Monte Carlo step

In HyperChem, the atoms are not chosen randomly, but instead, each atom is moved once per Monte Carlo step. 

Figure 12.5. (a) Lattice model showing a polymer chain of 200 beads , originally in a random configuration, after 10,000 Monte Carlo steps. The full model has 90% of lattice sites occupied by chains and 10% vacant, (b) Half of a lattice model eontaining two similar chain populations placed in contact. The left-hand side population is shown after 50,0000 Monte Carlo steps the short lines show the loeation of the original polymer interface (courtesy K. Anderson). 

At each Monte Carlo step (MCS), either a dimer is formed from two adjacent monomers or a monomer is added or deleted from the chain end. The transition probabilities are... 

Typical runs consist of 100 000 up to 300 000 MC moves per lattice site. Far from the phase transition in the lamellar phase, the typical equilibration run takes 10 000 Monte Carlo steps per site (MCS). In the vicinity of the phase transitions the equilibration takes up to 200 000 MCS. For the rough estimate of the equihbration time one can monitor internal energy as well as the Euler characteristic. The equilibration time for the energy and Euler characteristic are roughly the same. For go = /o = 0 it takes 10 000 MCS to obtain the equilibrium configuration in which one finds the lamellar phase without passages and consequently the Euler characteristic is zero. For go = —3.15 and/o = 0 (close to the phase transition) it takes more than 50 000 MCS for the equihbration and here the Euler characteristic fluctuates around its mean value of —48. 

Fig. 21 Time evolution of lateral lamellar dimension as a function of temperature at C = 0.0005. Time is in units of 104 Monte Carlo steps...

Fig. 23 Initial lamellar seed of Ro = 7 grows into a large lamella of R = 60 in t = 150 x 104 Monte Carlo steps for kT = 0.09 and C = 0.0005...

Fig. 3.7. Schematic comparison of the multicanonical and iterative transition-matrix methods. The light gray box indicates a single iteration of several thousand or more individual Monte Carlo steps. The acceptance criterion includes the configuration-space density of the original ensemble, p, and is presented for symmetric moves such as single-particle displacements...

Determine the work distribution. After every time (or Monte Carlo) step during the simulations, change the Hamiltonian according to the prescribed path A (f) and accumulate the work performed as part of that change. Collect the work values from the N runs for analysis. 

Just as in a conventional Monte Carlo simulation, correct sampling of the transition path ensemble is enforced by requiring that the algorithm obeys the detailed balance condition. More specifically, the probability n [ZW( ) - z(n)( )]2 to move from an old path z ° 22) to a new one " (2/ ) in a Monte Carlo step must be exactly balanced by the probability of the reverse move from 22) to z<,J> 22)... 

This detailed balance condition makes sure that the path ensemble sg[z )] is stationary under the action of the Monte Carlo procedure and that therefore the correct path distribution is sampled [23, 25]. The specific form of the transition matrix tt[z(° 2 ) -> z(n, 9-) depends on how the Monte Carlo procedure is carried out. In general, each Monte Carlo step consists of two stages in the first stage a new path is generated from an old one with a certain generation probability... 

In the second stage of each Monte Carlo step the new (or trial) pathway is accepted with a certain acceptance probability Pacc H -) The total proba-... 

Fig. 24. Contour plot of the structure factor (the kinematic LEED intensity) of a x y/i monolayer in a triangular lattice gas with nearest-neighbor repulsion, at a temperature k TIi = 0.355 (about 5% above T ) and a chemical potential // = 1.5 (0c = 0.336 at the transition temperature.) Contour increments are in a (common) logarithmic scale separated by 0.1, starting with 3.2 at the outermost contour. Center of the surface Brillouin zon is to the left k, and k the radial and azimuthal components of kH, are in units of nlXla, a being the lattice spacing. Data are based on averages over 2x10 Monte Carlo steps per site. (From Bartelt et...

Fig. 1.1. Three stages in the island formation of A atoms with attractive NN interactions and coverage nA = 0.4 (a) t = 0 (b) t = 1000 MCS (c) t = 5000 MCS (MCS = 1 Monte Carlo step). Shown are 100 x 100 Sections of the 500 x 500 lattice used in the calculations.

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

We used NVT MD to sample the contents of the simulation cell, and an NpT MC algorithm to vary its shape and volume. The latter moves were carried out within a rigid-molecule framework as described previously [76], using the atomic positions at the end of the preceding flexible molecule NVT MD segment. In practice, 1 ps of NVT MD simulation was followed by a sequence of 100 NpT MC steps. The Monte Carlo step size for a given thermodynamic state was adjusted to yield an acceptance probability of 40-50%. 

Static properties of the system are obtained by averaging structural or any other properties of interest over the sampled configurations. The dynamics of the system can be simulated by association of the MC steps with an artificial time parameter. Naturally, the Monte Carlo steps do not correspond to a real time scale and the dynamics do not represent physical phenomena. 

Fig.12. Computation by Monte Carlo methods of the first four order parameters of an ensemble of 1000 chromophores (of dipole moment 13 Debye) existing in a medium of uniform dielectric constant. At the beginning of the calculation, the chromophores are randomly ordered thus, ==O. During the first 400 Monte Carlo steps, an electric poling field (600 V/micron) is on but the chromophore number density (=10 7 molecules/cc) is so small that intermolecular electrostatic interactions are unimportant. The order parameters quickly evolve to well-known equilibrium values obtained analytically from statistical mechanics (black dots in figure also see text). During steps 400-800 the chromophore number density is increased to 5xl020 and intermolecular electrostatic interactions act to decrease order parameters consistent with the results of equilibrium statistical mechanical calculations discussed in the text. Although Monte Carlo and equilibrium statistical mechanical approaches described in the text are based on different approximations and mathematical methods, they lead to the same result (i.e., are in quantitative agreement)...

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