Boltzmann distribution Monte Carlo simulation

In computer simulations, we are particularly interested in the properties of a system comprising a number of particles. An ensemble is a collection of such systems, as might be generated using a molecular d)mamics or a Monte Carlo simulation. Each member of the ensemble has an energy, and the distribution of the system within the ensemble follows the Boltzmann distribution. This leads to the concept of the ensemble partition function, Q. 

The electrostatic charges of surfactants seriously affect the localization of host molecules in the water pool. Monte Carlo simulation in which ionic reversed micelles are treated as spherical entities showed the presence of the electrical double layer in the interface of the water pool, and the distribution of counterions followed the Poisson-Boltzmann approximation [51]. Mancini and Schiavo [52] assumed recently, by the yield of halogenation, that the specific interactions between bromide or chloride ions and an ammonium head-group in cationic reversed micelles keep the ions in a defined position on the interface. 

Abstract The electric fields and potential in a pore filled with water are calculated, without using the Poisson-Boltzmann equation. No assumption of macroscopic dielectric behavior is made for the interior of the pore. The field and potential at any position in the pore are calculated for a charge in any other position in the pore, or the dielectric boundary of the pore. The water, represented by the polarizable PSPC model, is then placed in the pore, using a Monte Carlo simulation to obtain an equilibrium distribution. The water, charges, and dielectric boundary, together determine the field and potential distribution in the channel. The effect on an ion in the channel is then dependent on both the field, and the position and orientation of the water. The channel can exist in two major configurations open or closed, in which the open channel allows ions to pass. In addition, there may be intermediate states. The channel has a water filled pore, and a wall... 

But the Poisson-Boltzmann theory treats only the distribution of mobile ions as a function of distance from the macroion P, and does not treat the dumpiness of the counterion/co-ion distribution. It involves an implicit averaging over the distributions of the mobile ions w hich eliminates the discreteness of the small ions [1). The Poisson-Boltzmaim model works best for low ion concentations, and for monovalent mobile ions because the dumpiness is greatest for multivalent ions, z = 2 or z = 3, etc. Better approaches, such as integral equation treatments or Monte Carlo simulations, take into account all the charge interactions, but at the expense of simplicity [2]. 

When simulating from the Boltzmann distribution, this should agree with the thermodynamic equilibrium value of in the NVT (constant mole munber, voliune, and temperature) ensemble as As oo. For more on Monte Carlo simulation, consult Frenkel Smit (2002). 

In structure matching methods, potentials between the CG sites are determined by fitting structural properties, typically radial distribution functions (RDF), obtained from MD employing the CG potential (CG-MD), to those of the original atomistic system. This is often achieved by either of two closely related methods, Inverse Monte Carlo [12-15] and Boltzmann Inversion [5, 16-22], Both of these methods refine the CG potentials iteratively such that the RDF obtained from the CG-MD approaches the corresponding RDF from an atomistic MD simulation. 

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