The NVT ensemble

This expression serves as a precise mathematical definition of temperature. It is interesting to note that temperature, a variable with which we have intuitive and sensory familiarity, is defined based on entropy, one with which we may be less familiar. In fact, we shall see that entropy and temperature are intimately related in the concept of free energy, in which temperature determines the relative importances of energy and entropy in driving thermodynamic processes. 

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. 

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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],... 

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 original work of Andersen and of Parrinello and Rahman on the generation of the NPT or isothermal-isobaric ensemble using an extended phase space predates Nose s work on the NVT ensemble, as noted above. Applying the extended system method to generate the NPT ensemble involves the inclusion of the volume into the phase space as a dynamical variable along... 

We simulated the SPC/E basal ice/water interface with Na+ and Cl- ions at a temperature of 225 K, which was estimated as the melting point for the SPC/E model of water in our previous study of two-phase coexistence [19]. We have used a collection of 2304 rigid water molecules plus a single solute ion in the NVT ensemble. The time step was chosen to be 1.5 fs. 

Two approaches exist for estimation of the free energy of solute ion transfer across the interface. The first one is calculation of the potential of mean force, when the reaction coordinate for ion transfer is considered to be the 2S position of the ion. With respect to a reference state where the ion is at zq, the change in Helmholtz free energy (for simulations in the NVT ensemble) when the ion is located at s is given by the following [32],... 

This switches on the Langevin thermostat for the NVT ensemble, with temperature temp and friction coefficient gamma. The skin depth skin is a parameter for the link-cell system which tunes its performance, but cannot be discussed here. 

Our Monte Carlo (MC) simulation uses the Metropolis sampling technique and periodic boundary conditions with image method in a cubic box(21). The NVT ensemble is favored when our interest is in solvent effects as in this paper. A total of 344 molecules are included in the simulation with one solute molecule and 343 solvent molecules. The volume of the cube is determined by the density of the solvent and in all cases used here the temperature is T = 298K. The molecules are rigid in the equilibrium structure and the intermolecular interaction is the Lennard-Jones potential plus the Coulombic term... 

The motion equations have been solved by the Verlet Leap-frog algorithm subject to periodic boundary conditions in a cubic simulation cell and a time step of 2 fs. The simulations have been performed in the NVT ensemble with the Nose-Hoover thermostat [62]. The SHAKE constraints scheme [65] was used. The spherical cutoff radius comprises 1.2 nm. The Ewald sum method was used to treat long-range electrostatic interactions. 

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