Weak Galilean invariance as a selection principle for coarse-grained diffusive models.

Area of Science:

• Statistical Mechanics
• Mathematical Physics
• Computational Biology

Background:

• Galilean invariance, fundamental to classical mechanics, states that laws of motion are identical in all inertial frames.
• Real-world systems often use coarse-grained models that incorporate friction and stochastic forces, violating Galilean invariance.
• A principle to ensure physical consistency of stochastic models across inertial frames is lacking.

Purpose of the Study:

• To investigate how Galilean invariance is broken during the coarse-graining process in deriving stochastic equations.
• To establish a new principle, termed 'weak Galilean invariance', for validating stochastic models in different inertial frames.
• To provide a theoretical framework for selecting physically consistent stochastic models, particularly for biological systems.

Main Methods:

• Analysis of the Kac-Zwanzig Hamiltonian model generating Brownian motion.
• Derivation of stochastic equations and examination of Galilean invariance breakdown during coarse-graining.
• Development of a set of rules defining 'weak Galilean invariance' for general stochastic models.

Main Results:

• Demonstrated that Galilean invariance is violated in coarse-grained stochastic models.
• Introduced 'weak Galilean invariance' as a criterion for physical consistency across inertial frames.
• Identified the continuous-time random walk as a process requiring a corrected invariant description under this new principle.

Conclusions:

• The study provides a novel theoretical principle, weak Galilean invariance, to assess the physical consistency of stochastic models.
• This principle is crucial for accurately modeling systems, especially in biology, where coarse-grained descriptions are common.
• The findings offer a method to pre-validate stochastic models before experimental data comparison.