The purpose of joining this community is to have a blog dedicated to popular scientific writing. The biggest advantage of this venture is that it will teach me how to communicate vague and
intuitive physical ideas in a clear scientific language.
This article assumes a preliminary knowledge of axiomatic thermodynamics. But do proceed nevertheless!
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We’ll see if this way of looking at entropy makes it more tangible!
We’ll assume that we know all the physical definitions and equations from classical mechanics (or quantum for that matter). Quantities such as energy, volume, velocity, forces, interaction potential are known. While quantities such as pressure, temperature, entropy, free energy are yet to be defined. We’ll see why we need these new quantities if we hope to describe the system at hand in the least risky manner.
Consider a box which does not leak energy, through any mechanism. Let the box be filled with marbles in vacuum. Initially the marbles are stationary and their positions are known. Now the box is given to an
unknown person who shakes it vigorously and gives it back to you. You now know the total energy content of the box, but since you didn’t witness the shaking yourself you are not very sure about the velocity of each particle.
Now it is commons sense that we expect that some fraction of the energy of the box can be transferred into some other system. We do not yet know if this is entirely possible. But we can always hope, because the law of conservation of energy does not prohibit that (this is a classical mechanical law, hence we know its truth).
In a slight detour, let’s consider a system where we have a single marble situated at the center of a cubical box. The box is shaken such that the marble is now moving parallel to one of the walls. This is a perfectly deterministic system, the equation of this system can be written as:
f(U,V,N) == 0
where U is the energy of the system (marble), V is the confinement volume and N is the number of marbles (=1). We can bravely say that the system is deterministic. The energy transfer from such a system is going to be easy since we exactly know how it is behaving. This can be generalized to any number of marbles as long as we know exactly the velocity and initial positions of all of them.
The equation of this multiple marble deterministic system will be the usual Hamiltonian equation:
symbolically f(U,V,N,{p_i,q_i}) == 0
Clearly this simple situation differs from our more complicated box with more than 1 marbles. Let {p_i(t),q_i(t)}_n be the set of all possible trajectories of our system. Depending on the initial shaking, the system will choose one of these trajectories. So the actual equation of the system will be
f(U,V,N,{p_i,q_i}_m) == 0
The subscript m corresponds to the shaking procedure employed and will vary depending upon it. Since we do not know how the energy was transferred to the system, we do not know which m to choose. This is repetition of the same statement that we do not know the trajectory of each and every particle. It is intuitive to assume that the maximal amount of energy that can be transferred from our box to any other system is dependent upon the particular trajectories the marbles are following. For a given method, some trajectories will yield higher energy transfer than others.
Now, even if we don’t know all information about the system, we are required to estimate the maximal amount of energy transfer that is possible. We have no inclination towards any of the specific trajectories and all of them are. We’ve ignored (or we are unaware of) the actual state of the system and yet we hope to characterize it. We are not familiar with such lack of knowledge in classical mechanics. Let us hope to quantify this ignorance and call it Entropy S.
Hence instead of the old equation of state, our equation now looks like the following:
f(U,V,N,S) == 0
Notice the absence of a particular trajectory. It is also intuitive that since we do not have access to some of the information about the system. This new approach is going to be an approximate one. We can at the best hope that it applies .
If we further write f(U,V,N,S) == 0 as S == S(U,V,N) we arrive at the first postulate of classical thermodynamics. The postulate of existence of entropy. More properties are boldly ascribed to this entropy function (which it seems to obey! in experiments) in further postulates.
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In various good books on thermodynamics, the existence of entropy is either postulated as a mathematical fact or is justified in retrospect as a useful tool. I find this ignorance approach much more realistic and logical since it justifies the postulate for me.
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