# Does Fluctuation Theorem prove the 2nd Law of Thermodynamics?

Does Fluctuation Theorem or Crooks Fluctuation Theorem prove the 2nd Law of Thermodynamics from statistical point of view?

Not in general, but is very closely related.

The fluctuation theorems are about non-equilibrium evolution of systems in time, a topic that is outside the purview and capabilities of classical equilibrium thermodynamic theory. In this sense, the fluctuation theorems deal with description more detailed than 2nd law of thermodynamics, and under some assumptions, a conclusion similar to the 2nd law (probability that during a non-equilibrium process entropy increases in time is very high) can be derived from them.

On the other hand, it seems to me the derivations of the theorems rely on special assumptions about the microscopic trajectories - ergodicity. But 2nd law of thermodynamics is not dependent on such artificial assumptions, it is much more general than that. It is believed to apply to any processes that make the system change its state from one to another equilibrium state.

So, I would rather say the fluctuation theorems provide interesting and useful quantitative description of matter not in equilibrium, but they DO NOT derive the 2nd law in its general form.

If you are interested in a derivation of 2nd law of thermodynamics where no explicit assumption about microscopic trajectories is made (except for validity of classical Hamiltonian mechanics) but only experimental facts are assumed, I recommend Jaynes' work. In short, what he had shown is this:

If some equilibrium state A has changed into another equilibrium state B in an irreversible adiabatic process, constancy of information entropy (which follows from the Hamiltonian evolution) together with reproducibility of the resulting state (follows from experience) imply that Clausius entropy of the final state B is greater or the same as entropy of the initial state A.

See Jaynes, E. T., 1965, `Gibbs vs Boltzmann Entropies,' Am. J. Phys., 33, 391 http://bayes.wustl.edu/etj/articles/gibbs.vs.boltzmann.pdf sec. 4,5

I tried to explain his point in a more formal way here:

Understanding Gibbs $H$-theorem: where does Jaynes' "blurring" argument come from?

Yes, the fluctuation theorem gives (among other things) the probability that a system will evolve from a state of greater statistical entropy toward one of smaller entropy. The second law of thermodynamics is only probabilistically correct for large but finite systems, and the fluctuation theorem gives the correction to the second law for finite-size systems. As the number of degrees of freedom in a system goes to infinity, the probability of an entropy-reducing fluctuation goes to zero, which verifies the second law for systems of infinite size.