Relating thermodynamics to information theory : the equality of free energy and mutual information

In this thesis we uncover a new relation which links thermodynamics and information theory. We consider time as a channel and the detailed state of a physical system as a message. As the system evolves with time, ever present noise insures that the "message" is corrupted. Thermodynamic free energy measures the approach of the system toward equilibrium. Information theoretical mutual information measures the loss of memory of initial state. We regard the free energy and the mutual information as operators which map probability distributions over state space to real numbers. In the limit of long times, we show how the free energy operator and the mutual information operator asymptotically attain a very simple relationship to one another. This relationship is founded on the common appearance of entropy in the two operators and on an identity between internal energy and conditional entropy. The use of conditional entropy is what distinguishes our approach from previous efforts to relate thermodynamics and information theory.

Nonexistence of looping trajectories in hydromagnetic waves of finite amplitude. Breaking of waves in a cold collision-free plasma in a magnetic field. On stabilities of periodic waves in a cold collision-free plasma in a magnetic field.

This dissertation consists of three parts. In Part I, it is shown that looping trajectories cannot exist in finite amplitude stationary hydromagnetic waves propagating across a magnetic field in a quasi-neutral cold collision-free plasma. In Part II, time-dependent solutions in series expansion are presented for the magnetic piston problem, which describes waves propagating into a quasi-neutral cold collision-free plasma, ensuing from magnetic disturbances on the boundary of the plasma. The expansion is equivalent to Picard's successive approximations. It is then shown that orbit crossings of plasma particles occur on the boundary for strong disturbances and inside the plasma for weak disturbances. In Part III, the existence of periodic waves propagating at an arbitrary angle to the magnetic field in a plasma is demonstrated by Stokes expansions in amplitude. Then stability analysis is made for such periodic waves with respect to side-band frequency disturbances. It is shown that waves of slow mode are unstable whereas waves of fast mode are stable if the frequency is below the cutoff frequency. The cutoff frequency depends on the propagation angle. For longitudinal propagation the cutoff frequency is equal to one-fourth of the electron's gyrofrequency. For transverse propagation the cutoff frequency is so high that waves of all frequencies are stable.

Nearly free molecular heat transfer from a sphere

Consider a sphere immersed in a rarefied monatomic gas with zero mean flow. The distribution function of the molecules at infinity is chosen to be a Maxwellian. The boundary condition at the body is diffuse reflection with perfect accommodation to the surface temperature. The microscopic flow of particles about the sphere is modeled kinetically by the Boltzmann equation with the Krook collision term. Appropriate normalizations in the near and far fields lead to a perturbation solution of the problem, expanded in terms of the ratio of body diameter to mean free path (inverse Knudsen number). The distribution function is found directly in each region, and intermediate matching is demonstrated. The heat transfer from the sphere is then calculated as an integral over this distribution function in the inner region. Final results indicate that the heat transfer may at first increase over its free flow value before falling to the continuum level.

On-chip integrated label-free optical biosensing

This thesis investigates the design and implementation of a label-free optical biosensing system utilizing a robust on-chip integrated platform. The goal has been to transition optical micro-resonator based label-free biosensing from a laborious and delicate laboratory demonstration to a tool for the analytical life scientist. This has been pursued along four avenues: (1) the design and fabrication of high-$Q$ integrated planar microdisk optical resonators in silicon nitride on silica, (2) the demonstration of a high speed optoelectronic swept frequency laser source, (3) the development and integration of a microfluidic analyte delivery system, and (4) the introduction of a novel differential measurement technique for the reduction of environmental noise.

The optical part of this system combines the results of two major recent developments in the field of optical and laser physics: the high-$Q$ optical resonator and the phase-locked electronically controlled swept-frequency semiconductor laser. The laser operates at a wavelength relevant for aqueous sensing, and replaces expensive and fragile mechanically-tuned laser sources whose frequency sweeps have limited speed, accuracy and reliability. The high-$Q$ optical resonator is part of a monolithic unit with an integrated optical waveguide, and is fabricated using standard semiconductor lithography methods. Monolithic integration makes the system significantly more robust and flexible compared to current, fragile embodiments that rely on the precarious coupling of fragile optical fibers to resonators. The silicon nitride on silica material system allows for future manifestations at shorter wavelengths. The sensor also includes an integrated microfluidic flow cell for precise and low volume delivery of analytes to the resonator surface. We demonstrate the refractive index sensing action of the system as well as the specific and nonspecific adsorption of proteins onto the resonator surface with high sensitivity. Measurement challenges due to environmental noise that hamper system performance are discussed and a differential sensing measurement is proposed, implemented, and demonstrated resulting in the restoration of a high performance sensing measurement.

