This is a big “why”, one that is beyond the scope of this paper. However, for someone like me who spends some time investigating free energy and the free energy of complex systems (i.e., in complex systems with a small number of interacting agents and a large number of variables, we know that free energy has zero on a small number of occasions), this seems like an obvious question to ask in order to see what could have caused Gibbs’ result.
There are a number of assumptions in this paper that are important to understand, and the assumptions that Gibbs makes seem to be incorrect (in a general sense). Let me explain.
To start with, Gibbs states that when a complex system is free energy is a function of a number of things:
The total number of variables in the system. Some of these will interact positively.
The degree the variables have to interact.
The degree their interactions will be negatively directed.
The degrees of freedom are independent.
The total number of variables is zero.
In simple terms: Gibbs is assuming that for each variable in the system, the total number of variables can be a number of zero, and they don’t all have to interact to achieve a free energy. He is ignoring a number of important facts, such as:
The positive and negative directions of different interacting variables.
The direction of the total number of variables, and the direction of degree of freedom.
The degree of freedom of a complex system is independent of the degree of freedom of its individual variables (that is, their degree of freedom is independent of their total number of variable).
But these are a few of the things that would obviously lead to a larger number of free energy values than he has, and as far as I know, the other important ones (e.g., Gibbs’ first assumption) are true at least in the case of his example. A more complete list of his assumptions can be found in the paper.
However, I can’t help here and assume that the “free energy” that Gibbs (and, according to his paper, “most other writers since then”) have in common is not a number that is completely independent of the number of variables in the system. Therefore, I have not done a very nice job and have skipped over a few details in my argument and argument in favor of a deeper understanding. Nonetheless, in a nutshell, the assumptions that Gibbs makes are:
There are multiple
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