Microcanonical Ensemble NVE

Temperature effects are included explicitly in molecular dynamics simulations by including kinetic energy terms - the balls representing the atoms are now on the move The principles are simple. In the microcanonical ensemble (NVE) ... 

At variance from Xe, the presented properties for Kr require more computional efforts. In order to reach the small-4 range of S(q), large-scale molecular dynamics have been carried out in the microcanonical ensemble (NVE) with the usual periodic boundary conditions. The equations of motion are integrated in the same discrete form as for Xe. The time step At is the same as for Xe and g r) is extracted over a sample of 8000 time-independent configurations every lOAf. 

The use of non-Hamiltonian dynamical systems has a long history in mechanics [8] and they have recently been used to study a wide variety of problems in molecular dynamics (MD). In equilibrium molecular dynamics we can exploit non-Hamiltonian systems in order to generate statistical ensembles other than the standard microcanonical ensemble NVE) that is generated by traditional Hamiltonian dynamics. These ensembles, such as the canonical (NVT) and isothermal-isobaric (NPT) ensembles, are much better than the microcanonical ensemble for representing the actual conditions under which experiments are carried out. 

A standard MD computer simulation consists in the computation of the trajectory in the phase space of a system of N interacting bodies. The time evolution is determined by solving Newton s equations of motion of classical mechanics with finite difference methods. Such a model system corresponds to the microcanonical ensemble (NVE) of statistical mechanics with a constant number of particles N, volume V, and total energy E. In MD simulations the collective properties are then determined from the trajectory of all particles, i.e., from the time evolution of positions r = r, and momenta p = p,. The method relies on the assumption that stationary values of every average observable A can be defined as time integrals over the trajectory in the phase space ... 

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. 

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. 

An NVE system is also referred to as a microcanonical ensemble of particles. In addition to the NVE system, we will encounter NVT (canonical) and NPT (isobaric) systems. Sticking for now to the NVE system, let us imagine that for any given thermodynamic state, or macrostate, the many particles making up... 

While the NVE (microcanonical) ensemble theory is sound and useful, the NVT (canonical) ensemble (which fixes the number of particles, volume, and temperature while allowing the energy to vary) proves more convenient than the NVE for numerous applications. 

Constant-energy molecular dynamics simulations (NVE microcanonical ensemble) were performed assuming that the subsystems are rigid (quaternion formalism) the respective code [23] uses a fifth-order predictor-corrector formalism. 

In the NVE microcanonical ensemble all the systems have the same energy each system is individually isolated. Populations of various structures were obtained by long runs of MD. 

In the case of Nj-methyluracil dimer the highest population (evaluated within MD simulations in the NVE microcanonical ensemble) was found for the fifth local (stacked) structure. Its stabilization energy being considerably lower than the global structure corresponding to the H-bonded structure. Also other stacked... 

These two methods will be described by considering a collection of atoms or molecules (particles) in the canonical or microcanonical ensemble, i.e., we will assume that the number of particles, the volume and either the temperature (T) or total potential energy ( ) are held constant in the simulation. (NVT, NVE). 

Another way to view MD simulation is as a technique to probe the atomic positions and momenta that are available to a molecular system under certain conditions. In other words, MD is a statistical mechanics method that can be used to obtain a set of configurations distributed according to a certain statistical ensemble. The natural ensemble for MD simulation is the microcanonical ensemble, where the total energy E, volume V, and amount of particles N (NVE) are constant. Modifications of the integration algorithm also allow for the sampling of other ensembles, such as the canonical ensemble (NVT) with constant temperature... 

The above ensemble of systems with constant number of particles N, occupying constant volume V, with the total energy E conserved, is called the microcanonical, or NVE) ensemble. 

In the simplest version of MD, the Newton (21) or Lagrange equations (22) are integrated for a closed system, in which the volume, total energy, and number of particles are conserved. This simulates the microcanonical, NVE ensemble. Both kinetic and potential energies fluctuate in the microcanonical ensemble but their sum remains constant. 

Consider an ensemble of systems, each with constant number of particles N, in constant volume V, with constant energy E. This is called the NVE or microcanonical ensemble. Each member system of this ensemble corresponds to a point X = (p, in the 61V-dimensional phase space F. 

We will also explain why two different ensembles, the microcanon-ical, NVE, and the canonical, NVT, yield the same thermodynamic behavior for ideal gases, and indeed for any macroscopic system. 

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. 

It is clear from the derivation presented above that the phase space average, Equation (38), is exactly equal to the desired ensemble average. That is, all phase points with energy E are included with equal probability. Consider the phase space volume, f2 NVE), the number of states with energy E given physical volume, V, and N particles. As the phase space volume increases, obviously, the number of microstates increases and the entropy should increase. This suggest that we postulate that S NVE) = F f2 NVE)) where Q NVE) is now referred to as the microcanonical partion function and F must be a monotonically increasing, function to be determined. 

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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