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 timereversal asymmetric macroscopic laws with the timereversal 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 hypersurface. 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 thermodynamiclike way independent of the number of degrees of freedom of the system, indicating that a whole thermodynamiclike 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 informationtheoretic 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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Publisher Info: 
Publisher Name: University of North Texas
Place of Publication: Denton, Texas


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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 timereversal asymmetric macroscopic laws with the timereversal 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 hypersurface. 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 thermodynamiclike way independent of the number of degrees of freedom of the system, indicating that a whole thermodynamiclike 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 informationtheoretic 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: 
Name:
Doctor of Philosophy
Level:
Doctoral
Discipline:
Physics
Department:
Department of Physics
Grantor:
University of North Texas


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Subject(s): 


Keyword(s):  finite heat bath  Heat theorem  microcanonical AMSE transition  second law  
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Partner: 
UNT Libraries


Collection: 
UNT Theses and Dissertations


Identifier:  
Resource Type:  Thesis or Dissertation  
Format:  Text  
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Holder:
Campisi, Michele
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Copyright is held by the author, unless otherwise noted. All rights reserved.
