Papers by Willem van Leeuwen
arXiv (Cornell University), Mar 23, 2010
It is shown that a first-order cosmological perturbation theory for the open, flat and closed Fri... more It is shown that a first-order cosmological perturbation theory for the open, flat and closed Friedmann-Lemaître-Robertson-Walker universes admits one, and only one, gauge-invariant variable which describes the perturbation to the energy density and which becomes equal to the usual Newtonian energy density in the non-relativistic limit. The same holds true for the perturbation to the particle number density. Using these two new variables, a new manifestly gauge-invariant cosmological perturbation theory has been developed. Density perturbations evolve diabatically. Perturbations in the total energy density are gravitationally coupled to perturbations in the particle number density, irrespective of the nature of the particles. There is, in first-order, no back-reaction of perturbations to the global expansion of the universe. Small-scale perturbations in the radiation-dominated era oscillate with an increasing amplitude, whereas in older, less precise treatments, oscillating perturbations are found with a decreasing amplitude. This is a completely new and, obviously, important result, since it makes it possible to explain and understand the formation of massive stars after decoupling of matter and radiation.
arXiv (Cornell University), May 4, 2008
It is shown that a first-order cosmological perturbation theory for Friedmann-Lemaître-Robertson-... more It is shown that a first-order cosmological perturbation theory for Friedmann-Lemaître-Robertson-Walker universes admits one and only one gauge-invariant variable which describes the perturbation to the energy density and which becomes equal to the usual energy density of the Newtonian theory of gravity in the limit that all particle velocities are negligible with respect to the speed of light. The same holds true for the perturbation to the particle number density. A cosmological perturbation theory based on these particular gauge-invariant quantities is more precise than any earlier first-order perturbation theory. In particular, it explains star formation in a satisfactory way, even in the absence of cold dark matter. In a baryon-only universe, the earliest stars, the so-called Population III stars, are found to have masses between 400 and 100,000 solar masses with a peak around 3400 solar masses. If cold dark matter, with particle mass 10 times heavier than the proton mass, is present then the star masses are between 16 and 4000 solar masses with a peak around 140 solar masses. They come into existence between 100 Myr and 1000 Myr. At much later times, star formation is possible only in high density regions, for example within galaxies. Late time stars may have much smaller masses than early stars. The smallest stars that can be formed have masses of 0.2-0.8 solar mass, depending on the initial internal relative pressure perturbation. It is demonstrated that the Newtonian theory of gravity cannot be used to study the evolution of cosmological density perturbations.
Biologically Inspired Learning Controlling the Neuronal Activity
A layered neural net with adaptable synaptic weights and fixed threshold potentials is studied, i... more A layered neural net with adaptable synaptic weights and fixed threshold potentials is studied, in the presence of a global feedback signal that can only have two values, depending on whether the output of the network as a reaction to its input is right or wrong. On the basis of four biologically motivated assumptions, it is found that only two forms of learning are possible, Hebbian and Anti–Hebbian learning. It is shown that Hebbian learning memorizes input–output relations, while Anti– Hebbian learning does the opposite: it changes the input–output relations of the network. Hebbian learning should take place when the output is right, while there should be Anti–Hebbian learning when the output is wrong. A particular choice for the Anti–Hebbian part of the learning rule is made, which guarantees an adequate average neuronal activity. A network with non– zero threshold potentials is shown to perform its task of realizing the desired input–output relations best if it is sufficiently ...
Hierarchical Structure Formation in FLRW Universes in the Framework of Einstein's General Theory of Relativity
It is shown that a first-order cosmological perturbation theory for Friedmann-Lemaitre-Robertson-... more It is shown that a first-order cosmological perturbation theory for Friedmann-Lemaitre-Robertson-Walker universes admits one and only one gauge-invariant variable which describes the perturbation to the energy density and which becomes equal to the usual energy density of the Newtonian theory of gravity in the limit that all particle velocities are negligible with respect to the speed of light. The same holds true for the perturbation to the particle number density. A cosmological perturbation theory based on these particular gauge-invariant quantities is more precise than any earlier first-order perturbation theory. In particular, it explains star formation in a satisfactory way, even in the absence of cold dark matter. In a baryon-only universe, the earliest stars, the so-called Population III stars, are found to have masses between 400 and 100,000 solar masses with a peak around 3400 solar masses. If cold dark matter, with particle mass 10 times heavier than the proton mass, is p...
Relativistic Kinetic Theory: Principles and Applications
Preface. Historical background. Part A. Basic Equations. I. Elements of relativistic kinetic theo... more Preface. Historical background. Part A. Basic Equations. I. Elements of relativistic kinetic theory. II. Conservation laws and H-theorem. Part B. Derivation of the Transport Equation. III. Scalar particles. IV. Spin-1/2 particles. Part C. Linear Theory. V. First Chapman-Enskog approximation. VI. Transport coefficients. VII. Moment method. VIII. Propagation of sound waves. IX. Mathematical aspects of the linearized transport equation. Part D. Applications. X. Lepton systems. XI. Systems of hadrons. A Model. XII. Photon-Electron System. XIII. Reduction of the collision brackets. Bibliography. Author index. Subject index.
