# (Finite-Time) Thermodynamics, Hyperbolicity, Lorentz Invariance: Study of an Example

## TL;DR

This paper explores how thermodynamics, relativity, and hyperbolic conservation laws interact in a geological reactor example, focusing on finite time and resource constraints.

## Contribution

The paper introduces a novel framework linking finite-time thermodynamics, relativity, and hyperbolic conservation laws through a geological reactor example.

## Key findings

- Finite propagation speeds and finite time interact in hyperbolic problems, affecting chemical change propagation.
- Entropy production is Lorentz invariant and essential for selecting physically meaningful solutions in hyperbolic problems.
- Resources and resource flows behave similarly to space-time variables under relativistic transformations.

## Abstract

Our study lies at the intersection of three fields: finite-time thermodynamics, relativity theory, and the theory of hyperbolic conservation laws. Each of these fields has its own requirements and richness, and in order to link them together as effectively as possible, we have simplified each one, reducing it to its fundamental principles. The example chosen concerns the propagation of chemical changes in a very large reactor, as found in geology. We ask ourselves two sets of questions: (1) How do the finiteness of propagation speeds modeled by hyperbolic problems (diffusion is neglected) and the finiteness of the time allocated to transformations interact? (2) How do the finiteness of time and that of resources interact? The similarity in the behavior of the pairs of variables (x, t and resources, resource flows) in Lorentz relativistic transformations allows us to put them on the same level and propose complementary-type relationships between the two classes of finiteness. If times are finite, so are resources, which can be neither zero nor infinite. In hyperbolic problems, a condition is necessary to select solutions with a physical sense among the multiplicity of weak solutions: this is given by the entropy production, which is Lorentz invariant (and not entropy alone).