Title:
On entropy production before and after the over-damped limit
Abstract:
The set-up of classical thermodynamics is that the world is divided into three parts:
the System, the External System, and the Thermal Environment.
This has in the last decade been generalized to single mesoscopic systems,
so the three parts could be a large molecule (the System), an experimentalist
controlling conformation of the molecule by e.g. optical tweezers (the External System),
and the surrounding medium at room temperature (the Thermal Environment).
If the dynamics of the System is modeled by stochastic differential equations
(Langevin equations) the mesoscopic equivalents of heat, work and entropy
production are stochastic functionals of the System. In earlier work with
Carlos Mejia-Monasteiro, Paolo Muratore-Ginanneschi and others I have shown
that optimization of such functionals lead to not entirely trivial mathematical problems.
In this talk I want to consider another aspect. Heat and work are not absolute
concepts on the mesoscopic scale, but depend on what one can observe and control and
thus describe as "the System". Indeed, one modern definition states that "heat is the
work done by the retained degrees of freedom on the thermal environment representing
the other degrees of freedom" (Sekimoto, 2010). In the same manner, for different
Systems the total entropy production will be split in different ways between
"entropy of the System" and "entropy of the Environment". For well-defined concepts
the fluctuation relations such as Jarzynski's equality hold, but these only partially
constrain the entropy production on different scales. I will discuss the simple
example of the over-damped limit of a system described by an under-damped
Langevin equation, and show that when temperature depends on space the over-damped
limit of the average entropy production is not the average entropy production in the
over-damped limit. The difference between the two expressions has the same form as
the entropy production of a macroscopic fluid at rest in a temperature gradient
(Landau & Lifshitz, Fluid Mechanics ?49.6). These results are obtained by a
multi-scale calculation i.e. by systematic but not rigorous arguments. I will then
discuss possible generalizations of these results to other systems.
This is joint work with Antonio Celani, Stefano Bo and Ralf Eichhorn.