Statistical mechanics of finitely connected neural networks
Physical Perturbations in Cosmology
A natural and transparent way is proposed to construct new, manifestly gauge invariant quantities... more A natural and transparent way is proposed to construct new, manifestly gauge invariant quantities out of gauge dependent quantities occurring in the linearized Einstein equations. It is shown that these new quantities can be identified with physical, i.e., measurable, perturbations to the particle and energy densities. Next, it is shown that the linearized Einstein equations, which contain gauge functions, can be combined in such a way that the resulting equations essentially only contain these new gauge invariant quantities and do not contain any gauge function. In this way, we have arrived at a treatment of cosmological perturbations that is both conceptually transparent and manifestly gauge invariant. The new set of linearized Einstein equations for the new manifestly gauge invariant combinations constitute the main result of this paper. As an illustration, we consider these equations for a case that can be treated analytically, namely the radiation-dominated era of a flat Friedm...
Viscous Bianchi universes with large scale magnetic field
We propose a way to construct manifestly gauge independent quantities out of the gauge dependent ... more We propose a way to construct manifestly gauge independent quantities out of the gauge dependent quantities occurring in the linearized Einstein equations. Thereupon, we show that these gauge-invariant combinations can be identified with measurable perturbations to the particle and energy densities. In the radiation-dominated era we find, for small-scale perturbations, acoustic waves with an increasing amplitude, while standard treatments predict acoustic waves with a decaying amplitude. For large-scale perturbations we find exactly the same growth rates as in the standard literature. When considering the non-relativistic limit of the linearized Einstein equations we find the Poisson equation. It is shown, using the linearized Einstein equations, that the usual Newtonian treatment of density perturbations does not describe the evolution of density perturbations.
Transport coefficients of a neutrino gas
Lettere Al Nuovo Cimento Series 2, 1973
Il Nuovo Cimento A Series 11, 1975
Physica, 1969
The conservation and entropy laws are derived from the transport equation for a mixture of molecu... more The conservation and entropy laws are derived from the transport equation for a mixture of molecules with spherically symmetrical interaction without the use of symmetry properties for the collision rate. On the other hand, rotational invariance of the collision rate is used to obtain phenomenological relations in accordance with Curie's law for isotropic systems, while reciprocal Onsager relations between the transport phenomena are deduced from time reversal invariance alone.
Chapman-Enskog method for a relativistic ionized gas
Physics Letters A, 1976
Abstract A linearized transport equation is derived for a multicomponent ionized gas. The magneti... more Abstract A linearized transport equation is derived for a multicomponent ionized gas. The magnetic field is treated as a zeroth-order, the electric field as a first-order effect.
Physica, 1973
The following subjects are treated: scattering matrix, transition rate and cross section in relat... more The following subjects are treated: scattering matrix, transition rate and cross section in relation to relativistic kinetic theory; unitarity, bilateral normalization and H theorem; covariant oneparticle distribution function; relativistic transport equation.
On relativistic kinetic gas theory. III
Physica, 1969
Abstract It is discussed how the non-relativistic theory may be obtained as an approximation from... more Abstract It is discussed how the non-relativistic theory may be obtained as an approximation from the covariant theory by means of expansions in powers of c -1 ( c is the velocity of light) and by an appropriate treatment of the c 2 (rest mass energy) terms. One finds the non-relativistic expressions for the thermodynamic and transport properties as the terms of order c 0 . The first corrections to these results are all of the order c -2 .
Physics Letters A, 1968
Reci})rocal Onsager relations are derived from microscopic reversibility and the relativistic Bol... more Reci})rocal Onsager relations are derived from microscopic reversibility and the relativistic Boltzmann equation for a gas mixture in which heat conduction, diffusion, viscous flow and cross-effects occur. A system of n non-reacting components, in which heat conduction, diffusion, viscous phenomena and their cros.s-eff.ects may occur, is studied. Each component i = 1, 2,..., n is described by a distribution function ft(x,pt) which is assumed to obey the relativistic Boltzmann equation [1],
Relativistic entropy law for a gas mixture outside equilibrium
Physics Letters A, 1968
Abstract The Gibbs relation and the entropy source strength for a mixture (in which heat conducti... more Abstract The Gibbs relation and the entropy source strength for a mixture (in which heat conduction, diffusion, viscous flow and cross-effects occur) are derived in the first Enskog approximation of relativistic kinetic gas theory.
Diffusion and thermal diffusion coefficients of a relativistic gas mixture
Physics Letters A, 1974
Values up to order c-2 are given for the diffusion and thermal diffusion coefficients of binary r... more Values up to order c-2 are given for the diffusion and thermal diffusion coefficients of binary relativistic gas mixtures of hard spheres and Israel particles.
Physics Letters A, 1971
A relativistic expansion to all orders is given for the transport coefficients of a simple gas w!... more A relativistic expansion to all orders is given for the transport coefficients of a simple gas w!; arbitrary particle interaction. On its basis ultra-relativistic results, as well as expansions in c follow.
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Papers by Willem van Leeuwen