Time's Arrow: The Origins of Thermodynamic Behavior

by Michael C. Mackey

Overview

Written by a well-known professor of physiology at McGill University, this text presents an informative exploration of the basis of the Second Law of Thermodynamics, detailing the fundamental dynamic properties behind the construction of statistical mechanics. Topics include maximal entropy principles; invertible and noninvertible systems; ergodicity and unique equilibria; asymptotic periodicity and entropy evolution; and open discrete and continuous time systems. The author demonstrates that the black body radiation law can be deduced from maximal entropy principles; discusses sufficient conditions for the existence of at least one state of thermodynamic equilibrium; describes the behavior of entropy in asymptotically periodic systems and the necessary and sufficient condition for the evolution of entropy to a global maximum; and presents the three main types of ergodic theorems and their proofs. He also explores the potential of incomplete knowledge of dynamical variables, measurement imprecision, and the effects of noise in entropy increases. Geared toward physicists and applied mathematicians with an interest in the foundations of statistical mechanics, this text is suitable for advanced undergraduate and graduate courses.

Product Details

ISBN-13:	9780486152257
Publisher:	Dover Publications
Publication date:	11/30/2011
Sold by:	Barnes & Noble
Format:	eBook
Pages:	192
File size:	11 MB
Note:	This product may take a few minutes to download.

Table of Contents

	Preface	ix
Chapter 1.	Starters	1
A.	Thermodynamic Systems	1
B.	Dynamics	2
C.	Thermodynamic States	4
D.	Boltzmann-Gibbs Entropy	5
E.	Summary	8
Chapter 2.	Maximal Entropy Principles	9
A.	Microcanonical Ensembles	9
B.	Canonical Ensembles	11
C.	The Thermodynamic Connection	12
D.	Boltzmann and Other Statistics	16
E.	Blackbody Radiation	18
F.	General Entropy Maxima	19
G.	Maximal Entropy Principles	20
H.	Summary	20
Chapter 3.	Invertible and Noninvertible Systems	21
A.	Markov Operators	21
B.	Conditional Entropy	27
C.	Existence of Equilibria	34
D.	Summary	37
Chapter 4.	Ergodicity and Unique Equilibria	39
A.	Frobenius-Perron Operator	39
B.	The Koopman Operator	43
C.	Recurrence	45
D.	Densities and Flows	46
E.	Ergodicity	48
F.	Summary	57
Chapter 5.	Mixing	58
A.	Mixing	58
B.	Kolmogorov Systems	66
C.	The Decay of Correlations	67
D.	Summary	68
Chapter 6.	Asymptotic Periodicity and Entropy Evolution	70
A.	Asymptotic Periodicity	70
B.	Asymptotic Periodicity Illustrated	75
C.	The Weak Form of the Second Law	80
D.	Asymptotic Periodicity and Correlations	86
E.	Summary	88
Chapter 7.	Global Evolution of Entropy	89
A.	Exactness	89
B.	The Strong Form of the Second Law	98
C.	Summary	101
Chapter 8.	Coarse Graining	103
A.	Coarse Grained Entropy	103
B.	Coarse Graining of Mixing Systems	106
C.	Sampling	107
D.	Summary	109
Chapter 9.	Traces and Factors	110
A.	Traces	110
B.	Factors	112
C.	Coarse Graining and Traces	113
D.	Summary	114
Chapter 10.	Open Discrete Time Systems	115
A.	An Operator Equation	116
B.	Loosely Coupled Systems	117
C.	Strongly Coupled Systems	120
D.	Asymptotic Periodicity and Additive Perturbations	123
E.	Exactness and Additive Perturbations	129
F.	Parametric Perturbations	131
G.	Markov Operators and Deterministic Processes	136
H.	Summary	138
Chapter 11.	Open Continuous Time Systems	140
A.	Wiener Processes and Stochastic Differential Equations	140
B.	The Fokker-Planck Equation	142
C.	Solutions of the Fokker-Planck Equation	145
D.	The Behavior of Entropy	148
E.	Phase Transitions and Additive Perturbations	151
F.	Parametric Perturbations and Phase Transitions	156
G.	Summary	158
	References	159
	Index	167