Monte Carlo update

Favrin, G., Irback, A., Sjunnesson, F. Monte Carlo update for chain molecules biased Gaussian steps in torsional space. J. Chem. Phys. 2001, 114, 8154 8. 

Wolff, U. Collective Monte Carlo updating for spin systems. Phys. Rev. Lett. 1989, 62, 361-4. 

An alternative to these extended ensembles for the simulation of frustrated magnets is the parallel tempering or replica exchange Monte Carlo method [22-25], Instead of performing a single simulation at a fixed temperature, simulations are performed for M replicas at a set of temperatures Ti,T2,. .., Tm- In addition to standard Monte Carlo updates at a fixed temperature, exchange moves are proposed to swap two replicas between adjacent temperatures. These swaps are accepted with a probability... 

Figure 11 World-line configuration for the XXZ Hamiltonian of [38]. The world lines (thick lines) connect space-time points where the component of the spin points up. They can be either straight or cross the shaded squares, which show where the imaginary time evolution operators and act. The dotted line shows the configuration change after a local Monte Carlo update.

A Monte Carlo update corresponds to the discrete time step A to in the simulation process. In order to reduce correlations, typically a number of updates are performed between measurements of a quantity O. This series of updates is called a sweep and the time passed in a single sweep is At =ATAto if the sweep consists of N updates. Thus, if M sweeps are performed, the discrete time series is expressed by the vector (0(Tinit + At), 0(Tiiut + 2At),. .., 0(xaat + wAt),. .., 0(Ti it + MAt)) and represents the Monte Carlo trajectory. The period of equilibration Tjnjt sets the starting point of the measurement. For convenience, we use the abbreviation 0 = 0(Tjnjt -b wAt) and... 

If the Monte Carlo updates in each sample are performed completely randomly without memory, i.e., a new conformation is created independently of the one in the step before (which is a possible but typically very inefficient strategy), two measured values Om and 0 are uncorrelated, if m n. Then, the autocorrelation function simplifies to Amn = mn-Thus, the variances of the individual data and of the mean are related with each other by... 

In order to correctly satisfy the detailed balance condition (4.77) in a Monte Carlo simulation, we have to take into account that each Monte Carlo step consists of two parts. First, a Monte Carlo update of the current state is suggested and second, it has to be decided whether or not to accept it according to the chosen sampling strategy. In fact, both steps are independent of each other in the sense that each possible update can be combined with any sampling method. Therefore, it is useful to factorize the transition probability t(X X ) into the selection probability 5(X X ) for a desired update from X to X and the acceptance probability a(X X ) for this update ... 

The expression (4.80) for the acceptance probability naturally fulfills the detailed-balance condition (4.77). The selection ratio a(X, X ) is unity, if the forward and backward selection probabilities are identical. This is typically the case for simple local Monte Carlo updates. If, for example, the update is a translation of a Cartesian coordinate, x =x + Ax, where Ax e [—xq, - -xo] is chosen from a uniform random distribution, the forward selection for a translation by Ax is equally probable to the backward move, i.e., to translate the... 

The efficiency of Monte Carlo updates in continuous polymer and protein models strongly depends on the model chosen and can hardly be generalized. For this reason, we will describe only a few of the most basic and popular variants. 

This is an instructive example, because it tells us that by ignoring the integral measure in the Monte Carlo update processes we effectively would have simulated a different model with modified enfropic properties The term S(uj, v w,) = k ln/u(i/ v w,) is a geometric entropy that accounts for the non-uniformity in the distribution of values of the coordinates of the rth monomer in the chosen coordinate system. It is not a physical entropy, but rather a mathematical counter-term that assigns the physical entropy its correct value. 

After this extended but necessary excursion to end up with a proper definition of the torsion angle, we can return to the torsional Monte Carlo update. The goal of this update is to perform a bond rotation by an offset angle A4> about b2 that effectively rotates... 

An important exception are atomic simulations of proteins in explicit solvent, where individual particles (e,g, water molecules) represent the solvent Monte Carlo updates of the system would too frequently collide with solvent particles which causes unacceptably high rejection rates of trial moves. In this case, the cooperative many-body motion is more efficiently simulated by integrating the equations of motion of each particle step by step. It can even be most efficient to combine Monte Carlo and nwlecular dynamics. The most prominent hybrid method is replica-exchange molecular dynamics (REMD), where Langevin simulations ran in different threads at various temperatures and the replicas are exchanged after some time steps with the exchange transition probability (4.92), which effectively is a Monte Carlo step. 

Conventional, generalized-ensemble Monte Carlo methods can also be employed, of course, but require sophisticated Monte Carlo updates to be efiScient [306,309],... 

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