Microcanonical Thermodynamics : Phase Transitions in Small Systems

Phase Transitions in Small Systems

By: Dieter H E Gross

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Hardcover | 1 December 1999

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Boltzmann's formula S = In(W(E) defines the microcanonical ensemble. The usual textbooks on statistical mechanics start with the microensemble but rather quickly switch to the canonical ensemble introduced by Gibbs. This has the main advantage of easier analytical calculations, but there is a price to pay -- for example, phase transitions can only be defined in the thermodynamic limit of infinite system size. The question how phase transitions show up from systems with, say, 100 particles with an increasing number towards the bulk can only be answered when one finds a way to define and classify phase transitions in small systems. This is all possible within Boltzmann's original definition of the microcanonical ensemble. Starting from Boltzmann's formula, the book formulates the microcanonical thermodynamics entirely within the frame of mechanics. This way the thermodynamic limit is avoided and the formalism applies to small as well to other nonextensive systems like gravitational ones. Phasetransitions of first order, continuous transitions, critical lines and multicritical points can be unambiguously defined by the curvature of the entropy S(E, N). Special attention is given to the fragmentation of nuclei and atomic clusters as a peculiar phase transition of small systems controlled, among others, by angular momentum. The dependence of the liquid-gas transition of small atomic clusters under prescribed pressure is treated. Thus the analogue to the bulk transition can be studied. New insights into the many facets of the many-body physics of the critical point are presented. The book also describes the microcanonical statistics of the collapse of a self-gravitating system under large angular momentum.

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Preface	p. vii
Who is addressed, and why	p. vii
A necessary clarification	p. viii
Acknowledgment	p. ix
Introduction	p. 1
Phase transitions and thermodynamics in "Small" Systems	p. 1
Boltzmann gives the key	p. 4
Micro-canonical Thermodynamics describes non-extensive systems	p. 7
Some realistic systems: Nuclei and atomic clusters	p. 13
Plan of this book	p. 14
The Mechanical Basis of Thermodynamics	p. 19
Basic definitions	p. 19
The thermodynamic limit, the global concavity of s(e, n)	p. 21
Phase transitions micro-canonically	p. 22
Second Law of Thermodynamics and Boltzmann's entropy	p. 29
Micro-canonical thermodynamics of Phase Transitions studied in the Potts model	p. 35
Introduction	p. 35
The surface tension in the Potts model. [GEZ50]	p. 36
The topology of the entropy surface S(E, N) for Potts lattice gases [GV99]	p. 45
On the origin of isolated critical points and critical lines	p. 66
Liquid-gas transition and surface tension under constant pressure	p. 93
Andersen's constant pressure ensemble	p. 93
Micro-canonical ensemble with given pressure; The enthalpy	p. 94
Liquid-gas transition in realistic metal systems	p. 97
The relation to the method of the Gibbs-ensemble	p. 110
Alternative microscopic methods to calculate the surface tension	p. 112
Criticism and necessary improvements of the computational method	p. 113
Conclusion	p. 114
Statistical Fragmentation under Repulsive Forces of Long Range	p. 119
Introduction	p. 119
Three dimensional stress of long range: The Coulomb force	p. 121
Two dimensional stress of long range: Rapidly rotating hot nuclei [BG95b]	p. 178
Conclusion	p. 186
The collapse transition in self-gravitating systems First model-studies	p. 191
1 - and 2 - dim. Hamiltonian Mean Field Model, a caricature of phase transitions under self-gravitation	p. 192
Collapse of non-extensive (gravitating) systems under conserved angular momentum	p. 203
On the historical development of statistical nuclear multifragmentation models	p. 215
Mathematical partition vs. quantum partition	p. 219
The micro-canonical ensemble of Na-clusters	p. 221
Micro-canonical Metropolis sampling	p. 226
The basic micro-canonical weights	p. 228
The Metropolis pass	p. 238
Some General Technical Aspects of Micro-canonical Monte Carlo Simulation on a Lattice	p. 241
Example: The diluted Potts model	p. 246
Bibliography	p. 249
Index	p. 265
Table of Contents provided by Syndetics. All Rights Reserved.

Microcanonical Thermodynamics : Phase Transitions in Small Systems

Phase Transitions in Small Systems

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