Stochastic geometry and dynamics of infinitely many particle systems—random matrices and interacting Brownian motions in infinite dimensions

We explain the general theories involved in solving an infinite-dimensional stochastic differential equation (ISDE) for interacting Brownian motions in infinite dimensions related to random matrices. Typical examples are the stochastic dynamics of infinite particle systems with logarithmic interaction potentials such as the sine, Airy, Bessel, and also for the Ginibre interacting Brownian motions. The first three are infinite-dimensional stochastic dynamics in one-dimensional space related to random matrices called Gaussian ensembles. They are the stationary distributions of interacting Brownian motions and given by the limit point processes of the distributions of eigenvalues of these random matrices. The sine, Airy, and Bessel point processes and interacting Brownian motions are thought to be geometrically and dynamically universal as the limits of bulk, soft edge, and hard edge scaling. The Ginibre point process is a rotation- and translation-invariant point process on R 2 \mathbb {R}^2 , and an equilibrium state of the Ginibre interacting Brownian motions. It is the bulk limit of the distributions of eigenvalues of non-Hermitian Gaussian random matrices. When the interacting Brownian motions constitute a one-dimensional system interacting with each other through the logarithmic potential with inverse temperature β = 2 \beta = 2 , an algebraic construction is known in which the stochastic dynamics are defined by the space-time correlation function. The approach based on the stochastic analysis (called the analytic approach) can be applied to an extremely wide class. If we apply the analytic approach to this system, we see that these two constructions give the same stochastic dynamics. From the algebraic construction, despite being an infinite interacting particle system, it is possible to represent and calculate various quantities such as moments by the correlation functions. We can thus obtain quantitative information. From the analytic construction, it is possible to represent the dynamics as a solution of an ISDE. We can obtain qualitative information such as semi-martingale properties, continuity, and non-collision properties of each particle, and the strong Markov property of the infinite particle system as a whole. Ginibre interacting Brownian motions constitute a two-dimensional infinite particle system related to non-Hermitian Gaussian random matrices. It has a logarithmic interaction potential with β = 2 \beta = 2 , but no algebraic configurations are known.The present result is the only construction.

Nonlinear Supratransmission in Quartic Hamiltonian Lattices With Globally Interacting Particles and On-Site Potentials

We investigate a family of one-dimensional (1D) Hamiltonian semi-infinite particle lattices whose interactions involve exclusively terms of fourth order in the potential. Our aim is to examine their distinct role in the dynamics, in the absence of quadratic (harmonic) interactions, which are typically included in most studies, as they are known to play an important role in many physical phenomena. We also include in our potentials on-site terms of the sine-Gordon type, which are also considered in many studies in connection with localization effects. Our 1D lattices are subjected to sinusoidal perturbation on one end and an absorbing boundary on the other. To simulate a semi-infinite chain, we will consider a relatively long chain with string coupling. Using reliable finite difference discretization schemes, we establish the existence of nonlinear supratransmission for both short-range and long-range interactions, and demonstrate that the presence of quadratic interactions is not necessary for a system to show nonlinear supratransmission. Additionally, we provide diagrams depicting novel relations between the critical amplitude at which supratransmission is triggered versus driving frequency and a parameter measuring the length of the interactions. Our investigation also shows that the presence of on-site potentials is also not crucial for the system to present supratransmission.

Electron phase-space hole transverse instability at high magnetic field

Analytic treatment is presented of the electrostatic instability of an initially planar electron hole in a plasma of effectively infinite particle magnetization. It is shown that there is an unstable mode consisting of a rigid shift of the hole in the trapping direction. Its low frequency is determined by the real part of the force balance between the Maxwell stress arising from the transverse wavenumber $k$ and the kinematic jetting from the hole’s acceleration. The very low growth rate arises from a delicate balance in the imaginary part of the force between the passing-particle jetting, which is destabilizing, and the resonant response of the trapped particles, which is stabilizing. Nearly universal scalings of the complex frequency and $k$ with hole depth are derived. Particle in cell simulations show that the slow-growing instabilities previously investigated as coupled hole–wave phenomena occur at the predicted frequency, but with growth rates 2 to 4 times greater than the analytic prediction. This higher rate may be caused by a reduced resonant stabilization because of numerical phase-space diffusion in the simulations.

The bizarre anti-de Sitter spacetime

Anti-de Sitter spacetime is important in general relativity and modern field theory. We review its geometrical features and properties of light signals and free particles moving in it. By applying only the elementary tools of tensor calculus, we derive ab initio of all these properties and show that they are really weird. One finds superluminal velocities of light and particles, infinite particle energy necessary to escape at infinite distance and spacetime regions inaccessible by a free fall, though reachable by an accelerated spaceship. Radial timelike geodesics are identical to the circular ones and actually all timelike geodesics are identical to one circle in a fictitious five-dimensional space. Employing the latter space, one is able to explain these bizarre features of anti-de Sitter spacetime; in this sense the spacetime is not self-contained. This is not a physical world.

AN INVARIANCE PRINCIPLE FOR THE TAGGED PARTICLE PROCESS IN CONTINUUM WITH SINGULAR INTERACTION POTENTIAL

We consider the dynamics of a tagged particle in an infinite particle environment moving according to a stochastic gradient dynamics. For singular interaction potentials this tagged particle dynamics was constructed first in Ref. 7, using closures of pre-Dirichlet forms which were already proposed in Refs. 13 and 24. The environment dynamics and the coupled dynamics of the tagged particle and the environment were constructed separately. Here we continue the analysis of these processes: Proving an essential m-dissipativity result for the generator of the coupled dynamics from Ref. 7, we show that this dynamics does not only contain the environment dynamics (as one component), but is, given the latter, the only possible choice for being the coupled process. Moreover, we identify the uniform motion of the environment as the reversed motion of the tagged particle. (Since the dynamics are constructed as martingale solutions on configuration space, this is not immediate.) Furthermore, we prove ergodicity of the environment dynamics, whenever the underlying reference measure is a pure phase of the system. Finally, we show that these considerations are sufficient to apply Ref. 4 for proving an invariance principle for the tagged particle process. We remark that such an invariance principle was studied before in Ref. 13 for smooth potentials, and shown by abstract Dirichlet form methods in Ref. 24 for singular potentials. Our results apply for a general class of Ruelle measures corresponding to potentials possibly having infinite range, a non-integrable singularity at 0 and a nontrivial negative part, and fulfill merely a weak differentiability condition on ℝd\{0}.