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## Liouville Equation (liouville + equation)
## Selected Abstracts## Time-domain approach to linearized rotational response of a three-dimensional viscoelastic earth model induced by glacial-isostatic adjustment: I. Inertia-tensor perturbations GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2005k MartinecSUMMARY For a spherically symmetric viscoelastic earth model, the movement of the rotation vector due to surface and internal mass redistribution during the Pleistocene glaciation cycle has conventionally been computed in the Laplace-transform domain. The method involves multiplication of the Laplace transforms of the second-degree surface-load and tidal-load Love numbers with the time evolution of the surface load followed by inverse Laplace transformation into the time domain. The recently developed spectral finite-element method solves the field equations governing glacial-isostatic adjustment (GIA) directly in the time domain and, thus, eliminates the need of applying the Laplace-domain method. The new method offers the possibility to model the GIA-induced rotational response of the Earth by time integration of the linearized Liouville equation. The theory presented here derives the temporal perturbation of the inertia tensor, required to be specified in the Liouville equation, from time variations of the second-degree gravitational-potential coefficients by the MacCullagh's formulae. This extends the conventional approach based on the second-degree load Love numbers to general 3-D viscoelastic earth models. The verification of the theory of the GIA-induced rotational response of the Earth is performed by using two alternative approaches of computing the perturbation of the inertia tensor: a direct numerical integration and the Laplace-domain method. The time-domain solution of both the GIA and the induced rotational response of the Earth is readily combined with a time-domain solution of the sea level equation with a time-varying shoreline geometry. In a follow-up paper, we derive the theory for the case when GIA-induced perturbations in the centrifugal force affect not only the distribution of sea water, but also deformations and gravitational-potential perturbations of the Earth. [source] ## Field-free molecular alignment of CO2 mixtures in presence of collisional relaxation JOURNAL OF RAMAN SPECTROSCOPY, Issue 6 2008T. VieillardAbstract The present work explores the extension of the concept of short-pulse-induced alignment to dissipative environments within quantum mechanical density matrix formalism (Liouville equation) from the weak to the strong field regime. This is illustrated within the example of the CO2 molecule in mixture with Ar and He, at room temperature, for which a steep decrease of the alignment is observed at moderate pressure because of the collisional relaxation. The field-free alignment is measured by a polarization technique where the degree of alignment is monitored in the time domain by measuring the resulting transient birefringence with a probe pulse Raman induced polarization spectroscopy (RIPS) Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Appropriate SCF basis sets for orbital studies of galaxies and a ,quantum-mechanical' method to compute them MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 3 2008Constantinos KalapotharakosABSTRACT We address the question of an appropriate choice of basis functions for the self-consistent field (SCF) method of simulation of the N -body problem. Our criterion is based on a comparison of the orbits found in N -body realizations of analytical potential,density models of triaxial galaxies, in which the potential is fitted by the SCF method using a variety of basis sets, with those of the original models. Our tests refer to maximally triaxial Dehnen ,-models for values of , in the range 0 ,,, 1, i.e. from the harmonic core up to the weak cusp limit. When an N -body realization of a model is fitted by the SCF method, the choice of radial basis functions affects significantly the way the potential, forces or derivatives of the forces are reproduced, especially in the central regions of the system. We find that this results in serious discrepancies in the relative amounts of chaotic versus regular orbits, or in the distributions of the Lyapunov characteristic exponents, as found by different basis sets. Numerical tests include the Clutton-Brock and the Hernquist,Ostriker basis sets, as well as a family of numerical basis sets which are ,close' to the Hernquist,Ostriker basis set (according to a given definition of distance in the space of basis functions). The family of numerical basis sets is parametrized in terms of a quantity , which appears in the kernel functions of the Sturm,Liouville equation defining each basis set. The Hernquist,Ostriker basis set is the ,= 0 member of the family. We demonstrate that grid solutions of the Sturm,Liouville equation yielding numerical basis sets introduce large errors in the variational equations of motion. We propose a quantum-mechanical method of solution of the Sturm,Liouville equation which overcomes these errors. We finally give criteria for a choice of optimal value of , and calculate the latter as a function of the value of ,, i.e. of the power-law exponent of the radial density profile at the central regions of the galaxy. [source] ## Carrier transport in nanodevices: revisiting the Boltzmann and Wigner distribution functions PHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 7 2009Fons BrosensAbstract In principle, transport of charged carriers in nanometer sized solid-state devices can be fully characterized once the non-equilibrium distribution function describing the carrier ensemble is known. In this light, we have revisited the Boltzmann and the Wigner distribution functions and the framework in which they emerge from the classical respectively quantum mechanical Liouville equation. We have assessed the method of the characteristic curves as a potential workhorse to solve the time dependent Boltzmann equation for carriers propagating through spatially non-uniform systems, such as nanodevices. In order to validate the proposed solution strategy, we numerically solve the Boltzmann equation for a one-dimensional conductor mimicking the basic features of a biased low-dimensional transistor operating in the on-state. Finally, we propose a computational scheme capable of extending the benefits of the above mentioned solution strategy when it comes to solve the Wigner,Liouville equation. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Manifestly covariant classical correlation dynamics I. General theory ANNALEN DER PHYSIK, Issue 10-11 2009C. TianAbstract In this series of papers we substantially extend investigations of Israel and Kandrup on nonequilibrium statistical mechanics in the framework of special relativity. This is the first one devoted to the general mathematical structure. Based on the action-at-a-distance formalism we obtain a single-time Liouville equation. This equation describes the manifestly covariant evolution of the distribution function of full classical many-body systems. For such global evolution the Bogoliubov functional assumption is justified. In particular, using the Balescu-Wallenborn projection operator approach we find that the distribution function of full many-body systems is completely determined by the reduced one-body distribution function. A manifestly covariant closed nonlinear equation satisfied by the reduced one-body distribution function is rigorously derived. We also discuss extensively the generalization to general relativity especially an application to self-gravitating systems. [source] ## Manifestly covariant classical correlation dynamics I. General theory ANNALEN DER PHYSIK, Issue 10-11 2009C. TianAbstract In this series of papers we substantially extend investigations of Israel and Kandrup on nonequilibrium statistical mechanics in the framework of special relativity. This is the first one devoted to the general mathematical structure. Based on the action-at-a-distance formalism we obtain a single-time Liouville equation. This equation describes the manifestly covariant evolution of the distribution function of full classical many-body systems. For such global evolution the Bogoliubov functional assumption is justified. In particular, using the Balescu-Wallenborn projection operator approach we find that the distribution function of full many-body systems is completely determined by the reduced one-body distribution function. A manifestly covariant closed nonlinear equation satisfied by the reduced one-body distribution function is rigorously derived. We also discuss extensively the generalization to general relativity especially an application to self-gravitating systems. [source] ## A hierarchy of Sturm,Liouville problems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2003Paul BindingSturm,Liouville equations will be considered where the boundary conditions depend rationally on the eigenvalue parameter. Such problems apply to a variety of engineering situations, for example to the stability of rotating axles. Classesof these problems will be isolated with a rather rich spectral structure, for example oscillation, comparison and completeness properties analogous to thoseof the ,usual' Sturm,Liouville problem which has constant boundary conditions. In fact it will be shown how these classes can be converted into each other, andinto the ,usual' Sturm,Liouville problem, by means of transformations preserving all but finitely many eigenvalues. Copyright © 2003 John Wiley & Sons, Ltd. [source] |