There are many ways to study approaches to equilibrium, which is obvious as there are many ways to drive a system out of equilibrium.
So there is really no unique answer to your question.
However, various universal results are known.
These include various fluctuation theorems.
The most famous of which is usually called just the fluctuation theorem which relates the probability of time-averaged entropy production, $\Sigma_t=A$ over time $t$ to $\Sigma_t=-A$,
$$
\frac{P(\Sigma_t=A)}{P(\Sigma_t=-A)}=e^{A t},
$$
which shows that positive entropy production is exponentially more likely than negative entropy production. Note that the second law follows from this theorem. The fluctuation-dissipation relation may also be derived from it.

There is also, for instance, the Crooks fluctuation theorem which relates the work done on a system, $W$, during a non-equilibrium transformation to the free energy difference, $\Delta F$, between the final and the initial state of the system,
$$
\frac{P_{A \rightarrow B} (W)}{P_{A
\leftarrow B}(- W)} = ~ \exp[\beta (W - \Delta F)],
$$
where $\beta$, is the inverse temperature, $A \rightarrow B$ denotes a forward transformation, and vice versa.

A lot of research has been done in this area so for more information I suggest reading some review articles, such as,

Esposito, M., Harbola, U., and Mukamel, S. (2009). Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Reviews of Modern Physics, **81**(4), 1665. (arxiv)

and

Campisi, M., Hänggi, P., and Talkner, P. (2011). Colloquium: Quantum fluctuation relations: Foundations and applications. Reviews of Modern Physics, **83**(3), 771. (arxiv)

There are various master equations (e.g., Fokker-Planck type, Boltzmann, Lindblad, etc.)in physics which will give you more information than theorems like these, but they are derived using various approximations and/or assumptions or are system specif. So, like I said there is no universal answer to your question.

EDIT: Deriving Fourier law is difficult. In fact there is an article from 2000 F. Bonetto, J.L. Lebowitz and L. Rey-Bellet, Fourier's Law: a Challenge for Theorists (arxiv) which states in the abstract: "There is however at present no rigorous mathematical derivation of Fourier's law..."

This post imported from StackExchange Physics at 2015-06-15 19:26 (UTC), posted by SE-user Bubble