Connection between dissipative and resonant conservative nonlinear oscillators

It is shown how to add dissipation to the resonant nonlinear oscillators studied by Ford and Lunsford in such a way that the system remains on the energy surface. In the dissipative system, the energy surface is stable in some directions and neutrally stable in other directions. The dissipative oscillators are special cases of the general type investigated by Sherman and McLaughlin. The connection between resonant conservative nonlinear oscillators and dissipative oscillators may make it easier to extend the theorem of Arnol'd to dissipative systems.     

Coherent Solid-State Phase Transitions with Atomic Diffusion: A Thermomechanical Treatment

Using the framework of modern continuum thermomechanics, we develop sharp- and diffuse-interface theories for coherent solid-state phase transitions. These theories account for atomic diffusion and for deformation. Of essential importance in our formulation of the sharp-interface theory are a system of configurational forces and an associated configurational force balance. These forces, which are distinct from standard Newtonian forces, describe the intrinsic material structure of a body. The configurational balance, when restricted to the interface, leads to a generalization of the classical Gibbs–Thomson relation, a generalization that accounts for the orientation dependence of the interfacial energy density and also for a broad spectrum of dissipative transition kinetics. Our diffuse-interface theory involves nonstandard microforces and an associated microforce balance. These forces arise naturally from an interpretation of the atomic densities as macroscopic parameters that describe atomistic kinematics distinct from the motion of material particles. When supplemented by thermodynamically consistent constitutive relations, the microforce balance yields a generalization of the Cahn–Hilliard relation giving the chemical potentials as variational derivatives of the total free energy with respect to the atomic densities. A formal asymptotic analysis (thickness of the transition layer approaching zero) demonstrates the correspondence between versions of our theories specialized to the case of a single mobile species for situations in which the time scale for interface propagation is small compared to that for bulk diffusion. While the configurational force balance is redundant in the diffuse-interface theory, when integrated over the transition layer, the limit of this balance is the interfacial configurational force balance (i.e., generalized Gibbs–Thomson relation) of the sharp-interface theory.     

A mode-mode coupling theory of chemical reaction in a dense fluid

A study is made of the coupling between chemical reaction and diffusion in a dense fluid. Our analysis utilizes the projection operator formalism and a generalized Langevin equation that is based on irreversible, phenomenological equations of motion instead of conventional Hamiltonian mechanics. It also is shown that this same non-Hamiltonian theory provides a simple way of deriving Kawasaki's mode-mode coupling theory of diffusion.This research was supported by a grant from the National Science Foundation.     

Correlation functions for quasi-linear response theory

A quasi-linear regression formula is derived by an expansion around quasi-static equilibrium. It relates the relaxation of thermodynamic forces to the regression of correlations of thermodynamic coordinates in quasi-static equilibrium. Correlation functions and memory kernels can be introduced in almost complete analogy to linear response theory. A non-linear, non-Markovian kinetic equation is derived. The kinetic coefficients are given in terms of correlation functions of stochastic forces in quasi-static equilibrium similar to the linear theory.     

Hamilton and the Law of Varying Action Revisited

According to history texts, philosophers searched for a unifying natural law whereby natural phenomena and numbers are related. More than 2300 years ago, Aristotle postulated that nature requires minimum energy. More than 220 years ago, Euler applied the minimum energy postulate. More than 200 years ago, Lagrange provided a mathematical proof of the postulate for conservative systems. The resulting Principle of Least Action served only to derive the differential equations of motion of a conservative system. Then, 170 years ago, Hamilton presented what he claimed to be a general method in dynamics. Hamilton's resulting Law of Varying Action was supposed to apply to both conservative and non-conservative systems and was supposed to yield either the differential equations of motion or the integrals of those differential equations. However, no direct evaluation of the integrals of motion ever resulted from Hamilton's law of varying action. In 1975, a scant 29 years ago, following five years of controversy with engineer mechanicians, Dr. Wolfgang Yourgrau, Editor, Foundations of Physics, published my first paper based on Aristotle's postulate, without mathematical proof. That and subsequent papers present, through applications, a true general method in dynamics. In this essay, I present the mathematical proof that is missing from my 1975 and subsequent papers. Six fundamental integrals of analytical mechanics are derived from Aristotle's postulate. First, however, Hamilton must be revisited to show why his H function and his force function prevents the law of varying action from being the general method in dynamics that he claimed it to be. I have found that Hamiltons Law of Varying Action (HLVA), as Hamilton presented it, cannot be applied to systems for which the force function is non-integrable. In 1972, Dr. B.E. Gatewood and Dr. D.P. Beres (then a graduate student) discovered that the end-point term associated with the principle of least action does not vanish. I named the new equation, the general energy equation. In 1973, because I was doing with it what Hamilton claimed could be done with HLVA, I simply assumed that this new equation was HLVA. I gave the new equation the misnomer HLVA. In 2001, I learned that I had made a grave mistake. I found that HLVA is at most a special case of the general energy equation. My interpretation of Aristotle's postulate permits one to by-pass the differential equations of motion completely for both conservative and non-conservative systems (no calculus of variations).     

