The NVE ensemble

Thermodynamic principles arise from a statistical treatment of matter by studying different idealized ensembles of particles that represent different thermodynamic systems. The first ensemble that we study is that of an isolated system a collection of N particles confined to a volume V, with total internal energy E. A system of this sort is referred to as an NVE system or ensemble, as N, V, and E are the three thermodynamic variables that are held constant. N, V, and E are extensive variables. That is, their values are proportional to the size of the system. If we combine NVE subsystems into a larger system, then the total N, V, and E are computed as the sums of N, V, and E of the subsystems. Temperature, pressure, and chemical potential are intensive variables, for which values do not depend on the size 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 

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. 

Applications in statistical mechanics are based on constructing expressions for Q(N, V, E) (and other partition functions for various ensembles) based on the nature of the interactions of the particles in a given system. To understand how thermodynamic principles arise from statistics, however, it is not necessary to worry about how one might go about computing Q(N, V, E), or how Q might depend on N, V, and E for particular systems (classical or quantum mechanical). It is necessary simply to appreciate that the quantity Q(N, V, E) exists for an NVE system. 

Now we introduce a fundamental postulate of statistical thermodynamics at a given Nx, Vx, and Ex, system 1 is equally likely to be in any one of its 121 microstates similarly system 2 is equally likely to be in any one of its 122 microstates (more on this assumption later). The combined system, consisting of systems 1 and 2, has associated with it a total partition function 120(.Ei, E2), which represents the total number of possible microstates. The number 120(.Ei, E2) may be expressed as the multiplication  

---

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... 

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 constant energy MD simulations in the NVE ensemble using AMBER 4.1 potential... 

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 partition function Z, which normalizes the density, is effectively a function of N, V and E it represents the number of microstates available under given conditions. As this ensemble is associated to constant particle number N, volume V and energy E, it is often referred to as the NVE-ensemble, and when we speak of NVE simulation, we mean simulation that is meant to preserve the microcanonical distribution this, most often, would be based on approximating Hamiltonian dynamics, e.g. using the Verlet method or another of the methods introduced in Chaps. 2 and 3, and assuming the ergodic property. For a discussion of alternative stochastic microcanonical methods see [126]. 

For all simulations, we used the LAMMPS simulation package [20,21], and the computational scheme was the following Initially, the system was allowed to equilibrate for a period of 1 to 2 ns in the NPT ensemble and then a total of 5x10 time steps were performed in the NVE ensemble. The number of H2O molecules in the simulations of the aqueous phase was equal to 2,000 with 1 CO2 molecule, while for the CO2 rieh phase the number of CO2 molecules was 1,000 with 1 H2O molecule. The composition of the mixtures simulated here corresponds to infinite dilution limit. 

For the flexible model simulations two different approaches were used when calculating those properties the first one was performing the simulations in the NVT ensemble, as has been done in several previous works, even they do not used the Green-Kubo relations the calculation but other statistical equivalent methods, and the second one was by simulating the system in the NVE ensemble, as previously done by some authors (Rey-Castro Vega, 2006 Rey-Castro et al., 2007). 

Fig. 10. Diffusion coefficient for the [C2-rriim]+ (circles) and Ch (squares) ions calculated by the Green-Kubo relations using the rigid model (full symbols) and the flexible model from the NVE ensemble (open symbols). Results obtained for the flexible model in the NVT ensemble were very similar to the NVE ensemble and are not presented here. The symbols represent MD simulation results while the lines are just guides to the eyes.

In Equation 4.22, E is the size of phase space occupied by the NVE ensemble of microscopic states. The Dirac 8 function simply selects the iVVE points and includes them in E. In other words, all points that represent systems with energy other than E are not included in the integral. The area E is then a (67V — l)-dimensional surface, because not all degrees of freedom can change independently while satisfying the constant energy constraint. 

From the size of the available phase space E(7V, V, E) we can define the number of microscopic states f2(7V, V, E) that comprise the NVE ensemble. Simply,... 

The parameter 0, can be thought of as a thermal inertia that controls the rate of temperature fluctuations inside the system. It is an adjustable parameter that can be empirically determined. If it is too high then the energy transfer between the system and the hath is slow, with the limit of 0, oo returning the NVE ensemble. On the other hand, if 0, is too small then there will be long-lived, weakly damped oscillations of the energy in the system. 

Obtain Moldyn from the textbook website (http //statthermo.sourceforge.net/). Moldyn is a set of codes which simulates point mass particles that interact with Lennard-Jones forces. Moldyn uses the velocity-Verlet integration algorithm in the NVE ensemble. Read the README file on how to run the programs and do this homework problem. 

---