About Monte Carlo Simulations

If one is interested in equilibrium canonical (fixed temperature) properties of liquid interfaces, an approach to sample phase space is the Monte Carlo (MC) method. Here, only the potential energy function l/(ri,r2,. .., rjy) is required to calculate the probability of accepting random particle displacement moves (and additional moves depending on the ensemble type ). All of the discussion above regarding the boundary conditions, treatment of long-range interactions, and ensembles applies to MC simulations as well. Because the MC method does not require derivatives of the potential energy function, it is simpler to implement and faster to run, so early simulations of liquid interfaces used However, dynamical information is not available with 

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The situation of concern is when Monte Carlo simulation is used to quantify uncertainty in quantities in order to inform decisions. It is important to keep in mind that a Monte Carlo simulation should not be associated to a particular framework. Introductory text books and common computer programs for Monte Carlo simulation describe methods to quantify uncertainty in input parameters and provide tools to describe the uncertainty in output. Vose (2008) present different statistical principles and argue for uncertainty being subjective, at the same time as the impression is that principles from classical statistical statistics are seen as the first choice. The tutorial for RISK (Palisade Corporation, 2013) rightly present the quantification and interpretation of uncertainty in a neutral way e.g. by talking about Monte Carlo simulation as a technique to combine all the uncertainties you identify in your modeling situation leading to results presented in the form ofprobability distributions . However, statements like you can explicitly... 

Path integral Monte Carlo simulations were performed [175] for the system with Hamiltonian (Eq. (25)) for uj = ujq/J = A (where / = 1) with N = 256 particles and a Trotter dimension P = 64 chosen to achieve good computer performance. It turned out that only data with noise of less than 0.1% led to statistically reliable results, which were only possible to obtain with about 10 MC steps. The whole study took approximately 5000 CPU hours on a CRAY YMP. 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

RAT grinding operations. This surface layer was removed except for a remnant in a second grind. Spectra - both 14.4 keV and 6.4 keV - were obtained on the undisturbed surface, on the bmshed surface and after grinding. The sequence of spectra shows that nanophase Oxide (npOx) is eiu-iched in the surface layer, while olivine is depleted. This is also apparent from a comparison of 14.4 keV spectra and 6.4 keV spectra [332, 346, 347]. The thickness of this surface layer was determined by Monte-Carlo (MC)-Simulation to about 10 pm. Our Monte Carlo simulation program [346, 347] takes into account all kinds of absorption processes in the sample as well as secondary effects of radiation scattering. For the MC-simulation, a simple model of the mineralogical sample composition was used, based on normative calculations by McSween [355]. 

A Monte Carlo simulation (Fig. 3) can be made as usual (that is, without constraints on the output age), in which case only about 24% of the trials will yield ratios corresponding to a finite age, and a younger limit of >821 ka (95% confidence) or >531 ka (68% conf) is indicated. If, however, the a priori assumption of a closed system with no initial °Th is made, the failed trials can be ignored (since they violate the a priori constraints), and solution of both age and age-error (630 +370/-210 ka at 95% conf., or +150/-140 ka at 68% conf) can be obtained from the Monte Carlo simulations. 

In both solvents, the variational transition state (associated with the free energy maximum) corresponds, within the numerical errors, to the dividing surface located at rc = 0. It has to be underlined that this fact is not a previous hypothesis (which would rather correspond to the Conventional Transition State Theory), but it arises, in this particular case, from the Umbrella Sampling calculations. However, there is no information about which is the location of the actual transition state structure in solution. Anyway, the definition of this saddle point has no relevance at all, because the Monte Carlo simulation provides directly the free energy barrier, the determination of the transition state structure requiring additional work and being unnecessary and unuseful. 

Figure 1. Spatial distribution of NSs in the Galaxy. The data was calculated by a Monte-Carlo simulation. The kick velocity was assumed following Arzoumanian et al. (2002). NSs were born in a thin disk with a semithickness 75 pc. Those NS that were bom inside R = 2 kpc and outside R = 16 kpc were not taken into account. NS formation rate was assumed to be constant in time and proportional to the square of the ISM density at the birthplace. Results were normalized to have in total 5 x 108 NSs born in the described region. Density contours are shown with a step 0.0001 pc 3. At the solar distance from the center close to the galactic plane the NS density is about 2.8 1CT4 pc 3. From Popov et al. (2003a).

A theoretical analysis of the stability of such colloidal crystals of spherical latex particles has been carried out by Marcel ja et al (28.). They employ the Lindemann criterion that a crystal will be stable if the rms thermal displacement of the particles about their equilibrium positions is a small fraction f of the lattice spacing R. Comparison with Monte Carlo simulations shows that f is about 0.1 for "hard crystals, and 0.08 for "soft crystals stabilized by long-ranged electrostatic forces. This latter criterion translates into a critical ratio... 

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