Metropolis-Based Stochastic Sampling

It should also be mentioned that the measure-preserving thermostatting methods discussed in [373] refer rather to the underlying measure of the phase space and not the thermodynamic equilibrium distribution those methods are deterministic and, like Nose-Hoover based schemes, will exhibit thermodynamic errors due to both lack of ergodicity and discretization error, in addition to sampling errors. 

One approach to providing ergodicity to deterministic systems is to introduce random fluctuations via a Monte-Carlo technique [268]. Several Monte-Carlo methods are described in Appendix C. Randomized steps are taken and then an accept-reject mechanism is introduced in order to ensure that the steps are consistent with the canonical distribution. It is possible to combine the Metropolis-Hastings concept with timestepping procedures in a variety of ways, which are often subsumed under the title Monte-Carlo Markov Chain methods , these include 

In a KMC method, it is typically assumed that various possible state-to-state transitions from a given state are well modelled by the Arrenhius law and then molecular dynamics is used to calculate the prefactor A and energy difference AE in order to understand the timescales and relative probabilities of different rare events. A Markov state model can be developed to help understand the global dynamics and simplify the model as a whole. For references on many interesting approaches to this important topic, the reader is referred to [36,42,137,149,391]. Andersen Thermostat. Of particular interest is the simple and useful Andersen thermostat [11]. This method works by selecting atoms at random and randomly perturbing their momenta in a way consistent with prescribed thermodynamic conditions. It has been rigorously proven to sample the canonical distribution [114], 

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Figure 6 Diagram illustrating kth step of the construction of a square king lattice of L X L spins with the stochastic models (SM) method solid circles denote lattice sites already filled with spins ( 1) in preceding steps of the process open circles denote the still empty lattice sites. The linear nature of the buildup construction is achieved by using spiral boundary conditions (i.e., the first spin in a row interacts with the last spin of the preceding row). Whereas all the L uncovered spins (at sites k - L,k — L + 1,..., k - ) determine the transition probability for selecting spin k, the spins in close proximity to k k - 1, k - L, etc.) have the largest effect. The local states method is based on the SM construction. Thus, the transition probabilities for spin k are obtained from a Metropolis Monte Carlo sample by calculating the number of occurrences of the various local states, (a, a) = n k-v k-2 k-L k-L v k-L 2 k-L 3 l- transition probability is jS(cT d ) = (cr, ff)/[ (a = 1,ct) -I- n(a = These transition...

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

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