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15
The Gibbs Paradox
, 1996
"... : We point out that an early work of J. Willard Gibbs (1875) contains a correct analysis of the "Gibbs Paradox" about entropy of mixing, free of any elements of mystery and directly connected to experimental facts. However, it appears that this has been lost for 100 years, due to some obscurities in ..."
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: We point out that an early work of J. Willard Gibbs (1875) contains a correct analysis of the "Gibbs Paradox" about entropy of mixing, free of any elements of mystery and directly connected to experimental facts. However, it appears that this has been lost for 100 years, due to some obscurities in Gibbs' style of writing and his failure to include this explanation in his later Statistical Mechanics. This "new" understanding is not only of historical and pedagogical interest; it gives both classical and quantum statistical mechanics a different status than that presented in our textbooks, with implications for current research. CONTENTS 1. INTRODUCTION 2 2. THE PROBLEM 3 3. THE EXPLANATION 4 4. DISCUSSION 6 5. THE GAS MIXING SCENARIO REVISITED 7 6. SECOND LAW TRICKERY 9 7. THE PAULI ANALYSIS 10 8. WOULD GIBBS HAVE ACCEPTED IT? 11 9. GIBBS' STATISTICAL MECHANICS 13 10. SUMMARY AND UNFINISHED BUSINESS 17 11. REFERENCES 18 y In Maximum Entropy and Bayesian Methods, C. R. Smith, G. J. E...
TimeReversal and Entropy
, 2002
"... There is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of timereversal symmetry, and the entropy production in such processes. We explain on an elementary mathematical level the relations between entropy production, phasespace contraction and timer ..."
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There is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of timereversal symmetry, and the entropy production in such processes. We explain on an elementary mathematical level the relations between entropy production, phasespace contraction and timereversal starting from a deterministic dynamics. Both closed and open systems, in the transient and in the steady regime, are considered. The main result identifies under general conditions the statistical mechanical entropy production as the source term of timereversal breaking in the path space measure for the evolution of reduced variables. This provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful (in)equalities. We also discuss the Markov approximation. Important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior.
On causally asymmetric versions of Occam’s Razor and their relation to thermodynamics
, 2007
"... and their relation to thermodynamics ..."
A selection of nonequilibrium issues
 In Lecture notes in Mathematics
"... Summary. We give a pedagogical introduction to a selection of recently discussed topics in nonequilibrium statistical mechanics, concentrating mostly on formal structures and on general principles. Part I contains an overview of the formalism of lattice gases that we use to explain various symmetrie ..."
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Summary. We give a pedagogical introduction to a selection of recently discussed topics in nonequilibrium statistical mechanics, concentrating mostly on formal structures and on general principles. Part I contains an overview of the formalism of lattice gases that we use to explain various symmetries and inequalities generally valid for nonequilibrium systems, including the fluctuation symmetry, Jarzynski equality, and the direction of currents. That mostly concerns the timeantisymmetric part of dynamical fluctuation theory. We also briefly comment on recent attempts to combine that with the timesymmetric sector in a Langrangian or extended OnsagerMachlup approach. In Part II we concentrate on the macroscopic state and how entropy provides a bridge between microscopic dynamics and macroscopic irreversibility; included is a construction of quantum macroscopic states and a result on the equivalence of ensembles. Part I. Fluctuations in stochastic lattice gases 1
HTheorems from Autonomous Equations
, 2005
"... Abstract: The Htheorem is an extension of the Second Law to a timesequence of states that need not be equilibrium ones. In this paper we review and we rigorously establish the connection with macroscopic autonomy. If for a Hamiltonian dynamics for many particles, at all times the present macrostat ..."
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Abstract: The Htheorem is an extension of the Second Law to a timesequence of states that need not be equilibrium ones. In this paper we review and we rigorously establish the connection with macroscopic autonomy. If for a Hamiltonian dynamics for many particles, at all times the present macrostate determines the future macrostate, then its entropy is nondecreasing as a consequence of Liouville’s theorem. That observation, made since long, is here rigorously analyzed with special care to reconcile the application of Liouville’s theorem (for a finite number of particles) with the condition of autonomous macroscopic evolution (sharp only in the limit of infinite scale separation); and to evaluate the presumed necessity of a Markov property for the macroscopic evolution. KEY WORDS: Htheorem, entropy, irreversible equations 1
Inferential Scattering
"... : Some difficult conceptual problems that have plagued Statistical Mechanics from the start, are explained by reference to a very simple experiment, which also explains why the MAXENT formalism gives reliable predictions. Then some of the inner workings of MAXENT are revealed by a general perturbati ..."
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: Some difficult conceptual problems that have plagued Statistical Mechanics from the start, are explained by reference to a very simple experiment, which also explains why the MAXENT formalism gives reliable predictions. Then some of the inner workings of MAXENT are revealed by a general perturbation theorem, showing how a prediction is modified by adding a new constraint. It is illustrated by the example of Rayleigh scattering in acoustics. Here it appears rather like Schwinger's Source Theory in that multiple scattered waves of arbitrarily high order appear already in the first order of the MAXENT perturbation scheme. The result holds in much more general problems of "inferential scattering" in which any statistical inference is modified by new information. 1. HISTORICAL BACKGROUND 2 2. INFERENTIAL SCATTERING 2 3. GENERALIZED FIRST LAW 4 4. THE BASIC DILEMMA 5 5. THOSE NUMBERS 8 6. SO WHY DOES MAXENT WORK? 9 7. MACROSCOPIC PREDICTIONS 10 8. MEANING OF THE GIBBS CONVEXITY 12 9. MITCH...
Note On Thermal Heating Efficiency
, 1996
"... Kelvin showed the maximum efficiency with which heat can be converted into work; but there is a dual theorem about the maximum efficiency with which heat at one temperature can be converted into heat at another temperature. It has some surprising implications, in particular that the efficiency with ..."
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Kelvin showed the maximum efficiency with which heat can be converted into work; but there is a dual theorem about the maximum efficiency with which heat at one temperature can be converted into heat at another temperature. It has some surprising implications, in particular that the efficiency with which we heat our buildings could in principle be improved by a large factor. This long known  but still little known  fact is of current pedagogical interest and practical importance.