Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles

Description:

This dissertation aims at addressing two important theoretical questions which are still debated in the statistical mechanical community. The first question has to do with the outstanding problem of how to reconcile time-reversal asymmetric macroscopic laws with the time-reversal symmetric laws of microscopic dynamics. This problem is addressed by developing a novel mechanical approach inspired by the work of Helmholtz on monocyclic systems and the Heat Theorem, i.e., the Helmholtz Theorem. By following a line of investigation initiated by Boltzmann, a Generalized Helmholtz Theorem is stated and proved. This theorem provides us with a good microscopic analogue of thermodynamic entropy. This is the volume entropy, namely the logarithm of the volume of phase space enclosed by the constant energy hyper-surface. By using quantum mechanics only, it is shown that such entropy can only increase. This can be seen as a novel rigorous proof of the Second Law of Thermodynamics that sheds new light onto the arrow of time problem. The volume entropy behaves in a thermodynamic-like way independent of the number of degrees of freedom of the system, indicating that a whole thermodynamic-like world exists at the microscopic level. It is also shown that breaking of ergodicity leads to microcanonical phase transitions associated with nonanalyticities of volume entropy. The second part of the dissertation deals with the problem of the foundations of generalized ensembles in statistical mechanics. The starting point is Boltzmann's work on statistical ensembles and its relation with the Heat Theorem. We first focus on the nonextensive thermostatistics of Tsallis and the associated deformed exponential ensembles. These ensembles are analyzed in detail and proved (a) to comply with the requirements posed by the Heat Theorem, and (b) to interpolate between canonical and microcanonical ensembles. Further they are showed to describe finite systems in contact with finite heat baths. Their mechanical and information-theoretic foundation, are highlighted. Finally, a wide class of generalized ensembles is introduced, all of which reproduce the Heat Theorem. This class, named the class of dual orthodes, contains microcanonical, canonical, Tsallis and Gaussian ensembles as special cases.

Creator(s): Campisi, Michele
Creation Date: May 2008
Partner(s):
UNT Libraries
Collection(s):
UNT Theses and Dissertations
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Total Uses: 406
Past 30 days: 5
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Publisher Info:
Publisher Name: University of North Texas
Place of Publication: Denton, Texas
Date(s):
  • Creation: May 2008
  • Digitized: July 25, 2008
Description:

This dissertation aims at addressing two important theoretical questions which are still debated in the statistical mechanical community. The first question has to do with the outstanding problem of how to reconcile time-reversal asymmetric macroscopic laws with the time-reversal symmetric laws of microscopic dynamics. This problem is addressed by developing a novel mechanical approach inspired by the work of Helmholtz on monocyclic systems and the Heat Theorem, i.e., the Helmholtz Theorem. By following a line of investigation initiated by Boltzmann, a Generalized Helmholtz Theorem is stated and proved. This theorem provides us with a good microscopic analogue of thermodynamic entropy. This is the volume entropy, namely the logarithm of the volume of phase space enclosed by the constant energy hyper-surface. By using quantum mechanics only, it is shown that such entropy can only increase. This can be seen as a novel rigorous proof of the Second Law of Thermodynamics that sheds new light onto the arrow of time problem. The volume entropy behaves in a thermodynamic-like way independent of the number of degrees of freedom of the system, indicating that a whole thermodynamic-like world exists at the microscopic level. It is also shown that breaking of ergodicity leads to microcanonical phase transitions associated with nonanalyticities of volume entropy. The second part of the dissertation deals with the problem of the foundations of generalized ensembles in statistical mechanics. The starting point is Boltzmann's work on statistical ensembles and its relation with the Heat Theorem. We first focus on the nonextensive thermostatistics of Tsallis and the associated deformed exponential ensembles. These ensembles are analyzed in detail and proved (a) to comply with the requirements posed by the Heat Theorem, and (b) to interpolate between canonical and microcanonical ensembles. Further they are showed to describe finite systems in contact with finite heat baths. Their mechanical and information-theoretic foundation, are highlighted. Finally, a wide class of generalized ensembles is introduced, all of which reproduce the Heat Theorem. This class, named the class of dual orthodes, contains microcanonical, canonical, Tsallis and Gaussian ensembles as special cases.

Degree:
Level: Doctoral
Discipline: Physics
Language(s):
Subject(s):
Keyword(s): finite heat bath | Heat theorem | microcanonical AMSE transition | second law
Contributor(s):
Partner:
UNT Libraries
Collection:
UNT Theses and Dissertations
Identifier:
  • OCLC: 265054086 |
  • ARK: ark:/67531/metadc6128
Resource Type: Thesis or Dissertation
Format: Text
Rights:
Access: Public
License: Copyright
Holder: Campisi, Michele
Statement: Copyright is held by the author, unless otherwise noted. All rights reserved.