期刊
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
卷 115, 期 22, 页码 5714-5719出版社
NATL ACAD SCIENCES
DOI: 10.1073/pnas.1717292115
关键词
stochastic processes; Galilean invariance; anomalous transport; coarse-graining; fractional calculus
资金
- Postgraduate Research Fund - Queen Mary University of London
- Science Research Fellowship - Royal Commission for the Exhibition of 1851
- Office of Naval Research Global
- London Mathematical Laboratory
- Engineering and Physical Sciences Research Council [EP/L020955/1]
- EPSRC [EP/L020955/1] Funding Source: UKRI
How does the mathematical description of a system change in different reference frames? Galilei first addressed this fundamental question by formulating the famous principle of Galilean invariance. It prescribes that the equations of motion of closed systems remain the same in different inertial frames related by Galilean transformations, thus imposing strong constraints on the dynamical rules. However, real world systems are often described by coarse-grained models integrating complex internal and external interactions indistinguishably as friction and stochastic forces. Since Galilean invariance is then violated, there is seemingly no alternative principle to assess a priori the physical consistency of a given stochastic model in different inertial frames. Here, starting from the Kac-Zwanzig Hamiltonian model generating Brownian motion, we show how Galilean invariance is broken during the coarse-graining procedure when deriving stochastic equations. Our analysis leads to a set of rules characterizing systems in different inertial frames that have to be satisfied by general stochastic models, which we call weak Galilean invariance. Several well-known stochastic processes are invariant in these terms, except the continuous-time random walk for which we derive the correct invariant description. Our results are particularly relevant for the modeling of biological systems, as they provide a theoretical principle to select physically consistent stochastic models before a validation against experimental data.
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