Force biasing

Having specified the interactions (i.e., the model of the system), the actual simulation then constructs a sequence of states (or the system trajectory) in some statistical mechanical ensemble. Simulations can be stochastic (Monte Carlo (MC)) or deterministic (MD), or they can combine elements of both, such as force-biased MC, Brownian dynamics, or generalized Lan-gevin dynamics. It is usually assumed that the laws of classical mechanics (i.e., Newton s second law) may adequately describe the atoms and molecules in the physical system. 

Brownian dynamics (BD), which is stochastic dynamics in the over-damped limit, can just as well be understood as force-biased (dynamic) MC employing collective moves only [100,101]. 

Note that in the MC methodology, only die energy of the system is computed at any given point. In MD, by contrast, forces are the fundamental variables. Pangali, Rao, and Berne (1978) have described a sampling scheme where forces are used to choose the direction(s) for molecular perturbations. Such a force-biased MC procedure leads to higher acceptance rates and greater statistical precision, but at tlie cost of increased computational resources. 

Molecular Dynamics simulation is one of many methods to study the macroscopic behavior of systems by following the evolution at the molecular scale. One way of categorizing these methods is by the degree of determinism used in generating molecular positions [134], On the scale from the completely stochastic method of Metropolis Monte Carlo to the pure deterministic method of Molecular Dynamics, we find a multitude and increasingly diverse number of methods to name just a few examples Force-Biased Monte Carlo, Brownian Dynamics, General Langevin Dynamics [135], Dissipative Particle Dynamics [136,137], Colli-sional Dynamics [138] and Reduced Variable Molecular Dynamics [139]. 

Watanabe, Y.S., Kim, J.G., Fukunishi, Y., Nakamura, H. Free energy landscapes of small peptides in an implicit solvent model determined by force-biased multicanonical molecular dynamics simulation. Chem. Phys. Lett. 2004,400,258-63. 

Two types of classical simulations were used in the course of the present work. Monte Carlo and molecular dynamics "annealing" calculations were used to determine the equilibrium structures, while isoenergetic molecular dynamics runs at a wide range of energies were used to determine the melting temperatures. Our Monte Carlo calculations used the Pangali et al. force-biased modification of the basic Metropolis et al. ... 

Mezei M (1991) Distance-scaled force biased Monte Carlo simulation for solutions containing a strongly interacting solute. Mol Simul 5 405 08... 

Limonova M, Groenewegen J, Thijsse BJ (2010) Modeling diffusion and phase transitions by a uniform-acceptance force-bias Monte Carlo method. Phys Rev B 81 (14) 144107 Neyts EC, Thijsse BJ, Mees MJ, Bal KM, Pourtois G (2012) Establishing uniform acceptance in force biased Monte Carlo simulations. J Chem Theory Comput 8 1865-1869 Rossky P, Doll J, Eriedman H (1978) Brownian dynamics as smart Monte-Carlo simulation. J Chem Phys 69(10) 4628 633... 

Neyts EC, Shibuta Y, Van Duin ACT, Bogaerts A (2010) Catalyzed growth of carbon nanotube with definable chirality by hybrid molecular dynamics-force biased Monte Carlo simulations. ACS Nano 4(11) 6665-6672... 

The thyristor is a semiconductor device made of germanium or silicon wafers and comprises three or more Junctions, which can be switched from the OFF state to the ON state or vice versa. Basically it is a ptipn junction, as shown in Figure 6.20(a) and can be considered as composed of two transistors with npn and pnpjunctions, as illustrated in Figure 6.20(b). It does not turn ON when it is forward biased, unlike a diode, unless there is a gate firing pulse. Thyristors are forced commutated (a technique... 

The diffusion constant should be small enough to damp out inertial motion. In the presence of a force the diffusion is biased in the direction of the force. When the friction constant is very high, the diffusion constant is very small and the force bias is attenuated— the motion of the system is strongly overdamped. The distance that a particle moves in a short time 8t is proportional to... 

FIG. 1 Schematic representation of the operation of the scanning polarization force microscope (SPFM). An electrically biased AFM tip is attracted toward the surface of any dielectric material. The polarization force depends on the local dielectric properties of the substrate. SPFM images are typically acquired with the tip scanning at a height of 100-300 A. (From Ref. 32.)... 

Based on the RIS Ansatz, the embedding algorithm benefits from a great flexibility in the choice of the input parameters that account for the local chain energy configuration the input for the correlations of torsion angles along the chain backbones can be either calculated with the help of a force field, or extracted from measurements, or even biased in order to study any thinkable structural properties of the macromolecules. 

These difficulties can be circumvented by using the adaptive biasing force (ABF) method of Darve, Pohorille, and coworkers [18, 28, 29], which is based on unconstrained molecular dynamics simulations. This is a very efficient approach which begins by establishing a simple formula to calculate d,4/d from regular molecular dynamics in which is not constrained. This derivative represents the mean force acting on . Therefore if we remove this force from the system we obtain... 

Let us first provide an expression to compute the derivative of the free energy for unconstrained simulations. Then, we will discuss the calculation of the biasing force and the algorithmic implementation of the method. 

When biasing the system, an external force AV is applied to improve the sampling along . Since this force is added, the calculation of the derivative of — to must be modified. Consider (4.35). The first term does not require any correction since x and x are sampled according to the correct distribution. However x includes the ABF force whose contribution needs to be removed to compute the free energy derivative. The correction is equal to... 

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