Metropolis rule

Another monomer is chosen at random and, if the bond to the right of it exists, it is attempted to break it according to the standard Metropolis rule [53]. 

This condition can be satisfied using the Metropolis rule [24]... 

The thermalization of the system follows the well-known Metropolis rules (importance sampling procedure) [14] obeying the hard core overlap, but no additional interactions. The degree of thermalization can be controlled by observing the melting factor distribution. The melting factor is a means of judging if the model system has already reached thermodynamic equilibrium. For our purposes we defined... 

Ef+ 0 from the target function are calculated according to Eq. (10). The proposed configuration ff+l r) is accepted with the probability p given by the Metropolis rule... 

Bond switches are made according to the usual Monte Carlo Metropolis rules. Switches are made at random on a trial basis and the energies of the two structures are compared. If the new structure is of lower energy it is accepted. If it is of higher energy it is accepted with probability... 

It should be stressed that the the random walk in state space need not be generated using this rule - in fact, for the random walk, the original Metropolis rule is usually better suited. The difference in the variance of the energy of WRMC and standard MC is ... 

In practice, the Metropolis rule is implemented in the following way. First, the new path is generated. If the new path is not reactive, it is rejected. Otherwise, the ratio P[n]Pgen[n — o]/P[o]Pgen[o —> n] is computed. If this ratio is larger... 

Any attempted movement consistent with the nonoverlapping restriction is accepted by considering the Metropolis rule [64] y < exp(—Aii(r)/ BT), where y is a random number uniformly distributed between zero and unity. According to this condition, the process evolves until a thermal equilibrium state is reached (see Figure 4.8). 

The simulation is performed on a cubic lattice of dimensions L = 44, with periodic boundary conditions in all directions (5). Reptation (7) and the extended Verdier-Stockmayer moves (5) are used to convert one replica into another, with the Metropolis rules (P) employed for acceptance of new replicas. Each of the 200 diblock copolymers contains Na beads of A and Nb beads of B. Vacant lattice sites are considered to be occupied by solvent, S. 

Insertion of this form of the transition probability into the detailed balance condition provides a condition for the acceptance probability that can be satisfied with the so-called Metropolis rule [174]. The resulting acceptance probability is given by [124] ... 

We can now take one of two approaches (1) construct a probabilistic CA along lines with the Metropolis Monte Carlo algorithm outlined above (see section 7.1.3.1), or (2) define a deterministic but reversible rule consistent with the microcanonical prescription. As we shall immediately see, however, neither approach yields the expected results. 

A more interesting problem is that the Metropolis Monte Carlo studies used a different (physically simplified) kinetic rate law for atomic motion than the KMC work. That is, the rules governing the rate at which atoms jump from one configuration to the next were fundamentally different. This can have serious implications for such dynamic phenomena as step fluctuations, adatom mobility, etc. In this paper, we describe the physical differences between the rate laws used in the previous work, and then present results using just one of the simulation techniques, namely KMC, but comparing both kinds of rate laws. 

From the present state of the system (denoted by the symbol o, a trial move is attempted to a trial state n. In the Metropolis scheme, the (stochastic) rule for the generation of these trial moves is such that the probability Oon to attempt a trial move to n, given that the system is initially in o, is equal to the probability a o to generate a trial move to o, given that the system is initially in n. 

The crucial point to note is that the above average combines information about both the accepted and the rejected state of a trial move. Note that the Monte Carlo algorithm used to generate the random walk among the states n need not be the same as the one corresponding to TTnm- For instance, we could use standard Metropolis to generate the random walk, and use the S3Tnmetric rule [4]... 

The most widely used acceptance/rejection rule was proposed by Metropolis et al. [14] almost a half a century ago. It is a rejection scheme based on the principle of reversibility between successive states of an equilibrium system. Consider proposing some arbitrary transition from configuration r N to configuration rN (where r represents the coordinates of the N particles in the system). The Metropolis criterion prescribes that a trial move be accepted with probability... 

Various acceptance rules have been devised for importance sampling, the simplest of which is the so-called Metropolis algorithm [45]. In this case. 

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