Equation of Motion for Thermodynamic Systems and Their Dynamical Equivalence with Mechanical and Electrical Systems

Abstract

Changes in the internal energy of a thermodynamic system are fundamentally described by the first law of thermodynamics. However, the first law has a highly compact form that does not explicitly reveal the dynamics of the system or provide a corresponding equation of motion. While equations of motion can be written explicitly for mechanical and electrical systems, an analogous dynamical formulation for thermodynamic systems has, to our knowledge, not been established directly from the first law. In this work, we show that, through dimensional reduction, the first law can be transformed from its compact thermodynamic form into an explicit equation of motion for an open thermodynamic system. Remarkably, the resulting equation has the same mathematical structure as the equations governing a driven damped harmonic oscillator and a driven series RLC circuit. Detailed dimensional analyses of the derived equation and the effective coefficients further confirm the dimensional consistency of the formulation. We further perform a systematic comparison of the mechanical, electrical, and thermodynamic systems and show that, despite their distinct physical origins, they share the same universal dynamical structure. Finally, we construct a Lagrangian formulation of the thermodynamic system and establish its correspondence with the Lagrangian descriptions of the mechanical and electrical systems. These results demonstrate that an explicit equation of motion can be generated directly from the first law of thermodynamics and reveal a common dynamical framework underlying thermodynamic, mechanical, and electrical systems.