Force random

The solution Xh(t) of the linearized equations of motion can be solved by standard NM techniques or, alternatively, by explicit integration. We have experimented with both and found the second approach to be far more efficient and to work equally well. Its handling of the random force discretization is also more straightforward (see below). For completeness, we describe both approaches here. 

The initial conditions of system (20) coincide with those for the original equations X/,(0) = X" and V/i(0) = V . Appropriate treatments, as discussed in [72], are essential for the random force at large timesteps to maintain thermal equilibrium since the discretization S(t — t ) => 6nml t is poor for large At. This problem is alleviated by the numerical approach below because the relevant discretization of the Dirac function is the inner timestep At rather than a large At. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

The Boltzmann constant is ks and T the absolute temperature. — is the Dirac delta function. Below we assume for convenience (equation (5)) that the delta function is narrow, but not infinitely narrow. The random force has a zero mean and no correlation in time. For simplicity we further set the friction to be a scalar which is independent of time or coordinates. 

We further note that the Langevin equation (which will not be discussed in detail here) is an intermediate between the Newton s equations and the Brownian dynamics. It includes in addition to an inertial part also a friction and a random force term ... 

A related algorithm can be written also for the Brownian trajectory [10]. However, the essential difference between an algorithm for a Brownian trajectory and equation (4) is that the Brownian algorithm is not deterministic. Due to the existence of the random force, we cannot be satisfied with a single trajectory, even with pre-specified coordinates (and velocities, if relevant). It is necessary to generate an ensemble of trajectories (sampled with different values of the random force) to obtain a complete picture. Instead of working with an ensemble of trajectories we prefer to work with the conditional probability. I.e., we ask what is the probability that a trajectory being at... 

The stochastic differential equation and the second moment of the random force are insufficient to determine which calculus is to be preferred. The two calculus correspond to different physical models [11,12]. It is beyond the scope of the present article to describe the difference in details. We only note that the Ito calculus consider r t) to be a function of the edge of the interval while the Stratonovich calculus takes an average value. Hence, in the Ito calculus using a discrete representation rf t) becomes r] tn) i]n — y n — A i) -I- j At. Developing the determinant of the Jacobian -... 

Tlere, y Is the friction coefficien t of the solven t. In units of ps, and Rj is th e random force im parted to th e solute atom s by the solvent. The friction coefficien t is related to the diffusion constant D oflh e solven l by Em stem T relation y = k jT/m D. Th e ran doin force is calculated as a ratulom number, taken from a Gaussian distribn-... 

A number of simulation methods based on Equation (7.115) have been described. Thess differ in the assumptions that are made about the nature of frictional and random forces A common simplifying assumption is that the collision frequency 7 is independent o time and position. The random force R(f) is often assumed to be uncorrelated with th particle velocities, positions and the forces acting on them, and to obey a Gaussiar distribution with zero mean. The force F, is assumed to be constant over the time step o the integration. 

The average random force over the time step is taken from a Gaussian with a varianc 2mk T y(St). Xj is one of the 3N coordinates at time step i E and R are the relevan components of the frictional and random forces at that time n, is the velocity component. 

The third and final force acting between any pair of beads is a random force ... 

Both the dissipative force and the random force act along the line joining the pair of beads and also conserve linear and angular momentum. The model thus has two unknown functions vP rij) and w Yij) and two unknown constants 7 and a. In fact, only one of the two weight functions can be chosen arbitrarily as they are related [Espanol and Warren 1995]. Moreover, the temperature of the system relates the two constants ... 

The usual choice for the weight functions is to make the random force the same as the conservative force ... 

The conceptual forerunner to mesoscale dynamics is Brownian dynamics. Brownian simulations used equations of motion modified by a random force... 

Dissipative particle dynamics (DPD) is a technique for simulating the motion of mesoscale beads. The technique is superficially similar to a Brownian dynamics simulation in that it incorporates equations of motion, a dissipative (random) force, and a viscous drag between moving beads. However, the simulation uses a modified velocity Verlet algorithm to ensure that total momentum and force symmetries are conserved. This results in a simulation that obeys the Navier-Stokes equations and can thus predict flow. In order to set up these equations, there must be parameters to describe the interaction between beads, dissipative force, and drag. 

Langevin dynamics simulates the effect of molecular collisions and the resulting dissipation of energy that occur in real solvents, without explicitly including solvent molecules. This is accomplished by adding a random force (to model the effect of collisions) and a frictional force (to model dissipative losses) to each atom at each time step. Mathematically, this is expressed by the Langevin equation of motion (compare to Equation (22) in the previous chapter) ... 

Here, y is the friction coefficient of the solvent, in units of ps and Rj is the random force imparted to the solute atoms by the solvent. The friction coefficient is related to the diffusion constant D of the solvent by Einstein s relation y = kgT/mD. The random force is calculated as a random number, taken from a Gaussian distribu-... 

There was a logical progression of technology development from continuous to piezoelecttic ink jet. Designers of continuous ink-jet systems ensure that the ink stream breaks into drops of constant size and frequency by applying vibrational energy with piezoelecttic crystals at the natural frequency of drop formation. This overcomes the effects of any random forces from noise, vibrations, or air currents. 

In the presence of a potential U(r) the system will feel a force F(rj,) = — ViT/(r) rj,. There will also be a stochastic or random force acting on the system. The magnitude of that stochastic force is related to the temperature, the mass of the system, and the diffusion constant D. For a short time, it is possible to write the probability that the system has moved to a new position rj,+i as being proportional to the Gaussian probability [43]... 

In Langevin dynamics, we simulate the effect of a solvent by making two modifications to equation 15.1. First of all, we take account of random collisions between the solute and the solvent by adding a random force R. It is usual to assume that there is no correlation between this random force and the particle velocities and positions, and it is often taken to obey a Gaussian distribution with zero mean. 

The random force is taken from a Gaussian distribution with zero mean and variance... 

For most systems in thermal equilibrium, it is sufficient to regard fB as random forces which follows a Gaussian distribution function with mean value = 0 and standard deviation = 2kBT 8(i — j) 8(t — t ) [44],... 

The motion of particles of the film and substrate were calculated by standard molecular dynamics techniques. In the simulations discussed here, our purpose is to calculate equilibrium or metastable configurations of the system at zero Kelvin. For this purpose, we have applied random and dissipative forces to the particles. Finite random forces provide the thermal motion which allows the system to explore different configurations, and the dissipation serves to stabilize the system at a fixed temperature. The potential energy minima are populated by reducing the random forces to zero, thus permitting the dissipation to absorb the kinetic energy. 

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