Exploring fundamental laws of classical mechanics via predicting the orbits of planets based on neural networks

Neural networks have provided powerful approaches to solve various scientific problems.Many of them are even difficult for human experts who are good at accessing the physical laws from experimental data.We investigate whether neural networks can assist us in exploring the fundamental laws of classical mechanics from data of planetary motion.Firstly,we predict the orbits of planets in the geocentric system using the gate recurrent unit,one of the common neural networks.We find that the precision of the prediction is obviously improved when the information of the Sun is included in the training set.This result implies that the Sun is particularly important in the geocentric system without any prior knowledge,which inspires us to gain Copernicus'heliocentric theory.Secondly,we turn to the heliocentric system and make successfully mutual predictions between the position and velocity of planets.We hold that the successful prediction is due to the existence of enough conserved quantities(such as conservations of mechanical energy and angular momentum)in the system.Our research provides a new way to explore the existence of conserved quantities in mechanics system based on neural networks.

Retainability of canonical distributions for a Brownian particle controlled by a time-dependent harmonic potential

The retainability of canonical distributions for a Brownian particle controlled by a time-dependent harmonic potential is investigated in the overdamped and underdamped situations, respectively. Because of different time scales, the overdamped and underdamped Langevin equations(as well as the corresponding Fokker-Planck equations) lead to distinctive restrictions on protocols maintaining canonical distributions. Two special cases are analyzed in details: First, a Brownian particle is controlled by a time-dependent harmonic potential and embedded in medium with constant temperature; Second, a Brownian particle is controlled by a timedependent harmonic potential and embedded in a medium whose temperature is tuned together with the potential stiffness to keep a constant effective temperature of the Brownian particle. We find that the canonical distributions are usually retainable for both the overdamped and underdamped situations in the former case. However, the canonical distributions are retainable merely for the overdamped situation in the latter case. We also investigate general time-dependent potentials beyond the harmonic form and find that the retainability of canonical distributions depends sensitively on the specific form of potentials.

Escape rate of an active Brownian particle in a rough potential

We discuss the escape problem with the consideration of both the activity of particles and the roughness of potentials.We derive analytic expressions for the escape rate of an active Brownian particle in two types of rough potentials by employing the effective equilibrium approach and the Zwanzig method.We find that activity enhances the escape rate,but both the oscillating perturbation and the random amplitude hinder escaping.