Chopping Time of the FPU alpha-Model

We study, both numerically and analytically, the time needed to observe the breaking of an FPU \u3b1-chain in two or more pieces, starting from an unbroken configuration at a given temperature. It is found that such a \u201cchopping\u201d time is given by a formula that, at low temperatures, is of the Arrhenius-Kramers form, so that the chain does not break up on an observable time-scale. The result explains why the study of the FPU problem is meaningful also in the ill-posed case of the \u3b1-model

THE IMPLEMENTATION OF CHARACTER EDUCATION ON ISLAMIC RELIGIOUS EDUCATION SUBJECTS AT SENIOR HIGH SCHOOL 1 BENTENG, SELAYAR

This article discusses the Implementation of Character Education in Islamic Religious Education Subjects at Senior High School 1 (SMA 1) Benteng, Selayar. This study uses qualitative research by analyzing existing information data, based on facts in the field. Data obtained through observation, interviews,    documentation and drawing conclusions so that they can unravel the problem completely in accordance with the problems discussed. Implementation of planting character values to students. So the most important thing is the teacher's expertise in internalizing the character education. into all aspects of subjects that will be given to students in schools because character education is one of the main elements of religion that cannot be separated from Islamic teachings. That is why character education is integrated in Islamic Religious Education subjects

The Fermi-Pasta-Ulam problem and its underlying integrable dynamics: an approach through Lyapunov Exponents

FPU models, in dimension one, are perturbations either of the linear model or of the Toda model; perturbations of the linear model include the usual $\beta$-model, perturbations of Toda include the usual $\alpha+\beta$ model. In this paper we explore and compare two families, or hierarchies, of FPU models, closer and closer to either the linear or the Toda model, by computing numerically, for each model, the maximal Lyapunov exponent $\chi$. We study the asymptotics of $\chi$ for large $N$ (the number of particles) and small $\epsilon$ (the specific energy $E/N$), and find, for all models, asymptotic power laws $\chi\simeq C\epsilon^a$, $C$ and $a$ depending on the model. The asymptotics turns out to be, in general, rather slow, and producing accurate results requires a great computational effort. We also revisit and extend the analytic computation of $\chi$ introduced by Casetti, Livi and Pettini, originally formulated for the $\beta$-model. With great evidence the theory extends successfully to all models of the linear hierarchy, but not to models close to Toda

LEXICAL AND CONTEXTUAL MEANINGS CONTAINED IN RELIGIOUS SONG LYRICS AT QUIVER CENTER ACADEMY (QCA)

ABSTRACTThe purpose of this research is to study the theme, lexical and contextual meaning in the lyrics of the songs frequently sung in Quiver Center Academy (QCA) school, located in Gading Serpong, Tangerang city. This research uses descriptive analysis method. The analysis  result in the conclusion that noun is the most category word that has lexical and contextual meaning in the songs. The main reason why these words have lexical and contextual meaning is to know the meaning of the song in a deeper way. Moreover, they also support the understanding of the theme in QCA’s are divinity, destiny, integrity, victory, struggle, and dedication.Keyword: theme, lexical meaning, and contextual meaning

Burgers Turbulence in the Fermi-Pasta-Ulam-Tsingou Chain

We prove analytically and show numerically that the dynamics of the Fermi-Pasta-Ulam-Tsingou chain is characterized by a transient Burgers turbulence regime on a wide range of time and energy scales. This regime is present at long wavelengths and energy per particle small enough that equipartition is not reached on a fast timescale. In this range, we prove that the driving mechanism to thermalization is the formation of a shock that can be predicted using a pair of generalized Burgers equations. We perform a perturbative calculation at small energy per particle, proving that the energy spectrum of the chain Ek decays as a power law, Ek & SIM; k-zeta ot thorn , on an extensive range of wave numbers k. We predict that zeta ot thorn takes first the value 8=3 at the Burgers shock time, and then reaches a value close to 2 within two shock times. The value of the exponent zeta 1/4 2 persists for several shock times before the system eventually relaxes to equipartition. During this wide time window, an exponential cutoff in the spectrum is observed at large k, in agreement with previous results. Such a scenario turns out to be universal, i.e., independent of the parameters characterizing the system and of the initial condition, once time is measured in units of the shock time

