scholarly journals Flexible constraints: An adiabatic treatment of quantum degrees of freedom, with application to the flexible and polarizable mobile charge densities in harmonic oscillators model for water

2002 ◽  
Vol 116 (22) ◽  
pp. 9602-9610 ◽  
Author(s):  
Berk Hess ◽  
Humberto Saint-Martin ◽  
Herman J. C. Berendsen
Keyword(s):  
Degrees Of Freedom ◽  
Harmonic Oscillators ◽  
Mobile Charge ◽  
Charge Densities

scholarly journals A mobile charge densities in harmonic oscillators (MCDHO) molecular model for numerical simulations: The water–water interaction

2000 ◽  
Vol 113 (24) ◽  
pp. 10899-10912 ◽  
Author(s):  
Humberto Saint-Martin ◽  
Jorge Hernández-Cobos ◽  
Margarita I. Bernal-Uruchurtu ◽  
Iván Ortega-Blake ◽  
Herman J. C. Berendsen
Keyword(s):  
Numerical Simulations ◽  
Molecular Model ◽  
Harmonic Oscillators ◽  
Mobile Charge ◽  
Charge Densities ◽  
Water Interaction

Analysis of a Class of Coupled Nonlinear Oscillators With an Application to Flow Induced Vibrations

2001 ◽  
Author(s):  
T. I. Haaker
Keyword(s):  
Normal Mode ◽  
Mixed Mode ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Original System ◽  
Harmonic Oscillators ◽  
Resonant System ◽  
Coupled Equations ◽  
Averaged Equations ◽  
Internal Resonant

Abstract We consider in this paper the following system of coupled nonlinear oscillatorsx..+x-k(y-x)=εf(x,x.),y..+(1+δ)y-k(x-y)=εf(y,y.). In this system we assume ε to be a small parameter, i.e. 0 < ε ≪ 1. A coupling between the two oscillators is established through the terms involving the positive parameter k. The coupling may be interpreted as a mutual force depending on the relative positions of the two oscillators. For both ε and k equal to zero the two oscillators are decoupled and behave as harmonic oscillators with frequencies 1 and 1+δ, respectively. The parameter δ may therefore be viewed as a detuning parameter. Finally, the term ε f represents a small force acting upon each oscillator. Note that this force depends only on the position and velocity of the oscillator upon which the force is acting. To analyse the system’s dynamic behaviour we use the method of averaging. When k and δ are choosen such that no internal resonance occurs, one typically observes the following behaviour. If the trivial solution is unstable, solutions asymptotically tend to one of the two normal modes or to a mixed mode solution. For the special case with δ = 0 a system of two identical oscillators is found. If in addition k is O(ε) we obtain a 1 : 1 internal resonant system. The averaged equations may then be reduced to a system of three coupled equations — two for the amplitudes and one for the phase difference. Due to the fact that we consider identical oscillators there is a symmetry in the averaged equations. The normal mode solutions, as found for the non-resonant case, are still present. New mixed mode solutions appear. Moreover, Hopf bifurcations in the averaged system lead to limit cycles that correspond to oscillations in the original system with periodically modulated amplitudes and phases. We also consider the case with δ = O(ε), i.e. the case with nearly identical oscillators. If k = O(ε) again a 1 : 1 internal resonant system is found. Contrary to the previous cases the normal mode solutions no longer exist. Moreover, different bifurcations are observed due to the disappearance of the symmetry present in the system for s = 0. We apply some of the results obtained to a model describing aeroelastic oscillations of a structure with two-degrees-of-freedom.

scholarly journals Classicalization of Quantum Fluctuations at the Planck Scale in the Rh=ct Universe

2021 ◽  
Author(s):  
FULVIO MELIA
Keyword(s):  
Degrees Of Freedom ◽  
Field Potential ◽  
Planck Scale ◽  
Quantum Fluctuations ◽  
Harmonic Oscillators ◽  
New Developments ◽  
Active Mass ◽  
Primordial Power Spectrum ◽  
Classical Transition ◽  
Planck Time

