Michele Battezzati
This book is intended to illustrate fluctuating behaviour and relevant properties of a small mechanical system in tight interaction with a heat bath, thereby experiencing position dependent frictional forces and stochastic random forces. It is focused on a one-dimensional classical mechanical system described by Langevin equation including inertia, driven by Gaussian white noise. The equation of motion is reduced to first order in time by solving the appropriate Hamilton-Jacobi equation with friction. The velocity is split into two components related to drift and diffusion respectively. The coefficients of the diffusion equation are evaluated up to third order in the asymptotic expansion for large friction. The Onsager-Machlup functional yielding the two-time transition probability density follows from Feynman-Kac path integral. In the alternative approach, the free energy variation of the small system is evaluated in terms of mechanical variable displacements, from which fluctuation probabilities follow by Einstein principle. The corresponding functional integral is recast into the form of OM functional by imposition of a constraint on the mean value of kinetic energy.
