I'm trying to intuitively understand what temperature is for a system of classical particles. The usual definitions via Gibbs measure or entropy appear very unintuitive me. But, as for ideal gas temperature is proportional to mean kinetic energy of its molecules (in the rest frame of gas' center of mass), I thought it's a good place to start.
So, suppose we have a system $\Lambda$ of interacting classical particles, such that there's always 3 degrees of freedom per particle (so that molecules would be modelled as multiple bound particles). Assume that their center of mass is at rest. In general they'll not behave as an ideal gas — they could condensate etc.. But when brought to contact with an ideal gas $\Gamma$ (with heat capacity much smaller than that of $\Lambda$) initially at $T_\Gamma=0\,\mathrm K$, the gas $\Gamma$ would ultimately assume the temperature of the system: $T_\Gamma\to T_\Lambda$ as $t\to\infty$. This actually means that the mean kinetic energy of the gas molecules will be proportional to the temperature of the system $\Lambda$.
So, I'd assume that the increase in mean kinetic energy of $\Gamma$ was due to the kinetic energy present in $\Lambda$. Then, isn't mean kinetic energy of the particles in $\Lambda$ actually proportional to the temperature, even despite $\Lambda$ being not an ideal gas, and not even a gas in general?