In their paper Comment on ”Irreversibility and Fluctuation Theorem in Stationary Time Series” C. Van den Broeck and B. Cleuren notice that the FT holds for the functional

where x denotes any random variable or stocastic process, which might be for example a Markov chain or a classical trajectory, and T is any involutory functional over such trajectories. Then the FT follows in a very straightforward way by integrating \exp \sigma(x) over all x’s that yeald a given value of \sigma(x), by simply making a change in variable x \to Tx with the absolute value of the determinant being 1. One gets

They observe that this makes the FTs much weaker from a physical point of view. This impression is shared by the author of this post. The great (even greater now) generality of these results make it a nice conceptual framework, but they hardly could serve as a means for deeper understanding. It is rather significative, for example, that they are already contained in the Girsanov and Feynman-Kac formulae of stocastica analysis as very special cases.

However, I cannot think of any relevant involution regarding stocastic processes other than time inversion, but have to make some thinking about it. It might be useful to explore spatial symmetries in some peculiar system, so that the theorem could be a useful tool to understand, for example, symmetry breaking. The fact that the demonstration is rather straightforward was very well known, being the papers by Jarzynski and Crooks rather elementary. What is not obvious at all is the thermodynamical interpretation of the functionals at work in the various setups, expecially the network theory one by Gaspard et al.

- - -

Let’s try to develop some observation. What about having multiple involutions?

so that one gets

      (1)

a sort of extended FT, where the order in which the involutions have to be performed must be inverted. We also have the chain rule

    (2)

It can be easily shown that

Now let’s go into the usual demonstration of the FT. One has to consider the probability distribution of \sigma:

Now I am too tired to worry about that determinant, which should be one in my view, so that we should be obtaining the FT as usual. This might be quite interesting as N grows, eventually defining an infinitesimal generator for the transformation and trying to make physical sense of all this in terms of symmetries. The rather upsetting thing though is that \sigma has no clear thermodynamical meaning, but using result (1) one can maybe hope to always get back to entropy. Fo example consider I and T, the first being a general involution and the second time-inversion. Then one has the FT

That can be worked out to become

      (5)

 

where the second passage is a Radon-Nicodym derivative. That result looks pretty nice.

One should define the I-entropy, besides the usual entropy, so that at the denominator one is summing the entropy of a trajectory and the I-entropy of the inverse trajectory and confronting its probability to that of the T-entropy of the trajectory plus the entropy of the I-symmetric trajectory. Does this mean something?

A big question remains: what is the thermodynamical meaning of all this? If \sigma_T measures the time-asymmetry, than \sigma_I measures the I-asymmetry, whatever it is. Cen we give it a special meaning? Can we give a meaning to \epsilon? It is central to our considerations. It obvioously depends on the system. Examples must be worked out. We also should try to give a generalization of formula (5).