### Bin Min

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.