FAREWELL TO ENTROPY,A: Statistical Thermodynamics Based on Information

by Arieh Ben-naim

Overview

The principal message of this book is that thermodynamics and statistical mechanics will benefit from replacing the unfortunate, misleading and mysterious term “entropy” with a more familiar, meaningful and appropriate term such as information, missing information or uncertainty. This replacement would facilitate the interpretation of the “driving force” of many processes in terms of informational changes and dispel the mystery that has always enshrouded entropy.It has been 140 years since Clausius coined the term “entropy”; almost 50 years since Shannon developed the mathematical theory of “information” — subsequently renamed “entropy”. In this book, the author advocates replacing “entropy” by “information”, a term that has become widely used in many branches of science.The author also takes a new and bold approach to thermodynamics and statistical mechanics. Information is used not only as a tool for predicting distributions but as the fundamental cornerstone concept of thermodynamics, held until now by the term “entropy”.The topics covered include the fundamentals of probability and information theory; the general concept of information as well as the particular concept of information as applied in thermodynamics; the re-derivation of the Sackur-Tetrode equation for the entropy of an ideal gas from purely informational arguments; the fundamental formalism of statistical mechanics; and many examples of simple processes the “driving force” for which is analyzed in terms of information.

Product Details

ISBN-13:	9789814338288
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/18/2008
Sold by:	Barnes & Noble
Format:	eBook
Pages:	412
File size:	7 MB

Table of Contents

List of Abbreviations     xiii
Preface     xv
Introduction     1
A Brief History of Temperature and Entropy     1
The Association of Entropy with Disorder     9
The Association of Entropy with Missing Information     19
Elements of Probability Theory     33
Introduction     33
The Axiomatic Approach     36
The sample space, denoted [Omega]     36
The field of events, denoted F     37
The probability function, denoted P     39
The Classical Definition     43
The Relative Frequency Definition     45
Independent Events and Conditional Probability     50
Conditional probability and subjective probability     58
Conditional probability and cause and effect     62
Conditional probability and probability of joint events     64
Bayes' Theorem     65
A challenging problem     72
A more challenging problem: The three prisoners' problem     74
Random Variables, Average, Variance and Correlation     76
Some Specific Distributions     86
The binomial distribution     86
The normal distribution     90
The Poissondistribution     93
Generating Functions     94
The Law of Large Numbers     100
Elements of Information Theory     103
A Qualitative Introduction to Information Theory     104
Definition of Shannon's Information and Its Properties     110
Properties of the function H for the simplest case of two outcomes     112
Properties of H for the general case of n outcomes     114
The consistency property of the missing information (MI)     125
The case of an infinite number of outcomes     130
The Various Interpretations of the Quantity H     138
The Assignment of Probabilities by the Maximum Uncertainty Principle     144
The Missing Information and the Average Number of Binary Questions Needed to Acquire It     149
The False Positive Problem, Revisited     170
The Urn Problem, Revisited     172
Transition from the General MI to the Thermodynamic MI     177
MI in Binding Systems: One Kind of Information     178
One ligand on M sites     179
Two different ligands on M sites     179
Two identical ligands on M sites     182
Generalization to N ligands on M sites     183
Some Simple Processes in Binding Systems      186
The analog of the expansion process     187
A pure deassimilation process     190
Mixing process in a binding system     194
The dependence of MI on the characterization of the system     196
MI in an Ideal Gas System: Two Kinds of Information. The Sackur-Tetrode Equation     201
The locational MI     201
The momentum MI     204
Combining the locational and the momentum MI     205
Comments     207
The Structure of the Foundations of Statistical Thermodynamics     211
The Isolated System; The Micro-Canonical Ensemble     213
System in a Constant Temperature; The Canonical Ensemble     220
The Classical Analog of the Canonical Partition Function     228
The Re-interpretation of the Sackur-Tetrode Expression from Informational Considerations     232
Identifying the Parameter [beta] for an Ideal Gas     235
Systems at Constant Temperature and Chemical Potential; The Grand Canonical Ensemble     236
Systems at Constant Temperature and Pressure; The Isothermal Isobaric Ensemble     242
The Mutual Information due to Intermolecular Interactions     244
Some Simple Applications     251
Expansion of an Ideal Gas     252
Pure, Reversible Mixing; The First Illusion     255
Pure Assimilation Process; The Second Illusion     257
Fermi-Dirac (FD) statistics; Fermions     259
Bose-Einstein (BE) statistics; Bosons     260
Maxwell-Boltzmann (MB) statistics     261
Irreversible Process of Mixing Coupled with Expansion     265
Irreversible Process of Demixing Coupled with Expansion     268
Reversible Assimilation Coupled with Expansion     270
Reflections on the Processes of Mixing and Assimilation     272
A Pure Spontaneous Deassimilation Process     284
A Process Involving only Change in the Momentum Distribution     287
A Process Involving Change in the Intermolecular Interaction Energy     290
Some Baffling Experiments     293
The Second Law of Thermodynamics     298
Appendices     317
Newton's binomial theorem and some useful identities involving binomial coefficients     317
The total number of states in the Fermi-Dirac and the Bose-Einstein statistics     319
Pair and triplet independence between events     321
Proof of the inequality [vertical bar]R(X, Y)[vertical bar] [less than or equal] 1 for the correlation coefficient     322
The Stirling approximation     326
Proof of the form of the function H     327
The method of Lagrange undetermined multipliers     331
Some inequalities for concave functions     334
The MI for the continuous case     340
Identical and indistinguishable (ID) particles     343
The equivalence of the Boltzmann's and Jaynes' procedures to obtain the fundamental distribution of the canonical ensemble     350
An alternative derivation of the Sackur-Tetrode equation     352
Labeling and un-labeling of particles     355
Replacing a sum by its maximal term     356
The Gibbs paradox (GP)     360
The solution to the three prisoners' problem     363
References     373
Index     381