Convergence of thermodynamic quantities and work fluctuation theorems in presence of random protocols

Abstract: Recently many results namely the Fluctuation theorems (FT), have been discovered for systems arbitrarily away from equilibrium. Many of these relations have been experimentally tested. The system under consideration is usually driven out of equilibrium by an external time-dependent parameter which follows a particular {\it protocol}. One needs to perform several iterations of the same experiment in order to find statistically relevant results. Since the systems are microscopic, fluctuations dominate. Studying the convergence of relevant thermodynamics quantities with number of realizations is also important as it gives a rough estimate of number of iterations one needs to perform. In each iteration the protocol follows a predetermined {\it identical/fixed} form. However, the protocol itself may be prone to fluctuations. In this work we are interested in looking at a simple non-equilibrium system namely a Brownian particle trapped in a harmonic potential. The center of the trap is then dragged according to a protocol. We however lift the condition of fixed protocol. In our case the protocol in each realization is different. We consider one of the parameters of the protocol as a random variable, chosen from some known distribution. We study the systems analytically as well as numerically. We specifically study the convergence of the average work and free energy difference with number of realizations. Interestingly, in several cases, randomness in the protocol does not seem to affect the convergence when compared to fixed protocol results. We study symmetry functions. A Brownian particle in a double well potential is also studied numerically. We believe that our results can be experimentally verified.