What evidence do we have of the second law of thermodynamics?

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.

The First Law of Thermodynamics is:

dU=dq+dw

where the differentials refer to changes of the system. By convention we have defined a gain of energy or heat by the system as positive, work done on the system as positive, and work done by the system on the surroundings as negative.

Without loss of generality, we consider pressure-volume work. The work done by the system is quantified by the amount of work done in the surroundings, and so the relevant pressure is the external pressure Pext in the surroundings that the system is pushing against. Then, the work done by the system is

dw=−PextdV

If the internal pressure of the system is greater than the external pressure of the surroundings,

Pint≥Pext

then according to classical mechanics the system will expand against the surroundings, i.e. dV≥0.

For a reversible change, the internal and external pressures are equal (Pint=Pext), and so the work done by the system in a reversible process is

dwrev=−PintdV

Therefore,

PintdV−PintdVdwrev≥PextdV≤−PextdV≤dw

which means that the magnitude of work done by the system on the surroundings is maximal during a reversible process. Combining this result with the First Law gives:

dqrev≥dq

We now define the state function entropy S classically as

dS=dqrevT

From the previous inequality for reversible heat, we see that

dS=dqrevT≥dqT

which is the generalized Clausius inequality. This is a complete mathematical statement of the Second Law of Thermodynamics. All consequences of the Second Law can be derived from it, including the proposition that heat always spontaneously flows from hot to cold.

The one missing part is that we did not establish that entropy S is a state function for an ideal gas, but this can be found in any introductory thermodynamics treatment (e.g. [1]).

H- theorem

This proof may be adjusted to particular or general physical systems, both classical ones and quantum ones. The H-theorem is a proof that the thermodynamic arrow of time - the direction of time in which the entropy increases - is inevitably aligned with the logical arrow of time - the direction in which one is allowed to make assumptions (the past) in order to evolve or predict other phenomena (in the future).

Every Universe of our type has to have a globally well-defined logical arrow of time: it has to know that the future is being directly evolving (although probabilistically, but with objectively calculable probabilities) from the past. So any universe has to distinguish the future and the past logically, it has to have a logical arrow of time, which is also imprinted to our asymmetric reasoning about the past and the future. Given these qualitative assumptions that are totally vital for the usage of logic in any setup that works with a time coordinate, the H-theorem shows that a particular quantity can't be decreasing, at least not by macroscopic amounts, for a closed system.