Abstract The quantum to classical transition of fluctuations in the early universe is still not completely understood. Some headway has been made incorporating the effects of decoherence and the squeezing of states, though the methods and procedures continue to be challenged. But new developments in the analysis of the most recent Planck data suggest that the primordial power spectrum has a cutoff associated with the very first quantum fluctuation to have emerged into the semi-classical universe from the Planck domain at about the Planck time. In this paper, we examine the implications of this result on the question of classicalization, and demonstrate that the birth of quantum fluctuations at the Planck scale would have been a `process' supplanting the need for a `measurement' in quantum mechanics. Emerging with a single wavenumber, these fluctuations would have avoided the interference between different degrees of freedom in a superposed state. Moreover, the implied scalar field potential had an equation-of-state consistent with the zero active mass condition in general relativity, allowing the quantum fluctuations to emerge in their ground state with a time-independent frequency. They were therefore effectively quantum harmonic oscillators with classical correlations in phase space from the very beginning.

scholarly journals A simple derivation of the Lindblad equation

2013 ◽  
Vol 35 (1) ◽  
Author(s):  
Carlos Alexandre Brasil ◽  
Felipe Fernandes Fanchini ◽  
Reginaldo de Jesus Napolitano
Keyword(s):  
Undergraduate Students ◽  
Degrees Of Freedom ◽  
Harmonic Oscillators ◽  
Primary System ◽  
Specific Situation ◽  
Markov Approximation ◽  
Zero Temperature ◽  
Simple Derivation ◽  
Lindblad Equation ◽  
Quantum Dynamical

We present a derivation of the Lindblad equation -an important tool for the treatment of nonunitary evolutions -that is accessible to undergraduate students in physics or mathematics with a basic background on quantum mechanics. We consider a specific case, corresponding to a very simple situation, where a primary system interacts with a bath of harmonic oscillators at zero temperature, with an interaction Hamiltonian that resembles the Jaynes-Cummings formato We start with the Born-Markov equation and, tracing out the bath degrees of freedom, we obtain an equation in the Lindblad formo The specific situation is very instructive, for it makes it easy to realize that the Lindblads represent the effect on the main system caused by the interaction with the bath, and that the Markov approximation is a fundamental condition for the emergence of the Lindbladian operator. The formal derivation of the Lindblad equation for a more general case requires the use of quantum dynamical semi-groups and broader considerations regarding the environment and temperature than we have considered in the particular case treated here.

scholarly journals Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1130
Author(s):  
Eugene Oks
Keyword(s):  
Infinite Number ◽  
Spherical Harmonic ◽  
Degrees Of Freedom ◽  
Hamiltonian Dynamics ◽  
Harmonic Oscillators ◽  
General Description ◽  
Stationary States ◽  
Vector Integral ◽  
Classical Formalism ◽  
Atomic And Molecular Physics

Dirac’s Generalized Hamiltonian Dynamics (GHD) is a purely classical formalism for systems having constraints: it incorporates the constraints into the Hamiltonian. Dirac designed the GHD specifically for applications to quantum field theory. In one of our previous papers, we redesigned Dirac’s GHD for its applications to atomic and molecular physics by choosing integrals of the motion as the constraints. In that paper, after a general description of our formalism, we considered hydrogenic atoms as an example. We showed that this formalism leads to the existence of classical non-radiating (stationary) states and that there is an infinite number of such states—just as in the corresponding quantum solution. In the present paper, we extend the applications of the GHD to a charged Spherical Harmonic Oscillator (SHO). We demonstrate that, by using the higher-than-geometrical symmetry (i.e., the algebraic symmetry) of the SHO and the corresponding additional conserved quantities, it is possible to obtain the classical non-radiating (stationary) states of the SHO and that, generally speaking, there is an infinite number of such states of the SHO. Both the existence of the classical stationary states of the SHO and the infinite number of such states are consistent with the corresponding quantum results. We obtain these new results from first principles. Physically, the existence of the classical stationary states is the manifestation of a non-Einsteinian time dilation. Time dilates more and more as the energy of the system becomes closer and closer to the energy of the classical non-radiating state. We emphasize that the SHO and hydrogenic atoms are not the only microscopic systems that can be successfully treated by the GHD. All classical systems of N degrees of freedom have the algebraic symmetries ON+1 and SUN, and this does not depend on the functional form of the Hamiltonian. In particular, all classical spherically symmetric potentials have algebraic symmetries, namely O4 and SU3; they possess an additional vector integral of the motion, while the quantal counterpart-operator does not exist. This offers possibilities that are absent in quantum mechanics.

scholarly journals Anharmonic motionsversusdynamic disorder at the Mg ion from the charge densities in pyrope (Mg3Al2Si3O12) crystals at 30 K: six of one, half a dozen of the other

