Searcharxiv⌕ Search

arXiv subjects

Michael Joyce

Publications and source records attributed to Michael Joyce.

At least 37 records · Page 2Linked to original sources
arXiv2016

Attractor non-equilibrium stationary states in perturbed long-range interacting systems

Isolated long-range interacting particle systems appear generically to relax to non-equilibrium states ("quasi-stationary states" or QSS) which are stationary in the thermodynamic limit. A fundamental open question concerns the "robustness" of these states when the system is not isolated. In this paper we explore, using both analytical and numerical approaches to a paradigmatic one dimensional model, the effect of a simple class of perturbations. We call them "internal local perturbations" in that the particle energies are perturbed at collisions in a way which depends only on the local properties. Our central finding is that the effect of the perturbations is to drive all the very different QSS we consider towards a unique QSS. The latter is thus independent of the initial conditions of the system, but determined instead by both the long-range forces and the details of the perturbations applied. Thus in the presence of such a perturbation the long-range system evolves to a unique non-equilibrium stationary state, completely different to its state in absence of the perturbation, and it remains in this state when the perturbation is removed. We argue that this result may be generic for long-range interacting systems subject to perturbations which are dependent on the local properties (e.g. spatial density or velocity distribution) of the system itself.

cond-mat.stat-mech↗
arXiv2015

Finite $N$ corrections to Vlasov dynamics and the range of pair interactions

We explore the conditions on a pair interaction for the validity of the Vlasov equation to describe the dynamics of an interacting $N$ particle system in the large $N$ limit. Using a coarse-graining in phase space of the exact Klimontovich equation for the $N$ particle system, we evaluate, neglecting correlations of density fluctuations, the scalings with $N$ of the terms describing the corrections to the Vlasov equation for the coarse-grained one particle phase space density. Considering a generic interaction with radial pair force $F(r)$, with $F(r) \sim 1/r^γ$ at large scales, and regulated to a bounded behaviour below a "softening" scale $\varepsilon$, we find that there is an essential qualitative difference between the cases $γ< d$ and $γ> d$, i.e., depending on the integrability at large distances of the pair force. In the former case the corrections to the Vlasov dynamics for a given coarse-grained scale are essentially insensitive to the softening parameter $\varepsilon$, while for $γ> d$ the amplitude of these terms is directly regulated by $\varepsilon$, and thus by the small scale properties of the interaction. This corresponds to a simple physical criterion for a basic distinction between long-range ($γ\leq d $) and short range ($γ> d$) interactions, different to the canonical one ($γ\leq d +1$ or $γ> d +1$ ) based on thermodynamic analysis. This alternative classification, based on purely dynamical considerations, is relevant notably to understanding the conditions for the existence of so-called quasi-stationary states in long-range interacting systems.

cond-mat.stat-mech↗
arXiv2015

On the generation of triaxiality in the collapse of cold spherical self-gravitating systems

Initially cold and spherically symmetric self-gravitating systems may give rise to a virial equilibrium state which is far from spherically symmetric, and typically triaxial. We focus here on how the degree of symmetry breaking in the final state depends on the initial density profile. We note that the most asymmetric structures result when, during the collapse phase, there is a strong injection of energy preferentially into the particles which are localized initially in the outer shells. These particles are still collapsing when the others, initially located in the inner part, are already re-expanding; the motion of particles in a time varying potential allow them to gain kinetic energy --- in some cases enough to be ejected from the system. We show that this mechanism of energy gain amplifies the initial small deviations from perfect spherical symmetry due to finite $N$ fluctuations. This amplification is more efficient when the initial density profile depends on radius, because particles have a greater spread of fall times compared to a uniform density profile, for which very close to symmetric final states are obtained}. These effects lead to a distinctive correlation of the orientation of the final structure with the distribution of ejected mass, and also with the initial (very small) angular fluctuations.

