Simulation transition probabilities

A consideration of the transition probabilities allows us to prove that microscopic reversibility holds, and that canonical ensemble averages are generated. This approach has greatly extended the range of simulations that can be perfonned. An early example was the preferential sampling of molecules near solutes [77], but more recently, as we shall see, polymer simulations have been greatly accelerated by tiiis method. 

Now run this experiment three times and note the numbers of A and B remaining after each run. Let [A] equal the number of A remaining at the end of the run and [B] the number of B. From your three runs, determine average values and standard deviations for these numbers. We can define the equilibrium constant for the interconversion of A and B as Wgq = [B]/[A]. Determine an average value and standard deviation for this ratio based on your results. What value would you expect for Wgq based on the transition probabilities Does your calculated value from the simulations agree with this value ... 

We now discuss in more detail the time unit, tmc(T) [Eq. (5.15)], for the dynamic MC simulation. Remember that the Metropolis algorithm [7,12] only involves a transition probability, and in converting that into a transition prob-... 

Monte Carlo heat flow simulation, 69-70 nonequilibrium statistical mechanics, microstate transitions, 44 46 nonequilibrium thermodynamics, 7 time-dependent mechanical work, 52-53 transition probability, 53-57 Angular momentum, one- vs. three-photon... 

Fortunately for us, measurement of the macroscopic transition probabilities is straightforward. We could accomplish this, for example, by counting the number of times moves are made between every I and J macrostate in our simulation. The estimate for 7 ( / — J) would then be the number of times a move from I to J occurred, divided by the total number of attempted moves from I. The latter is simply given by the sum of counts for transitions from I to any state. A more precise procedure that retains more information than simple counts is to record the acceptance probabilities themselves, regardless of the actual acceptance of the moves [46, 47]. In this case, one adds a fractional probability to the running tallies, rather than a count (the number one). This data is stored in a matrix, which we will notate C(I, J) and which initially contains all zeros. With each move, we then update C as... 

Transition-matrix estimators are typically more accurate than their histogram counterparts [25,26,46], and they offer greater flexibility in accumulating simulation data from multiple state conditions. This statistical improvement over histograms is likely due to the local nature of transition probabilities, which are more readily equilibrated than global measures such as histograms [25], Fenwick and Escobedo... 

Fitzgerald, M. Picard, R. R. Silver, R. N., Canonical transition probabilities for adaptive metropolis simulation, Eur. Phys. Lett. 1999, 46, 282-287... 

Transition matrix estimators have received less attention than the multicanonical and Wang-Landau methods, but have been applied to a small collection of informative examples. Smith and Bruce [111, 112] applied the transition probability approach to the determination of solid-solid phase coexistence in a square-well model of colloids. Erring ton and coworkers [113, 114] have also used the method to determine liquid-vapor and solid-liquid [115] equilibria in the Lennard-Jones system. Transition matrices have also been used to generate high-quality data for the evaluation of surface tension [114, 116] and for the estimation of order parameter weights in phase-switch simulations [117]. 

Simulations were performed for both cost functions. Target trajectories in range and Doppler were randomly created. The maneuvers for the trajectories were generated using a given transition probability matrix. We identified four maneuvers 0 acceleration 10m/s2 acceleration 50m/s2 acceleration —10m/s2 acceleration. 

Eqs. 2, 3 and 4 can be used to simulate a powder spectrum, taking into consideration the correction for the transition probability necessary in a field-swept ESR spectrum" . In most cases an orientation dependent linewidth is assumed in generating the spectrum ... 

In order to test the small x assumptions in our calculations of condensed phase vibrational transition probabilities and rates, we have performed model calculations, - for a colinear system with one molecule moving between two solvent particles. The positions ofthe solvent particles are held fixed. The center of mass position of the solute molecule is the only slow variable coordinate in the system. This allows for the comparison of surface hopping calculations based on small X approximations with calculations without these approximations. In the model calculations discussed here, and in the calculations from many particle simulations reported in Table II, the approximations made for each trajectory are that the nonadiabatic coupling is constant that the slopes of the initial and final... 

In Figs. 4.1 and 4.2, the broken lines do not represent the sample paths of the process X(t), but join the outcoming states of the system observed at a discrete set of times f, t2,.. . , tn. To understand the behavior of X(t), it is necessary to know the transition probability. In Fig. 4.3 are given numerical simulations of a Wiener process W(t) (Brownian motion) and a Cauchy process C(t), both supposed one dimensional, stationary, and homogeneous. Their transitions functions are defined... 

Except the kinetic equations, now various numerical techniques are used to study the dynamics of surfaces and gas-solid interface processes. The cellular automata and MC techniques are briefly discussed. Both techniques can be directly connected with the lattice-gas model, as they operate with discrete distribution of the molecules. Using the distribution functions in a kinetic theory a priori assumes the existence of the total distribution function for molecules of the whole system, while all numerical methods have to generate this function during computations. A success of such generation defines an accuracy of simulations. Also, the well-known molecular dynamics technique is used for interface study. Nevertheless this topic is omitted from our consideration as it requires an analysis of a physical background for construction of the transition probabilities. This analysis is connected with an oscillation dynamics of all species in the system that is absent in the discussed kinetic equations (Section 3). 

When a reverse transition probability Pt(B,A) for the transition B —> A is included, the model simulates the first-order equilibrium ... 

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