Abstract
The evolution of a microcanonical statistical ensemble of states of isolated systems from order to disorder as determined by increasing entropy, is compared to an alternative evolution that is determined by mixing character. The fact that the partitions of an integer N are in one-to-one correspondence with macrostates for N distinguishable objects is noted. Orders for integer partitions are given, including the original order by Young and the Boltzmann order by entropy. Mixing character (represented by Young diagrams) is seen to be a partially ordered quality rather than a quantity (See Ruch, 1975). The majorization partial order is reviewed as is its Hasse diagram representation known as the Young Diagram Lattice (YDL).Two lattices that show allowed transitions between macrostates are obtained from the YDL: we term these the mixing lattice and the diversity lattice. We study the dynamics (time evolution) on the two lattices, namely the sequence of steps on the lattices (i.e., the path or trajectory) that leads from low entropy, less mixed states to high entropy, highly mixed states. These paths are sequences of macrostates with monotonically increasing entropy. The distributions of path lengths on the two lattices are obtained via Monte Carlo methods, and surprisingly both distributions appear Gaussian. However, the width of the path length distribution for diversity is the square root of the mixing case, suggesting a qualitative difference in their temporal evolution. Another surprising result is that some macrostates occur in many paths while others do not. The evolution at low entropy and at high entropy is quite simple, but at intermediate entropies, the number of possible evolutionary paths is extremely large (due to the extensive branching of the lattices). A quantitative complexity measure associated with incomparability of macrostates in the mixing partial order is proposed, complementing Kolmogorov complexity and Shannon entropy.
Highlights
The evolution from order to disorder involves at least three readily recognizable stages.System complexions near complete order are simple, but as the system ages the simple complexions evolve into intricate structures
Several alternative paths leading from completely ordered macrostates with zero entropy to the equilibrium macrostate with maximum entropy have been presented and were shown to predict three different dynamics for the evolution of isolated thermodynamic systems
As a function of particle number N, the evolution based on entropy alone allowed exponential steps to equilibrium, the evolution based on mixing showed an increase in steps to equilibrium scaling as N 1.375, and a third evolution showed a linear increase in steps to equilibrium
Summary
The evolution from order to disorder involves at least three readily recognizable stages. In order to connect statistical physics characterizations of entropy with the evolution of a system, it is customary to use the Boltzmann entropy It was originally introduced in statistical mechanics to describe the macrostate of a gas by its characterization in a six dimensional position—momentum phase space for each of the molecules. In this paper we will explore two types of sequences of macrostates that lead to equilibrium, both based on fundamental criteria for the simplest transition from one (more ordered) state to the (less ordered) state While these approaches to equilibrium do not in any way violate the second law of thermodynamics, they do not require that entropy uniquely determines the transition from a lower to a higher entropy macrostate.
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