Canonical ensemble molecular dynamics simulations

We present two major classes of NVT algorithms. These are widely used methods in molecular dynamics simulations, but are by no means the only ones. The interested reader is referred to a number of excellent textbooks in Further reading that detail simulation methodologies. 

These methods continuously scale the velocities so that the desired temperature Tbath is attained. The algorithmic steps at any time t during the simulation are as follows  

Multiply all velocities by to re-scale the temperature to the correct value. 

This is a simple method that ensures the temperature is always at the correct value, but it results in unphysical, abrupt, discontinuous changes in particle velocities. A preferable rescaling alternative borrows from systems control theory. The algorithmic steps at any time f during the simulation are now the following  

Determine a scaled difference between the actual temperature and the targeted one, 

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Fig. 2 Snapshots of canonical ensemble molecular dynamic simulations of water adsorbed on a model mesoporous silica thin film in (a) layer-adsorption state and (b) pore-filling state at 298 K...

Lynch GC, Pettitt BM (1997) Grand canonical ensemble molecular dynamics simulations Reformulation of extended system dynamics approaches. J Chem Phys 107 8594-8610 Madura JD, Pettitt BM, Calef DF (1988) Water under high pressure. Mol Phys 64 325 Mahoney MW, Jorgensen WL (2000) A five-site model for liquid water and the reproduction of the density anomaly by rigid, nonpolarizable potential functions. J ChemPhys 112 8910-8922 March RP, Eyring H (1964) Application of significant stmcture theory to water. J Phys Chem 68 221-228 Martin MG, Chen B, Siepman JI (1998) A novel Monte Carlo algorithm for polarizable force fields. 

Tobias D J, Martyna G J and Klein M L 1993 Molecular dynamics simulations of a protein In the canonical ensemble J. Phys. Chem. 9712959-66... 

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]. 

The mean transition temperatures obtained in molecular dynamics simulations of a canonical (Can) and a microcanonical (Mic) ensembles differ slightly... 

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. 

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 ... 

Nose S 1984 A Molecular Dynamics Method for Simulations in the Canonical Ensemble. Molecular Physics 53-255-268. 

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. 

Ruocco, G. Sampoli, M. (1995) Computer Simulation of Polarizable Fluids On the Determination of the Induced Dipoles, Molecular Simulations 15, 281-300 Nose, S. (1984) A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255-268... 

Shroll RM, Smith DE (1999a) Molecular dynamics simulations in the grand canonical ensemble Formulation of a bias potential for umbrella sampling. J Chem Phys 110 8295-8302 Shroll RM, Smith DE (1999b) Molecular dynamics simulations in the grand canonical ensemble Application to clay mineral swelling. J Chem Phys 111 9025-9033 Smith DE, Haymet ADJ (1992) Strucmre and dynamics of water and aqueous solutions The role of flexibility. J Chem Phys 96 8450-8459... 

Once the force field is chosen, a proper simulation method needs to be selected. Molecular dynamics simulations are applied to determine the solvation behaviour of ionic liquids by means of solving the Newtonian equations of motion for all molecules in the presence of a gradient in potential energy. Ionic liquid phase equilibria are determined by using Monte Carlo simulations in the isothermal isobaric Gibbs ensemble, grand canonical ensemble or osmotic ensemble with clever sampling schemes. 

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. 

Shroll, R. M. Smith, D. E. (1999). Molecular Dynamics Simulations in the Grand Canonical Ensemble Application to Clay Mineral Swelling. The Journal of Chemical Physics, 111, 9025. 

Shroll, R.M., and D.E. Smith. 1999. Molecular dynamics simulations in the grand canonical ensemble Application to clay mineral swelling. J. Chem. Phys. 111 9025-9033. 

Nose, S. (1984b). A molecular dynamics method for simulations in the canonical ensemble. Molecular Physics, 52, 255. 

It is usually desired to perform molecular dynamics simulations at constant temperature. In the simulations considered here, constant temperature is accomplished in one of two ways. In the first, stochastic forces and associated frictional forces are introduced which act individually on each atom of the system. This approach is hereafter referred to as stochastic dynamics (SD) simulation. The mean-square magnitude of the stochastic forces, which are purely random and Gaussian, is proportional to the temperature of the system, as described in Ref. 24. A canonical ensemble is simulated if the friction coefficient 7 for the frictional forces is chosen such that 1 /7 is much smaller than the total simulation time. The resulting damping forces do influence dynamic quantities, however, and this is the primary drawback of the SD method. The SD method was used in Refs 24, 25 and 31. The second method for constant temperature simulations involves either direct scaling of atomic velocities, as in Refs 29 and 30, or the inclusion of an additional temperature degree of freedom to the system by the Nose method,as appUed in Refs 31 and 32. Such simulations are hereafter referred to as molecular dynamics (MD) simulations. Of the MD methods considered, only the Nose method yields a true canonical ensemble. 

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