Temperature & Equilibrium
Temperature describes thermal state. When systems can exchange energy, a temperature difference drives net heat transfer until thermal equilibrium is reached.
What disappears at thermal equilibrium, and what does not?
At equilibrium, the temperature difference disappears, so there is no net heat transfer between the systems. Microscopic particles do not stop moving. Equilibrium is a stable macroscopic balance, not microscopic stillness.
For a monatomic ideal gas, absolute temperature is proportional to average translational kinetic energy per particle. Temperature is not the total energy of the whole sample.
Bring two equal samples into thermal contact.
Doubling the amount of the same material at the same thermal state does not double its temperature.
Two systems are in thermal equilibrium when contact between them produces no net heat transfer.
Equal temperatures mean matching thermal state, not identical particle-by-particle velocities and certainly not zero motion.
If A is in thermal equilibrium with C, and B is in thermal equilibrium with C, then A and B are in thermal equilibrium with each other. This is what makes temperature a consistent measurable property.
A bathtub and a cup of water can have the same temperature while storing very different amounts of internal energy because the amount and kind of matter differ.