecule is certainly a single classical particle and it can be associated with a temperature.

The important point here is what information do we know about the "blue" particle, the only information we know is that the system of "red" and "blue" particles has been set up to be interact with a heat bath of temperature T. Assuming the particles are described by an ideal gas, you can indeed associate an average energy of $\frac{1}{2}KT$ with each degree of freedom, including for the single "blue" molecule that is present. We can also associate it with any statistical quantity such $E_{avg}$, $P_{avg}$, S etc..

Why do we say that a single particle cannot be associated with a temperature? It would again depend on the context. It should be clear if we know the exact co-ordinates and momenta of the "blue" particle we cannot associate it with any statistical uncertainty and certainly not a temperature. It is usually in this context we say we cannot associate a single particle with a temperature. Applying statistical concepts to a single particle one not only needs imperfect information about the description of the particle. But also the probabilities associated with imperfect information should be of the precise form given by the Partition function. It is only in this condition one can associate a single particle with a temperature.

It would be interesting to think about if one can apply the concept of temperature for arbitrary statistical distributions to my knowledge no such definition that is also useful exists.

Let us go one layer deeper, can one determine the temperature of the gas by a single observation of the "blue" particle in bath of "red" particles with temperature T? The answer is NO. It would require a very large number of observations on identically prepared systems to check if the distribution is Maxwell-Boltzmann and also to determine the temperature. For this reason only in the thermodynamic limit one can have meaningful definition of temperature using only a single macroscopic measurement.