astro-ph.GA↗
arXiv2014

Chains in Weak Order Posets Associated to Involutions

The W-set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties. We explicitly describe in a purely combinatorial manner the W-sets of the weak order posets of three different sets of involutions in the symmetric group, namely, the set of all involutions, the set of all fixed point free involutions, and the set of all involutions with signed fixed points (or "clans"). These distinguished sets of involutions parameterize Borel orbits in the classical symmetric spaces associated to the general linear group. In particular, we give a complete characterization of the maximal chains of an arbitrary lower order ideal in any of these three posets.

math.CO↗
arXiv2014

Self-similarity and stable clustering in a family of scale-free cosmologies

We study non-linear gravitational clustering from cold gaussian power-law initial conditions in a family of scale-free EdS models, characterized by a free parameter $κ$ fixing the ratio between the mass driving the expansion and the mass which clusters. As in the "usual" EdS model, corresponding to $κ=1$, self-similarity provides a powerful instrument to delimit the physically relevant clustering resolved by a simulation. Likewise, if stable clustering applies, it implies scale-free non-linear clustering. We derive the corresponding exponent $γ_{sc} (n, κ)$ of the two point correlation function. We then report the results of extensive N-body simulations, of comparable size to those previously reported in the literature for the case $κ=1$, and performed with an appropriate modification of the GADGET2 code. We observe in all cases self-similarity in the two point correlations, down to a lower cut-off which decreases monotonically in time in comoving coordinates. The self-similar part of the non-linear correlation function is fitted well in all cases by a single power-law with an exponent in good agreement with $γ_{sc} (n, κ)$. Our results thus indicate that stable clustering provides an excellent approximation to the non-linear correlation function over the resolved self-similar scales, at least down to $γ_{sc} (n, κ) \approx 1$, corresponding to the case $n=-2$ for $κ=1$. We conclude, in contrast notably with the results of Smith et al. (2003), that a clear identification of the breakdown of stable clustering in self-similar models - and the possible existence of a "universal" region in which non-linear clustering becomes independent of initial conditions - remains an important open problem, which should be addressed further in significantly larger simulations.

astro-ph.CO↗
arXiv2014

Particle ejection during mergers of dark matter halos

Dark matter halos are built from accretion and merging. During merging some of the dark matter particles may be ejected with velocities higher than the escape velocity. We use both N-body simulations and single-particle smooth-field simulations to demonstrate that rapid changes to the mean field potential are responsible for such ejection, and in particular that dynamical friction plays no significant role in it. Studying a range of minor mergers, we find that typically between 5-15% of the particles from the smaller of the two merging structures are ejected. We also find that the ejected particles originate essentially from the small halo, and more specifically are particles in the small halo which pass later through the region in which the merging occurs.

astro-ph.CO↗
arXiv2014

Scaling quasi-stationary states in long range systems with dissipation

Hamiltonian systems with long-range interactions give rise to long lived out of equilibrium macroscopic states, so-called quasi-stationary states. We show here that, in a suitably generalized form, this result remains valid for many such systems in the presence of dissipation. Using an appropriate mean-field kinetic description, we show that models with dissipation due to a viscous damping or due to inelastic collisions admit "scaling quasi-stationary states", i.e., states which are quasi-stationary in rescaled variables. A numerical study of one dimensional self-gravitating systems confirms both the relevance of these solutions, and gives indications of their regime of validity in line with theoretical predictions. We underline that the velocity distributions never show any tendency to evolve towards a Maxwell-Boltzmann form.

