Stochastic heat engine acting like a weakly nonlinear wave ensemble

We have shown that a stochastic heat engine which is modelled by an over-damped random particle confined in an externally driven time-varying logarithmic-harmonic potential could behave like the wave amplitude of a system of weakly interacting waves. The system of weakly interacting waves may thus serve as an empirical testing ground of the stochastic heat engine. In addition, we have proposed a simple Lie-algebraic method to solve the time evolution equation for the probability density function (p.d.f.) of the system of weakly interacting waves by exploiting its dynamical symmetry. This Lie-algebraic approach has the advantage of generating both the p.d.f. and the generating function in a straightforward manner.