2017 ◽  
Vol 73 (4) ◽  
pp. 722-736 ◽  
Author(s):  
Riccardo Destro ◽  
Riccardo Ruffo ◽  
Pietro Roversi ◽  
Raffaella Soave ◽  
Laura Loconte ◽  
...  
Keyword(s):  
Charge Density ◽  
Low Temperatures ◽  
Degrees Of Freedom ◽  
Phonon Modes ◽  
Dynamic Disorder ◽  
Charge Densities ◽  
Atomic Displacements ◽  
Maximum Resolution ◽  
Full Analysis ◽  
Experimental Findings

The possible occurrence of static/dynamic disorder at the Mg site in pyrope (Mg3Al2Si3O12), with or without anharmonic contribution to the thermal vibrations even at low temperatures, has been largely debated but conclusions were contrasting. Here a report is given on the experimental charge density distribution, ρEXP, of synthetic pyrope atT= 30 K, built through a Stewart multipolar expansion up tol= 5 and based on a very precise and accurate set of in-home measured single-crystal X-ray diffraction amplitudes with a maximum resolution of 0.44 Å. Local and integral topological properties of ρEXPare in substantial agreement with those of ρTHEO, the corresponding DFT-grade quantum charge density of an ideal pyrope crystal, and those derived from synchrotron investigations of chemical bonding in olivines. Relevant thermal atomic displacements, probably anharmonic in nature, clearly affect the whole structure down to 30 K. No significant (> 2.5σ) residual Fourier peaks are detectable from the ρEXPdistribution around Mg, after least-squares refinement of a multipole model with anharmonic thermal motion at the Mg site. Experimental findings were confirmed by a full analysis of normal vibration modes of the DFT-optimized structure of the perfect pyrope crystal. Mg undergoes wide displacements from its equilibrium position even at very low temperatures, as it is allocated in a ∼ 4.5 Å large dodecahedral cavity and involved in several soft phonon modes. Implications on the interplay among static/dynamic disorder of Mg and lattice vibrational degrees of freedom are discussed.

scholarly journals On entropy and information in gene interaction networks

2018 ◽  
Vol 35 (5) ◽  
pp. 815-822 ◽  
Author(s):  
Z S Wallace ◽  
S B Rosenthal ◽  
K M Fisch ◽  
T Ideker ◽  
R Sasik
Keyword(s):  
Degrees Of Freedom ◽  
Association Studies ◽  
Gene List ◽  
Interaction Network ◽  
Gene Interaction ◽  
Interaction Networks ◽  
Harmonic Oscillators ◽  
Genome Wide Association Studies ◽  
Gene Enrichment ◽  
Gene Sets

Abstract Motivation Modern biological experiments often produce candidate lists of genes presumably related to the studied phenotype. One can ask if the gene list as a whole makes sense in the context of existing knowledge: Are the genes in the list reasonably related to each other or do they look like a random assembly? There are also situations when one wants to know if two or more gene sets are closely related. Gene enrichment tests based on counting the number of genes two sets have in common are adequate if we presume that two genes are related only when they are in fact identical. If by related we mean well connected in the interaction network space, we need a new measure of relatedness for gene sets. Results We derive entropy, interaction information and mutual information for gene sets on interaction networks, starting from a simple phenomenological model of a living cell. Formally, the model describes a set of interacting linear harmonic oscillators in thermal equilibrium. Because the energy function is a quadratic form of the degrees of freedom, entropy and all other derived information quantities can be calculated exactly. We apply these concepts to estimate the probability that genes from several independent genome-wide association studies are not mutually informative; to estimate the probability that two disjoint canonical metabolic pathways are not mutually informative; and to infer relationships among human diseases based on their gene signatures. We show that the present approach is able to predict observationally validated relationships not detectable by gene enrichment methods. The converse is also true; the two methods are therefore complementary. Availability and implementation The functions defined in this paper are available in an R package, gsia, available for download at https://github.com/ucsd-ccbb/gsia.

scholarly journals Toppling Pencils—Macroscopic Randomness from Microscopic Fluctuations

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 1046
Author(s):  
Thomas Dittrich ◽  
Santiago Peña Martínez
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Harmonic Oscillators ◽  
Bistable System ◽  
Asymptotic State ◽  
Initial State ◽  
Central System ◽  
Bistable Systems ◽  
Stable Equilibria ◽  
The Right

We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for N=1, the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the Kolmogorov-Arnol’d-Moser (KAM) scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

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