The instrument developed in this work represents an adaptable and cost-effective platform capable of various sensitive, label-free measurements relevant to the study of biophysics, biomolecular interactions, cell signaling, and a wide range of other life science fields. Further development is necessary for it to be capable of binding assays, or thermodynamic and kinetics measurements; however, this work has laid the foundation for the demonstration of these applications.

Nuclear weak interaction rates during stellar evolution and collapse

Nuclear weak interaction rates, including electron and positron emission rates, and continuum electron and positron capture rates , as well as the associated v and –/v energy loss rates are calculated on a detailed grid of temperature and density for the free nucleons and 226 nuclei with masses between A = 21 and 60. Gamow-Teller and Fermi discrete-state transition matrix element systematics and the Gamow-Teller T^< →/← T^> resonance transitions are discussed in depth and are implemented in the stellar rate calculations. Results of the calculations are presented on an abbreviated grid of temperature and density and comparison is made to terrestrial weak transition rates where possible. Neutron shell blocking of allowed electron capture on heavy nuclei during stellar core collapse is discussed along with several unblocking mechanisms operative at high temperature and density. The results of one-zone collapse calculations are presented which suggest that the effect of neutron shell blocking is to produce a larger core lepton fraction at neutrino trapping which leads to a larger inner-core mass and hence a stronger post-bounce shock.

I. Impact spallation experiments: fracture patterns and spall velocities. II. Craters in carbonate rocks: an electron paramagnetic resonance analysis of shock damage

This work is divided into two independent papers.

PAPER 1.

Spall velocities were measured for nine experimental impacts into San Marcos gabbro targets. Impact velocities ranged from 1 to 6.5 km/sec. Projectiles were iron, aluminum, lead, and basalt of varying sizes. The projectile masses ranged from a 4 g lead bullet to a 0.04 g aluminum sphere. The velocities of fragments were measured from high-speed films taken of the events. The maximum spall velocity observed was 30 m/sec, or 0.56 percent of the 5.4 km/sec impact velocity. The measured velocities were compared to the spall velocities predicted by the spallation model of Melosh (1984). The compatibility between the spallation model for large planetary impacts and the results of these small scale experiments are considered in detail.

The targets were also bisected to observe the pattern of internal fractures. A series of fractures were observed, whose location coincided with the boundary between rock subjected to the peak shock compression and a theoretical "near surface zone" predicted by the spallation model. Thus, between this boundary and the free surface, the target material should receive reduced levels of compressive stress as compared to the more highly shocked region below.

PAPER 2.

Carbonate samples from the nuclear explosion crater, OAK, and a terrestrial impact crater, Meteor Crater, were analyzed for shock damage using electron para- magnetic resonance, EPR. The first series of samples for OAK Crater were obtained from six boreholes within the crater, and the second series were ejecta samples recovered from the crater floor. The degree of shock damage in the carbonate material was assessed by comparing the sample spectra to spectra of Solenhofen limestone, which had been shocked to known pressures.

The results of the OAK borehole analysis have identified a thin zone of highly shocked carbonate material underneath the crater floor. This zone has a maximum depth of approximately 200 ft below sea floor at the ground zero borehole and decreases in depth towards the crater rim. A layer of highly shocked material is also found on the surface in the vicinity of the reference bolehole, located outside the crater. This material could represent a fallout layer. The ejecta samples have experienced a range of shock pressures.

It was also demonstrated that the EPR technique is feasible for the study of terrestrial impact craters formed in carbonate bedrock. The results for the Meteor Crater analysis suggest a slight degree of shock damage present in the β member of the Kaibab Formation exposed in the crater walls.

