The Stochastic Piston Problem

We obtain analytical solutions for the perturbed shock paths induced by time-varying random motions of a piston moving inside an adiabatic tube of con-stant area. The variance of the shock location grows quadratically with time for early times and switches to linear growth for longer times. The analytical results are con¯rmed by stochastic numerical simulations, and deviations for large random piston motions are estab-lished. The English physicist James Joule was perhaps the ¯rst to use the concept of a moving piston in order to demonstrate the mechanical equivalent of heat in his pioneering studies, almost two centuries ago. The moving piston has also been used extensively in funda-mental studies of °uid mechanics and shock disconti-nuities in the last century, and this now classical prob-lem has been solved analytically in one- and also higher space dimensions (1; 2). It is well known that a shock wave propagating into a stationary °uid sets it into motion and raises its pressure, temperature and den-sity. This situation can be physically realized by a planar, cylindrical or spherical piston moving at spec