  We study fluctuating dynamics of a freely movable piston that separates an
infinite cylinder into two regions filled with ideal gas particles at the same
pressure but different temperatures. To investigate statistical properties of
the time-averaged velocity of the piston in the long-time limit, we
perturbatively calculate the large deviation function of the time-averaged
velocity. Then, we derive an infinite number of effective Langevin equations
yielding the same large deviation function as in the original model. Finally,
we provide two possibilities for uniquely determining the form of the effective
model.
