We indicate some of the emerging thematic connections between strongly nonlinear effects in conde... more We indicate some of the emerging thematic connections between strongly nonlinear effects in condensed matter and biological materials. These are illustrated with model studies of: (1) structural phase transitions in anisotropic lattices; and (2) finite temperature effects on self-trapped states in vibron-phonon models of ..alpha..-helix proteins. 13 refs., 8 figs.
The Hubbard Hamiltonian is the simplest model that describes interacting electrons on a lattice. ... more The Hubbard Hamiltonian is the simplest model that describes interacting electrons on a lattice. In this work, we use the properties of stochastic matrices to examine the ground state with an even number of lattice sites and one electron less than half-filling. We show that there exists a highly symmetric state with energy −2 (in units where t = 1) at all U. At U = ∞ this state becomes the lowest energy state, consistent with the established lower energy bound. 1 Using this result, several properties of the strongly coupled ground state are derived, including the chemical potential and momentum distribution. This method may be applicable to other models as well. Disagreements between our results and previous work are examined.
We have constructed a lattice-dynamical model which possesses many of the features occurring at f... more We have constructed a lattice-dynamical model which possesses many of the features occurring at first-order structural phase transitions in solids. The model includes an asymmetric nonlinear on-site potential and anharmonic interparticle interactions. The anharmonicity in the interaction is introduced in a way which lowers the phonon frequencies in the high-temperature, metastable phase. The interaction provides a mechanism for a vibrational-entropy-driven first-order phase transition. We present results from molecular-dynamics calculations which show (i) clear evidence in the thermodynamic functions for the existence of a first-order phase transition produced by heating from low temperature, and (ii) unusual properties for the position probability distribution and the dynamic structure factor. These spectral functions have significant intensity in the quasielastic region, and this contribution is strongly maximized near the transition temperature. The wave-vector dependence of this "central peak" clearly points to the existence of propagating nonlinear modes.
Dynamics of Sine-Gordon Soliton Interactions with Impurities
Springer Proceedings in Physics, 1992
Molecular dynamics simulations of the sine-Gordon system containing static impurities are describ... more Molecular dynamics simulations of the sine-Gordon system containing static impurities are described. A pictorial device which serves as a “soliton detector” shows that the impurities can reflect solitons and breathers, dynamically trap and release them, and cause sufficiently large local distortions to produce trapped solitons. These effects of the impurities are exhibited in the dynamical scattering functions as a very sharp static “central peak” superimposed on the dynamical central peaks produced by soliton and breather propagation.
Self-Consistent-Field Approach to Lattice Dynamics
Physical Review B, 1970
The self-consistent-field theory of lattice dynamics is examined with particular emphasis on the ... more The self-consistent-field theory of lattice dynamics is examined with particular emphasis on the physical assumptions entering this approach. The solution of the basic equation is generalized beyond that of earlier treatments to include damping and the corresponding frequency ...
Computer Simulation of Disordered Electron-Phonon Systems
ABSTRACT
Nucleation Rates in an Extended System with a Symmetric Local Potential
We have derived a formula for the thermal decay rate from the metastable state for a spatially ex... more We have derived a formula for the thermal decay rate from the metastable state for a spatially extended 1-d system with a very general local potential(A. J. Graham and W. C. Kerr, Phys. Rev. E (to be published)). In this talk we give numerical results from this formula for the phi^6 potential. The ingredients of our calculation are an analytical solution for the saddle point in phase space (the so-called critical nucleus) and an evaluation of the spectrum of small oscillations around that saddle point. The eigenvalue equation determining the spectrum is reduced to a Heun equation. In this case it has three bound state solutions, including the translation mode, in addition to a continuum spectrum. We obtain a solution of the (functional) Fokker-Planck equation in the full phase space of the system, and for moderate to large damping. We present results for the nucleation rate as a function of temperature and damping. For this potential there are no results from computer simulation wit...
European Journal of Applied Physiology and Occupational Physiology, 1996
At similar levels of carbon dioxide production (I?CO2) and oxygen consumption 0202), runners have... more At similar levels of carbon dioxide production (I?CO2) and oxygen consumption 0202), runners have been shown to have a greater minute ventilation (I?E) during running as compared to walking. The mechanism responsible for these differences has yet to be identified. To determine if these differences are a result of differences in acid-base status, potassium (K § norepinephrine and/or epinephrine levels, seven welltrained runners completed walk and run tests at similar l?Oa and gco2 levels. The occurrence of entrainment of the breathing and stride frequencies during both walking and running was also determined. I/E was significantly greater during the run as compared to the walk, 73.7 (2.2) versus 68.6 (2.0)1' min-1, respectively, despite the similarity in NO 2 and VCO 2 levels. Alveolar ventilation was not significantly different between the run and the walk, 60.4 (4.7) versus 59.6 (4.4) 1' min-1, respectively. Dead space ventilation was found to be significantly greater during running as compared to walking, 13.3 (3.2) versus 9.0 (4.7) 1' min-1, respectively. The increases in l?E were due to increases in breathing frequency and decreases in tidal volume during the run as compared to the walk. Arterial partial pressures of CO2 (PaCO2) were not significantly different when comparing walking and running to rest values nor when comparing walking and running. Arterial pH was significantly lower during walking as compared to rest and running. Bicarbonate levels were significantly lower during walking as compared to rest. Lactate was M.J. Berry ([:~).