cond-mat.stat-mech↗
arXiv2013

Introduction of interactive learning into French university physics classrooms

We report on a project to introduce interactive learning strategies (ILS) to physics classes at the Université Pierre et Marie Curie (UPMC), one of the leading science universities in France. In Spring 2012, instructors in two large introductory classes, first-year, second-semester mechanics, and second-year introductory E&M, enrolling approximately 500 and 250 students respectively, introduced ILS into some sections of each class. The specific ILS utilized were Think-Pair-Share questions and Peer Instruction in the main lecture classrooms, and UW Tutorials for Introductory Physics in recitation sections. Pre- and post-instruction assessments (FCI and CSEM respectively) were given, along with a series of demographics questions. We were able to compare the results of the FCI and CSEM between interactive and non-interactive classes taught simultaneously with the same curriculum. We also analyzed final exam results, as well as the results of student and instructor attitude surveys between classes. In our analysis, we argue that Multiple Linear Regression modeling is superior to other common analysis tools, including normalized gain. Our results show that ILS are effective at improving student learning by all measures used: research-validated concept inventories and final exam scores, on both conceptual and traditional problem-solving questions. Multiple Linear Regression analysis reveals that interactivity in the classroom is a significant predictor of student learning, showing a similar or stronger relationship with student learning than such ascribed characteristics as parents' education, and achieved characteristics such as GPA and hours studied per week. Analysis of student and instructors attitudes shows that both groups believe that ILS improve student learning in the physics classroom, and increases student engagement and motivation.

physics.ed-ph↗
arXiv2013

Exponents of non-linear clustering in scale-free one dimensional cosmological simulations

One dimensional versions of cosmological N-body simulations have been shown to share many qualitative behaviours of the three dimensional problem. They can resolve a large range of time and length scales, and admit exact numerical integration. We use such models to study how non-linear clustering depends on initial conditions and cosmology. More specifically, we consider a family of models which, like the 3D EdS model, lead for power-law initial conditions to self-similar clustering characterized in the strongly non-linear regime by power-law behaviour of the two point correlation function. We study how the corresponding exponent γdepends on the initial conditions, characterized by the exponent n of the power spectrum of initial fluctuations, and on a single parameter κcontrolling the rate of expansion. The space of initial conditions/cosmology divides very clearly into two parts: (1) a region in which γdepends strongly on both n and κand where it agrees very well with a simple generalisation of the so-called stable clustering hypothesis in three dimensions, and (2) a region in which γis more or less independent of both the spectrum and the expansion of the universe. We explain the observed location of the boundary in (n, κ) space dividing the "stable clustering" region from the "universal" region. We compare and contrast our findings to results in three dimensions, and discuss in particular the light they may throw on the question of "universality" of non-linear clustering in this context.

astro-ph.CO↗
arXiv2013

Landen transforms as families of (commuting) rational self-maps of projective space

The classical (m,k)-Landen transform F_{m,k} is a self-map of the field of rational functions C(z) obtained by forming a weighted average of a rational function over twists by m'th roots of unity. Identifying the set of rational maps of degree d with an affine open subset of P^{2d+1}, we prove that F_{m,0} induces a dominant rational self-map R_{d,m,0} of P^{2d+1} of algebraic degree m, and for 0 < k < m, the transform F_{m,k} induces a dominant rational self-map R_{d,m,k} of algebraic degree m of a certain hyperplane in P^{2d+1}. We show in all cases that R_{d,m,k} extends nicely to a map of P^{2d+1} over Spec(Z), and that {R_{d,m,0} : m \ge 0} is a commuting family of maps.

math.AG↗
arXiv2013

Unipotent Invariant Matrices

We describe the variety of fixed points of a unipotent operator acting on the space of matrices. We compute the determinant and the rank of a generic (symmetric, or anti-symmetric) matrix in the fixed variety, yielding information about the generic singular locus of the corresponding bilinear form.

math.AG↗
arXiv2012

Evolution of isolated overdensities as a control on cosmological N body simulations

Beyond convergence studies and comparison of different codes, there are essentially no controls on the accuracy in the non-linear regime of cosmological N body simulations, even in the dissipationless limit. We propose and explore here a simple test which has not been previously employed: when cosmological codes are used to simulate an isolated overdensity, they should reproduce, in physical coordinates, those obtained in open boundary conditions without expansion. In particular, the desired collisionless nature of the simulations can be probed by testing for stability in physical coordinates of virialized equilibria. We investigate and illustrate the test using a suite of simulations in an Einstein de Sitter cosmology from initial conditions which rapidly settle to virial equilibrium. We find that the criterion of stable clustering allows one to determine, for given particle number N in the "halo" and force smoothing, a maximum red-shift range over which the collisionless limit may be represented with desired accuracy. We also compare our results to the so-called Layzer Irvine test, showing that it provides a weaker, but very useful, tool to constrain the choice of numerical parameters. Finally we outline in some detail how these methods could be employed to test the choice of the numerical parameters used in a cosmological simulation.