Noised-induced phase transition in an oscillatory system with dynamical traps

A new type of noised induced phase transitions is proposed. It occurs in noisy systems with dynamical traps. Dynamical traps are regions in the phase space where the regular forces are depressed substantially. By way of an example, a simple oscillatory system with additive white noise is considered and its dynamics is analyzed numerically. The dynamical trap region is assumed to be located near the x-axis where the velocity v of the system becomes sufficiently low. The meaning of this assumption is discussed. The observed phase transition is caused by the asymmetry in the residence time distribution in the vicinity of zero value velocity. This asymmetry is due to a cooperative effect of the random Langevin force in the trap region and the regular force not changing the direction of action when crossing the trap region.Received: 25 April 2003, Published online: 19 November 2003PACS: 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.45.-a Nonlinear dynamics and nonlinear dynamical systems - 05.70.Fh Phase transitions: general studies     

Effective Hamiltonians from first principles

The equations of motion for a quantum system of the electromagnetic field interacting with matter are derived from the first principle Hamiltonian (two-level atomic systems are assumed) and from the effective Hamiltonian. The derivation is based on the memoryfunction theory of Mori and on the theory of random frequency modulation of Tokuyama and Mori. On the ground of comparison of the equations of motion from the first principle Hamiltonian with those from the effective Hamiltonian a justification of the use of effective Hamiltonians is discussed.     

Optical second-harmonic generation of metal-island films

In metal-island films consisting of nanometer particles on a transparent substrate irradiated light fields can be locally enhanced by electron-plasma resonances. Therefore, nonlinear optical processes should be enhanced dramatically. However, second-order nonlinear processes as second-harmonic generation occuring in the surface are strongly reduced by the centrosymmetric shape of the metal particles. It is found that this surface-specific contribution to second-harmonic generation is less enhanced, as is expected from the field enhancement. The bulk contribution, at smooth metal surfaces known to be much weaker than the contribution from the real surface, is strongly enhanced by the plasma resonances without symmetry restrictions and becomes comparably important.Paper presented at the 129th WE-Heraeus-Seminar on Surface Studies by Nonlinear Laser Spectroscopies, Kassel, Germany, May 30 to June 1, 1994.     

Quantum mechanical coherence in human red blood cells

Fritz London predicted that the behavior of the quantum fluids ...might prove useful for an understanding of the macromolecular systems of biology which behave... much more simply than would be expected in view of the apparent great complexity of their structure. The Fröhlich theory is of an energy-driven laserlike process in living cells which should drive cellular phonons into coherence. Fröhlich's theory predicts specific ultra-long-range forces which can explain the presently mysterious, ordered tensor interactions within and without the living cell. Several different types of experiments demonstrate a specific ultralong-range interaction between mammalian red blood cells which accords with the postulates of the Fröhlich theory. One phenomenon seems to be compatible with processes analogous to self-focusing and trapping in nonlinear optics. As work progresses more and more biological mechanisms appear to be similar to those known in condensed matter physics.     