Label-free detection of single biological molecules using microtoroid optical resonators

Being able to detect a single molecule without the use of labels has been a long standing goal of bioengineers and physicists. This would simplify applications ranging from single molecular binding studies to those involving public health and security, improved drug screening, medical diagnostics, and genome sequencing. One promising technique that has the potential to detect single molecules is the microtoroid optical resonator. The main obstacle to detecting single molecules, however, is decreasing the noise level of the measurements such that a single molecule can be distinguished from background. We have used laser frequency locking in combination with balanced detection and data processing techniques to reduce the noise level of these devices and report the detection of a wide range of nanoscale objects ranging from nanoparticles with radii from 100 to 2.5 nm, to exosomes, ribosomes, and single protein molecules (mouse immunoglobulin G and human interleukin-2). We further extend the exosome results towards creating a non-invasive tumor biopsy assay. Our results, covering several orders of magnitude of particle radius (100 nm to 2 nm), agree with the `reactive' model prediction for the frequency shift of the resonator upon particle binding. In addition, we demonstrate that molecular weight may be estimated from the frequency shift through a simple formula, thus providing a basis for an ``optical mass spectrometer'' in solution. We anticipate that our results will enable many applications, including more sensitive medical diagnostics and fundamental studies of single receptor-ligand and protein-protein interactions in real time. The thesis summarizes what we have achieved thus far and shows that the goal of detecting a single molecule without the use of labels can now be realized.

Microstructure control and iodine doping of bismuth telluride

On the materials scale, thermoelectric efficiency is defined by the dimensionless figure of merit zT. This value is made up of three material components in the form zT = Tα²/ρκ, where α is the Seebeck coefficient, ρ is the electrical resistivity, and κ is the total thermal conductivity. Therefore, in order to improve zT would require the reduction of κ and ρ while increasing α. However due to the inter-relation of the electrical and thermal properties of materials, typical routes to thermoelectric enhancement come in one of two forms. The first is to isolate the electronic properties and increase α without negatively affecting ρ. Techniques like electron filtering, quantum confinement, and density of states distortions have been proposed to enhance the Seebeck coefficient in thermoelectric materials. However, it has been difficult to prove the efficacy of these techniques. More recently efforts to manipulate the band degeneracy in semiconductors has been explored as a means to enhance α.

The other route to thermoelectric enhancement is through minimizing the thermal conductivity, κ. More specifically, thermal conductivity can be broken into two parts, an electronic and lattice term, κ_e and κ_l respectively. From a functional materials standpoint, the reduction in lattice thermal conductivity should have a minimal effect on the electronic properties. Most routes incorporate techniques that focus on the reduction of the lattice thermal conductivity. The components that make up κ_l (κ_l = 1/3Cνl) are the heat capacity (C), phonon group velocity (ν), and phonon mean free path (l). Since the difficulty is extreme in altering the heat capacity and group velocity, the phonon mean free path is most often the source of reduction.

Past routes to decreasing the phonon mean free path has been by alloying and grain size reduction. However, in these techniques the electron mobility is often negatively affected because in alloying any perturbation to the periodic potential can cause additional adverse carrier scattering. Grain size reduction has been another successful route to enhancing zT because of the significant difference in electron and phonon mean free paths. However, grain size reduction is erratic in anisotropic materials due to the orientation dependent transport properties. However, microstructure formation in both equilibrium and nonequilibrium processing routines can be used to effectively reduce the phonon mean free path as a route to enhance the figure of merit.

This work starts with a discussion of several different deliberate microstructure varieties. Control of the morphology and finally structure size and spacing is discussed at length. Since the material example used throughout this thesis is anisotropic a short primer on zone melting is presented as an effective route to growing homogeneous and oriented polycrystalline material. The resulting microstructure formation and control is presented specifically in the case of In₂Te₃-Bi₂Te₃ composites and the transport properties pertinent to thermoelectric materials is presented. Finally, the transport and discussion of iodine doped Bi₂Te₃ is presented as a re-evaluation of the literature data and what is known today.

