Collective motion algorithm

An algorithm that incorporates large nonlocal moves of bonds and works for dense polymer systems (even without any vacancies, = 1) is the collective motion algorithm where one transports beads from kinks or chain ends along the chain contour to another position along the chain, for several chains simultaneously, so that in this way this rearrange-... 

Faster computers and development of better numerical algorithms have created the possibility to apply RPA in combination with semiempirical Hamiltonian models to large molecular sterns. Sekino and Bartlett - derived the TDHF expressions for frequency-dependent off-resonant optical polarizabilities using a perturbative expansion of the HF equation (eq 2.8) in powers of external field. This approacii was further applied to conjugated polymer (iialns. The equations of motion for the time-dependent density matrix of a polyenic chain were first derived and solved in refs 149 and 150. The TDHF approach based on the PPP Hamiltonian - was subsequently applied to linear and nonlinear optical response of neutral polyenes (up to 40 repeat units) - and PPV (up to 10 repeat units). " The electronic oscillators (We shall refer to eigenmodes of the linearized TDHF eq with eigenfrequencies Qv as electronic oscillators since they represent collective motions of electrons and holes (see Section II))... 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

A few details on the basic algorithms used to integrate the equations of motion are collected in Appendix B. 

The general principle of MD [1] is the numerical solution of Newton s laws of motion for a collection of particles with given initial position and velocity, and empirical potentials desaibing the interactions between particles. This is achieved by approximating the motion with a series of small timesteps the smaller the timesteps the better the approximation. Although there are a number of different methods for calculating the future position, velocity and acceleration of a particle, the one employed in this work is the velocity Verlet algorithm [1], which works as follows ... 

The objective of a simulation is to generate particle motion by using appropriate algorithms and to obtain an adequate distribution function, and thus, the macroscopic properties. In thermod5mamics, we use this method to calculate the energy of interaction of a collection of molecules, the part of configuration of the translational partition function and the radial distribution function. 

Cross Correlation Displacement of particles in PIV image pairs is calculated based on correlation approach in contrast to the particle tracking algorithm, where particle path is followed. Here, the average motion of a small group of particles contained in the interrogation spot is calculated by spatial autocorrelation or cross correlation. Autocorrelation is performed when images for both laser pulses are recorded on the same sensor, while in cross correlation, each pulse is collected into separate frames. Cross-correlation calculation can be carried out faster... 

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