Fundamental postulate. Entropy
Of our world the following facts are known:
- An isolated system has a set of quantum states.
- The isolated system ceaselessly flips from one quantum state to another.
- A system isolated for a long time is equally probable to be in any one of its quantum states.
Thus, an isolated system behaves like a fair die. The following notes remind you what an isolated system is, and translate the theory of probability of rolling a fair die to the thermodynamics of an isolated system.
We then describe the procedure to apply the fundamental postulate to study experimental phenomena. For a given phenomenon, we construct an isolated system. As always, the isolated system flips among a set of quantum states. We then identify an internal variable of the isolated system, and use the internal variable to dissect the set of quantum states into subsets, each subset being a macrostate of the isolated system, with a specific value of the internal variable. The fundamental postulate says that a system isolated for a long time is equally probable in any one of its quantum states. Consequently, the most probable macrostate has the largest number of quantum states. This statement leads to an algebraic equation to determine the internal variable in equilibrium, or an ordinary differential equation to evolve the internal variable in time. We illustrate this procedure by considering a half glass of wine.
The fundamental postulate implies the second law of thermodynamics. The latter is an incomplete expression of the former.
We give the phrase "the logarithm of the number of quantum states" a symbol, S, and call it entropy.
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Re. definitions of fundamental thermodynamic terms
I am not sure if I can be happy with the definition of a system as "a part of the world." Thermodynamic statements can be made about the entire world taken as a system too.
Therefore, I think the focus of the definition should be put on the *objects* under study---by which the way they are put together also is included. (The principles or rules governing the way the objects are put together, do have thermodynamic entailments.) Sometimes, the set of objects under study may in principle include all objects in the world. The definition would remain unstrained even for application to this case.
The whole thing might seem hair-splitting at this stage, but consider the next defintion.
"An isolated system is one that does not interact with the rest of the world." Fine. What about the entire world taken as a system? Is this system open? closed? isolated? It is usually said to be isolated. But why? If we try to apply the given definition, doesn't it involve making a reference to nothing?
Instead, how about this definition (just a small change): "An isolated system is one which does not interact with any other object/system."
Here, the tricky issue of whether we know if there is any object left outside of the system under consideration or not, is made silent, i.e. a non-issue. I hope it does.
The whole idea is that with with open systems, you have to put theoretical sentries at the borders---you have to worry about all sorts of things that get exchanged at those borders---matter, energy (heat), work. In contrast, with closed systems, you know that mass (practically, the same as matter) won't get exchanged at the borders but energy and work might. In contrast to *both* these, with isolated systems, you simply don't bother putting a sentry in the first place---the system formulation has been so abstracted away from the concrete reality that you don't have to bother about that part.
To conclude, ask the question: What is it that an isolated system is isolated from? The usual answer is: From everything else (aka the enviornment).
But following my speculation, I think another, epistemologically deeper, perspective is possible. The answer can be: from certain theoretically complicating aspects of the concrete reality, via a process of abstraction that removes interactions out of the theory via appropriate limiting processes. Thus, the isolation in question is rather about abstracting away than about "air-tighting", "energy-tighting", "work-tighting", in short, "everything-tighting" away. Once you understand this part, a lot of questions arising in physical application---whether in cosmology or concerning accurate experimental realization---begin to better fall in place.
Just some food for thought. I think I am onto something right, but won't argue about it. (And, yes, it's easily possible that others have said exactly the same points---including the epistemological points---before me.)
--Ajit
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[E&OE]
More on entropy
See http://philsci-archive.pitt.edu/8592/1/EntropyPaperFinal.pdf