The geodesic dispersion phenomenon in random fields dynamics

TL;DR

The paper investigates whether time irreversibility in Gaussian random fields stems from the intrinsic geometry of their parameter space. It develops the information geometric framework by deriving the metric tensor and Christoffel symbols for the GRF manifold and introduces a novel MCMC augmented RK4 scheme to trace geodesics. A key finding is the geodesic dispersion phenomenon, where time-reversed geodesics diverge from forward paths during phase transitions, linking curvature dynamics to irreversibility. This work provides a computational geometry lens for analyzing random field dynamics and points to future exploration of higher curvature constructs such as the Ricci tensor to better characterize geodesic dispersion.

Abstract

Random fields are ubiquitous mathematical structures in physics, with applications ranging from thermodynamics and statistical physics to quantum field theory and cosmology. Recent works on information geometry of Gaussian random fields proposed mathematical expressions for the components of the metric tensor of the underlying parametric space, allowing the computation of the curvature in each point of the manifold. In this study, our hypothesis is that time irreversibility in Gaussian random fields dynamics is a direct consequence of intrinsic geometric properties (curvature) of their parametric space. In order to validate this hypothesis, we compute the components of the metric tensor and derive the twenty seven Christoffel symbols of the metric to define the Euler-Lagrange equations, a system of partial differential equations that are used to build geodesic curves in Riemannian manifolds. After that, by the application of the fourth-order Runge-Kutta method and Markov Chain Monte Carlo simulation, we numerically build geodesic curves starting from an arbitrary initial point in the manifold. The obtained results show that, when the system undergoes phase transitions, the geodesic curve obtained by time reversing the computational simulation diverges from the original curve, showing a strange effect that we called the geodesic dispersion phenomenon, which suggests that time irreversibility in random fields is related to the intrinsic geometry of their parametric space.

Figures (8)

• Figure 1: First, second and third order neighborhood systems on a 2D lattice.
• Figure 2: Decomposition of $\Sigma_{p}$ into $\Sigma_{p}^{-}$ and $\vec{\rho}$ on a second-order neighborhood system ($\Delta=8$). By rewriting the components of the metric tensor in terms of Kronocker products, we can make numerical simulations faster.
• Figure 3: Geodesic curves in the Gaussian density manifold obtained by fourth-order Runge-Kutta and its time reverse simulations. A denotes the initial point and B denotes the final point in the regular simulation and vice-versa in the time reversed version.
• Figure 4: Geodesic curves, entropy and curvature evolution in the Gaussian random fields manifold obtained by the proposed iterative MCMC based fourth-order Runge-Kutta algorithm and its time reverse simulations (first row in Table \ref{['tab:ds2']}). The blue curve denotes the orignal geodesic and the red curve denotes its time reversed version.
• Figure 5: Geodesic curves, entropy and curvature evolution in the Gaussian random fields manifold obtained by the proposed iterative MCMC based fourth-order Runge-Kutta algorithm and its time reverse simulations (third row in Table \ref{['tab:ds2']}). The blue curve denotes the orignal geodesic and the red curve denotes its time reversed version.