Lennard-Jonesium

Valleau, J. P. Temperature-and-density-scaling Monte-Carlo methodology and the canonical thermodynamics of Lennard-Jonesium. Mol. Sim. 2005, 31, 223-253... 

The scheme just described (2.14), (2.15) for generating an effective sampling distribution % has some attractive features. For one, an improved n(rN) corresponds merely to improved estimates of /L4 , a quantity that is to be estimated in any case thus no extra computations are required while refining . And the fact that one knows that j3Aa is (very nearly) proportional to N means that there is no need to start from scratch when seeking n for a larger or smaller system size N. The scheme has proved simple and effective in practice, and was used unmodified to study Lennard-Jonesium [10] and the square-well fluid [11] by TDSMC. 

The hierarchical data structure of mBuild is composed of Compounds. Compounds maintain an ordered set of children which are other Compounds. Compounds at the bottom of an mBuild hierarchy, i.e., the leafs of the tree, are referred to as Particles and can be instantiated as, for example, Ij = mb. Particle (name= lennard-jonesium ). Note however, that this merely serves to illustrate that this Compound is at the bottom of the hierarchy Particle is an alias for Compound which can be used to clarify the intended role of an object you are creating. 

Fig. 8.7 We plot the results of an NPT simulation of 64 atoms of Lennard-Jonesium using the given discretization where yr = yp = y, with y set to zero (blue), y = 0.1 [green) or y = 1 [red). Although the distribution of instantaneous pressure 77 (centered on the target pressure P) looks similar for all three cases, we can see that without any stochasticity the simulation exhibits artificial periodic fluctuations in the temperature and volume...

The method works well as long as there are internal molecular structures that prevent a degeneracy (one cell dimension collapsing) as might happen, for example, if the scheme were applied to a simple liquid such as Lennard-Jonesium. 

Figure 6. (a) About 30% above the critical temperature, the correlation function of 1-component Lennard-Jonesium [9], at a low density p = 0 one nearest- neighbor peak at a density about 30% higher than critical p = 0.5 a second maximum and at a density near close-packing p =1 pronounced oscillations, (b) schematically, in a fluid near the critical point. (Reprinted from [21 Hg. 3, copyright 1994, with kind permission from Kluwer Academic Publishers)... 

This technique is most often used in lattice quantum chromodynamics ((JCD) simulations. Mehlig et al. [55] demonstrated its use by simulating Lennard-Jonesium as an example of a condensed matter system. Similarly, Clamp et al. [56] simulated a 2D Lennard-Jones fluid using both MD and hybrid MC and found that hybrid MC is more ergodic and samples phase space more efficiently than MD. A more realistic system was studied by Brotz et al. [57], employed hybrid MC to calculate the phase diagram of silicon. 

Figure 1. Density and structure factor profiles of a planar sheet of Lennard-Jonesium fluid in equilibrium with its own vapor at 110 K. The origin is chosen to be the Gibbs equimolecular dividing surface. The curves are obtained from Monte Carlo simulation using one million configurations. Circles denote the density profile and the crosses denote the structure factor, r.

The correlation function equals -1 as long as r is smaller than the diameter of the hard core, or if it is well within the Lennard-Jones diameter a. It falls off to 0 for large r. Fig. 6a shows the behavior of the coirelation function in supercritical Ixnnard-Jonesium, at f = kTIz = 2, about 30% above the critical point s is the Lennard-Jones well-depth and k Boltzmann s constant. It was calculated by De Boer in 1949, and is reported in [9]. 

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