I. A seismotectonic study of the Middle America Subduction Zone. II. Lithosphere and upper mantle structure of the Canadian Shield and Eastern North America

This thesis consists of two separate parts. Part I (Chapter 1) is concerned with seismotectonics of the Middle America subduction zone. In this chapter, stress distribution and Benioff zone geometry are investigated along almost 2000 km of this subduction zone, from the Rivera Fracture Zone in the north to Guatemala in the south. Particular emphasis is placed on the effects on stress distribution of two aseismic ridges, the Tehuantepec Ridge and the Orozco Fracture Zone, which subduct at seismic gaps. Stress distribution is determined by studying seismicity distribution, and by analysis of 190 focal mechanisms, both new and previously published, which are collected here. In addition, two recent large earthquakes that have occurred near the Tehuantepec Ridge and the Orozco Fracture Zone are discussed in more detail. A consistent stress release pattern is found along most of the Middle America subduction zone: thrust events at shallow depths, followed down-dip by an area of low seismic activity, followed by a zone of normal events at over 175 km from the trench and 60 km depth. The zone of low activity is interpreted as showing decoupling of the plates, and the zone of normal activity as showing the breakup of the descending plate. The portion of subducted lithosphere containing the Orozco Fracture Zone does not differ significantly, in Benioff zone geometry or in stress distribution, from adjoining segments. The Playa Azul earthquake of October 25, 1981, M_s=7.3, occurred in this area. Body and surface wave analysis of this event shows a simple source with a shallow thrust mechanism and gives M_o=1.3x10²⁷ dyne-cm. A stress drop of about 45 bars is calculated; this is slightly higher than that of other thrust events in this subduction zone. In the Tehuantepec Ridge area, only minor differences in stress distribution are seen relative to adjoining segments. For both ridges, the only major difference from adjoining areas is the infrequency or lack of occurrence of large interplate thrust events.

Part II involves upper mantle P wave structure studies, for the Canadian shield and eastern North America. In Chapter 2, the P wave structure of the Canadian shield is determined through forward waveform modeling of the phases P_nl, P, and PP. Effects of lateral heterogeneity are kept to a minimum by using earthquakes just outside the shield as sources, with propagation paths largely within the shield. Previous mantle structure studies have used recordings of P waves in the upper mantle triplication range of 15-30°; however, the lack of large earthquakes in the shield region makes compilation of a complete P wave dataset difficult. By using the phase PP, which undergoes triplications at 30-60°, much more information becomes available. The WKBJ technique is used to calculate synthetic seismograms for PP, and these records are modeled almost as well as the P. A new velocity model, designated S25, is proposed for the Canadian shield. This model contains a thick, high-Q, high-velocity lid to 165 km and a deep low-velocity zone. These features combine to produce seismograms that are markedly different from those generated by other shield structure models. The upper mantle discontinuities in S25 are placed at 405 and 660 km, with a simple linear gradient in velocity between them. Details of the shape of the discontinuities are not well constrained. Below 405 km, this model is not very different from many proposed P wave models for both shield and tectonic regions.

Chapter 3 looks in more detail at recordings of P_nl in eastern North America. First, seismograms from four eastern North American earthquakes are analyzed, and seismic moments for the events are calculated. These earthquakes are important in that they are among the largest to have occurred in eastern North America in the last thirty years, yet in some cases were not large enough to produce many good long-period teleseismic records. A simple layer-over-a-halfspace model is used for the initial modeling, and is found to provide an excellent fit for many features of the observed waveforms. The effects on P_nl of varying lid structure are then investigated. A thick lid with a positive gradient in velocity, such as that proposed for the Canadian shield in Chapter 2, will have a pronounced effect on the waveforms, beginning at distances of 800 or 900 km. P_nl records from the same eastern North American events are recalculated for several lid structure models, to survey what kinds of variations might be seen. For several records it is possible to see likely effects of lid structure in the data. However, the dataset is too sparse to make any general observations about variations in lid structure. This type of modeling is expected to be important in the future, as the analysis is extended to more recent eastern North American events, and as broadband instruments make more high-quality regional recordings available.

Band structure and transport properties of simple cubic tellurium-base alloys

The electrical transport properties and lattice spacings of simple cubic Te-Au, Te-Au-Fe, and Te-Au-Mn alloys, prepared by rapid quenching from the liquid state, hove been measured and correlated with a proposed bond structure. The variations of superconducting transition temperature, absolute thermoelectric power, and lattice spacing with Te concentration all showed related anomalies in the binary Te-Au alloys. The unusual behavior of these properties has been interpreted by using nearly free electron theory to predict the effect of the second Brillouin zone boundary on the area of the Fermi surface, and the electronic density of states. The behavior of the superconducting transition temperature and the lattice parameter as Fe and Mn ore added further supports the proposed interpretation as well as providing information on the existence of localized magnetic states in the ternary alloys. In addition, it was found that a very distinct bond structure effect on the transition temperatures of the Te-Au-Fe alloys could be identified.

