# Time average

The product sublimation and melting are both carried out on a noncontinuous basis. Thus time-averaged values have been taken. [c.334]

This method of running drilling operations has been very successfully applied in recent years and has resulted in considerable cost savings. Various systems are in operation, usually providing a bonus for better than average performance. The contractor agrees with the company the specifications for the well. Then the historic cost of similar wells which have been drilled in the past are established. This allows estimation of the costs expected for the new well. The contractor will be entirely in charge of drilling the well, and cost savings achieved will be split between company and contractor. [c.62]

It is of interest in the present context (and is useful later) to outline the statistical mechanical basis for calculating the energy and entropy that are associated with rotation [66]. According to the Boltzmann principle, the time average energy of a molecule is given by [c.582]

Dispersion forces caimot be explained classically but a semiclassical description is possible. Consider the electronic charge cloud of an atom to be the time average of the motion of its electrons around the nucleus. [c.192]

The thinking behind this was that, over a long time period, a system trajectory in F space passes tlirough every configuration in the region of motion (liere the energy shell), i.e. the system is ergodic, and hence the infinite time average is equal to the average in the region of motion, or the average over the microcanonical ensemble density. The ergodic hypothesis is meant to provide justification of the equal a priori probability postulate. [c.387]

It is a strong condition. For a system of particles, the infinitely long time must be much longer than O (e" ), whereas the usual observation time window is O (1). (When one writes y = O (x) and z = o (x), it implies that lim v/z = finite 0 and lim z/x = 0.) However, if by reasonable fiinctions one means large variables 0(N), then their values are nearly the same everywhere in the region of motion and the trajectory need not be truly ergodic for the time average to be equal to the ensemble average. Ergodicity of a trajectory is a difficult mathematical problem in mechanics. [c.387]

The Boltzmaim weight appears implicitly in the way the states are chosen. The fomi of the above equation is like a time average as calculated in MD. The MC method involves designing a stochastic algorithm for stepping from one state of the system to the next, generating a trajectory. This will take the fomi of a Markov chain, specified by transition probabilities which are independent of the prior history of the system. [c.2256]

The dipole force can be readily understood by considering tire light as a classical wave. It is simply tire time-averaged force arising from tire interaction of tire transition dipole, induced by tire oscillating electric field of tire light, witli tire gradient of tire electric field amplitude. Focusing tire light beam controls tire magnitude of tliis gradient and detuning tire optical frequency below or above tire atomic transition controls tire sign of tire force acting on tire atom. Tuning tire light below resonance attracts tire atom to tire centre of tire light beam while tuning above resonance repels it. The dipole force is a stimulated process in which no net exchange of energy between tire [c.2457]

The time-averaged force, equation (Cl.4.3), consists of two tenns tire first tenn is proportional to tire gradient of tire electric field amplitude tire second tenn is proportional to tire gradient of tire phase. Substituting equation (Cl.4.4) and equation (Cl.4.5) into equation (Cl.4.3), we have for tire two tenns. [c.2459]

In applications, one is often interested in approximating time averages over a time interval [0, T] via associated mean values of a , k = 1. ..Tfr. For T (or r) small enough, the above backward analysis may lead to much better error estimates than the worst case estimates of forward analysis. [c.101]

Abstract. Simulation of the dynamics of biomolecules requires the use of a time step in the range 0.5-1 fs to obtain acceptable accuracy. Nevertheless, the bulk of the CPU time is spent computing interactions, such as those due to long-range electrostatics, which vary hardly at all from one time step to the next. This unnecessary computation is dramatically reduced with the use of multiple time stepping methods, such as the Verlet-I/r-RESPA method, which is based on approximating slow forces as widely separated impulses. Indeed, numerical experiments show that time steps of 4 fs are possible for these slow forces but unfortunately also show that a long time step of 5 fs results in a dramatic energy drift. Moreover, this is less pronounced if one uses a yet larger long time step The cause of the problem can be explained by exact analysis of a simple two degree-of-freedom linear problem, which predicts numerical instability if the time step is just less than half the period of the fastest normal mode. To overcome this, a modification of the impulsive Verlet-I/r-RESPA method is proposed, called the mollified impulse method. The idea is that one modifies the slow part of the potential energy so that it is evaluated at time averaged values of the positions, and one uses the gradient of this modified potential for the slow part of the force. Various versions of the algorithm are implemented for water and numerical results are presented. [c.318]

Fig. 5 shows that all three time averaging methods succeed for a long timestep At of 5 fs. [c.328]