The nonsymmetric Kaluza-Klein (Jordan-Thiry) theory in the electromagnetic case

We present the nonsymmetric Kaluza-Klein and Jordan-Thiry theories as interesting propositions of physics in higher dimensions. We consider the five-dimensional (electromagnetic) case. The work is devoted to a five-dimensional unification of the NGT (nonsymmetric theory of gravitation), electromagnetism, and scalar forces in a Jordan-Thiry manner. We find interference effects between gravitational and electromagnetic fields which appear to be due to the skew-symmetric part of the metric. Our unification, called the nonsymmetric Jordan-Thiry theory, becomes the classical Jordan-Thiry theory if the skew-symmetric part of the metric is zero. It becomes the classical Kaluza-Klein theory if the scalar field=1 (Kaluza's Ansatz). We also deal with material sources in the nonsymmetric Kaluza-Klein theory for the electromagnetic case. We consider phenomenological sources with a nonzero fermion current, a nonzero electric current, and a nonzero spin density tensor. From the Palatini variational principle we find equations for the gravitational and electromagnetic fields. We also consider the geodetic equations in the theory and the equation of motion for charged test particles. We consider some numerical predictions of the nonsymmetric Kaluza-Klein theory with nonzero (and with zero) material sources. We prove that they do not contradict any experimental data for the solar system and on the surface of a neutron star. We deal also with spin sources in the nonsymmetric Kaluza-Klein theory. We find an exact, static, spherically symmetric solution in the nonsymmetric Kaluza-Klein theory in the electromagnetic case. This solution has the remarkable property of describing mass without mass and charge without charge. We examine its properties and a physical interpretation. We consider a linear version of the theory, finding the electromagnetic Lagrangian up to the second order of approximation with respect toh _v =g _v –n _v . We prove that in the zeroth and first orders of approximation there is no skewonoton interaction. We deal also with the Lagrangian for the scalar field (connected to the gravitational constant). We prove that in the zeroth and first orders of approximation the Lagrangian vanishes.     

Conserved moments in nonequilibrium field dynamics

We demonstrate with the example of Cahn-Hilliard dynamics that the macroscopic kinetics of first-order phase transitions exhibits an infinite number of constants of motion. Moreover, this result holds in any space dimension for a broad class of nonequilibrium processes whose macroscopic behavior is governed by equations of the form /t = W(), where is an order parameter,W is an arbitrary function of , and is a linear Hermitian operator. We speculate on the implications of this result.     

Quantum equilibrium and the origin of absolute uncertainty

The quantum formalism is a measurement formalism-a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from Bohmian mechanics, which is what emerges from Schrödinger's equation for a system of particles when we merely insist that particles means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We find that a Bohmian universe, though deterministic, evolves in such a manner that anappearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by = ¦¦ ². A crucial ingredient in our analysis of the origin of this randomness is the notion of the effective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate.This paper is dedicated to the memory of J. S. Bell.     

Multiatom interactions,order and stability in binary transition metal alloys

Interface delocalization or depinning transitions such as wetting or surface induced disorder are considered. At these transitions, the correlation length for transverse correlations parallel to the surface diverges. These correlations are studied in the framework of Landau theory. It is shown the t^–1/2 at all types of transitions for systems with short-range forces wheret measures the distance from bulk coexistence.     

One-electron parametric oscillator

The parametric oscillation of a trapped electron is studied and used to measure enhanced spontaneous emission. Hysteresis in this motion provides a one bit memory to store information about excitations made with the electron in the dark.Dedicated to H. Walther on the occasion of his 60th birthday     

Dewetting of polymeric films: unresolved issues

Since the early seminal theoretical work by Brochard and coworkers, and experiments by Reiter over a decade ago, considerable progress has been made toward the development of a comprehensive picture of the equilibrium and dynamic behavior of unstable thin polymeric films. Generally, theoretical work has carefully guided the design of many experiments conducted in this field. Recent experimental findings, however, raise new questions that could probably not have been foreseen by theory and now need to be revisited. In this paper we highlight three problems in two general areas, (1) the use of the effective interfacial potential describing film substrate interactions and (2) the dynamics of dewetting and the associated connection to slip phenomena and fingering instabilities. We suggest that in addition to experiments, analytical theory and simulations will play a critical role toward elucidating the ultimate goal of a universal picture of equilibrium and dynamic behavior of instabilities in thin films.Received: 1 August 2003PACS: 68.15. + e Liquid thin films - 61.30.Hn Surface phenomena: alignment, anchoring, anchoring transitions, surface-induced layering, surface-induced ordering, wetting, prewetting transitions, and wetting transitions     

Dynamical ensembles in stationary states

We propose, as a generalization of an idea of Ruelle's to describe turbulent fluid flow, a chaotic hypothesis for reversible dissipative many-particle systems in nonequilibrium stationary states in general. This implies an extension of the zeroth law of thermodynamics to nonequilibrium states and it leads to the identification of a unique distribution describing the asymptotic properties of the time evolution of the system for initial data randomly chosen with respect to a uniform distribution on phase space. For conservative systems in thermal equilibrium the chaotic hypothesis implies the ergodic hypothesis. We outline a procedure to obtain the distribution : it leads to a new unifying point of view for the phase space behavior of dissipative and conservative systems. The chaotic hypothesis is confirmed in a nontrivial, parameter-free, way by a recent computer experiment on the entropy production fluctuations in a shearing fluid far from equilibrium. Similar applications to other models are proposed, in particular to a model for the Kolmogorov-Obuchov theory for turbulent flow.     

