Is there a mathematical relationship between time and entropy? If there is a relation between time and entropy, what is it?
Are there limitations for this equation?
Or if there is no relationship between them, what is the current state of research?
 A: There is no relation in which you can find both the entropy $S$ you know from equilibrium thermodynamics (ETD) or statistical mechanics (ESM) and the time variable $t$ you know from dynamics.
The reason for this is that the concept of entropy, as everything else in ETD and ESM, only has meaning if the system is at equilibrium, and if the system is at equilibrium there is, by definition, no time evolution. 
In other words, what you can do in ETD and ESM is calculate the entropy of specific equilibrium states, and the entropy difference between equilibrium states, but you can never write an expression for the entropy when the system is evolving between those equilibrium states, because when the system is evolving it is not at equilibrium.
There is however a theorem by Boltzmann that comes very close to defining an "entropy" that depends on time: the famous H theorem.
What Boltzmann showed is that the functional
$$H [f] \equiv \int d \mathbf p \ f(\mathbf p, t) \log f(\mathbf p, t)$$
where $f(\mathbf p, t)$ is a solution of Boltzmann's transport equation, can only decrease with time or remain stationary:
$$\frac{dH}{dt} \leq 0$$
and that we obtain the "$=$" sign only when $f=f_0$, where $f_0$ is the Maxwell-Boltzmann distribution. It is possible to show (see for example K. Huang, Statistical Mechanics) that
$$H[f_0] = -\frac{S}{Vk_B}$$
where $k_B$ is Boltzmann's constant. The $H$ theorem would therefore apparently be a statement of the Second Law of thermodynamics in the special case of fixed volume. However, there are some issues:


*

*Even if $-H[f_0]\propto S$, it is not clear whether $-H[f]$ can be rigorously identified with the entropy we know from thermodynamics and statistical mechanics.

*To derive the H theorem, Boltzmann made a strong assumption, the assumption of molecular chaos, that effectively introduces a time asymmetry in the system, and it is not clear whether this assumption is physically justifiable or not.

