How large a closed system does the 2nd law of thermodynamics require?

Question

Does the second law of thermodynamics apply to a single electron in deep space? Or two electrons? Or 100? (assume the electrons are restricted in a finite area)

From which point we can start to talk about the entropy of a closed system?

Answer 1

The laws of statistical physics work in thermodynamic limit, i.e., they become exact as the number of particles goes to infinity, privided that the density remains constant: $N\rightarrow \infty, V\rightarrow\infty, N/V\rightarrow\text{const}$.

For a finite number of particles, the quantities predicted by statistical physics exhibit fluctuations, i.e., deviations from the average values predicted by the law, which scale as $$\propto\frac{1}{\sqrt{N}}$$ with the number of particles.
In other words, in a system of 100 particles the fluctuations can be as high as 10%, but in a system of $N_A\approx 10^{24}$ (Avogadro number of) particles they are one about trillionth of the magnitude of the quantity, and can be safely neglected for all the practical purposes

Update
In response to comments: indeed, it doesn't make sense to apply the above reasoning to a single particle. The scaling of cluctuations as $1/\sqrt{N}$ is useful for discussing the difference between a system of $N_A$ particles, where the thermodynamic laws are essentially exact, and that of, e.g., hundred particles, where the deviations from tehrmodynamic behavior (i.e., the fluctuations) are likely to be notcieable.

There are some basic assumptions in statistical physics which clearly do not hold for a system of one or just a few particles - most of these assumptions have to do with the conditions, where we can apply statistical description to the system, in terms of its integrals of motion (of which the most important one is energy.) Whether these hold for a system of a 100 of particles has to be explored on a case-by-case basis. E.g., one sometimes applies statistical description to big nuclei, where the numbers of nucleons are around 100.

Furthermore, one does use statistical description for a single particle (or an atom) coupled to a thermostat, i.e., when it is a part of a much bigger thermodynamic system, although of different nature. E.g., an atom coupled to thermal the electromagnetic field (black-body radiation).