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There is actually a very simple derivation of the Second Law in classical thermodynamics for an ideal gas, assuming only classical mechanics and the First Law. Here is a brief sketch -- whether this constitutes a "proof" depends largely on taste, the level of rigor desired, and how comfortable you are with thermo-style derivations.
A certain volume of space with a uniform distribution of particles has maximum entropy. However, the action of gravity would condense these particles, decreasing the entropy of the system, which would violate the second law of thermodynamics. My question is simply: what am I missing here? What is the solution to this contradiction?
I think that the second law of thermodynamics is a direct consequence of the symmetry of spacetime. Imagine you have two identical cubes of metal A and B with one common side, the first one warmer than the other. If you wait long enough, A and B will end up being at the same temperature. The same phenomenon would occur if B were warmer than A.
The second law of thermodynamics says that entropy can only increase, so if the early universe had been in a state of maximum entropy, then the cosmos would have experienced its heat death immediately after being born. This contradicts the observation that the present universe contains burning stars, heat engines, and life.
The Second Law of Thermodynamics is an approximation, it has statistical or probabilistic validity. Statistical Mechanics corrects the plain flat out version of it that says entropy never decreases to the following.
The second law of thermodynamics and the associated concept of entropy have been sources of confusion for thermodynamics students for centuries. The objective of the present development is to clear up much of this confusion.
2nd law in term of entropy: The second law of thermodynamics can be stated in terms of entropy. If a reversible process occurs,there is no net change in entropy. In an irreversible process, entropy always increases, so the change in entropy is positive. The total entropy of the universe is continually increasing
A good layperson's summary of why we have a second law of thermodynamics, how entropy is to some extent a subjective concept, and the discussion of this profound mystery is to be found in chapter 27 of Roger Penrose's "The Road to Reality". I would highly recommend you look at this reference.
$\begingroup$ @JánLalinský Fair, we do not know those. But then why are physicists so certain the 2nd law will never be violated for the whole universe when the FT (a generalization of the 2nd law) requires finite systems, there is a question whether energy is balanced for entire universe (calls into question requirement of being called isolated), etc, etc.
So I have this (possibly dumb) question about the implications of the second law of thermodynamics to the use of renewable energy for the world, so please bear with me. Here goes: Apart from the finiteness of fossil fuels (FFs), which is obviously an issue, the main problem for sustainability in burning FFs for energy is that it leads to the ...