Trial move

Selecting trial moves in an unbiased way typically means (a) choose an atom randomly , with equal probability from the complete set (b) displace it by random amounts in the v, y and z directions, chosen... 

It is usefiil to write down here the basic fomuilae for sampling with an additional weight fimction applied, sometimes called non-Boltzmaim or umbrella sampling, and for sampling when the selection of trial moves is done in a biased way, i.e., the a matrix is not syimnetrical. 

The usual MC procedure is adopted, with trial moves F probability... 

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

Closely related to the transition matrix is the stochastic matrix, whose elements are labelle a . TTiis matrix gives the probability of choosing the two states m and n between whic the move is to be made. It is often known as the underlying matrix of the Markov chain, the probability of accepting a trial move from m to n is then the probability of makir a transition from m to n (7r, ) is given by multiplying the probability of choosing states... 

The size of the move at each iteration is governed by the maximum displacement, Sr ax This is an adjustable parameter whose value is usually chosen so that approximately 50/i of the trial moves are accepted. If the maximum displacement is too small then mam moves will be accepted hut the states will be very similar and the phase space will onb he explored very slowly. Too large a value of Sr,, x and many trial moves will be rejectee because they lead to unfavourable overlaps. The maximum displacement can be adjuster automatically while the program is running to achieve the desired acceptance ratio bi keeping a running score of the proportion of moves that are accepted. Every so often thi maximum displacement is then scaled by a few percent if too many moves have beei accepted then the maximum displacement is increased too few and is reduced. 

The main difference between the force-bias and the smart Monte Carlo methods is that the latter does not impose any limit on the displacement that m atom may undergo. The displacement in the force-bias method is limited to a cube of the appropriate size centred on the atom. However, in practice the two methods are very similar and there is often little to choose between them. In suitable cases they can be much more efficient at covering phase space and are better able to avoid bottlenecks in phase space than the conventional Metropolis Monte Carlo algorithm. The methods significantly enhance the acceptance rate of trial moves, thereby enabling Icirger moves to be made as well as simultaneous moves of more than one particle. However, the need to calculate the forces makes the methods much more elaborate, and comparable in complexity to molecular dynamics. 

Randomly choose a trial move for the system. This could be moving all atoms, but it more often involves moving one atom or molecule for efficiency reasons. 

A trial move of an atom consisted of a small displacement in its position and momentum simultaneously. Step lengths of 0.9 in velocity and 0.09 in position gave an acceptance rate of about 50%. A cycle consisted of one trial move of all the atoms. 

Pgeniz ( ) — z(-I1 (i/ )]. For simplicity, this so-called trial move is often carried out such that the generation probability is symmetric, i.e., such that the probability to generate z nU.-9 ) from z , )( j equals the probability to generate z ( ) from z-n sr)... 

Pu depends on the quotient flj, / TT, the calculation of the configurational integral Z(N,V,T) is avoided. The change in potential energy of the system due to the trial move determines if the attempted new configuration is accepted. 

The main idea of the method is to assign a weight function Wi(y) to each solvent molecule y of the configuration i so that the probability that a given y is chosen for the trial move is ... 

A trial move is made by displacing the particle and also the cell edge ... 

To perform the isobaric-isothermal MC simulation [122], we perform Metropolis sampling on the scaled coordinates r, = L 1qi (qi are the real coordinates) and the volume V (here, the particles are placed in a cubic box of size L = /V). The trial moves from state x with the scaled coordinates r with volume V to state x with the scaled coordinate r and volume V are generated by uniform random numbers. The enthalpy is accordingly changed from Ti(E(r, V), V) to 7i E r, V), V) by these trial moves. The trial moves will be accepted with the probability... 

The line search is essentially an approximate one-dimensional minimization problem. It is usually performed by safeguarded polynomial interpola-tion.5 6>S4 56 That is, in a typical line step iteration, cubic interpolation is performed in a region of X that ensures that the minimum of /along p has been bracketed. The minimum of that polynomial then provides a new candidate for X. If the search directions are properly scaled, the initial trial point Xt = 1 produces a first reasonable trial move from xk. A simple illustration of such a first line search step is shown in Figure 9. The minimized one-dimensional function at the current point xk is defined by /(X) = f(xk + Xp ). The vectors corresponding to different values of X are set by x(X) = xk + Xp. ... 

When the trial moves out of the courtroom and to the site of the crime, a profound sense of consternation predominates 195... 

The transfer of particles between the two boxes forces equality of chemical potentials. The probability of accepting a trial move in which a molecule of type i is transferred to or from volume Vp is, respectively. 

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