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Maximum Entropy

Maximum entropy and non-equilibrium physics

 

There exists well-established statistical theory for systems that are in equilibrium and close to equilibrium. Unfortunately, no such formalism is unanimously believed to exist for systems that are arbitrarily far from equilibrium. A solution to this was proposed by E. T. Jaynes, in the guise of establishing a connection between entropy and missing information, as codified in the principle of maximum entropy (the Gibbs algorithm). We seek to demonstrate the validity of extending the Gibbs algorithm for attaining probability distributions into non-equilibrium settings. Instead of assuming an equal a priori distribution of states over which the entropy is maximized (principle of maximum entropy), the time-dependent trajectories of objects are the quantity over which the entropy (here, it is called the "caliber") is maximized (principle of maximum caliber).
Using optical tweezers, we test the Gibbs-Jaynes algorithm by watching a microscopic glass bead diffuse on a 1D potential landscape. An inverted double-Gaussian potential was generated by rapidly scanning a laser beam at two different points, and stochastic forces impart the necessary animus for the object to switch between the two minima of the Gaussians. By specifying the fluxes and switching behavior, we construct and tune the trajectories of the non-equilibrium "ensemble". Using such an approach, we explore the principle of maximum caliber and speculate at its' correspondence with dynamical kinetic equations.

MaxEnt

Left: The experimental setup. Right: We can control constants of the non-equilibrium trajectory by landscaping the effective potential (some examples of the probability distribution in position space).