When the Physicist Mr. Jourdain meets Martin Gale

Synopsis

Martingales are perhaps best known as a gambling strategy of repeatedly doubling the stakes of a bet, in the hope that eventually a single win will cover previous losses and return a profit.  In physics, though little known, martingales are also widely used, sometimes just under different pseudonyms and/or in a hidden way.   In my talk, I will start with a historical prelude on the history of the notion of martingales, illustrated by an elementary example of a random walk.  Then I will present recent contributions of martingales to stochastic thermodynamics [1,2], where the notion of fluctuating entropy is strongly linked to martingales and leads to a family of strong Second Laws, and where the breaking of “martingality” turns out to be equivalent to a Maxwell demon.

[1] Chetrite et al., J Stat Phys 2011,  Two Refreshing Views of Fluctuation Theorems Through Kinematics Elements and Exponential Martingale.
[2] Roldan et al., Arxiv 2022,  ''Martingales for Physicists".