# On the additivity of entropy

In Chapter 1 of his famous textbook on thermodynamics, Callen gives (among various other posulates) the following postulate:

Postulate III The entropy of a composite system is additive over the constituent subsystems. The entropy is continuous and differentiable and is a monotonically increasing function of the energy.

I am trying to understand why we can say the entropy is additive. At this early point, I guess one could say that my question is: does the additivity of entropy apply only if the constituent subsystems do not interact in a given equilibrium state (i.e. imagine given internal constraints separating the constituent subsystems)?

An important clue is Callen's definition of a composite system:

Given two or more simple systems, they may be considered as constituting a single composite system.

I interpret the above as saying that if we have two or more simple systems, then (and only then?) we can define an overall "composite" system as all of these. A simple system is further defined as below:

Simple systems are systems that are macroscopically homogeneous, isotropic, and uncharged, that are large enough so that surface effects can be neglected, and that are not acted on by electric, magnetic, or gravitational fields.

My question about the postulate can be then posed, perhaps, by an example: suppose we begin with two different fluids in two different containers (these are ostensibly reasonably taken as simple systems if they are large vats etc.). Call these systems $$A$$ and $$B$$. By the postulate, if we define the composite system $$C$$ as the "union" of these two simple systems, the entropy of this composite system is additive: $$S_C = S_A+S_B$$. Now mix these two fluids. Is it correct to say that the postulate cannot be applied to each of $$A$$ and $$B$$? That is, after mixing, I suppose I cannot say that if I "focus" on only the particles associated with $$A$$, that $$A$$ is still well-definable as a simple system (and similarly $$B$$)? If I can't do this, can you explain precisely why given Callen's definitions? If indeed I cannot do this then it would not be correct to continue to insist that $$S_C = S_A+S_B$$ (intuitively, of course, I know that this is not the case).

I suppose then that, later in the development, Callen will show we can still cleverly use the postulates to analyze the overall system even though it is no longer a union over simple systems?

Entropy is not an additive quantity, entropy is sub-additive meaning that if two bodies are isolated individually each having by themselves $$S_A$$ and $$S_B$$ entropies then when they are interacting and are still isolated together the joint entropy is not less than the sum: $$S_{A \cup B} \ge S_A+S_B$$
We cannot define a macroscopic system by focusing on specific particles, e.g. "particles that before mixing belonged to fluid $$A$$." We may only define it in terms of macroscopic quantities, such as its volume, its number of particles of various types, and its total energy.