November 12, 2013
Stochastic energetics and adiabatic elimination for non-Gaussian processes: Insight from Marcus stochastic calculus
With the development of nanotechnologies, there is increasing interest in understanding small systems in which fluctuations would play an important role. Stochastic energetics is a powerful theory to deal with these small systems, where thermodynamic quantities can be meaningfully defined trajectory by trajectory. One key point of stochastic energetics is to choose an appropriate stochastic integral to fulfill the first law of thermodynamics on the level of individual trajectories. For Gaussian noises, it is found that the Stratonovich integral is the very integral. For non-Gaussian noises which have been recently demonstrated in some biological systems, Kanazawa et al. found that a new stochastic integral should be used. In the talk, I will demonstrate that this so-called new integral is exactly the Marcus integral where each jump can be characterized by a hidden mapping. This hidden mapping could largely facilitate the computation of not only the trajectory of the stochastic dynamics but also the thermodynamic quantities associated with the trajectory. I will argue that this hidden mapping should be considered as an indispensable part of the trajectory, without which it would be very hard to determine the thermodynamic quantities. Furthermore, noticing that this Marcus integral can be obtained by Wong-Zakai smoothing limit, I will discuss the adiabatic elimination for non-Gaussian processes. I will show that depending on the ratio of the fractional relaxation time to the noise correlation time, three different regimes including Ito, Marcus and one crossover type integral would arise in the limiting equation.