# A theory in search of experiment

A small provocation/proposal. I will be very sketchy, postponing a more thorough discussion to later posts. Of course, the contents of this post represents my current state of ignorance on the topic.

As far as I don’t know, “Stochastic Thermodynamics” is a relatively recent tag for a growing body of literature dedicated to the theoretical study of the energetic and entropic aspects of stochastic processes. It is well-rooted in a long history of connections between stochastic processes and the laws of thermodynamics. While most of its core techniques are relatively old, recent advancements gave rise a very prolific and consistent scientific body, that encompasses older theories of nonequilibrium thermodynamics in at least three respects:

1) Far-from equilibrium phenomena;
2) Systems subject to fluctuations;
3) Information as a source of power/dissipation.

I understand that many insiders dislike this tag on the basis that its techniques heavily rely on known facts in the theory of stochastic processes, and as often is the case in science the problem of attribution of specific results sometimes led to acrimony. My personal view is that, today, Stochastic Thermodynamics has one great potentiality and vocation: To finally provide a much needed rigorous formulation of thermodynamics, ranging from theory to experiment, and flexible enough to embrace any open system subject to dissipation.

The cornerstone of Stochastic Thermodynamics is the celebrated Fluctuation Theorem that states that the probability of observing a stochastic trajectory that produces positive entropy is exponentially favored with respect to that producing negative entropy:

$\frac{P(\sigma)}{P(-\sigma)} = e^{\sigma}$

The theorem follows quite straightforwardly once a reasonable definition of the entropy production $\sigma$ along a stochastic trajectory is given. In general, the crucial step in Stochastic Thermodynamics is to properly define energetic and entropic quantities related to stochastic degrees of freedom, bearing a clear physical interpretation.

Following the first formulations of the Fluctuation Theorem (today we count as many formulations as there are authors working on it…), several experimental tests of it have been performed, with a host of techniques: RNA pulling, colloidal particles in laser beams, noisy electronic systems, quantum dots, etc. The results are always very clean and precise, thus the FT is a story of great success.

Can we consider these experiments to be tests of Stochastic Thermodynamics? I think that, to make this claim, we should go one step further. The Fluctuation Theorem follows without further assumptions from the underlying Brownian motion / Markovian nature of the stochastic degrees of freedom. To my knowledge, all the systems employed to verify the FT are very specific and controllable; very often people can describe with great precision all the details of such systems, including the sources of noise. Hence, testing the FT means, indirectly, testing Brownian motion. The fact that the particular observable satisfying the Fluctuation Theorem is called “entropy production” or “work” has little relevance. For example, in the case of the colloidal particle in a laser trap, what one is actually measuring is the displacement of the particle from the center of the trap; from knowledge of its position one can define work, heat etc. in a way that is consistent with our intuition of what work, heat etc. should be, but this does not provide any further prediction to be tested.

So, what would a real experimental test of ST look like? Which goes along asking: what predictions does ST make? My proposal is the following: to clearly claim that we have a successful theory of thermodynamics at microscopic scales, we should find a way perform some sort of calorimetry, or thermometry. That is, we should obtain an indirect measurement of coarser degrees of freedom in a larger system where the definition of heat is uncontroversial, and check that this measure coincides with what expected from the microscopic theory. That is, an experiment should test the consistency of thermodynamics at different scales, up to the macroscopic scale where thermodynamics was born. Very much in the spirit of the connection between the gas laws and kinetic theory.

What exactly this calorimetry/thermometry should consist of I don’t know. Maybe, in the case of a colloidal particle, we should find a way to measure the power consumption of the laser beam, or the temperature of the fluid where the Brownian particle is floating. It’s easy to foresee that this task will be extremely complicated, much more complicated than testing Brownian motion. In the end, it’s a test of consistency of the physics of open systems at different scales.