This procedure is illustrated with an example consisting of a much simplified situation (Figure 8-8). Consider a 2D space, with only two points (Pj and P ), and a predetermined mirror axis. In order to find the nearest mirror-symmetric configuration (Pj and Pj). the identity operation is performed on Pj, and the mirror reflection is performed on P2. yielding Pj and P2 respectively. The two new points. Pi and, P2 are then averaged. The resulting point, Pi, is mirror-reflected, resulting in P2, and obtaining the mirror-symmetric configuration. [c.419]

Time Averages, Ensemble Averages and Some Historical Background [c.317]

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. [c.63]

The solid phase of matter offers a very different enviromnent to examine the chemical bond than does a gas or liquid [1, 2, 3, 4 and 5]. The obvious difference involves describing the atomic positions. In a solid state, one can often describe atomic positions by a static configuration, whereas for liquid and gas phases this is not possible. The properties of the liquids and gases can be characterized only by considering some time-averaged ensemble. This difference between phases offers advantages in describing the solid phase, especially for crystalline matter. Crystals are characterized by a periodic synnnetry that results in a system occupying all space [6]. Periodic, or translational, synnnetry of crystalline phases greatly simplifies discussions of the solid state smce knowledge of the atomic structure within a ftmdamental subunit of the crystal, called the unit cell, is sufficient to describe the entire system encompassing all space. For example, if one is interested in the spatial distribution of electrons in a crystal, it is sufficient to know what this distribution is within a unit cell. [c.86]

It was assumed that, apart from a vanishingly small number of exceptions, the initial conditions do not have an effect on these averages. However, since tire limitmg value of the time averages caimot be computed, an [c.387]

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors [c.1503]

In writing equation (Cl. 4.3) we have made use of tire fact tliat tire time-average dipole has in-phase and inquadrature components. [c.2459]

Magnetic resonance techniques, while powerful spectroscopic probes of molecular stmcture (sections B1.12-B1.16), typically have quite low sensitivities, and direct detection of single nuclear or electron spins has yet to be demonstrated. ITowever, electron spin resonance at the single-molecule level has been demonstrated tlirough the indirect teclmique of optically detected magnetic resonance. The original experiments exploited the dependence of the time-averaged fluorescence intensity on the rate of intersystem crossing from the fluorescent singlet to the essentially nonemissive triplet state. The splittings among the magnetic components of the triplet state were detected by sweeping the RF field while measuring the total fluorescence intensity from a single molecule selected out of the inlromogeneous ensemble by its fluorescence excitation frequency [132, 133]. This idea has subsequently been extended to examine isotope effects on the rf resonant linewidths [134, 135] and to demonstrate single-spin coherence and spin echo phenomena [136, 137]. [c.2497]

Abstract. The paper presents basic concepts of a new type of algorithm for the numerical computation of what the authors call the essential dynamics of molecular systems. Mathematically speaking, such systems are described by Hamiltonian differential equations. In the bulk of applications, individual trajectories are of no specific interest. Rather, time averages of physical observables or relaxation times of conformational changes need to be actually computed. In the language of dynamical systems, such information is contained in the natural invariant measure (infinite relaxation time) or in almost invariant sets ("large finite relaxation times). The paper suggests the direct computation of these objects via eigenmodes of the associated Probenius-Perron operator by means of a multilevel subdivision algorithm. The advocated approach is different from both Monte-Carlo techniques on the one hand and long term trajectory simulation on the other hand in our setup long term trajectories are replaced by short term sub-trajectories, Monte-Carlo techniques are connected via the underlying Probenius-Perron structure. Numerical experiments with the suggested algorithm are included to illustrate certain distinguishing properties. [c.98]

In fact, numerical observations show that the average of the total energy is nearly constant over rather long time spans for large stepsizes, say r ss 1 fs. However, this desirable property does not carry over to other averages, where stepsizes much smaller than desirable (r

[c.133]

An example of a time averaging function -4(a ) is the formula termed LongAverage in [7] [c.325]

If the fastest forces irfastest consist only of bond stretching and angle bending, then it is possible to define a time averaging A as a projection onto the equilibrium value of the bond lengths and bond angles. This technique appears to have better stabilizing properties. Flirther details are to be provided elsewhere [11]. Here this time averaging is called Equilibrium. [c.327]

Fig. 4. Total pseudoenergy (in kcal/moi) vs. sirauiation time (in fs) for time averaging, Equilibrium, and impuise methods. (At for all methods equals 8 fs.) |

Fig. 5. Total pseudoenergy (in kcal/mol) vs. simulation time (in fs) for time averaging, Equilibrium, and impulse methods. At for all methods equals 5 fs.) |

See pages that mention the term

**Time average**:

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Modelling molecular structures (2000) -- [ c.59 ]