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The latter equation expresses that the states of the two systems are independent.

From these equations it follows:

$\phi (W_{1}\cdot W_{2})=\phi _{1}(W_{1})+\phi _{2}(W_{2})$

and hence finally

${\begin{aligned}\phi _{1}(W_{1})&=C\lg(W_{1})+const.\ ,\\\phi _{2}(W_{2})&=C\lg(W_{2})+const.\ ,\\\phi (W)&=C\lg(W)+const.\end{aligned}}$

The quantity C is also a universal constant; it follows from kinetic gas theory, where the constants R and N have the same meaning as above. Denoting the entropy at a particular starting state as S₀, and the relative probability of a state with entropy S as W we have in general:

$S-S_{0}={\frac {R}{N}}\lg W.$

We now consider the following special case. Let a number (n) of movable points (for example molecules) be present in a volume v₀, these points will be the subject of our considerations. Other than these, arbitrarily many other movable points can be present. As to the law that describes how the considered points move around in the space the only assumption is that no part of the space (and no direction) is favored over others. The number of the (first-mentioned) points that we are considering is so small that mutual interactions are negligible.

The system considered, which can be for example an ideal gas or a diluted solution, has a certain entropy. We take a part of the volume v₀ with a size of v and we think of all n movable points displaced to that volume v, with otherwise no change of the system. Clearly this state has another entropy (S), and here we want to determine that entropy difference with the help of Boltzmann's principle.