Canonical solution of a system of long-range interacting rotators on a lattice

The canonical partition function of a system of rotators (classical X-Y spins) on a lattice, coupled by terms decaying as the inverse of their distance to the power alpha, is analytically computed. It is also shown how to compute a rescaling function that allows to reduce the model, for any d-dimensional lattice and for any alpha<d, to the mean field (alpha=0) model.Comment: Initially submitted to Physical Review Letters: following referees' Comments it has been transferred to Phys. Rev. E, because of supposed no general interest. Divided into sections, corrections in (5) and (20), reference 5 updated. 8 pages 1 figur

Canonical Solution of Classical Magnetic Models with Long-Range Couplings

We study the canonical solution of a family of classical $n-vector$ spin models on a generic $d$-dimensional lattice; the couplings between two spins decay as the inverse of their distance raised to the power $\alpha$, with $\alpha<d$. The control of the thermodynamic limit requires the introduction of a rescaling factor in the potential energy, which makes the model extensive but not additive. A detailed analysis of the asymptotic spectral properties of the matrix of couplings was necessary to justify the saddle point method applied to the integration of functions depending on a diverging number of variables. The properties of a class of functions related to the modified Bessel functions had to be investigated. For given $n$, and for any $\alpha$, $d$ and lattice geometry, the solution is equivalent to that of the $\alpha=0$ model, where the dimensionality $d$ and the geometry of the lattice are irrelevant.Comment: Submitted for publication in Journal of Statistical Physic

Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems

Classical spin systems with nonadditive long-range interactions are studied in the microcanonical ensemble. It is expected that the entropy of such a system is identical to that of the corresponding mean-field model, which is called "exactness of the mean-field theory". It is found out that this expectation is not necessarily true if the microcanonical ensemble is not equivalent to the canonical ensemble in the mean-field model. Moreover, necessary and sufficient conditions for exactness of the mean-field theory are obtained. These conditions are investigated for two concrete models, the \alpha-Potts model with annealed vacancies and the \alpha-Potts model with invisible states.Comment: 23 pages, to appear in J. Stat. Phy

1-d gravity in infinite point distributions

The dynamics of infinite, asymptotically uniform, distributions of self-gravitating particles in one spatial dimension provides a simple toy model for the analogous three dimensional problem. We focus here on a limitation of such models as treated so far in the literature: the force, as it has been specified, is well defined in infinite point distributions only if there is a centre of symmetry (i.e. the definition requires explicitly the breaking of statistical translational invariance). The problem arises because naive background subtraction (due to expansion, or by "Jeans' swindle" for the static case), applied as in three dimensions, leaves an unregulated contribution to the force due to surface mass fluctuations. Following a discussion by Kiessling, we show that the problem may be resolved by defining the force in infinite point distributions as the limit of an exponentially screened pair interaction. We show that this prescription gives a well defined (finite) force acting on particles in a class of perturbed infinite lattices, which are the point processes relevant to cosmological N-body simulations. For identical particles the dynamics of the simplest toy model is equivalent to that of an infinite set of points with inverted harmonic oscillator potentials which bounce elastically when they collide. We discuss previous results in the literature, and present new results for the specific case of this simplest (static) model starting from "shuffled lattice" initial conditions. These show qualitative properties (notably its "self-similarity") of the evolution very similar to those in the analogous simulations in three dimensions, which in turn resemble those in the expanding universe.Comment: 20 pages, 8 figures, small changes (section II shortened, added discussion in section IV), matches final version to appear in PR

Relaxation to thermal equilibrium in the self-gravitating sheet model

We revisit the issue of relaxation to thermal equilibrium in the so-called "sheet model", i.e., particles in one dimension interacting by attractive forces independent of their separation. We show that this relaxation may be very clearly detected and characterized by following the evolution of order parameters defined by appropriately normalized moments of the phase space distribution which probe its entanglement in space and velocity coordinates. For a class of quasi-stationary states which result from the violent relaxation of rectangular waterbag initial conditions, characterized by their virial ratio R_0, we show that relaxation occurs on a time scale which (i) scales approximately linearly in the particle number N, and (ii) shows also a strong dependence on R_0, with quasi-stationary states from colder initial conditions relaxing much more rapidly. The temporal evolution of the order parameter may be well described by a stretched exponential function. We study finally the correlation of the relaxation times with the amplitude of fluctuations in the relaxing quasi-stationary states, as well as the relation between temporal and ensemble averages.Comment: 37 pages, 24 figures; some additional discussion of previous literature and other minor modifications, final published versio

Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach

We investigate the laws that rule the behavior of the largest Lyapunov exponent (LLE) in many particle systems with long range interactions. We consider as a representative system the so-called Hamiltonian alpha-XY model where the adjustable parameter alpha controls the range of the interactions of N ferromagnetic spins in a lattice of dimension d. In previous work the dependence of the LLE with the system size N, for sufficiently high energies, was established through numerical simulations. In the thermodynamic limit, the LLE becomes constant for alpha greater than d whereas it decays as an inverse power law of N for alpha smaller than d. A recent theoretical calculation based on Pettini's geometrization of the dynamics is consistent with these numerical results (M.-C. Firpo and S. Ruffo, cond-mat/0108158). Here we show that the scaling behavior can also be explained by a random matrix approach, in which the tangent mappings that define the Lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble.Comment: 5 pages, no figure

Effects of a 12-week suspension versus traditional resistance training program on body composition, bioimpedance vector patterns, and handgrip strength in older men: A randomized controlled trial

This investigation aimed to compare the effects of suspension training versus traditional resistance exercise using a combination of bands and bodyweight on body composition, bioimpedance vector patterns, and handgrip strength in older men. Thirty-six older men (age 67.4 ± 5.1 years, BMI 27.1 ± 3.3 kg/m2) were randomly allocated into suspension training (n = 12), traditional training (n = 13), or non-exercise (n = 11) groups over a 12-week study period. Body composition was assessed using conventional bioelectrical impedance analysis and classic and specific bioelectric impedance vector analysis, and handgrip strength was measured with a dynamometer. Results showed a significant (p &lt; 0.05) group by time interaction for fat mass, fat-free mass, total body water, skeletal muscle index, classic and specific bioelectrical resistance, classic bioelectrical reactance, phase angle, and dominant handgrip strength. Classic and specific vector displacements from baseline to post 12 weeks for the three groups were observed. Handgrip strength increased in the suspension training group (p &lt; 0.01, ES: 1.50), remained stable in the traditional training group, and decreased in the control group (p &lt; 0.01, ES: −0.86). Although bodyweight and elastic band training helps to prevent a decline in muscle mass and handgrip strength, suspension training proved more effective in counteracting the effects of aging in older men under the specific conditions studied

Implications of the Unitarity Triangle `uc' for J, δ\delta and ∣VCKM∣|V_{CKM}| elements

The Jarlskog rephasing invariant parameter $|J|$ is evaluated using one of the six Unitarity Triangles involving well known CKM matrix elements \vud, \vus,~\rub, ~\vcd, ~\vcs~ and ~\vcb. With PDG2000 values of \vud~ etc. as input, we obtain $|J|=(2.71 \pm 1.12) \times 10^{-5}$, which in the PDG representation of CKM matrix leads to the range $21^o~to~159^o$ for the CP violating phase $\delta$. The CKM matrix elements evaluated using this range of $\delta$ are in agreement with the PDG CKM matrix. The implications of refinements in the input on $|J|$, $\delta$ and CKM matrix elements have also been studied.Comment: 14 pages, 3 figures (eps), updated in the light of latest PDG2000 dat

Kenya model: Development and implementation of an overseas study course on African wildlife ecology and management

The brochure declares: What better place to study a diversity of wildlife species and ecosystems than Kenya\u27s spectacular National Parks and Conservation Areas? Enticing! Exhilarating! A once in a life time experience! African Wildlife Ecology and Management in Kenya is an intensive two and a half week overseas study program offered by Michigan State University\u27s (MSU) Department of Fisheries and Wildlife. Through this hands-on experience, students apply wildlife management principles to issues in Kenya\u27s National Parks and Conservation Areas. Planning and coordination of this course requires a year\u27s worth of thoughtful preparation in order to provide students with a dynamic yet placid in-country experience. To better aid other educators and coordinators in development and implementation of similar courses, we present a detailed account of the history and evolution of African Wildlife Ecology and Management in Kenya. How was this course conceived? How was support garnered from the University? What is required for developing such a course? Furthermore, we present information on why different sites within Kenya were selected and how the order of visitation to these sites allows for a logical progression and increasingly more elaborate acquisition of knowledge of course material. Finally, we describe the various projects assigned to students and the rational for assigning them; the basis for using student groups throughout the in-country experience; the use of alternative forms of assessment to evaluate student learning; assigned readings and course packet development and contents; and implications of limited time and lack of technology while in-country