Randomness before Probability, Quantised Gas Laws Directly from Objective Martin-Lof Randomness of Detailed Data

TL;DR

This paper demonstrates that objective Martin-Lof randomness and Kolmogorov complexity of detailed data are fundamental for deriving thermodynamic properties of gases, providing a formal basis that precedes traditional probability-based approaches.

Contribution

It introduces a novel framework linking algorithmic randomness directly to thermodynamics, deriving gas laws and quantum statistics without relying on conventional probability assumptions.

Findings

Martin-Lof randomness determines Helmholtz free energy

Kolmogorov complexity relates to thermodynamic entropy

Fermi-Dirac statistics emerge from randomness conditions

Abstract

We show that objective Martin-Lof randomness and Kolmogorov complexity of instantaneous detailed data lists for $N$ helium gas atoms on $M$ possible energies is necessary and sufficient to directly write down its Helmholtz free energy and thus all thermostatics of the gas. We show that such theory formally precedes application of probability and statistics. Each datum in a list is distinct if $N,M$ are formally well defined and with passive monitoring of thermostatic variables only, each is to be intrinsic. In this introductory paper we consider a low density cool gas of noninteracting He atoms in quantum and classical regimes. Objective definitions of detailed disorder and of thermostatic entropy arise for gas in spontaneous detailed motion along with new insights into irreversible processes and an objective Second Law. Algorithmic probability is rigorously associated with Kolmogorov complexity. A condition for equal a priori probability naturally arises and Fermi-Dirac quantum statistics follows from Martin-Lof randomness.