# Tightening the uncertainty principle for the currents

We have a new paper out in the arXiv! It’s a rather technical paper on how to improve on certain  thermodynamic uncertainty principles recently proved for stochastic dynamics.

The uncertainty principle of Heisenberg is one of the coolest concepts in all of physics, so it is not surprising that fields of research close and far to quantum mechanics have searched for similar principles, at times successfully. It has recently been the case for Stochastic Thermodynamics, the theory that formulates in a proper manner thermodynamics, extending it to nonequilibrium and fluctuating systems. The thermodynamic uncertainty principle roughly states that

“The more precise the measurement of a thermodynamic current, the greater the total mean dissipation”.

The more significant of our contributions to this subject is that we managed to replace the “total mean dissipation” by the “least possible mean dissipation in a system that can sustain that particular current” (given a whole bunch of other assumptions, of course…). A sort of minimum entropy production principle. Furthermore, we argued that uncertainty relation for the so-called Fano factor the dissipation rate $\sigma$

$\frac{\mathrm{var}\;\sigma}{\mathrm{mean}\;\sigma} \geq 2$

is the more significant of these uncertainty relations.

But that’s not what I want to talk about here. I’d rather collect a few random thoughts on quantum vs. stochastic uncertainties.

Uncertainty relations in quantum physics relate the variances of two canonically conjugate observables, such as position and momentum. The thermodynamic uncertainty displayed above relates the variance of the dissipation rate and its mean. This is indeed quite different: as usually the case, the temptation to reconnect quantum and stochastic phenomena hits against a square too much: probabilities are squared* wave functions, variances are squared* means etc. (*of course, not literally!).

However, in quantum physics a number of very different uncertainty relations have been derived. In particular I like the energy-time uncertainty relation. Strictly speaking, since time is not an operator but a parameter, there is no time-energy uncertainty in the same sense that there is position-momentum uncertainty. A quote attributed to Landau says that “To violate the time-energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch.” However, one can indeed make sense of such a relation in the following form:

“The uncertainty in time is expressed as the average time taken, starting in state ψ, for the expectation of some arbitrary operator A to change by its standard deviation.”

The quote is taken from the always excellent diary of John Baez. Now, that makes things much closer in spirit to the thermodynamic uncertainty principle, in that we are not talking about the variance of time, but of the actual time it takes to do something. And moreover, it confronts the variance and the expectation value of the same observable. This starts to echo to thermodynamic uncertainty principle. In particular, in this recent paper

E. Roldan et al., Decision making in the arrow of time, Phys. Rev. Lett. 115, 250602 (2015)

it is stated that

“The steady state entropy production rate of a stochastic process is inversely proportional to the minimal time needed to decide on the direction of the arrow of time”.

I find that things might actually converge somehow…