Hydrodynamical Reformulation and Quantum Limit of The Barut–Zanghi Theory

One of the most satisfactory pictures for spinning particles is the Barut-Zanghi (BZ) classical theory for the relativistic extended-like electron, that relates spin to zitterbewegung (zbw). The BZ motion equations constituted the starting point for recent works about spin and electron structure, co-authored by us, which adopted the Clifford algebra language. This language results to be actually suited for a hydrodynamical reformulation of the BZ theory. Working out a probabilistic fluid, we are allowed to reinterpret the original classical spinors as quantum wave-functions for the electron. We can pass to quantize the BZ theory: by employing this time the tensorial language, more popular in first-quantization. Quantizing the BZ theory, however, does notlead to the Dirac equation, but rather to a nonlinear, Dirac–like equation, which can be regarded as the actual quantum limit of the BZ classical theory. Moreover, a new variational approach to the BZ probabilistic fluid shows that it is a typical Weyssenhoff fluid, while the Hamilton-Jacobi equation (linking mass, spin,and zbw frequency together) appears to be nothing but a special case of the de Broglie energy–frequency relation. Finally, after having discussed the remarkable relation existing between the gauge transformation U(1) and ageneral rotation on the spin plane, we clarify and comment on the two-valuedeness nature of the fremionic wave-function, as well as on the parity and charge conjugation transformations.     

Muon states in metals: Recent progress

We report on our results in two interesting questions related to muon spin rotation studies in condensed matter: (i) energetics of muons in metals, including lattice relaxation and zero point motion in self-trapping phenomena, and (ii) systematics of Knight shifts and hyperfine fields.In the former topic, a comprehensive theory is developed which entails the construction of the muon potential energy field in terms of the effective-medium or quasi-atom theory first introduced by Zaremba, Stott, NØrskov and Lang. The muon wave function is then solved by numerical (3-D) relaxation techniques. From this the forces exerted by the muon on the neighbouring lattice atoms are calculated, and the ensuing relaxations are evaluated by lattice Green's function techniques. These in turn modify the potential energy field, and the calculation is iterated to self-consistency. We search for the stable trapping sites in bcc and fcc metals, calculate self-trapping energies, diffusion barriers and excitation energies. Other hydrogenic imputies are also considered, and isotopic effects in e.g. heats of solution are investigated.In the latter topic, the spin-density functional theory is applied, including in the Knight shift calculation both the contact spin density and the diamagnetic shielding. The lattice potential is described in terms of the spherical solid model. A systematic behaviour as a function of the electron density and the host valency is found in good agreement with the experimental results.     

On learning and energy-entropy dependence in recurrent and nonrecurrent signed networks

Learning of patterns by neural networks obeying general rules of sensory transduction and of converting membrane potentials to spiking frequencies is considered. Any finite number of cellsA can sample a pattern playing on any finite number of cells without causing irrevocable sampling bias ifA = orA =. Total energy transfer from inputs ofA to outputs of depends on the entropy of the input distribution. Pattern completion on recall trials can occur without destroying perfect memory even ifA = by choosing the signal thresholds sufficiently large. The mathematical results are global limit and oscillation theorems for a class of nonlinear functional-differential systems.The preparation of this work was supported in part by the National Science Foundation (GP 9003), the Office of Naval Research (N00014-67-A-024-OQ16), and the A.P. Sloan Foundation.     

Relativistic transformations of thermodynamic quantities

A unique solution is proposed to the problem of how thermodynamic processes between thermodynamic systems at relative rest appear to a moving observer. Assuming only transformations for entropy, pressure, and volume and the invariance of the fundamental thermodynamic equation, one can derive transformations for (thermodynamic) energy and temperature. The invariance of the first and second laws entails transformations for work and heat. All thermodynamic relations become Lorentz-invariant. The transformations thus derived are in principle equivalent to those of Einstein and Planck, except that our expressions for energy and work do not include the mass motion energy. This results in a simpler formulation and endows our transformations (especially that of temperature) with a straightforward physical interpretation.