Canonical phase-space sampling

Here the canonical phase-space sampling is performed via the Monte Carlo method (for details see [6]). As the cluster in the beam is oriented statistically, we have to perform three different calculations,... 

A virtue of this simple method is that it deterministically samples phase space in a manner consistent with the canonical ensemble, 4io while providing a quantity, similar to a total energy, which is conserved. The latter prop-... 

We now have established a framework for the thermodynamic properties of a model system. What would be desirable is that we could approximate the force terms arising in the bulk, without the need to simulate them directly. It should be apparent that constant-energy dynamics (designed to sample the microcanonical ensemble with constant energy E) will not sample the canonical distribution in the absence of the heat bath such Newtonian trajectories cannot access regions of the phase space where H z) H(zo), where zo is the initial condition. 

In this section, we discuss how one, guided by the principles of nonequilibrium thermodynamics, can use the Monte Carlo technique to drive an ensemble of system configurations to sample statistically appropriate steady-state nonequilibrium phase-space points corresponding to an imposed external field [161,164,193-195]. For simplicity, we limit our discussion to the case of an unentangled polymer melt. The starting point is the probability density function of the generalized canonical... 

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... 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

Monte Carlo algorithms can be developed to sample the phase space of ensembles other than the canonical. We discuss here the puTT or grand canonical Monte Carlo (GCMC). With GCMC, phase equilibria and the volumetric properties of matter can be efficiently determined. 

The particles positions are initialized on a 5 x 5 x 5 grid, with equidistant grid spacing 2 / the minimum of the Lennard-Jones potential cpu(r). We will sample canonically at reciprocal temperature P = 10, with a force cutoff chosen equal to the side length of the periodic box. The size of the box defines the particle density of the simulation and hence the phase of matter that we shall study. After some experimentation, a cubic box with side length 7 was found to give conditions for a liquid state. 

Because of the dominance of a certain restricted space of microstates in ordered phases, it is obviously a good idea to primarily concentrate in a simulation on a precise sampling of the microstates that form the macrostate under given external parameters such as, for example, the temperature. The canonical probability distribution functions clearly show that within the certain stable phases, only a limited energetic space of microstates is noticeably populated, whereas the probability densities drop off rapidly in the tails. Thus, an efficient sampling of this state space should yield the relevant information within comparatively short Markov chain Monte Carlo runs. This strategy is called importance sampling. 

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