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

TL;DR

This work reframes equilibrium gas thermodynamics by deriving the Helmholtz free energy and thermostatics from objective randomness, asserting that Martin-Löf randomness and Kolmogorov complexity of detailed data lists precede probabilistic descriptions. It introduces quantised complexity mechanics ($\mathbb{QKM}$) and defines a disorder measure $D^{-}$ built from the complexities of intrinsic data lists, yielding the free energy $F(T,V,N) = -k_B T D^{-}$ and a complete set of state equations without ensemble averaging. By contrasting $\mathbb{QKM}$ with conventional quantum statistical mechanics ($\mathbb{QSM}$), the paper argues that ML-random energy lists imply equal a priori probability and recover Fermi-Dirac combinatorics through an algorithmic partition function $z_C = 2^{D^{-}/\ln 2}$, thus linking randomness to thermodynamic behavior. The framework also treats reversible and irreversible processes via $S^{rev}$ and $\Delta S^{irrev}$, providing an objective Second Law grounded in the growth of algorithmic disorder, and suggests a broader micro-to-macro hierarchical view with potential implications for nonequilibrium dynamics and cosmology.

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.