However, the universe also cools down, meaning that the average energy per particle is getting smaller with time.

In particular, in the first equation v refers to the empty space available per particle.

At a statistical mechanical level, this results due to the change in available volume per particle with mixing.

At low densities, the number of available quantum states per particle is large, and this difference becomes irrelevant.

Rather, it is represented in configuration space, with three dimensions per particle.

According to current models, nuclear reactions, such as transmutations, require large amounts of energy per particle.

By definition, a gas is ideal when its temperature is only a function of the internal energy per particle, not the volume.

As a consequence, the quantum state of a system of N particles is described by a function with four variables per particle, e.g.

Removes the factor of 1/2 in the nondimensionalized equation for the thermal energy per particle per degree of freedom.

However, the Universe also cools down, meaning that the average energy per particle gets smaller.