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 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.

翻译：随机场是物理学中普遍存在的数学结构，其应用涵盖热力学、统计物理、量子场论及宇宙学等领域。近期关于高斯随机场信息几何的研究提出了底层参数空间度规张量分量的数学表达式，从而能够计算流形各点的曲率。本研究的假设是：高斯随机场动力学中的时间不可逆性是其参数空间几何特性（曲率）的直接结果。为验证该假设，我们计算了度规张量分量，推导了度规的27个克里斯托费尔符号，从而定义出欧拉-拉格朗日方程——这一偏微分方程组用于构建黎曼流形中的测地线。随后，通过四阶龙格-库塔法和马尔可夫链蒙特卡洛模拟，我们从流形中任意初始点出发数值构建测地线。结果表明：当系统经历相变时，通过计算模拟时间反转获得的测地线与原始曲线产生偏离，展现出我们称之为“测地线弥散现象”的奇特效应，这暗示随机场的时间不可逆性与其参数空间的本征几何相关。