Molecular dynamics simulation thermodynamical ensembles

In the canonical ensemble (P2) = 3kBTM and p M. In the microcanonical ensemble (P2) = 3kgT i = 3kBTMNm/(M + Nm) [49]. If the limit M —> oo is first taken in the calculation of the force autocorrelation function, then p = Nm and the projected and unprojected force correlations are the same in the thermodynamic limit. Since MD simulations are carried out at finite N, the study of the N (and M) dependence of (u(t) and the estimate of the friction coefficient from either the decay of the momentum or force correlation functions is of interest. Molecular dynamics simulations of the momentum and force autocorrelation functions as a function of N have been carried out [49, 50]. 

Surface tension is one of the most basic thermodynamic properties of the system, and its calculation has been used as a standard test for the accuracy of the intermolecular potential used in the simulation. It is defined as the derivative of the system s free energy with respect to the area of the interface[30]. It can be expressed using several different statistical mechanical ensemble averages[30], and thus we can use the molecular dynamics simulations to directly compute it. An example for such an expression is ... 

Molecular dynamics simulations entail integrating Newton s second law of motion for an ensemble of atoms in order to derive the thermodynamic and transport properties of the ensemble. The two most common approaches to predict thermal conductivities by means of molecular dynamics include the direct and the Green-Kubo methods. The direct method is a non-equilibrium molecular dynamics approach that simulates the experimental setup by imposing a temperature gradient across the simulation cell. The Green-Kubo method is an equilibrium molecular dynamics approach, in which the thermal conductivity is obtained from the heat current fluctuations by means of the fluctuation-dissipation theorem. Comparisons of both methods show that results obtained by either method are consistent with each other [55]. Studies have shown that molecular dynamics can predict the thermal conductivity of crystalline materials [24, 55-60], superlattices [10-12], silicon nanowires [7] and amorphous materials [61, 62]. Recently, non-equilibrium molecular dynamics was used to study the thermal conductivity of argon thin films, using a pair-wise Lennard-Jones interatomic potential [56]. 

The third step is to estimate the —> Conformational Ensemble Profile (GEP) for each compound by molecular dynamic simulation this profile encodes those conformations selected on the basis of the Boltzmann distribution. Then, different alignments are selected to compare the molecules of the training set. In the following step, each conformation of a molecule is placed in the reference grid space on the basis of the alignment scheme being explored and the thermodynamic probability of each grid cell occupied by each I PE type is computed. 

Molecular dynamic simulation methods, in addition to being essential for interpreting NMR data at the atomic level, also augment experimental studies in a number of other ways [101] modeling techniques can (i) yield structural information where experimental data has not yet been acquired, (ii) expand on experimental data through simulations that yield dynamic trajectories whose analysis provides unique information on lesion mobility, and (iii) provide thermodynamic insights by ensemble analysis using statistical mechanical methods. Furthermore, reaction mechanisms can now be determined with some confidence by combined quantum mechanical and molecular mechanical methods [104, 105],... 

In the last section we have assumed that we perform our simulation for a fixed number, N, of particles at constant temperature, T, and volume, V, the canonical ensemble. A major advantage of the Monte Carlo technique is that it can be easily adapted to the calculation of averages in other thermodynamic ensembles. Most real experiments are performed in the isobaric-isothermal (constant- ) ensemble, some in the grand-canonical (constant-pFT) ensemble, and even fewer in the canonical ensemble, the standard Monte Carlo ensemble, and near to none in the microcanonical (constant-NFE) ensemble, the standard ensemble for molecular-dynamics simulations. 

Molecular dynamics is traditionally performed in the constant NVE (or NVEP) ensemble. Although thermodynamic results can be transformed between ensembles, this is strictly only possible in the limit of infirate system size ( the thermodjmamic limit ). It may therefore be desired to perform the simulation in a different ensemble. The two most common alternative ensembles are the constant NVT and the constant NPT ensembles. In this section we will therefore consider how molecular dynamics simulations can be performed under conditions of constant temperature and/ or constant pressure. 

As with a molecular dynamics simulation, a Monte Carlo simulation comprises an equilibration phase followed by a production phase During equilibration, appropriate thermodynamic and structural quantities such as the total energy (and the partitioning of the energy among the various components), mean square displacement and order parameters (as appropriate) are monitored until they achieve stable values, whereupon the production phase can commence. In a Monte Carlo simulation from the canonical ensemble, the temperature and volume are, of course, fixed. In a constant pressure simulation the volume will change and should therefore also be monitored to ensure that a stable system density is achieved. 

There are several conditions in which molecular dynamics simulations can take place. These include the microcanonical ensemble (NV ), canonical ensemble (NVT), isothermal-isobaric ensemble (NPT), and grand canonical ensemble (p-VT), where N is the number of particles, V is the volume, E is the energy, P is the pressure, T is the temperature, and x is the chemical potential. In each of these ensembles, the thermodynamic variables held constant are designated by the appropriate letters. 

The implementation of the thermodynamic perturbation methods is relatively straightforward. An ensemble generated by a Monte Carlo simulation or the time trajectory generated by a molecular dynamics simulation for a system described by Hamiltonian "Kq is used to evaluate the ensemble average of expi-AM/k T). The free energy difference between the reference system described by Hamiltonian o> for which the ensemble is generated, and the perturbed system with Hamiltonian + A% is found using Eq, [17],... 

Molecular dynamics simulations have generally a great advantage of allowing the study of time-dependent phenomena. However, if thermodynamic and structural properties alone are of interest, Monte Carlo methods might be more useful. On the other hand, with the availability of ready-to-use computer simulation packages (e g.. Molecular Simulations Inc. 1999), the implementation of particular statistical ensembles in molecular dynamics simulations becomes nowadays much less problematic even for an end user without deep knowledge of statistical mechanics. 

If there is no time-dependent external force, the dynamics of a molecular system will evolve on a constant-energy surface. Therefore, a natural choice of the statistical ensemble in molecular dynamics simulation is the micro-canonical ensemble (NVE). Other types of ensembles, such as the canonical ensemble (NVT) and the isothermal-isobaric ensemble (NPT), can also be realized by controlling corresponding thermodynamic variables. For the last two ensembles, the temperature of the ensemble needs to be controlled and four different control mechanisms, namely differential control, proportional control, integral control and stochastic control, have been developed in the literature. As an example, a proportional thermostat for the NVT ensemble will be briefly discussed as follows. 

As mentioned in the introduction, nanoparticles have sizes below the thermodynamic limit, implying that their properties do not scale with their size. Obviously, it is therefore of interest to determine how their properties then scale. Eryiirek and Giiven have recently presented one theoretical study devoted to this aspect. They studied the thermodynamic properties of finite clusters with N = 39-55 atoms for which the interatomic interactions were modeled with a Lennard-Jones potential, eqn (2). To this end they performed molecular-dynamics simulations within the microcanonical ensemble for the different clusters, meaning that the total energy and the number of particles were kept fixed. From trajectories of 2 x 10 steps they determined various averaged values from which they subsequently determined, e.g., the temperature of the cluster,... 

The presented mechanism for PEM water sorption is consistent with thermodynamic principles, while predicting experimentally found trends correctly. Currently, full-scale molecular dynamics simulations would not be able to capture wall charge density effects and elastic effects in an ensemble of pores. 

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