Logarithmic Oscillations Research Articles

Thermostated Hamiltonian Dynamics with Log Oscillators

With this work, we present two new methods for the generation of thermostatted, manifestly Hamiltonian dynamics and provide corresponding illustrations. The basis for this new class of thermostats is the peculiar thermodynamics as exhibited by logarithmic oscillators. These two schemes are best suited when applied to systems with a small number of degrees of freedom.

Comment on “Logarithmic Oscillators: Ideal Hamiltonian Thermostats”

A Comment on the Letter by M. Campisi et al. Phys. Rev. Lett. 108, 250601 (2012). The authors of the Letter offer a Reply.Received 1 June 2012DOI:https://doi.org/10.1103/PhysRevLett.110.028901© 2013 American Physical Society

Logarithmic Oscillators: Ideal Hamiltonian Thermostats

A logarithmic oscillator (in short, log-oscillator) behaves like an ideal thermostat because of its infinite heat capacity: When it weakly couples to another system, time averages of the system observables agree with ensemble averages from a Gibbs distribution with a temperature T that is given by the strength of the logarithmic potential. The resulting equations of motion are Hamiltonian and may be implemented not only in a computer but also with real-world experiments, e.g., with cold atoms.

On sensitive elliptic singular perturbation problems and logarithmic oscillations: the case of shells with edges

This work deals with singular perturbation problems depending on small positive parameter ϵ. The limit problem as ϵ → 0 has no solution within the classical theory of PDEs, which uses distribution theory. A very particular and less‐known phenomenon appears: large oscillations. These problems exhibit some kind of instability; very small and smooth variations of the data imply large singular perturbations of the solution. That kind of problems appears in elasticity for highly compressible two‐dimensional bodies and thin shells with elliptic middle surface with a part of the boundary free. Here, we consider certain properties of that oscillations and extend the theory to shells with edges. Copyright © 2012 John Wiley & Sons, Ltd.

Logarithmic mean oscillation on the polydisc, endpoint results for multi-parameter paraproducts, and commutators on BMO

We study boundedness properties of a class ofmultiparameter paraproducts on the dual space of the dyadic Hardy space H 1 d (T N), the dyadic product BMO space BMOd (T N). For this, we introduce a notion of logarithmic mean oscillation on the polydisc. We also obtain a result on the boundedness of iterated commutators on BMO [0, 1]N).

Discrete to continuum transition in multifractal spacetimes

We outline a field theory on a multifractal spacetime. The measure in the action is characterized by a varying Hausdorff dimension and logarithmic oscillations governed by a fundamental physical length. A fine hierarchy of length scales identifies different regimes, from a microscopic structure with discrete symmetries to an effectively continuum spacetime. Thanks to general arguments from fractal geometry, this scenario explicitly realizes two indirect or conjectured features of most quantum gravity models: a change of effective spacetime dimensionality with the probed scale, and the transition from a fundamentally discrete quantum spacetime to the continuum. It also allows us to probe ultramicroscopic scales where spectral methods based on ordinary geometry typically fail. Consequences for noncommutative field theories are discussed.

Anomalous diffusion with log-periodic modulation in a selected time interval

On certain self-similar substrates the time behavior of a random walk is modulated by logarithmic-periodic oscillations on all time scales. We show that if disorder is introduced in a way that self-similarity holds only in average, the modulating oscillations are washed out but subdiffusion remains as in the perfect self-similar case. Also, if disorder distribution is appropriately chosen the oscillations are localized in a selected time interval. Both the overall random walk exponent and the period of the oscillations are analytically obtained and confirmed by Monte Carlo simulations.

Log-periodic oscillations for diffusion on self-similar finitely ramified structures

Under certain circumstances, the time behavior of a random walk is modulated by logarithmic-periodic oscillations. Using heuristic arguments, we give a simple explanation of the origin of this modulation for diffusion on a substrate with two properties: self-similarity and finite ramification order. On these media, the time dependence of the mean-square displacement shows log-periodic modulations around a leading power law, which can be understood on the basis of a hierarchical set of diffusion constants. Both the random walk exponent and the period of oscillations are analytically obtained for a pair of examples, one is fractal and the other is nonfractal, and confirmed by Monte Carlo simulations. The last example shows that the anomalous diffusion can arise from substrates without holes of all sizes.

Ballistic deposition on deterministic fractals: Observation of discrete scale invariance

The growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations. First, we attempt the description of the evolving interface of the aggregates by applying the well-established Family-Vicsek dynamic scaling approach. Systematic deviations from that standard scaling law are observed, suggesting that significant scaling corrections have to be introduced in order to achieve a more accurate understanding of the behavior of the interface. Subsequently, we study the internal structure of the growing aggregates that can be rationalized in terms of the scaling behavior of frozen trees, i.e., structures inhibited for further growth, lying below the growing interface. It is shown that the rms height (h_{s}) and width (w_{s}) of the trees of size s obey power laws of the form h_{s} proportional, variants;{nu_{ parallel}} and w_{s} proportional, variants;{nu_{ perpendicular}} , respectively. Also, the tree-size distribution (n_{s}) behaves according to n_{s} approximately s;{-tau} . Here, nu_{ parallel} and nu_{ perpendicular} are the correlation length exponents in the directions parallel and perpendicular to the interface, respectively. Also, tau is a critical exponent. However, due to the interplay between the discrete scale invariance of the underlying fractal substrates and the dynamics of the growing process, all these power laws are modulated by logarithmic periodic oscillations. The fundamental scaling ratios, characteristic of these oscillations, can be linked to the (spatial) fundamental scaling ratio of the underlying fractal by means of relationships involving critical exponents. We argue that the interplay between the spatial discrete scale invariance of the fractal substrate and the dynamics of the physical process occurring in those media is a quite general phenomenon that leads to the observation of logarithmic-periodic modulations of physical observables.

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H 1 and give a new proof of the old result stating that the Hankel operator H a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H a to be compact on H 1. The Fredholm properties of Toeplitz operators on H 1 are studied for symbols in a Banach algebra similar to C + H ∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H 1 and H 2.

Dipole oscillator strength distributions, properties and dispersion energies for the dimethyl, diethyl and methyl–propyl ethers1

Isotropic dipole oscillator strength distributions (DOSDs) have been constructed for the dimethyl, diethyl and methyl–propyl ether molecules through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength data. The constraints are furnished by molar refractivity data and the Thomas–Reiche–Kuhn sum rule. The DOSDs are used to obtain recommended values for a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for the molecules. Pseudo-DOSDs for the ethers are also constructed and used to obtain reliable results for the isotropic dipole–dipole dispersion energy coefficients for all two-body interactions of the ethers with each other and with fifty other species. In addition reliable results are also obtained for the triple–dipole dispersion energy coefficients for all three-body interactions involving the ethers. 1Dedicated to Anthony Stone, an excellent scientist and friend, on the occasion of his 70th birthday.

Dipole oscillator strength distributions, properties, and dispersion energies for ethylene, propene, and 1-butene

A recommended isotropic dipole oscillator strength distribution (DOSD) has been constructed for the ethylene molecule through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength (DOS) data; the DOS data employed are recent experimental results not available at the time of the original constrained DOSD analysis of this molecule. The constraints are furnished by molar refractivity data and the Thomas–Reiche–Kuhn sum rule. The DOSD is used to evaluate a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for ethylene. Pseudo-DOSDs for this molecule, and for propene and 1–butene, which are based on an earlier constrained DOSD analysis for these molecules, are developed. They are used to obtain reliable results for the isotropic dipole–dipole dispersion-energy coefficients C6, for the interactions of the alkenes with each other and with 47 other species, and the triple-dipole dispersion-energy coefficients C9 for interactions involving any triple of molecules taken from ethylene, propene, and 1–butene.Key words: alkenes, dipole properties, pseudo-states, dipole–dipole and triple-dipole dispersion energies, long-range additive, non-additive interaction energies.

Boundedness of Hankel operators on [formula omitted

We prove that the Hankel operator h b associated to the Szegö projection on the unit ball B n is bounded on the Hardy space H 1 ( B n ) if and only if its symbol b has logarithmic mean oscillation on the unit sphere. To cite this article: A. Bonami et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Discrete scale invariance effects in the nonequilibrium critical behavior of the Ising magnet on a fractal substrate

The nonequilibrium critical dynamics of the Ising magnet on a fractal substrate, namely the Sierpinski carpet with Hausdorff dimension d(H)=1.7925, has been studied within the short-time regime by means of Monte Carlo simulations. The evolution of the physical observables was followed at criticality, after both annealing ordered spin configurations (ground state) and quenching disordered initial configurations (high temperature state), for three segmentation steps of the fractal. We have obtained evidence showing that during these relaxation processes both the growth and the fragmentation of magnetic domains become influenced by the hierarchical structure of the substrate. In fact, the interplay between the dynamic behavior of the magnet and the underlying fractal leads to the emergence of a logarithmic-periodic oscillation, superimposed to a power law, which has been observed in the time dependence of both the decay of the magnetization and its logarithmic derivative. These oscillations have been carefully characterized in order to determine the critical temperature of the second-order phase transition and the critical exponents corresponding to the short-time regime. The effects of the substrate can also be observed from the dependence of the effective critical exponents on the segmentation step. The exponent theta of the initial increase of the magnetization has also been obtained and the results suggest that it would be almost independent of the fractal dimension of the substrate, provided that d(H) is close enough to d=2. The oscillations have been discussed within the framework of the discrete scale invariance of the substrate.

Dipole Oscillator Strength Distributions and Properties for Methanol, Ethanol and Propan-1-ol and Related Dispersion Energies

Recommended isotropic dipole oscillator strength distributions (DOSDs) have been constructed for the methanol and ethanol molecules through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength (DOS) data; the DOS data employed are recent experimental results not available at the time of the original constrained DOSD analysis of these molecules. The constraints are furnished by molar refractivity data and the Thomas-Reiche-Kuhn sum rule. The DOSDs are used to evaluate a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for the molecules. Pseudo-DOSDs for these molecules, and for propan-1-ol based on an earlier constrained DOSD analysis for this molecule, are also presented. They are used to obtain reliable results for the isotropic dipole-dipole dispersion energy coefficients C6, for the interactions of the alcohols with each other and with 36 other species, and the triple-dipole dispersion energy coefficients C9for interactions involving any triple of molecules involving methanol, ethanol and propan-1-ol.

