Boltzmann: the probabilistic interpretation of entropy

2020 ◽  
pp. 570-595
Author(s):  
Robert T. Hanlon
Keyword(s):  
Probability Theory ◽  
Statistical Mechanics ◽  
Gas Phase ◽  
Probabilistic Interpretation ◽  
Atoms And Molecules ◽  
Collective Work ◽  
Novel Approach ◽  
Probable State ◽  
Probable Distribution ◽  
Most Probable Distribution

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.

Application of Boltzmann Statistical Mechanics in the Validation of the Gaussian Summit-Height Distribution in Rough Surfaces

1997 ◽  
Vol 119 (4) ◽  
pp. 846-850 ◽  
Author(s):  
M. Leung ◽  
C. K. Hsieh ◽  
D. Y. Goswami
Keyword(s):  
Surface Roughness ◽  
Statistical Mechanics ◽  
Theoretical Basis ◽  
Rough Surfaces ◽  
Electrical Contact ◽  
Height Distribution ◽  
Probable Distribution ◽  
Theoretical Predictions ◽  
Most Probable Distribution ◽  
Roughness Measurements

In theoretical modeling of contact mechanics, a homogeneously, isotropically rough surface is usually assumed to be a flat plane covered with asperities of a Gaussian summit-height distribution. This assumption yields satisfactory results between theoretical predictions and experimental measurements of the physical characteristics, such as thermal/electrical contact conductance and friction coefficient. However, lack of theoretical basis of this assumption motivates further study in surface modeling. This paper presents a theoretical investigation by statistical mechanics to determine surface roughness in terms of the most probable distribution of surface asperities. Based upon the surface roughness measurements as statistical constraints, the Boltzmann statistical model derives a distribution equivalent to Gaussian. The Boltzmann statistical mechanics derivation in this paper provides a rigorous validation of the Gaussian summit-height assumption presently in use for study of rough surfaces.

Boltzmann Entropy & Equilibrium in Non-Isolated Systems

2017 ◽  
pp. 74-92 ◽  
Author(s):  
Alberto Gianinetti
Keyword(s):  
Statistical Mechanics ◽  
Stable Condition ◽  
Microscopic Level ◽  
Boltzmann Entropy ◽  
Microscopic Approach ◽  
Macroscopic Level ◽  
Probable Distribution ◽  
Isolated Systems ◽  
Most Probable Distribution

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

2014 ◽  
Vol 20 (2) ◽  
pp. 237-270
Author(s):  
Javier Segovia
Keyword(s):  
Statistical Mechanics ◽  
Mathematical Problem ◽  
Statistical Thermodynamics ◽  
Economic Systems ◽  
Global Behavior ◽  
Probable Distribution ◽  
Or Economics ◽  
Artificial Ecosystems ◽  
Multi Agent ◽  
Most Probable Distribution

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.

Introduction

2017 ◽  
Author(s):  
Jochen Rau
Keyword(s):  
Probability Theory ◽  
Phase Transitions ◽  
Statistical Mechanics ◽  
Conservation Laws ◽  
Law Of Large Numbers ◽  
Macroscopic Scale ◽  
Classical Probability ◽  
Large Numbers ◽  
Microscopic World ◽  
New Phenomena

Statistical mechanics concerns the transition from the microscopic to the macroscopic realm. On a macroscopic scale new phenomena arise that have no counterpart in the microscopic world. For example, macroscopic systems have a temperature; they might undergo phase transitions; and their dynamics may involve dissipation. How can such phenomena be explained? This chapter discusses the characteristic differences between the microscopic and macroscopic realms and lays out the basic challenge of statistical mechanics. It suggests how, in principle, this challenge can be tackled with the help of conservation laws and statistics. The chapter reviews some basic notions of classical probability theory. In particular, it discusses the law of large numbers and illustrates how, despite the indeterminacy of individual events, statistics can make highly accurate predictions about totals and averages.

Method of most probable distribution: New solutions and results

Pramana ◽  
1989 ◽  
Vol 33 (4) ◽  
pp. 455-465 ◽  
Author(s):  
V J Menon ◽  
D C Agrawal
Keyword(s):  
Probable Distribution ◽  
Most Probable Distribution

The Emergence of Statistical Mechanics

2017 ◽  
Author(s):  
Olivier Darrigol ◽  
Jürgen Renn
Keyword(s):  
Kinetic Theory ◽  
Statistical Mechanics ◽  
Statistical Physics ◽  
Second Law Of Thermodynamics ◽  
Boltzmann Distribution ◽  
Probabilistic Interpretation ◽  
Specific Heats ◽  
Mechanical Models ◽  
History Of ◽  
Thermal Phenomena

This article traces the history of statistical mechanics, beginning with a discussion of mechanical models of thermal phenomena. In particular, it considers how several circumstances, including the establishment of thermodynamics in the mid-nineteenth century, led to a focus on the model of heat as a motion of particles. It then describes the concept of heat as fluid and the kinetic theory before turning to gas theory and how it served as a bridge between mechanics and thermodynamics. It also explores gases as particles in motion, the Maxwell–Boltzmann distribution, the problem of specific heats, challenges to the second law of thermodynamics, and the probabilistic interpretation of entropy. Finally, it examines how the results of the kinetic theory assumed a new meaning as cornerstones of a more broadly conceived statistical physics, along with Josiah Willard Gibbs and Albert Einstein’s development of statistical mechanics as a synthetic framework.

The polymerization of styrene by sulphuric acid - III. The molecular weight distribution

1961 ◽  
Vol 263 (1312) ◽  
pp. 75-81 ◽  
Keyword(s):  
Molecular Weight ◽  
Sulphuric Acid ◽  
Molecular Weight Distribution ◽  
Weight Distribution ◽  
High Conversion ◽  
Chain Growth ◽  
Characteristic Parameters ◽  
Relative Probability ◽  
Probable Distribution ◽  
Most Probable Distribution

The theory of the molecular weight distribution in polystyrenes initiated by sulphuric acid, described in part I, has been tested by preparation and fractionation of suitable low-yield polymer samples. The expected ‘most probable’ distribution is found in these samples, but not in a high-conversion polymer. The characteristic parameters of the distributions the relative probability of chain growth—agree with values calculated from the kinetic con­stants measured in part II.

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