Acceptance probability

A configurational Monte Carlo algorithm based on uniform random trial moves and the acceptance probability... 

When the integrator used is reversible and symplectic (preserves the phase space volume) the acceptance probability will exactly satisfy detailed balance and the walk will sample the equilibrium distribution... 

In J-walking [20] the periodic MC trial probability for a simulation at temperature T is taken to be a Boltzmann distribution at a high temperature, Tj ( 3j = IkTj), The jumping temperature, 7), is sufficiently high that the Metropolois walk can be assumed to be ergodic. This results in the acceptance probability. 

Detonation pressure may be computed theoretically or measured exptly. Both approaches are beset with formidable obstacles. Theoretical computations depend strongly on the choice of the equation of state (EOS) for the detonation products. Many forms of the EOS have been proposed (see Vol 4, D269—98). So.far none has proved to be unequivocally acceptable. Probably the EOS most commonly, used for pressure calcns are the polytropic EOS (Vol 4, D290-91) and the BKW EOS (Vol 4, D272-74 Ref 1). A modern variant of the Lennard Jones-Devonshire EOS, called JCZ-3, is now gaining some popularity (Refs -11. 14). Since there is uncertainty about the correct form of the detonation product EOS there is obviously uncertainty in the pressures computed via the various types of EOS ... 

However, since and -5 asymptote to the same function, one might approximate (U) = S dJ) in (3.57) so that the acceptance probability is a constant.3 The procedure allows trial swaps to be accepted with 100% probability. This general parallel processing scheme, in which the macrostate range is divided into windows and configuration swaps are permitted, is not limited to density-of-states simulations or the WL algorithm in particular. Alternate partition functions can be calculated in this way, such as from previous discussions, and the parallel implementation is also feasible for the multicanonical approach [34] and transition-matrix calculations [35],... 

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... 

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-... 

The detailed balance condition (7.12) can now be satisfied by selecting an appropriate acceptance probability. By inserting the product in (7.14) into the detailed balance condition we find that for a symmetric generation probability the acceptance probabilities for the forward and the reverse move must be related by... 

The momentum displacements dp are, for instance, selected from a Gaussian distribution whose width can be used to tune the acceptance probability. Note that it is not strictly necessary to draw the momentum displacements from a Gaussian distribution any other isotropic distribution can be used as well for this purpose. Starting from the new state z[n which consists of the old positions and the new momenta,... 

It is, of course, possible to bias the selection toward certain segments of the path. In this case, the bias must be properly taken into account in the acceptance probability. 

For a microcanonical distribution of initial conditions all initial conditions have the same energy and the acceptance probability becomes... 

For Hamiltonian dynamics with a canonical or microcanonical distribution of initial conditions the acceptance probability for pathways generated with the shifting algorithm is particularly simple. Provided forward and backward shifting moves are carried out with the same probability the acceptance probability from (7.23) reduces to... 

As in the case of conventional Monte Carlo simulations depending on the cost of rejected moves the optimum acceptance probability may be lower than 40% [23]. 

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