Cycle symmetry, limit theorems, and fluctuation theorems for diffusion processes on the circle

Cyclic structure and dynamics are of great interest in both the fields of stochastic processes and nonequilibrium statistical physics. In this paper, we find a new symmetry of the Brownian motion named as the quasi-time-reversal invariance. It turns out that such an invariance of the Brownian motion is the key to prove the cycle symmetry for diffusion processes on the circle, which says that the distributions of the forming times of the forward and backward cycles, given that the corresponding cycle is formed earlier than the other, are exactly the same. With the aid of the cycle symmetry, we prove the strong law of large numbers, functional central limit theorem, and large deviation principle for the sample circulations and net circulations of diffusion processes on the circle. The cycle symmetry is further applied to obtain various types of fluctuation theorems for the sample circulations, net circulation, and entropy production rate.Comment: 28 page

Reveal flocking of birds flying in fog by machine learning

We study the first-order flocking transition of birds flying in low-visibility conditions by employing three different representative types of neural network (NN) based machine learning architectures that are trained via either an unsupervised learning approach called "learning by confusion" or a widely used supervised learning approach. We find that after the training via either the unsupervised learning approach or the supervised learning one, all of these three different representative types of NNs, namely, the fully-connected NN, the convolutional NN, and the residual NN, are able to successfully identify the first-order flocking transition point of this nonequilibrium many-body system. This indicates that NN based machine learning can be employed as a promising generic tool to investigate rich physics in scenarios associated to first-order phase transitions and nonequilibrium many-body systems.Comment: 7 pages, 3 figure