Detailed balance algorithm

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)... 

For this algorithm, one can prove that detailed balance is guaranteed and the exact average of any configuration-dependent property over the accessible space is obtained. Two key issues determine the detailed balance. The first is the fact that the trial probability to pick the displacement vector Dfc to go from the fcth to the Zth e-sphere equals the trial probability to pick the displacement vector D fc for the reverse step. The second issue is that the trial probability for a local MC step that moves the walker from a point inside an e-sphere to a point outside that sphere is the same as for the reverse move i.e., (1 - / ) times what it would be in a walk restricted to local moves. 

After the momenta are selected from the distribution (8.39), the dynamics is propagated by a standard leapfrog algorithm (any symplectic and time-reversible integrator is suitable). The move is then accepted or rejected according to a criterion based on the detailed balance condition... 

GEMC utilizes two simulation subsystems ( boxes ) though physically separate, the two boxes are thermodynamically coupled through the MC algorithm, which allows them to exchange both volume and particles subject to the constraint that the total volume and number of particles remain fixed. Implementing these updates (in a way that respects detailed balance) ensures that the two systems will come to equilibrium at a common temperature, pressure, and chemical potential. The temperature is fixed explicitly in the MC procedure but the procedure itself selects the chemical potential and pressure that will secure equilibrium. 

Viewing the SA algorithm in terms of Markov chains, Greene and Supowit [8] pointed out that any type of function may be used for the decision making process about acceptance of new configurations, provided the detailed balance equation for the Markov process is satisfied. 

In contrast to the Wang-Landau algorithm the weights W(E) are modified only after a batch of N sweeps, thereby ensuring detailed balance between successive moves at all times. 

Show that the Metropolis algorithm satisfies detailed balance. 

Now we have N mass balances, (J - N) aqueous reaction equilibrium relations, K solubility-prodnct-constant equations, (I - 1) cation-exchange on matrix eqnilibrinm relations and one electroneutralinity condition, (M - 1) cation-exchange on micelle eqnilibrinm relations and one electroneutralinity condition for foe micelles, giving a total of (J -F K -F I -F M) independent eqna-tions to solve the same nnmber of concentration unknowns. For foe detailed calcnlation algorithm, see Bhnyan (1989). 

In the Metropolis algorithm, a single move of a walker starting at R can be split into two steps as follows first a possible final point R is selected then an acceptance/rejection step is executed. If the first step is taken with a transition probability T(R -+ R ), and if we denote for the acceptance/ rejection step the probability yl(R - R ) that the attempted move from R to R is accepted, then the total probability that a walker moves from R to R is T(R -> R )4(R -> R ). Since we seek the distribution P(R) using such a Markov process, we note that at equilibrium (and in an infinite ensemble), the fraction of walkers going from R to R, P(R)T(R - R )A(R -> R ), must be equal to the fraction of walkers going from R to R, namely P(R )T(R -+ R)A(R R). This condition, called detailed balance, is a sufficient condition to reach the desired steady state, and provides a constraint on the possible forms for T and A. For a given P(R)... 

For a given choice of , infinitely many choices can be made for A that satisfy detailed balance but the choice given above is the one with the largest acceptance. We note that if the preceding algorithm is used, then X-rfS,) in the sum in Eq. (3.4), can be replaced by the expectation value conditional on S" having been proposed ... 

Just as with the Metropolis algorithm, the optimal choice of the two acceptance ratios is then to make the larger of the two equal to one, and choose the smaller to satisfy this equation. If we do this, we again achieve detailed balance, and we now have an algorithm which, as E0 is made larger and... 

The proof that this algorithm satisfies detailed balance is exactly the same as it was for the Ising model. If we consider two states g and v of the system which differ by the changing of just one cluster, then the ratio g(g -> v)/ g(v -> g) of the selection probabilities for the moves between these states depends only on the number of bonds broken m and the number made n around the edges of the cluster. This gives us an equation of detailed balance, which reads... 

For the choice of Padd given above, this is just equal to one. Equation (4.3) is satisfied by making the acceptance ratios equal to 1, and with this choice the algorithm satisfies detailed balance. 

Readers may like to demonstrate for themselves that, with this choice, the ratio of the selection probabilities g(ji -> v) and g(v - ) is equal to e E, where AE is the change in energy in going from a state to a state v by flipping a single cluster. Thus, detailed balance is obeyed as in the Ising case by an algorithm for which the acceptance probability for the cluster flip is... 

After finishing the simulation phase, the ARMS sampled values are used for inferring about the marginal distribution of each unknown variable of interest and any additional information from the approximate functions g is neglected. In the present paper, instead, it is proposed a reasonably accurate approximate function g (due to ASR) which can be improved at every new evaluation of the target FCD (the step 3 of ARMS and MH algorithms). Thus, besides guarantying detailed balance to GGS, a better approximate function is provided. 

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