astro-ph.CO↗
arXiv2012

Ordered Bell numbers, Hermite polynomials, Skew Young Tableaux, and Borel orbits

We give three interpretations of the number $b$ of orbits of the Borel subgroup of upper triangular matrices on the variety $\ms{X}$ of complete quadrics. First, we show that $b$ is equal to the number of standard Young tableaux on skew-diagrams. Then, we relate $b$ to certain values of a modified Hermite polynomial. Third, we relate $b$ to a certain cell decomposition on $\ms{X}$ previously studied by De Concini, Springer, and Strickland. Using these, we give asymptotic estimates for $b$ as the dimension of the quadrics increases.

math.CO↗
arXiv2012

Weak Order on Complete Quadrics

Using an action of the Richardson-Springer monoid on involutions, we study the weak order on the variety of complete quadrics. Maximal chains in the poset are explicitly determined. Applying results of Brion, our calculations describe certain cohomology classes in the complete flag variety.

math.CO↗
arXiv2011

SL(2)-regular Subvarieties of Complete Quadrics

We determine SL(n)-stable, SL(2)-regular subvarieties of the variety of complete quadrics. We extend the results of Akyıldız and Carrell on Kostant-Macdonald identity by computing the Poincar{é} polynomials of these regular subvarieties.

math.AG↗
arXiv2011

Quasi-stationary states in the self-gravitating sheet model

We study quasi-stationary states (QSS) resulting from violent relaxation in the one-dimensional self-gravitating "sheet model", revisiting in particular the question of the adequacy of the theory of Lynden-Bell (LB) to describe them. For "waterbag" initial conditions characterized by a single phase space density, the prediction of this theory is, in this model, a function of only one parameter, which can conveniently be chosen to be the ratio of the energy to that in the degenerate limit. Studying a class of such initial conditions in which the shape of the initial waterbag is varied, we find that the LB predictions are reasonably good always in the low energy region, while at higher energies (i.e. in the non-degenerate limit) they are generally not even qualitatively correct, although certain initial conditions can still be found where they are as good as at low energy. We find notably that, in line with what has been observed by Levin et al. in some other models, when LB theory does not work the QSS are always characterized by the presence of a {\it degenerate} core, which these authors explain as the result of dynamical resonances. In short LB theory appears to be a good approximation only when violent relaxation is sufficiently "gentle", and otherwise a degenerate core-halo structure results.

cond-mat.stat-mech↗
arXiv2011

Non-linear gravitational clustering of cold matter in an expanding universe: indications from 1D toy models

Studies of a class of infinite one dimensional self-gravitating systems have highlighted that, on the one hand, the spatial clustering which develops may have scale invariant (fractal) properties, and, on the other, that they display "self-similar" properties in their temporal evolution. The relevance of these results to three dimensional cosmological simulations has remained unclear. We show here that the measured exponents characterizing the scale-invariant non-linear clustering are in excellent agreement with those derived from an appropriately generalized "stable-clustering" hypothesis. Further an analysis in terms of "halos" selected with a friend-of-friend algorithm reveals that such structures are, statistically, virialized across the range of scales corresponding to scale-invariance. Thus the strongly non-linear clustering in these models is accurately described as a virialized fractal structure, very much in line with the "clustering hierarchy" which Peebles originally envisaged qualitatively as associated with stable clustering. If transposed to three dimensions these results would imply, notably, that cold dark matter halos (or even subhalos) are 1) not well modeled as smooth objects, and 2) that the supposed "universality" of their profiles is, like apparent smoothness, an artefact of poor numerical resolution.

astro-ph.CO↗