Strong atom-light interactions along nanostructures: Transition from free-space to nanophotonic interfaces

An exciting frontier in quantum information science is the integration of otherwise "simple'' quantum elements into complex quantum networks. The laboratory realization of even small quantum networks enables the exploration of physical systems that have not heretofore existed in the natural world. Within this context, there is active research to achieve nanoscale quantum optical circuits, for which atoms are trapped near nano-scopic dielectric structures and "wired'' together by photons propagating through the circuit elements. Single atoms and atomic ensembles endow quantum functionality for otherwise linear optical circuits and thereby enable the capability of building quantum networks component by component. Toward these goals, we have experimentally investigated three different systems, from conventional to rather exotic systems : free-space atomic ensembles, optical nano fibers, and photonics crystal waveguides. First, we demonstrate measurement-induced quadripartite entanglement among four quantum memories. Next, following the landmark realization of a nanofiber trap, we demonstrate the implementation of a state-insensitive, compensated nanofiber trap. Finally, we reach more exotic systems based on photonics crystal devices. Beyond conventional topologies of resonators and waveguides, new opportunities emerge from the powerful capabilities of dispersion and modal engineering in photonic crystal waveguides. We have implemented an integrated optical circuit with a photonics crystal waveguide capable of both trapping and interfacing atoms with guided photons, and have observed the collective effect, superradiance, mediated by the guided photons. These advances provide an important capability for engineered light-matter interactions, enabling explorations of novel quantum transport and quantum many-body phenomena.

Free algebras in Von Neumann-Bernays-Gӧdel set theory and positive elementary inductions in reasonable structures

This thesis consists of two independent chapters. The first chapter deals with universal algebra. It is shown, in von Neumann-Bernays-Gӧdel set theory, that free images of partial algebras exist in arbitrary varieties. It follows from this, as set-complete Boolean algebras form a variety, that there exist free set-complete Boolean algebras on any class of generators. This appears to contradict a well-known result of A. Hales and H. Gaifman, stating that there is no complete Boolean algebra on any infinite set of generators. However, it does not, as the algebras constructed in this chapter are allowed to be proper classes. The second chapter deals with positive elementary inductions. It is shown that, in any reasonable structure ᶆ, the inductive closure ordinal of ᶆ is admissible, by showing it is equal to an ordinal measuring the saturation of ᶆ. This is also used to show that non-recursively saturated models of the theories ACF, RCF, and DCF have inductive closure ordinals greater than _ω.

Characterization and modeling of premixed turbulent n-heptane flames in the thin reaction zone regime

n-heptane/air premixed turbulent flames in the high-Karlovitz portion of the thin reaction zone regime are characterized and modeled in this thesis using Direct Numerical Simulations (DNS) with detailed chemistry. In order to perform these simulations, a time-integration scheme that can efficiently handle the stiffness of the equations solved is developed first. A first simulation with unity Lewis number is considered in order to assess the effect of turbulence on the flame in the absence of differential diffusion. A second simulation with non-unity Lewis numbers is considered to study how turbulence affects differential diffusion. In the absence of differential diffusion, minimal departure from the 1D unstretched flame structure (species vs. temperature profiles) is observed. In the non-unity Lewis number case, the flame structure lies between that of 1D unstretched flames with "laminar" non-unity Lewis numbers and unity Lewis number. This is attributed to effective Lewis numbers resulting from intense turbulent mixing and a first model is proposed. The reaction zone is shown to be thin for both flames, yet large chemical source term fluctuations are observed. The fuel consumption rate is found to be only weakly correlated with stretch, although local extinctions in the non-unity Lewis number case are well correlated with high curvature. These results explain the apparent turbulent flame speeds. Other variables that better correlate with this fuel burning rate are identified through a coordinate transformation. It is shown that the unity Lewis number turbulent flames can be accurately described by a set of 1D (in progress variable space) flamelet equations parameterized by the dissipation rate of the progress variable. In the non-unity Lewis number flames, the flamelet equations suggest a dependence on a second parameter, the diffusion of the progress variable. A new tabulation approach is proposed for the simulation of such flames with these dimensionally-reduced manifolds.