On the role of the Integrable Toda model in one-dimensional molecular dynamics

We prove that the common Mie-Lennard-Jones (MLJ) molecular potentials, appropriately normalized via an affine transformation, converge, in the limit of hard-core repulsion, to the Toda exponential potential. Correspondingly, any Fermi-Pasta-Ulam (FPU)-like Hamiltonian, with MLJ-type interparticle potential, turns out to be $1/n$-close to the Toda integrable Hamiltonian, $n$ being the exponent ruling repulsion in the MLJ potential. This means that the dynamics of chains of particles interacting through typical molecular potentials, is close to integrable in an unexpected sense. Theoretical results are accompanied by a numerical illustration; numerics shows, in particular, that even the very standard 12--6 MLJ potential is closer to integrability than the FPU potentials which are more commonly used in the literature.Comment: 22 pages, 14 figures, Submitted in Journal of Statistical Physic

Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

We consider the Fermi\u2013Pasta\u2013Ulam\u2013Tsingou (FPUT) chain composed by N 6b 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature \u3b2- 1. Given a fixed 1 64 m 6a N, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order \u3b2, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics

Tsallis Ensemble as an Exact Orthode

We show that Tsallis ensemble of power-law distributions provides a mechanical model of nonextensive equilibrium thermodynamics for small interacting Hamiltonian systems, i.e., using Boltzmann's original nomenclature, we prove that it is an exact orthode. This means that the heat differential admits the inverse average kinetic energy as an integrating factor. One immediate consequence is that the logarithm of the normalization function can be identified with the entropy, instead of the q-deformed logarithm. It has been noted that such entropy coincides with Renyi entropy rather than Tsallis entropy, it is non-additive, tends to the standard canonical entropy as the power index tends to infinity and is consistent with the free energy formula proposed in [S. Abe et. al. Phys. Lett. A 281, 126 (2001)]. It is also shown that the heat differential admits the Lagrange multiplier used in non-extensive thermodynamics as an integrating factor too, and that the associated entropy is given by ordinary nonextensive entropy. The mechanical approach proposed in this work is fully consistent with an information-theoretic approach based on the maximization of Renyi entropy.Comment: 5 pages. Added connection with Renyi entrop

Energy Localization in the Peyrard-Bishop DNA model

We study energy localization on the oscillator-chain proposed by Peyrard and Bishop to model the DNA. We search numerically for conditions with initial energy in a small subgroup of consecutive oscillators of a finite chain and such that the oscillation amplitude is small outside this subgroup for a long timescale. We use a localization criterion based on the information entropy and we verify numerically that such localized excitations exist when the nonlinear dynamics of the subgroup oscillates with a frequency inside the reactive band of the linear chain. We predict a mimium value for the Morse parameter $(\mu >2.25)$ (the only parameter of our normalized model), in agreement with the numerical calculations (an estimate for the biological value is $\mu =6.3$). For supercritical masses, we use canonical perturbation theory to expand the frequencies of the subgroup and we calculate an energy threshold in agreement with the numerical calculations

Low-dimensional q-Tori in FPU Lattices: Dynamics and Localization Properties

This is a continuation of our study concerning q-tori, i.e. tori of low dimensionality in the phase space of nonlinear lattice models like the Fermi-Pasta-Ulam (FPU) model. In our previous work we focused on the beta FPU system, and we showed that the dynamical features of the q-tori serve as an interpretational tool to understand phenomena of energy localization in the FPU space of linear normal modes. In the present paper i) we employ the method of Poincare - Lindstedt series, for a fixed set of frequencies, in order to compute an explicit quasi-periodic representation of the trajectories lying on q-tori in the alpha model, and ii) we consider more general types of initial excitations in both the alpha and beta models. Furthermore we turn into questions of physical interest related to the dynamical features of the q-tori. We focus on particular q-tori solutions describing low-frequency `packets' of modes, and excitations of a small set of modes with an arbitrary distribution in q-space. In the former case, we find formulae yielding an exponential profile of energy localization, following an analysis of the size of the leading order terms in the Poincare - Lindstedt series. In the latter case, we explain the observed localization patterns on the basis of a rigorous result concerning the propagation of non-zero terms in the Poincare - Lindstedt series from zeroth to subsequent orders. Finally, we discuss the extensive (i.e. independent of the number of degrees of freedom) properties of some q-tori solutions.Comment: To appear in Physica D, 34 pages, 9 figure