Statistical Analysis of Random Motion and Energetic Behavior of Counting: Gibbs' Theory Revisited

Following a recently reformulated Gibbs' statistical chemical thermodynamic theory on discrete state space, we present a treatment on statistical measurements of random mechanical motions in continuous space. In particular, we show how the concept of temperature and an ideal gas/solution law arise from a statistical analysis of a collection of independent and identically distributed complex particles without relying on Newtonian mechanics, nor the very concept of mechanical energy. When sampling from an ergodic system, the data ad infinitum limit elucidates how the entropy function characterizes randomness among measurements with the emergence of a novel energetic representation for the statistics and an internal energy additivity. This generalization of Gibbs' theory is applicable to statistical measurements on single living cells and other complex biological organisms, one individual at a time.