Dissipated Work

Dissipation: The Phase-Space Perspective

R. Kawai, J.M.R. Parrondo, C. Van den Broeck

PRL 98, 080602 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e080602

I attended a couple of interesting focus sessions on non-equilibrium thermodynamics at this year's APS March Meeting in Denver. This is the first paper I've read on the subject since returning.

The authors show how the dissipated work (the work you have to do in addition to the free energy change) for a non-equilibrium process can be calculated. They imagine a Hamiltonian with a single control parameter evolving from H(A) to H(B). By considering the trajectory in phase space and the time-reverse of the same trajectory, they show how one can calculate the dissipated work, even though the system does not remain in equilibrium.

Their derivation makes use of three ideas:
⁃ When the system is in equilibrium, the density of states in phase space is determined by the Hamiltonian and the partition function.
⁃ The phase space density is conserved along a Hamiltonian trajectory. (I believe this is Liouville's Theorem.)
⁃ Since the evolution is deterministic, the entire trajectory of the system through phase space is determined by its location in phase space at some particular time. (Sounds similar to Hamilton's principle.)

The third idea implies that if we know location of the system in phase space at some particular time, we can extrapolate backward to the initial by following the trajectory that passes through this state. The second idea implies that the density of states is the same at both of these points. The first means that if the evolution began in an equilibrium state, we can relate the density of state at any point along the trajectory to the initial density of states. Putting this all together, if we know the density of states at any time during the evolution, we can determine the Hamiltonian of the initial state of the system.

In order to calculate the work, we need to know the total change in the Hamiltonian over the course of the evolution:

W = H(B) - H(A)

If the system remained in equilibrium, or if it was in equilibrium at the end of the process, we could determine the Hamiltonian from the density of states. But this is not the case, in general.

To get around this, the authors consider the time reverse of the evolution. Starting in equilibrium with the final state, the state is evolved backward in time to the initial state. For this process, knowledge of the density of states at any point along the trajectory allows one to determine H(B).

By taking the ratio of the density of states for the forward and reverse paths at the same point in time, one can calculate the dissipated work for the process connecting the initial and final states, even if the system does not remain in equilibrium. To quote the authors, "The dissipated work is fully revealed by the phase-space density of forward and backward processes at any intermediate time of the experiment."

The authors establish the equivalence of their calculation and a well-known concept (to experts in the field) called the relative entropy. They go on to show that their algorithm can place a lower bound on the dissipated work even if one does not have perfect knowledge of the density of states. I.e., coarse graining the density of states decreases the calculated quantity. It provides a better estimate of the dissipated work than the second law of thermodynamics, which says the dissipated work is greater than or equal to zero.