Norbornenyl- and nortricyclyl free radicals

The number, symmetry, and product-forming capabilities of the intermediates in the photoinitiated reductions of endo- and exo-5- bromonorbornene and 2-bromonortricyclene with tri-n-butyltin hydride at temperatures between -10° and 22° were investigated.

Three mechanisms were evaluated:

1. The 5-norbornenyl- and 2-nortricyclyl radicals isomerize reversibly with the former producing nortricyclene by abstraction of hydrogen from tri-n-butyltin hydride.

2. The 5-norbornenyl- and 2-nortricyclyl radicals isomerize reversibly, but some norbornene can be formed from the 2-nortricyclyl radical or some nortricyclene can be formed from the 5-norbornenyl radical by abstraction of hydrogen.

3. There is intervention of a "bridged" radical which may be for med reversibly or irreversibly from the 5-norbornenyl- and 2-nortricyclyl radicals.

Within small error limits, the ratios of norbornene to nortricyclene as a function of the concentration of tri-n-butyltin hydride are consistent with the first mechanism.

In the reductions with tri-n-butyltin deuteride, primary deuterium isotope effects of 2. 3 and 2. 1 for the abstraction of deuterium by the 2-nortricyclyl- and 5-norbornenyl radicals, respectively, were found. The primary deuterium isotope effects were invariant with the concentration of tri-n-butyltin deuteride, although the ratios of norbornene to nortricyclene changed appreciably over this range. This is consistent with the first mechanism, and can accommodate the formation of either product from more than one intermediate only if the primary kinetic deuterium isotope effects are nearly equal for all reactions leading to the single product.

The reduction of endo-5-bromonorbornene-5, 6, 6-d₃ with tri-n-butyltin hydride or tri-n-butyltin deuteride leads to both unrearranged and rearranged norbornenes. The ratios of unrearranged to rearranged norbornene require that the 5-norbornenyl-5, 6, 6-d₃ radical isomerize to an intermediate with the symmetry expected of a nortricyclyl free radical. The results are consistent with mechanism 1, but imply a surprising normal secondary kinetic deuterium isotope effect of about 1.25 for the abstraction of hydrogen by the 5-norbornenyl- 5, 6, 6-d₃ radical.

Approximate calculations show that there does not appear to be any substantial difference in the stabilities of the 5-norbornenyl and 2-nortricyclyl radicals.

Although the results can not exclude a small contribution by a mechanism other than mechanism 1, no such contribution is required to adequately explain the results.

Part I: Latent heat of vaporization of n-decane. Part II: Heat and mass transfer from a cylinder placed in a turbulent air stream - Sherwood and Frössling numbers as a function of free stream turbulence level and Reynolds number

Part I

The latent heat of vaporization of n-decane is measured calorimetrically at temperatures between 160° and 340°F. The internal energy change upon vaporization, and the specific volume of the vapor at its dew point are calculated from these data and are included in this work. The measurements are in excellent agreement with available data at 77° and also at 345°F, and are presented in graphical and tabular form.

Part II

Simultaneous material and energy transport from a one-inch adiabatic porous cylinder is studied as a function of free stream Reynolds Number and turbulence level. Experimental data is presented for Reynolds Numbers between 1600 and 15,000 based on the cylinder diameter, and for apparent turbulence levels between 1.3 and 25.0 per cent. n-heptane and n-octane are the evaporating fluids used in this investigation.

Gross Sherwood Numbers are calculated from the data and are in substantial agreement with existing correlations of the results of other workers. The Sherwood Numbers, characterizing mass transfer rates, increase approximately as the 0.55 power of the Reynolds Number. At a free stream Reynolds Number of 3700 the Sherwood Number showed a 40% increase as the apparent turbulence level of the free stream was raised from 1.3 to 25 per cent.

Within the uncertainties involved in the diffusion coefficients used for n-heptane and n-octane, the Sherwood Numbers are comparable for both materials. A dimensionless Frössling Number is computed which characterizes either heat or mass transfer rates for cylinders on a comparable basis. The calculated Frössling Numbers based on mass transfer measurements are in substantial agreement with Frössling Numbers calculated from the data of other workers in heat transfer.