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 Gaussian curvature in each point of the manifold that represents the space of all possible parameter values that define such mathematical model. A key result in the dynamics of these random fields concerns the curvature effect, a series of variations in the curvature that happens in the parametric space when there are significant increase/decrease in the inverse temperature parameter. In this paper, we propose a numerical algorithm for the computation of geodesic curves in the Gaussian random fields manifold by deriving the 27 Christoffel symbols of the metric required for the definition of the Euler-Lagrange equations. The fourth-order Runge-Kutta method is applied to solve the Euler-Lagrange equations using an iterative approach based in Markov Chain Monte Carlo simulation. Our results reveal that, when the system undergoes phase trasitions, the geodesic dispersion phenomenon emerges: the geodesic curve obtained by reversing the system of differential equations in time, diverges from the original geodesic curve, as we move from zero curvature configurations (Euclidean geometry) to negative curvature configurations (hyperbolic-like geometry), and vice-versa. This phenomenon suggest that, time irreversibility in random field dynamics can be a direct consequence of the geometry of the underlying parametric space.

翻译：随机场是物理学中普遍存在的数学结构，其应用涵盖热力学与统计物理、量子场论及宇宙学等多个领域。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，使得能够计算代表该数学模型所有可能参数值流形上各点的高斯曲率。这些随机场动力学的一个关键结果涉及曲率效应——当逆温度参数显著增减时，参数空间发生的曲率系列变化。本文通过推导定义欧拉-拉格朗日方程所需的27个克里斯托费尔符号，提出了计算高斯随机场流形测地线曲线的数值算法。采用基于马尔可夫链蒙特卡洛模拟的迭代方法，应用四阶龙格-库塔法求解欧拉-拉格朗日方程。我们的结果表明，当系统经历相变时，测地线弥散现象显现：当从零曲率构型（欧几里得几何）过渡到负曲率构型（类双曲几何）及其逆过程时，通过时间反向求解微分方程组得到的测地线曲线会偏离原始测地线曲线。该现象表明，随机场动力学中的时间不可逆性可能是底层参数空间几何的直接结果。