Ph.D. Theses (Mathematics)
http://hdl.handle.net/1957/15739
2014-04-23T18:16:03ZCanonical states in quantum statistical mechanics
http://hdl.handle.net/1957/47523
Canonical states in quantum statistical mechanics
Kvarda, Robert Edward
This report presents a characterization of the quantum mechanical
analog of the Gibbs canonical density. The approach is
based on a method developed by D.S. Carter for the case of classical
statistical mechanics, which considers composite mechanical systems
composed of mechanically and statistically independent components.
After a brief introductory chapter, Chapter II outlines how
the case of classical mechanics may be described in terms of the
usual measure theoretic treatment of probability. The necessary
statistical background of quantum mechanics is then discussed in
Chapter III, relying on the classic treatment of J. von Neumann and
the more recent work of G. W. Mackey. The basic idea of probability
measure in quantum mechanics differs from that in classical measure
theory, for the measure is defined on a non-Boolean lattice consisting
of all closed linear subspaces of a Hilbert space. Because of this
difference, the classical theory of product measures does not apply. Chapter IV presents a detailed treatment of probability measures for
composite quantum systems.
The analog of the Gibbs canonical density is characterized in
Chapter V, by considering a large collection Q of noninteracting
quantum systems, each of which is in an equilibrium statistical state.
The set Q, the Hamiltonian operator for each system, and the
equilibrium states are assumed to have certain properties which are
given as axioms.
The axioms require each Hamiltonian operator to have a pure
point spectrum. It is assumed, without loss of generality, that the
lowest characteristic value of each Hamiltonian is zero. The set Q
is assumed to be closed under the formation of pairwise mechanically
independent composite systems. This implies that the set D of all
Hamiltonian spectra is closed under addition. It is further assumed
that D is closed under positive differences. The final requirement
on the set Q is that it contain certain "harmonic oscillators".
More precisely, for each positive λεD, Q must contain a system
whose Hamiltonian has the spectrum {nλ : n=0,1,2,[superscript ...]}. The usual
assumption is made that each density operator is a function of the
system Hamiltonian. Finally, it is assumed that for each composite
system in Q, with two mechanically independent components, the
component systems are statistically independent.
It is shown that these assumptions imply that each member of Q is in a canonical state at a temperature which is the same for all
systems. The possibility of zero absolute temperature is included.
Graduation date: 1966
1965-08-26T00:00:00ZThe local conservation laws of the nonlinear Schrödinger equation
http://hdl.handle.net/1957/44749
The local conservation laws of the nonlinear Schrödinger equation
Barrett, Johnner
The nonlinear Schrödinger equation is a well-known partial differential equation that provides a successful model in nonlinear optic theory, as well as other applications. In this dissertation, following a survey of mathematical literature, the geometric theory of differential equations is applied to the nonlinear Schrödinger equation. The main result of this dissertation is that the known list of local conservation laws for the nonlinear Schrödinger equation is complete. A theorem is proved and used to produce a sequence of local conservation law characteristics of the nonlinear Schrödinger equation. The list of local conservation laws as given by Faddeev and Takhtajan and a theorem of Olver, which provides a one-to-one correspondence between equivalence classes of conservation laws and equivalence classes of their characteristics, are then used to prove the main result.
Graduation date: 2014; Access restricted to the OSU Community, at author's request, from Dec. 12, 2013 - June 12, 2014
2013-11-08T00:00:00ZAdiabatic and stable adiabatic times
http://hdl.handle.net/1957/40078
Adiabatic and stable adiabatic times
Bradford, Kyle B.
While the stability of time-homogeneous Markov chains have been extensively studied through the concept of mixing times, the stability of time-inhomogeneous Markov chains has not been studied as in depth. In this manuscript we will introduce special types of time-inhomogeneous Markov chains that are defined through an adiabatic transition. After doing this, we define the adiabatic and the stable adiabatic times as measures of stability these special time-inhomogeneous Markov chains. To construct an adiabatic transition one needs to make a transitioning convex combination of an initial and final probability transition matrix over the time interval [0, 1] for two time-homogeneous, discrete time, aperiodic and irreducible Markov chains. The adiabatic and stable adiabatic times depend on how this convex combinations transitions. In the most general setting, we suggested that as long as P : [0, 1] --> P[superscript ia][subscript n] is a Lipschitz continuous function with respect to the ‖ ·‖₁ matrix norm, then the adiabatic time is bounded above by a function of the mixing time of the final probability transition matrix [equation] For the stable adiabatic time, the most general result we achieved was for nonlinear adiabatic transitions P[subscript ø (t)] = (1-ø (t))P₀+ ø(t)P₁ where ø is a Lipschitz continuous functions that is piecewise defined over a finite partition of the interval [0, 1] so that on each subinterval ø is a bi-Lipschitz continuous function. In this setting we asymptotically bounded the stable adiabatic time by the largest mixing of P[subscript ø(t)] over all t∈[0, 1]. We found that [equation] We also have some additional results at bound the stable adiabatic time in this manuscript, but they are included to show the different attempts we took and highlight how important it is to pick the right variables to compare. We also provide examples to queueing and statistical mechanics.
Graduation date: 2013
2013-05-15T00:00:00ZContinued fractions and the divisor at infinity on a hyperelliptic curve : examples and order bounds
http://hdl.handle.net/1957/38663
Continued fractions and the divisor at infinity on a hyperelliptic curve : examples and order bounds
Daowsud, Katthaleeya
We use the theory of continued fractions over function fields in the setting of hyperelliptic curves of equation y²=f(x), with deg(f)=2g+2. By introducing a new sequence of polynomials defined in terms of the partial quotients of the continued fraction expansion of y, we are able to bound the sum of the degrees of consecutive partial quotients. This allows us both (1) to improve the known naive upper bound for the order N of the divisor at infinity on a hyperelliptic curve; and, (2) to apply a naive method to search for hyperelliptic curves of given genus g and order N. In particular, we present new families defined over ℚ with N=11 and 2 ≤ g ≤ 10.
Graduation date: 2013
2013-04-25T00:00:00Z