  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.
