Abstract
A model has two main aims: predicting the behavior of a physical system and understanding its nature, that is how it works, at some desired level of abstraction. A promising recent approach to model building consists in deriving a Langevin-type stochastic equation from a time series of empirical data. Even if the protocol is based upon the introduction of drift and diffusion terms in stochastic differential equations, its implementation involves subtle conceptual problems and, most importantly, requires some prior theoretical knowledge about the system. Here we apply this approach to the data obtained in a rotational granular diffusion experiment, showing the power of this method and the theoretical issues behind its limits. A crucial point emerged in the dense liquid regime, where the data reveal a complex multiscale scenario with at least one fast and one slow variable. Identifying the latter is a major problem within the Langevin derivation procedure and led us to introduce innovative ideas for its solution.
Highlights
The Langevin equation is surely one of the pillars of non equilibrium statistical mechanics [1]
Even if the protocol is based upon the introduction of drift and diffusion terms in stochastic differential equations, its implementation involves subtle conceptual problems and, most importantly, requires some prior theoretical knowledge about the system
In a nutshell the basic idea is the following: in a system with some slow variables it is possible to model the dynamics of these observables with an effective stochastic equation containing a systematic drift and a noisy term
Summary
The Langevin equation is surely one of the pillars of non equilibrium statistical mechanics [1] Such a stochastic process has been introduced more than one century ago by Langevin in his seminal paper on the Brownian motion of colloidal particles in a fluid. The study of Brownian motion played a crucial role to establish, in a conclusive way, the physical validity of the atomic hypothesis [2]. In his celebrated work Langevin had been able to write the Brownian evolution law with a deep intuition and a clever combination of macroscopic and microscopic ingredients (namely the Stokes law and energy equipartition).
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