NVE ensemble

By far the most common methods of studying aqueous interfaces by simulations are the Metropolis Monte Carlo (MC) technique and the classical molecular dynamics (MD) techniques. They will not be described here in detail, because several excellent textbooks and proceedings volumes (e.g., [2-8]) on the subject are available. In brief, the stochastic MC technique generates microscopic configurations of the system in the canonical (NYT) ensemble the deterministic MD method solves Newton s equations of motion and generates a time-correlated sequence of configurations in the microcanonical (NVE) ensemble. Structural and thermodynamic properties are accessible by both methods the MD method provides additional information about the microscopic dynamics of the system. 

A standard MD simulation generates an NVE ensemble, i.e. the temperature and pressure will flucmate. It is possible also to generate NVT or NET ensembles by MD techniques by modifying the velocities or positions in each time step. As indicated in eq. (16.12) the instant value of the temperamre is given by the average of the kinetic energy over the number of particles. If this is different from the desired temperature, all velocities may be scaled by a factor of (T desired/T actuai) in each time step to achieve... 

A series of articles were published by Ennari et al. on MD simulation of transport processes in Poly(Ethylene Oxide) and sulfonic acid-based polymer electrolyte.136,137 The work was started by the determination of the parameters for the ions missing from the PCFF forcefield made by MSI (Molecular Simulations Inc.), to create a new forcefield, NJPCFF. In the models, the proton is represented as a hard ball with a positive charge. Zhou et al. used the similar approach to model Nation.138 The repeating unit of Nafion (Fig. 17) was optimized using ab initio VAMP scheme. The protons were modeled with hydronium ions. Three unit cell or molecular models were used for the MD simulation. The unit cell contains 5000 atoms 20 pendent side chains, and branched Nafion backbone created with the repeating unit. Their water uptakes or water contents were 3, 13, or 22 IEO/SO3, which correspond to the room temperature water uptakes at 50% relative humidity (RH), at 100% RH, and in liquid water respectively.18 The temperature was initially set at a value between 298.15 and 423.15 K under NVE ensemble with constant particle number, constant volume (1 bar), and constant energy. 

In standard molecular dynamic simulations the temperature is not constant. The MD simulation samples the microcanonical ensemble, or NVE ensemble, as the volume (unit-cell size) is assumed to be constant. The control of temperature is on the other hand especially important in the simulation of chemical reactions, when the excess of heat dissipated or adsorbed during the reaction strongly influences the kinetic energy (temperature) of the system. 

For classical systems the microstates are not discrete and the number of possible states for a fixed NVE ensemble is in general not finite. To see this imagine a system of a single particle (N = 1) traveling in an otherwise empty box of volume V. There are no external force fields acting on the particle so its total energy is E = mv2. The particle could be found in any location within the box, and its velocity could be directed in any direction without changing the thermodynamic macrostate defined by the fixed values of N, V, and E. To apply ensemble theory to classical systems Q(N, V, E) is defined as the (appropriately scaled) total volume accessible by the state variables of position and momentum accessible by the particles in the system. 

We shall see that the variables fl and i) are related to the intensive thermodynamic quantities temperature and pressure, respectively. But before completing the picture of how macroscopic thermodynamics emerges from the NVE ensemble, we first have one more equilibration to consider - concentration equilibration. For this case, imagine that the partition between the chambers is perforated and particles are permitted to freely travel from one system to the next. The equilibrium statement for this system is... 

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... 

For the molecules investigated, the MD and MC methods furnish similar adsorption energies although the MD results are slightly better when compared to the experiments. Since the MC and MD simulations have been performed in different ensembles but with the same force field, the difference in the results of the two simulations may be partly due to finite size effects. However, temperature fluctuations during the MD simulations in the NVE ensemble may also contribute to this difference. For the linear alkanes the MM results are qualitatively correct but only when a specific force field is used to describe the zeolite. However, for the branched alkane the MM results are comparable to the MD and MC ones even when a generic force field, such as Dreiding n, is used to represent the zeolite. 

In the work of Edwards et al [36], which included interactions only to the third neighbor shell (approximately 4.4 A), the problem of energy instability for a 1000 K simulation did not arise the timestep was lO s at all temperatures. Since no analysis of the effects of timestep and cutoff was presented, it is difficult to determine the reasons for the differences in the behavior observed by Edwards et al and in the present study. Some possibilities are (a) They may have employed a minimum image approach for the periodic boundary conditions, whereas we have calculated both primaryprimary and primary-image interactions for each atom, (b) Their system was much larger than ours (2048 vs 256 atoms), (c) Edwards et al used the NVE ensemble for the production simulations, in contrast to our choice of the NPT ensemble. 

Relative populations in % from the constant energy MD simulations in NVE ensemble using AMBER 4.1 potential. 

In MD simulations we are interested in obtaining the properties of the system at certain macroscopic conditions (temperature T, volume V, pressure P, etc.). These parameters act as constraints and determine a statistical ensemble that is sampled. For instance, one commonly used ensemble is the microcanoni-cal or NVE ensemble, for which the number of particles N, volume of the... 

An ensemble may be characterized by parameters which are fixed, and those which can be derived from the simulation data, as shown in Table 16.1. Ensembles generated by MC techniques are naturally of the constant NVT type, while MD methods naturally generate a constant NVE ensemble. Both MC and MD methods, however, may be modified to simulate other ensembles, as described in Section 16.2. Of special... 

Integrators and thermostats ESPResSo can currently only perform MD simulations using a Velocity-Verlet integration scheme. Various ensembles can be obtained by different thermostats. For the NVE ensemble, no thermostat is used, for NVT, one can use either a Langevin or DPD thermostat. Constant pressure, i.e. NPT, simulations, can be performed using an algorithm by Diinweg et. al. [39]. 

Solid-state phase transitions of salts have been studied by fitting the pressure and internal energy of each phase to an equation of state and determining the temperature for which AG = 0 at each pressure. Simulations for each solid-state phase are performed separately in the NVE ensemble.[158] In general, the thermodynamic -integration method discussed in Sec. 4.1 can be used to study solid-state phase transitions as well. 

MD simulations use different physical principles to simulate actual laboratory conditions [27]. One is the NVE ensemble, in which the number of particles... 

The molecular dynamics and Monte Carlo simulation methods differ in a variety of ways. The most obvious difference is that molecular dynamics provides information about the time dependence of the properties of the system whereas there is no temporal relationship between successive Monte Carlo configurations. In a Monte Carlo simulation the outcome of each trial move depends only upon its immediate predecessor, whereas in molecular dynamics it is possible to predict the configuration of the system at any time in the future - or indeed at any time in the past. Molecular dynamics has a kinetic energy contribution to the total energy whereas in a Monte Carlo simulation the total energy is determined directly from the potential energy function. The two simulation methods also sample from different ensembles. Molecular dynamics is traditionally performed under conditions of constant number of particles (N), volume (V) and energy (E) (the microcanonical or constant NVE ensemble) whereas a traditional Monte Carlo simulation samples from the canonical ensemble (constant N, V and temperature, T). Both the molecular dynamics and Monte Carlo techniques can be modified to sample from other ensembles for example, molecular dynamics can be adapted to simulate from the canonical ensemble. Two other ensembles are common ... 

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