The theory of Brownian motion was developed by Einstein, Smoluchowski and Langevin a little over a century ago. A central result of this theory is the over-damped limit, which says that inertia can be ignored if mass is small, or friction is large. The motion of a Brownian particle, which obeys Newtonian mechanics and is driven by collisions and external forces, is thus described by a first-order diffusion equation.
With the advent of micromanipulation it has become possible to measure and control the positions of individual Browian particles and other small systems (Li et al, Science 2010; Blickle and Bechinger, Nature Physics 2011; Berut et al, Nature 2012). Thermodynamic concepts such as heat, work and entropy production have hence taken a meaning for single systems (Seifert 2005, Sekimoto 2010). I will show that in this setting the overdamped approximation fails, as soon as the temperature field varies in space.
Although the overdamped approximation correctly yields the trajectories of the Browian particles in space it incorrectly estimates the entropy production in the velocity directions of phase space, associated to the breaking of a symmetry of the overdamped dynamics for finite inertia, namely the joint reversal of time and particle velocity.
This is joint work with Antonio Celani, Stafano Bo and Ralf Eichhorn (arXiv:1206.1742, submitted).