This paper reports a method for studying correlation functions for simple classical liquids. One ... more This paper reports a method for studying correlation functions for simple classical liquids. One atom of the liquid is considered to be an external agent acting on the others, and Liouville' s equation is formally solved to obtain their response. From this solution an equation for the velocity autocorrelation function is derived. The method is also applied to the distinct-particIe part of the density-density correlation function, G~(x, t). The moment relations for Gd(x, t) are in this way made to depend on a hierarchy of equations relating the static correlation functions. When the basic equations derived by this method are approximated further, previously given approximations for G~(x, t) are obtained.
Generalized phase space version of Langevin equations and associated Fokker-Planck equations
The European Physical Journal B, 2000
Abstract. Generic Langevin equations are almost always given as first-order stochastic ordinary d... more Abstract. Generic Langevin equations are almost always given as first-order stochastic ordinary differential equations for the phase space variables of a system, with noise and damping terms in the equation of motion of every variable. In contrast, Langevin equations for ...
Analytic Solution of the Equation of Motion for an Optically-Torqued, Overdamped Nanorod
We examine the one-dimensional Hubbard model and derive expressions for the chemical potential an... more We examine the one-dimensional Hubbard model and derive expressions for the chemical potential and momentum distribution in the limit of strong electron-electron interactions to first order in perturbation theory. By showing that the perturbation matrix is stochastic, the chemical potential is found to depend only on the model parameters and not on the system size, consistent with the well-known result in the thermodynamic limit. The momentum distribution for electrons of a given spin is found to depend only on the fraction of electrons with that spin.
The paper presents a self-consistent mean-field theory for a lattice-dynamical model that exhibit... more The paper presents a self-consistent mean-field theory for a lattice-dynamical model that exhibits a firstorder structural phase transition. In this model the phase transition is produced because the high-energy structure has lower vibrational frequencies than those of the low-energy structure. This mechanism produces higher entropy in the higher-energy structure and thereby drives a phase transition. These structure-dependent frequencies are produced by anharmonicity in the interparticle interaction. The approximate theory of the transition given here reduces the exact coupled equations of motion to a single mean-field equation by replacing coupling terms between neighbors with appropriate averages. This step produces an effective potential that is used to calculate self-consistently the averages that appear in it. Thermodynamic properties calculated by this method show that the system has a first-order phase transition for sufficiently large strength of the interparticle anharmonicity. Further properties of the system obtained by this method include a discontinuous change in the shape of the average displacement and free-energy vs temperature relations as a function of the anharmonicity strength. This feature may be related to the hysteresis seen in previously performed computer simulations on the model. The effective potential also determines the displacement probability distribution function. For the parameter values studied here this distribution has a single maximum with only small asymmetry about this maximum. ͓S0163-1829͑96͒07729-6͔
The Berry's phase, although well-known since 1984, has received little attention among textbook a... more The Berry's phase, although well-known since 1984, has received little attention among textbook authors of solid state physics. We attempt to address this lack by showing how the presence of the Berry's phase significantly changes a standard concept (effective mass) found in most solid state texts. Specifically, we show that the presence of a non-zero Berry curvature in Bloch systems makes the traditional concept of an inverse effective mass tensor M -1 problematic, since a routine application of Newton's 2nd Law leads to a circular definition. As a consequence, the related concept of cyclotron effective mass m * also requires modification. It is shown that m * is magnetic-field dependent in non-inversion symmetric systems. This has important ramifications for cyclotron resonance experiments, since such experiments yield m * and thereby purportedly give the components of M -1 . This work represents a "case study" in how Berry's phase effects can modify "standard" solid-state topics in ways that students and instructors may find surprising.
Using a straightforward extension of the analysis of Lieb and Wu, we derive a simple analytic for... more Using a straightforward extension of the analysis of Lieb and Wu, we derive a simple analytic form for the ground state energy of a one-dimensional Hubbard ring at U/t = ∞. This result is valid for an arbitrary number of lattice sites L and electrons N ≤ L. Furthermore, our analysis provides insight into the degeneracy and spin properties of the ground states in this limit.
Temperature and domain size dependence of sickle cell hemoglobin polymer melting in high concentration phosphate buffer
Biophysical Chemistry, 1999
Deoxygenated sickle cell hemoglobin (Hb S) in 1.8 M phosphate buffer, and carbon monoxide (CO) sa... more Deoxygenated sickle cell hemoglobin (Hb S) in 1.8 M phosphate buffer, and carbon monoxide (CO) saturated buffer were rapidly mixed using a stopped-flow apparatus. The binding of the CO to the Hb S polymers and the polymer melting was measured by time resolved optical spectroscopy. Polymer melting was associated with decreased turbidity, and CO binding to deoxy-Hb S was monitored by observation of changes in the absorption profile. The reaction temperature was varied from 20 degrees C to 35 degrees C. Polymer domain size at 20 degrees C was also varied. The data for mixtures involving normal adult hemoglobin (Hb A) fit well to a single exponential process whereas it was necessary to include a second process when fitting data involving Hb S. The overall Hb S-CO reaction rate decreased with increasing temperature from 20 degrees C to 35 degrees C, and increased with decreasing domain size. In comparison, Hb A-CO reaction rates increased uniformly with increasing temperature. Two competing reaction channels in the Hb S-CO reaction are proposed, one involving CO binding directly to the polymer and the other involving CO only binding to Hb molecules in the solution phase. The temperature dependence of the contribution of each pathway is discussed.
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Papers by William C Kerr