Tuning the Heterogeneous Early Folding Dynamics of Phosphoglycerate Kinase

We recently reported stretched kinetics during the formation of a collapsed, long-lived intermediate state of the large two-domain enzyme phosphoglycerate kinase (PGK). It was postulated that intrinsic roughness of the energy landscape on the way downhill to the intermediate causes the lack of a single time-scale. Here, we investigate several alternative explanations for stretched refolding dynamics in more detail: tyrosine fluorescence, multiple tryptophan probes, and rate differences between independently folding domains. To this end, we systematically simplify PGK in several steps from the full protein with two tryptophan residues and all tyrosine residues probed, to a single domain with only one tryptophan residue and no tyrosine residue probed. The kinetics in the 10 μs to 10 ms range are revealed by laser-induced temperature-jump relaxation experiments. The isolated N-terminal domain forms an intermediate by nearly single-exponential kinetics, but the isolated C-terminal domain shows strongly non-exponential kinetics. Thus, domain interaction and a cis-proline residue between the two domains are ruled out as the sole contributors to heterogeneity during the earliest folding dynamics of the C-terminal domain. We apply two limiting models for the roughness of the energy landscape. A sequential three-state model lumps all the roughness into a single trap. The “strange kinetics” model with logarithmic oscillations developed by Klafter and co-workers distributes the roughness over a larger number of states. Both models explain our data about equally well, but the coincidental values of rate constants in all of our double-exponential fits, and the absence of a spectroscopic signature distinct from the endpoints of the folding process favors more roughness than can be explained by just a single trap.

Dipole oscillator strengths, dipole properties and dispersion energies for SiF4

A recommended isotropic dipole oscillator strength distribution (DOSD) has been constructed for the silicon tetrafluoride (SiF4) molecule through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength data. The constraints are furnished by experimental molar refractivity data and the Thomas-Reiche-Kuhn sum rule. The DOSD is used to evaluate a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums and mean excitation energies for the molecule. A pseudo-DOSD for SiF4 is also presented which is used to obtain reliable results for the isotropic dipole-dipole dispersion energy coefficients C6, for the interaction of SiF4 with itself and with 43 other species and the triple-dipole dispersion energy coefficient C9 for (SiF4)3.

Dipole oscillator strength properties and dispersion energies for SiH4

A recommended isotropic dipole oscillator strength distribution (DOSD) has been constructed for the silane (SiH4) molecule through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength data. The constraints are furnished by experimental molar refractivity data and the Thomas–Reiche–Kuhn sum rule. The DOSD is used to evaluate a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for the molecule. A pseudo-DOSD for SiH4 is also presented which is used to obtain reliable results for the isotropic dipole–dipole dispersion energy coefficients C6, for the interaction of silane with itself and with forty-four other species, and the triple–dipole dispersion energy coefficient C9 for ðSiH4Þ 3 . 2002 Elsevier Science B.V. All rights reserved.

Dipole oscillator strength properties and dispersion energies for CI2

A recommended isotropic dipole oscillator strength distribution (DOSD) has been constructed for the chlorine molecule through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength and molar refractivity data. It has been used to evaluate a variety of dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for the molecule. A pseudo-DOSD for C12 is also presented which is used to obtain reliable results for the isotropic dipole-dipole dispersion energy coefficients C6, for the interaction of Cl2 with itself and forty-two other species, and the triple-dipole dispersion energy coefficient C9 for (Cl2)3.

Dynamics of hierarchical folding on energy landscapes of hexapeptides

In this paper we apply the master equation approach to study the effects of the energy landscape topology and topography on the kinetics of folding, and on kinetic transitions of three alanine-hexapeptides analogs which involve polypeptides with neutral and charged groups and a cyclized polypeptide. We rely on the potential-energy landscapes of these molecular systems, which have been constructed using both a topological mapping analysis and a principal component analysis. It was found that the different topology and topography of the energy landscapes result in different “folding” time scales and that the systems with geometrical constraints (cyclization and opposite charges at the termini) “fold” more slowly than the unconstrained peptide. In addition, for each of the three polypeptide systems, the kinetics is nonexponential at the temperature range 400–600 K. The relaxation kinetics is characterized by logarithmic oscillations, which indicate hierarchical dynamics characterized by multiple time scales of fast (few ps) and slow (few μs) events. At higher temperatures, successive relaxation channels with similar characteristic time scales collapse into a single relaxation channel. While the kinetics of the unconstrained peptide at 600 K can be reasonably well described by a single exponential time scale, the kinetics of the constrained hexapeptides are inherently hierarchical and featured by multiple time scales even at high temperatures.