A Sparcely Confined Water Molecules Undergoing Finite-Time Cyclic Process: A Case Study in Extracting Equilibrium Information from Non-Equilibrium Process

A large number of water molecules are each placed on a lattice far apart so that they are very weakly interacting with each other and in contact with a heat bath at temperature $T$. A strong static electric field, $E_{0}$, is applied to these molecules  along a $z$-axis causing three level split energy values. A weak AC electric field that acts for a finite time $\tau$ applied in the $xy-$plane induces transitions between the three levels. This weak AC field acts as a protocol $\zeta(t)$, that is switched on at $t=0$ and switched off at $t=\tau$.The same cyclic process is repeated  for a large number of times. The data available for this finite-time non-equilibrium process allowed us to extract equilibrium thermodynamic quantities like free energy, which is what we call Jarzynski equality and its relation to the second law of thermodynamics. The work distributions of the three-level system in the optimum condition is obtained. Besides, the average work of the system as a function of $\omega$ and time around the optimum frequency are evaluated, where $\omega$ is the frequency of the AC electric field.