Economic Entropy and Sectoral Dynamics: A Thermodynamic Approach to Market Analysis

This paper presents an application of geometrothermodynamics (GTD) to the economic analysis of Bogot\'a's sports sector through the Satellite Account of Sport (CSDB). By establishing an analogy between thermodynamic systems and economic structures, we develop a mathematical framework where monetary flows behave analogously to energy, while economic entropy, temperature, and heat capacity acquire well-defined economic interpretations. The study focuses on two contrasting sectors: gambling and betting $\mathbb{S}_{15}$, and recreational and sports activities $\mathbb{S}_{16}$, analyzing data from 2018-2023. Our results demonstrate that $\mathbb{S}_{15}$ exhibits lower economic entropy than $\mathbb{S}_{16}$ , indicating a higher degree of organization and regulatory structure in the gambling sector compared to the more heterogeneous recreational sports sector. The heat capacity function reveals critical points that may signal phase transitions in economic dynamics, while Ricci and Kretschmann curvature scalars identify potential crisis points in the sectoral organization. Furthermore, the cross-income elasticity analysis shows distinct resource flow patterns between sectors, suggesting that gambling activities may serve as an economic driver for recreational sports. This thermodynamic approach provides a quantitative tool for analyzing resource redistribution policies and anticipating critical transitions in sectoral economics. The findings suggest that econophysics and statistical thermodynamics constitute powerful frameworks for understanding the sectoral dynamics of Bogot\'a's sports economy, with significant potential for developing prospective analysis tools in public policy design.