Molecular-dynamics at constant temperature

D.A. Gibson and E.A. Carter, Generalized valence bond molecular dynamics at constant temperature, Mol. Phys., 89(1996), 1265-1276. 

Several variations of ordinary MD have been developed. Andersen has proposed "molecular dynamics at constant temperature," in which an MD system is made to represent a canonical system, by altering the momentum of random particles at sequential random Instants of time. The new momentum is picked from a Boltzmann distribution, with a given parameter p. Since the motion of the system is no longer Hamiltonian, this procedure is a statistical sampling method. A combination technique was used by Wood and Erpenbeck, who ran a set of independent MD calculations, with the initial phase of each calculatlon glcked from a canonical, or mlcrocanonical, distribution. Andersen also described molecular dynamics at constant pressure," in which the pressure is a parameter of the Lagranglan, and the system volume fluctuates. 

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

Similar schemes to the above can be used in molecular dynamics simulations in other ensembles such as those at constant temperature or constant pressure (see Frenkel and Smit, and Allen and Tildesley (Further reading)). A molecular dynamics simulation is computationally much more intensive than an energy minimization. Typically with modern computers the real time sampled in a simulation run for large cells is of the order of nanoseconds (106 time steps). Dynamical processes operating on longer time-scales will thus not be revealed. 

With representative values for A, Cp, and po with Vq 50 cm/s, equation (4) gives S 10 cm. Therefore 5 is large compared with a molecular mean free path (about 10 cm), and the continum equations of fluid dynamics are valid within the deflagration wave but 3 is small compared with typical dimensions of experimental equipment (for example, the diameter of the burner mouth, and hence the radius of curvature of the flame cone, for experiments with Bunsen-type burners), and laminar deflagration waves may be approximated as discontinuities in many experiments. Since equations (3) and (4) imply that 3 at constant temperature, experimental... 

Recently Calderon introduced a surrogate process approximation (SPA) to improve the sampling in calculation of the JE. The scheme is applied to the study of the unravelling of deca-alanine at constant temperature in a steered molecular dynamics simulation. The distribution of the work is approximated by developing a model for the dynamics using a relatively small number of real trajectories in conjunction with stochastic differential equations selected to model the process. The... 

The continuous metadynamics algorithm can be applied to any system evolving under the action of a dynamics whose equilibrium distribution is canonical at an inverse temperature 1// . In a molecular dynamics scheme this requires that the evolution is carried out at constant temperature, by using a suitable thermostat [51]. In the continuous version of metad3mamics, Gaussians are added at every MD step and act directly on the microscopic variables. This generates at time t extra forces on x that can be written as... 

Twenty years ago Car and Parrinello introduced an efficient method to perform Molecular Dynamics simulation for classical nuclei with forces computed on the fly by a Density Functional Theory (DFT) based electronic calculation [1], Because the method allowed study of the statistical mechanics of classical nuclei with many-body electronic interactions, it opened the way for the use of simulation methods for realistic systems with an accuracy well beyond the limits of available effective force fields. In the last twenty years, the number of applications of the Car-Parrinello ab-initio molecular d3mam-ics has ranged from simple covalent bonded solids, to high pressure physics, material science and biological systems. There have also been extensions of the original algorithm to simulate systems at constant temperature and constant pressure [2], finite temperature effects for the electrons [3], and quantum nuclei [4]. 

Two unusual features can be observed in these plots (and, at least for the self-diffusion coefficient, this behaviour is common to all hydrogen-bonded liquids). This ratio is a function of temperature. At constant temperature and pressure, rotation and translation reveal the same isotope effect. From simple sphere dynamics one would expect the rotation to scale as the square root of the ratio of the moments of inertia (=1.38) while for translational mobility the square root of the ratio of the molecular masses ( = 1.05) should be found. This is clearly not the case, indicating that the dynamics of liquid water are really the dynamics of the hydrogen-bond network. The hydrogen bonds in D2O are stronger than those in H2O and thus the mobility in the D2O network decreases more rapidly as the temperature decreases. 

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. 

S. Nose, in Computer Simulation in Materials Science, M. Meyer and V. Pontikis, Eds., NATO ASl Series, Kluwer Academic Publishers, Dordrecht, 1991, Vol. 205, pp. 21-41. Molecular Dynamics Simulations at Constant Temperature and Pressure. 

We shall consider the medium to be incompressible v = 0, where v is the linear velocity) and at constant temperature t = Tj = 0). We shall assume further that the director is of constant magnitude. This implies that the external forces and fields responsible for elastic deformation, viscous flow, etc., are very much weaker than the molecular interactions giving rise to the spontaneous alignment of the neighbouring molecules. It is indeed a valid assumption in all the static and dynamic phenomena discussed in this chapter. We may therefore conveniently choose n to be a dimensionless unit vector (n[Pg.86]

For many problems, however, it is more convenient to keep the temperature, pressure, or chemical potential constant, instead of the total energy, volume, and number of particles. Generalizations of the molecular dynamics technique to virtually any ensemble have been developed, and they will be discussed in the following chapters. Of course, constant temperature MD does not conserve the total system energy, allowing it to fluctuate as is required at constant temperature. Similarly, volume is allowed to fluctuate in constant pressure molecular dynamics. The trick is to make these quantities fluctuate in a manner consistent with the probability distribution of the desired ensemble. 

Being a molecular dynamics technique, GEMD provides dynamical information such as diffusion coefficients in the coexisting phases [5]. Shown in Fig. 3 are the mean squared displacements of the carbon atoms in -hexane, calculated in coexisting phases with GEMD, compared to the results of the constant temperature molecular dynamics at the same conditions. This shows that diffusion coefficients and other dynamical information can be extracted from GEMD simulations, together with the thermodynamic properties. 

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