Boltzmann: the probabilistic interpretation of entropy

Boltzmann’s collective work was a mathemetical tour de force. Building on Clausius and Maxwell, he demonstrated that the distribution of gas phase atoms and molecules follows from probability theory. Atoms and molecules distribute themselves in space and momentum to the most probable distribution. Boltzmann used probability theory to quantify the most probable state and then demonstrated the connection between this state and its entropy. This novel approach, later validated by Sackur–Tetrode, led to the creation of statistical mechanics.

Full-Vectorial 3D Microwave Imaging of Sparse Scatterers through a Multi-Task Bayesian Compressive Sensing Approach

In this paper, the full-vectorial three-dimensional (3D) microwave imaging (MI) of sparse scatterers is dealt with. Towards this end, the inverse scattering (IS) problem is formulated within the contrast source inversion (CSI) framework and it is aimed at retrieving the sparsest and most probable distribution of the contrast source within the imaged volume. A customized multi-task Bayesian compressive sensing (MT-BCS) method is used to yield regularized solutions of the 3D-IS problem with a remarkable computational efficiency. Selected numerical results on representative benchmarks are presented and discussed to assess the effectiveness and the reliability of the proposed MT-BCS strategy in comparison with other competitive state-of-the-art approaches, as well.

Statistic thermodynamic analysis of fructans – Part 3: Relationship between polymer structure and mixing entropy at equilibrium conditions

The reduced mixing entropy, which is a concentration and unimer independent equivalent to the polymer mixing entropy defined by Flory, for various probabilistic distributed polymer distributions is calculated. The unbranched most probable distribution proves to reach an extremum value at any given number average degree of polymerization, clearly differentiating it from both broader and narrower polymer distributions with branching structures. Entropy driven polymerization reactions thus inevitably produce unbranched polymer structures as discussed for the case of inulin biosynthesis.

Boltzmann Entropy & Equilibrium in Non-Isolated Systems

The microscopic approach of statistical mechanics has developed a series of formal expressions that, depending on the different features of the system and/or process involved, allow for calculating the value of entropy from the microscopic state of the system. This value is maximal when the particles attain the most probable distribution through space and the most equilibrated sharing of energy between them. At the macroscopic level, this means that the system is at equilibrium, a stable condition wherein no net statistical force emerges from the overall behaviour of the particles. If no force is available then no work can be done and the system is inert. This provides the bridge between the probabilistic equilibration that occurs at the microscopic level and the classical observation that, at a macroscopic level, a system is at equilibrium when no work can be done by it.

Statistical Thermodynamics Concepts and Mathematical Tools for a Multi-Agent Ecosystem

Finding the distribution of systems over their possible states is a mathematical problem. One possible solution is the method of the most probable distribution developed by Boltzmann. This method has been instrumental in developing statistical mechanics and explaining the origin of many thermodynamics concepts, like entropy or temperature, but is also applicable in many other fields like ecology or economics. Artificial ecosystems have many features in common with ecological or economic systems, but surprisingly the method does not appear to have been very successful in this field of application. The hypothesis of this article is that this failure is due to the incorrect interpretation of the method's concepts and mathematical tools. We propose to review and reinterpret the method so that it can be correctly applied and all its potential exploited in order to study and characterize the global behavior of an artificial multi-agent ecosystem.