Maxwell's Equations from Electrostatics and Einstein's Gravitational Field Equation from Newton's Universal Law of Gravitation Using Tensors

Maxwell's Equations from Electrostatics and Einstein's Gravitational Field Equation from Newton's Universal Law of Gravitation Using Tensors

Date: May 2004
Creator: Burns, Michael E.
Description: Maxwell's equations are obtained from Coulomb's Law using special relativity. For the derivation, tensor analysis is used, charge is assumed to be a conserved scalar, the Lorentz force is assumed to be a pure force, and the principle of superposition is assumed to hold. Einstein's gravitational field equation is obtained from Newton's universal law of gravitation. In order to proceed, the principle of least action for gravity is shown to be equivalent to the maximization of proper time along a geodesic. The conservation of energy and momentum is assumed, which, through the use of the Bianchi identity, results in Einstein's field equation.
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Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor

Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor

Date: August 2005
Creator: Zheng, Yindong
Description: The de Broglie-Bohm (BB) approach to quantum mechanics gives trajectories similar to classical trajectories except that they are also determined by a quantum potential. The quantum potential is a "fictitious potential" in the sense that it is part of the quantum kinetic energy. We use quantum trajectories to treat quantum chaos in a manner similar to classical chaos. For the kicked rotor, which is a bounded system, we use the Benettin et al. method to calculate both classical and quantum Lyapunov exponents as a function of control parameter K and find chaos in both cases. Within the chaotic sea we find in both cases nonchaotic stability regions for K equal to multiples of π. For even multiples of π the stability regions are associated with classical accelerator mode islands and for odd multiples of π they are associated with new oscillator modes. We examine the structure of these regions. Momentum diffusion of the quantum kicked rotor is studied with both BB and standard quantum mechanics (SQM). A general analytical expression is given for the momentum diffusion at quantum resonance of both BB and SQM. We obtain agreement between the two approaches in numerical experiments. For the case of nonresonance the ...
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Two-Fold Role of Randomness: A Source of Both Long-Range Correlations and Ordinary Statistical Mechanics

Two-Fold Role of Randomness: A Source of Both Long-Range Correlations and Ordinary Statistical Mechanics

Date: December 1998
Creator: Rocco, A. (Andrea)
Description: The role of randomness as a generator of long range correlations and ordinary statistical mechanics is investigated in this Dissertation. The difficulties about the derivation of thermodynamics from mechanics are pointed out and the connection between the ordinary fluctuation-dissipation process and possible anomalous properties of statistical systems is highlighted.
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On Delocalization Effects in Multidimensional Lattices

On Delocalization Effects in Multidimensional Lattices

Date: May 1998
Creator: Bystrik, Anna
Description: A cubic lattice with random parameters is reduced to a linear chain by the means of the projection technique. The continued fraction expansion (c.f.e.) approach is herein applied to the density of states. Coefficients of the c.f.e. are obtained numerically by the recursion procedure. Properties of the non-stationary second moments (correlations and dispersions) of their distribution are studied in a connection with the other evidences of transport in a one-dimensional Mori chain. The second moments and the spectral density are computed for the various degrees of disorder in the prototype lattice. The possible directions of the further development are outlined. The physical problem that is addressed in the dissertation is the possibility of the existence of a non-Anderson disorder of a specific type. More precisely, this type of a disorder in the one-dimensional case would result in a positive localization threshold. A specific type of such non-Anderson disorder was obtained by adopting a transformation procedure which assigns to the matrix expressing the physics of the multidimensional crystal a tridiagonal Hamiltonian. This Hamiltonian is then assigned to an equivalent one-dimensional tight-binding model. One of the benefits of this approach is that we are guaranteed to obtain a linear crystal with a ...
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Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles

Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles

Date: May 2008
Creator: Campisi, Michele
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 ...
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The Nonadditive Generalization of Klimontovich's S-Theorem for Open Systems and Boltzmann's Orthodes

The Nonadditive Generalization of Klimontovich's S-Theorem for Open Systems and Boltzmann's Orthodes

Date: August 2008
Creator: Bagci, Gokhan Baris
Description: We show that the nonadditive open systems can be studied in a consistent manner by using a generalized version of S-theorem. This new generalized S-theorem can further be considered as an indication of self-organization in nonadditive open systems as prescribed by Haken. The nonadditive S-theorem is then illustrated by using the modified Van der Pol oscillator. Finally, Tsallis entropy as an equilibrium entropy is studied by using Boltzmann's method of orthodes. This part of dissertation shows that Tsallis ensemble is on equal footing with the microcanonical, canonical and grand canonical ensembles. However, the associated entropy turns out to be Renyi entropy.
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