Energy Conservation as the Constraint Precondition for Monotonic Entropy Increase, The Mathematical Structure Between the First and Second Laws of Thermodynamics

Abstract: Building upon the gradient-driven multiplicative entropy framework proposed by Zou (2025)[3], this paper reveals the intrinsic mathematical constraint structure between the first and second laws of thermodynamics. In this framework, the entropy of a system is defined as S = ∏ᵢ mᵢ, where mᵢ denotes the quantized energy value carried by each discrete node, subject to the strict conservation of total energy E = ∑ᵢ mᵢ. Energy transfer follows a rigorous gradient-driven rule: energy flows exclusively from higher-valued nodes to lower-valued nodes (mᵢ > mⱼ + 1). This paper demonstrates that, under this multiplicative entropy formulation, the first law is not merely an independent postulate standing alongside the second law, but rather serves as the constraint precondition for the monotonic increase of entropy—energy conservation, via the arithmetic-geometric mean inequality, directly yields the upper bound of entropy S ≤ (E/N)ᴺ, while the gradient-driven rule ensures that every step of energy transfer strictly increases the multiplicative entropy. In this manner, the two laws of thermodynamics are unified into a single dynamical process—namely, gradient-driven evolution under a conservation constraint—wherein entropy increase no longer depends on probabilistic or statistical assumptions, but is a necessary consequence jointly determined by energy conservation and the existence of energy gradients.

Keywords: multiplicative entropy; gradient-driven; energy conservation; first law of thermodynamics; second law of thermodynamics; non-probabilistic entropy; constraint-dynamics relation

The gradient-driven dynamics proposed here aligns with the thermodynamic formulation of time by Tuisku, Pernu, and Annila (2009), who describe physical evolution as a process driven by energy dispersal.

Graham, C., & Tokieda, T. (2020). Their work revealed the intrinsic connection between thermodynamics and the AM-GM inequality, which shares part of the mathematical structure underlying the multiplicative entropy framework I subsequently proposed.