NVT ensemble

This ensemble is a weighted superposition of NVT ensembles with different values, of N. As a rule of tiuiinb, a typical MC sweep consists of N attempted moves, each of which is chosen randomly to be (i) a displacement (liandled exactly as in constant-AfTMC) (ii) the creation of a new particle at a randomly selected position (iii) the destruction of a randomly selected particle from the system. The probabilities for attempting creation and destniction must be equal (for consistency with what follows), but they need not be equal to the probability for attempting displacement (although they often are). 

A set of flexible chains is considered in the canonical NVT ensemble. The density of chains, pch = - ch/ (- ch is a chosen number of chains), is the parameter of simulation. The system of flexible chains in question has been equilibrated, then during the productive part of the simulation run the pair... 

The heptane water and toluene water interfaces were simulated by the use of the DREIDING force field on the software of Cerius2 Dynamics and Minimizer modules (MSI, San Diego) [6]. The two-phase systems were constructed from 62 heptane molecules and 500 water molecules or 100 toluene molecules and 500 water molecules in a quadratic prism cell. Each bulk phase was optimized for 500 ps at 300 K under NET ensemble in advance. The periodic boundary conditions were applied along all three directions. The calculations of the two-phase system were run under NVT ensemble. The dimensions of the cells in the final calculations were 23.5 A x 22.6 Ax 52.4 A for the heptane-water system and 24.5 A x 24.3 A x 55.2 A for the toluene-water system. The timestep was 1 fs in all cases and the simulation almost reached equilibrium after 50 ps. The density vs. distance profile showed a clear interface with a thickness of ca. 10 A in both systems. The result in the heptane-water system is shown in Fig. 3. Interfacial adsorption of an extractant can be simulated by a similar procedure after the introduction of the extractant molecule at the position from where the dynamics will be started. 

The reduction in the number of degrees of freedom can lead to an incorrect pressure in the simulation of the coarse-grained systems in NVT ensembles or to an incorrect density in NPT ensembles [24], The pressure depends linearly on the pair-forces in the system, hence the effect of the reduced number of degrees of freedom can be accounted for during the force matching procedure [24], If T is the temperature, V the volume, N the number of degrees of freedom of the system, and kb the Boltzmann constant then the pressure P of a system is given by... 

The Car-Parrinello method is similar in spirit to the extended system methods [37] for constant temperature [38, 39] or constant pressure dynamics [40], Extensions of the original scheme to the canonical NVT-ensemble, the NPT-ensemble, or to variable cell constant-pressure dynamics [41] are hence in principle straightforward [42, 43]. The treatment of quantum effects on the ionic motion is also easily included in the framework of a path-integral formalism [44-47]. 

In the NVT ensemble one cannot compute the chemical potential or entropy of the system two properties which are of critical importance for interfacial systems. The choice of an ensemble also determines the sampling algorithm used to generate molecular configurations from random moves of the molecules. 

In recent years, substantial efforts have been made to develop a theoretical framework for understanding the nature of such corrections [93]. In the case of lattice models (i.e., models of strictly localized particles) in the NVT ensemble with periodic boundary conditions (PBCs), it has been established a priori [94] and corroborated in explicit simulation [95] that the corrections are exponentially small in the system size [96],... 

We have seen that a classical system is represented in phase space by a point, and that an ensemble of identical systems will therefore be represented by a cloud of points. The distribution of system points is determined by the constraints we have imposed on the system, e.g., constant number of particles, constant volume, and constant energy (AWE-ensemble) or constant number of particles, constant volume, and constant temperature (NVT-ensemble). 

Another parameter that can have a great influence on the results obtained is the type of the simulation performed. Generally, simulations are carried out at constant particle number (N). The volume (V) and energy (E) of the simulated system can be held constant, leading to a so-called NVE, or microcanonical, ensemble. When the volume and temperature are held constant, this yields a canonical or NVT ensemble. In both cases, the size of the simulated system is chosen in such a way as to represent the desired state of the phospholipid, mostly the liquid crystalline La phase. The surface per lipid and the thickness of the bilayer are set based on experimental values and remain unchanged during the simulation. Therefore, the system is not able to adjust its size and thickness. 

Various methodologies have been developed to determine the free energy barriers from MD simulations. One of the common approaches applied in ab initio MD, known as the potential of mean force method has been derived from the thermodynamic integration technique. 34,35 In a canonical (NVT) ensemble the free energy difference, AA, between the two states, 0 and 1, can be calculated as the integral... 

Our study of the NVT ensemble begins by treating a large heat reservoir thermally coupled to a smaller system using the NVE approach. The energy of the heat reservoir is denoted Er and the energy of the smaller subsystem, El. The composite system is assumed closed and the total energy is fixed Er + ) = E0 = constant. The composite system is assumed to be a closed NVE system and the subsystem is assumed to have constant N and V. 

A second approach to the NVT ensemble found in Feynman s lecture notes on statistical mechanics [55] is also based on the central idea from NVE ensemble theory that the probability of a microstate is proportional to the number of microstates available to the system. Thus... 

The equipartition theorem, which describes the correlation structure of the variables of a Hamiltonian system in the NVT ensemble, is a central component of the held of statistical mechanics. Although the intent of this chapter is to introduce aspects of statistical thermodynamics essential for the remainder of this book -and not to be a complete text on statistical mechanics - the equipartition theorem provides an interpretation of the intrinsic variable T that is useful in guiding our intuition about temperature in chemical reaction systems. 

The Monte Carlo simulations were performed in the NVT ensemble and, in order to minimize the possibility of sampling regions of the zeolitic stmcture not accessible to the molecules, the initial distribution of the hydrocarbon molecules were chosen to be idaitical to the corresponding one of the thermalized configuration, at 300 K. The Metropolis [33] algorithm was then used to generate up to 8000 configurations. Three different steps, with equal probability, were considered random translation of the center of mass, random rotation of the whole molecule [34] and a perturbation on any of the internal coordinates of the molecules. 

The organic linkers can rotate freely in the solid MOF at ambient conditions. The energy barrier of a linker rotation along the connection axis is AE ai 0.35 eV (IRMOF-1). The rotation can, therefore, be thermally activated, which we also observed in MD simulations. These we have performed within NVT ensembles at 300 K and 1200 K. The nearly free rotation of the linker was already observed at 300 K. These simulations indicate also good thermal stability of the MOFs even at 1200 K. 

Other derivations for free energy differences in ensembles other than canonical NVT ensemble have been presented,as well as the ratio of partition functions for systems that are in different ensembles. " The results have also been tested numerically. In our group, the difference in free energy has also been derived and verified in numerical simulations of realistic systems where the system of interest is thermostatted and/or barostatted externally. 

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