Galileo Galilei famously stated the principle of Galilean invariance, which links the equations of motion of closed systems as viewed in distinct inertial frames translating relative to one another at a constant velocity. This principle constrains the possible form of mathematical descriptions of classical systems. However, models for many systems are based not on microscopic equations of motion, but on effective descriptions on a mesoscopic level using random processes, such as stochastic Langevin equations or Fokker–Planck diffusion equations. Such equations capture the consequences on a coarse-grained level of microscopic interactions such as friction or noise. Galilean invariance does not apply to such systems, and so offers no help in assessing the consistency of a given stochastic model in different inertial frames.

In a new paper, LML Fellow Rainer Klages and colleagues explore how Galilean invariance is broken during the coarse-graining procedure of deriving stochastic equations. Their analysis leads to a different set of rules – a principle of “weak Galilean invariance” – linking general stochastic models in different inertial frames. While several standard stochastic processes are invariant in these terms, this is not true for the continuous-time random walk. In the paper, the authors derive the correct invariant description for this model. The work provides a theoretical principle to select physically consistent stochastic models well before any comparison with experimental data.

The paper is available at http://www.pnas.org/content/115/22/5714

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