Newtonian trajectories

In the classical picture developed above, the wavepacket is modeled by pseudo-particles moving along uncorrelated Newtonian trajectories, taking the electrons with them in the form of the potential along the Uajectory. In this spirit, a classical wavepacket can be defined as an incoherent (i.e., noninteracting) superposition of confignrations, X/(, t)tlt,(r, t)... 

Lechner et al.2n have used equation (7) in free energy calculations, taking advantage of the fact that the procedure does not need to sample physically realistic trajectories to replace solution of the Newtonian dynamics by an appropriate mapping. This improves the efficiency of the usual implementation of (7) where Newtonian trajectories are used in the transformation from state 1 to state 2. They study a system where a particle is moved through a dense fluid and find an order of magnitude improvement in the results compared to the standard fast-switching technique. 

The Newtonian trajectories are the special case in which C is equal to zero on the boundaries and everywhere else. There are however other solutions that may satisfy the variational principle and the boundary conditions. For illustration purposes we consider a... 

Since the frequency of the external force is exactly on resonance with the free oscillator, the amplitude of the forced solution is unbound. Hence, the second stationary trajectory of the Gauss action (in addition to the Newtonian trajectory) is a solution with a growing amplitude (and energy) as a function of time. This solution is unlikely to be present in a boundary value formulation in which both end points have physically sound values. Note that the energy growth here has a different origin when compared to the numerical instability induced by a large step in the difference equation (Sect. 12.3.3). 

After presenting the algorithm, we show that the proposed numerical protocol interpolates (as a function of the approximation quality) from a steepest descent path (a poor approximation to the dynamics) to an exact Newtonian trajectory. Finally, we present an efficient algorithm to compute relative rates that can be used with our formulation to compute experimental observables,... 

A complication we should keep in mind when comparing Sg to the usual classical action is that the Newtonian trajectory is not the only stationary solution of the Gauss action. A standard variation of Eq. (20) leads to a fourth order differential equation and hence to two more solutions in addition to the true classical trajectory (the two additional solutions are related by a time reversal operation). An example was discussed in details in Ref. 4 [see... 

We now have established a framework for the thermodynamic properties of a model system. What would be desirable is that we could approximate the force terms arising in the bulk, without the need to simulate them directly. It should be apparent that constant-energy dynamics (designed to sample the microcanonical ensemble with constant energy E) will not sample the canonical distribution in the absence of the heat bath such Newtonian trajectories cannot access regions of the phase space where H z) H(zo), where zo is the initial condition. 

An earlier suggestion of Fixman of how to mimic flexibility has been discussed with respect to a system with two degrees of freedom. Harmonic stretching and bending of the bonds of butane in liquid CCI4 has subsequently been allowed transition state theory does not exactly apply as many barrier crossings are reflected by solvent collisions. In a modified molecular dynamics examination of conformational isomerizations in butane the effect of solvent was expressed with a stochastic model in which the Newtonian trajectory was modified by random impulses. The frequency of these impulses, which have a frictional effect upon the trajectory, reduced the value of the transmission coefficient by inducing oscillatory motion at the col. At the inner bonds of decane isomerization rates are less than in butane. 

It is possible to parametarize the time-dependent Schrddinger equation in such a fashion that the equations of motion for the parameters appear as classical equations of motion, however, with a potential that is in principle more general than that used in ordinary Newtonian mechanics. However, it is important that the method is still exact and general even if the trajectories aie propagated by using the ordinary classical mechanical equations of motion. 

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... 

D[X t) is used to denote a path integral. Hence, equation (14) corresponds to a summation of all paths leading from X(0) to X t). The same expression is used for the Brownian trajectories and for Newtonian s trajectories with errors. The action is of course different in both cases. 

Simulations. In addition to analytical approaches to describe ion—soHd interactions two different types of computer simulations are used Monte Cado (MC) and molecular dynamics (MD). The Monte Cado method rehes on a binary coUision model and molecular dynamics solves the many-body problem of Newtonian mechanics for many interacting particles. As the name Monte Cado suggests, the results require averaging over many simulated particle trajectories. A review of the computer simulation of ion—soUd interactions has been provided (43). 

As with Newtonian molecular dynamics, a number of different algorithms have been developed to calculate the diffusional trajectories. An efficient algorithm for solving the Brownian equation of motion was introduced by Ermak and McCammon [21]. A detailed survey of this and other algorithms as well as their application can be found in Ref. 2. 

The highest probability paths will make the argument of the exponential small. That will be true for paths that follow Newtonian dynamics where mr = F(r). Olender and Elber [45] demonstrated how large values of the time step ht can be used in a way that projects out high frequency motions of the system and allows for the simulation of long-time molecular dynamics trajectories for macromolecular systems. 

For Newtonian dynamics and a canonical distributions of initial conditions one can reject or accept the new path before even generating the trajectory. This can be done because Newtonian dynamics conserves the energy and the canonical phase-space distribution is a function of the energy only. Therefore, the ratio plz ]/p z at time 0 is equal to the ratio p[.tj,n ]/p z ° at the shooting time and the new trajectory needs to be calculated only if accepted. For a microcanonical distribution of initial conditions all phase-space points on the energy shell have the same weight and therefore all new pathways are accepted. The same is true for Langevin dynamics with a canonical distribution of initial conditions. 

An alternative way to obtain the quantum trajectories is by formulating the Bohmian mechanics as a Newtonian-like theory. Then, Equation 8.29 gives rise to a generalized Newton s second law ... 

The basic principles are described in many textbooks [24, 26]. They are thus only sketchily presented here. In a conventional classical molecular dynamics calculation, a system of N particles is placed within a cell of fixed volume, most frequently cubic in size. A set of velocities is also assigned, usually drawn from a Maxwell-Boltzmann distribution appropriate to the temperature of interest and selected in a way so as to make the net linear momentum zero. The subsequent trajectories of the particles are then calculated using the Newtonian equations of motion. Employing the finite difference method, this set of differential equations is transformed into a set of algebraic equations, which are solved by computer. The particles are assumed to interact through some prescribed force law. The dispersion, dipole-dipole, and polarization forces are typically included whenever possible, they are taken from the literature. 

For a particle of mass m moving in the (x, y) plane with force per mass (X, Y), instantaneous motion is described by velocity v along the trajectory. An instantaneous radius of curvature p is defined by angular momentum i = mvp such that the centrifugal force mu2/p balances the true force normal to the trajectory. Hence, following Euler s derivation, Newtonian mechanics implies that... 

Molecular dynamics is frequently portrayed as a method based on the ergodicity hypothesis which states that the trajectory of a system propagating in time through the phase space following the Newtonian laws of motion given by the equations ... 

In principle, one can write down all of these forces and formulate the Newtonian equations of motion for the fluid this yields a complicated differential equation known as the Navier-Stokes equation [1-3]. A complete solution of the Navier-Stokes equation gives the exact trajectory and velocity of each fluid element. In practice, the calculations are often difficult because one must simultaneously account for all fluid elements and the interactions between these elements caused by the viscous drag forces. (The simultaneous motion of many interacting fluid elements is analogous to the simultaneous motion of many interacting mechanical objects, the latter being so complicated that it is described as the many body problem. ) However, in certain cases, the Navier-Stokes equation is reduced to a tractable form by the existence of steady low-velocity flow and high symmetry in the flow conduit (e.g., capillary tubes of circular cross section). We will examine such simple cases shortly. 

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