A Note on Energy and Probability

If the Universe has a finite total energy (ignoring fields, which appear to be unbounded in energy over time), and energy is quantized, then there is with certainty a distribution over the possible energies of systems. For example, there is only one instance of the entire Universe, which means there’s only one possible system with an energy equal to the total energy of the Universe, at a given moment in time. We can then use combinatorics to partition this total amount of energy, into different systems, and basic counting principles will imply a distribution of possible energies. However, of course, we can’t be certain all such energies are actually physically possible, but the notion does imply that there is in fact an underlying distribution that describes the densities of systems of a given energy.

In particular, assume that the total energy of the Universe comes in $N$ discrete chunks of energy. All systems must therefore be a subset of this total energy, and since we’re concerned only with total energy, let’s assume that there’s no order in which the particular chunks of energy are selected. This would reduce the number of possible instances of a system that consists of $K \leq N$ chunks of energy to ${N \choose K}$. You would then remove physically impossible energy levels from this set, which is obviously not trivial, but the point remains, that there must be such a distribution, if the total energy of the Universe is finite, and energy is itself quantized. That is, there are only a finite number of $N$ chunks of energy in the Universe, and therefore, a finite number of ways to select any $K$ of them. Some energy levels might not be physically possible, and we remove those.

Note we are not counting how many configurations there are of a given energy level (e.g., scrambling their positions in space), but instead, counting the number of ways to assemble a collection of discrete chunks of energy into a system with some total energy. This simple method provides a basis for the intuition that extraordinarily high energy systems, and extraordinarily low energy systems, are both rare. Obviously, there’s more to a system than its total energy, since the components will have for example, position, and what I call, “state”, and together, I show that using state and position alone, you can get everything you need to define all the elementary particles, and time-dilation. What I didn’t discuss, which I concluded later, is that particles can actually change their relationships to one another, independent of state and position, since, for example, an atom, is simply not the same as an unassociated group of subatomic particles at the same distances –

There’s a relationship between the particles in an atom, that defines new physics, for example, electron orbitals, that you just don’t find with free electrons. The point being that what actually constitutes a system, is at times, objectively real.

Finally, we can also think about the number of possible ways to partition the energy of the Universe, assuming again that it’s finite and quantized, which will give us a sense of how the energy of the Universe can be allocated to the systems of which it is comprised. This would instead be given by the distribution defined by the Bell Number of the number of discrete chunks of energy in the Universe. These partitions would allow us to consider the possible states of the Universe as a whole, looking only to the energies of systems, that together comprise that configuration of the Universe.