Extended system dynamics

Martyna G J, Tuckerman M, Tobias D J and Klein M L 1996 Explicit reversible Integrators for extended systems dynamics Mol. Phys. 87 1117-57... 

Martyna GJ, Tuckerman ME, Tobias DJ, Klein ML (1996) Explicit reversible integrators for extended systems dynamics. Mol Phys 87(5) 1117-1157... 

Equilibrium molecular dynamics was put on a firm theoretical ground with Andersen s seminal paper on extended system dynamics (to be discussed in great detail later). NEMD found its first success in this area with the advent of the so-called DOLL s (not an acronym) algorithm by Hoover and coworkers. ... 

Nose [147, 148] introduced a so-called extended system dynamics by giving an extra degree of freedom that can be used to control a variable in the system. Hoover [149] modified the time-scaling somewhat into a scheme known as Nose-Hoover thermostat. 

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. 

Martyna, G. J., etal. 1996). Explicit Reversible Integrators for Extended Systems Dynamics. Molecular Physies, 87(5), 1117-1157. 

In practice modifications are made to incorporate thermostats or barostats that may destroy the time-reversible and symplectic properties. While extended-system algorithms such as Nose dynamics [41] can be designed on the principles of the reversible operators, methods that use proportional velocity or coordinate scaling [42] cannot. Such methods arc very... 

Each state of the extended system that is generated by the molecular dynamics simulatic corresponds to a unique state of the real system. There is not, however, a dire correspondence between the velocities and the time in the real and the extended system The velocities of the atoms in the real system are given by ... 

Another popular approach to the isothennal (canonical) MD method was shown by Nose [25]. This method for treating the dynamics of a system in contact with a thennal reservoir is to include a degree of freedom that represents that reservoir, so that one can perform deterministic MD at constant temperature by refonnulating the Lagrangian equations of motion for this extended system. We can describe the Nose approach as an illustration of an extended Lagrangian method. Energy is allowed to flow dynamically from the reservoir to the system and back the reservoir has a certain thermal inertia associated with it. However, it is now more common to use the Nose scheme in the implementation of Hoover [26]. 

To construct Nose-Hoover constant-temperature molecular dynamics, an additional coordinate, s, and its conjugate momentum p, are introduced. The Hamiltonian of the extended system of the N particles plus extended degrees of freedom can be expressed... 

To include the volume as a dynamic variable, the equations of motion are determined in the analysis of a system in which the positions and momenta of all particles are scaled by a factor proportional to the cube root of the volume of the system. Andersen [23] originally proposed a method for constant-pressure MD that involves coupling the system to an external variable, V, the volume of the simulation box. This coupling mimics the action of a piston on a real system. The piston has a mass [which has units of (mass)(length) ]. From the Fagrangian for this extended system, the equations of motion for the particles and the volume of the cube are... 

An algorithm for performing a constant-pressure molecular dynamics simulation that resolves some unphysical observations in the extended system (Andersen s) method and Berendsen s methods was developed by Feller et al. [29]. This approach replaces the deterministic equations of motion with the piston degree of freedom added to the Langevin equations of motion. This eliminates the unphysical fluctuation of the volume associated with the piston mass. In addition, Klein and coworkers [30] present an advanced constant-pressure method to overcome an unphysical dependence of the choice of lattice in generated trajectories. 

An important advance in making explicit polarizable force fields computationally feasible for MD simulation was the development of the extended Lagrangian methods. This extended dynamics approach was first proposed by Sprik and Klein [91], in the sipirit of the work of Car and Parrinello for ab initio MD dynamics [168], A similar extended system was proposed by van Belle et al. for inducible point dipoles [90, 169], In this approach each dipole is treated as a dynamical variable in the MD simulation and given a mass, Mm, and velocity, p.. The dipoles thus have a kinetic energy, JT (A)2/2, and are propagated using the equations of motion just like the atomic coordinates [90, 91, 170, 171]. The equation of motion for the dipoles is... 

A further advantage of using Lagrangian dynamics is that we can easily impose boundary conditions and constraints by applying the method of Lagrangian multipliers. This is particularly important for the dynamics of the electronic degrees of freedom, as we will have to impose that the one-electron wavefunctions remain orthonormal during their time evolution. The Lex of our extended system can then be written as ... 

The Car-Parrinello method is similar in spirit to the extended system methods [37] for constant temperature [38, 39] or constant pressure dynamics [40], Extensions of the original scheme to the canonical NVT-ensemble, the NPT-ensemble, or to variable cell constant-pressure dynamics [41] are hence in principle straightforward [42, 43]. The treatment of quantum effects on the ionic motion is also easily included in the framework of a path-integral formalism [44-47]. 

The formulation described above provides a useful framework for treating feedback control of combustion instability. However, direct application of the model to practical problems must be exercised with caution due to uncertainties associated with system parameters such as and Eni in Eq. (22.12), and time delays and spatial distribution parameters bk in Eq. (22.13). The intrinsic complexities in combustor flows prohibit precise estimates of those parameters without considerable errors, except for some simple well-defined configurations. Furthermore, the model may not accommodate all the essential processes involved because of the physical assumptions and mathematical approximations employed. These model and parameter uncertainties must be carefully treated in the development of a robust controller. To this end, the system dynamics equations, Eqs. (22.12)-(22.14), are extended to include uncertainties, and can be represented with the following state-space model ... 

There have been a number of theoretical investigations of the structure and dynamics of heterogeneous clusters in which a single atom or molecule interacts with a rare-gas cluster Pair potentials are often a good candidate for providing a simplified treatment for the ground state of extended systems such as small van der Waals clusters like NaAr HgArn or Li Arn... 

All local concentrations C of particles entering the non-linear functions F in equation (2.1.40) are taken at the same space points, in other words, the chemical reaction is treated as a local one. Taking into account that for extended systems we shouldn t consider distances greater than the distinctive microscopic scale Ao, the choice of equation (2.1.40) means that inside infinitesimal volumes vo particles are well mixed and their reaction could be described by the phenomenological reaction rates earlier used for systems with complete reactant mixing. This means that Ao value must exceed such distinctive scales of the reaction as contact recombination radius, effective radius of a dynamical interaction and the particle hop length, which imposes quite natural limits on the choice of volumes v0 used for averaging. 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

Also pure density-functional methods combined with plane-wave basis sets and ultrasoft pseudopotentials [58] were used in our studies of extended systems [59]. The computational efficiency of these methods enables larger systems and to some extent dynamical processes to be studied. Generalized-gradient approximation (GGA) or spin-polarized GGA DFT functionals [60, 61] were employed in the electronic structure calculations. 

While the electronic structure calculations addressed in the preceding Section could in principle be used to construct the potential surfaces that are a prerequisite for dynamical calculations, such a procedure is in practice out of reach for large, extended systems like polymer junctions. At most, semiempirical calculations can be carried out as a function of selected relevant coordinates, see, e.g., the recent analysis of Ref. [44]. To proceed, we therefore resort to a different strategy, by constructing a suitably parametrized electron-phonon Hamiltonian model. This electron-phonon Hamiltonian underlies the two- and three-state diabatic models that are employed below (Secs. 4 and 5). The key ingredients are a lattice model formulated in the basis of localized Wannier functions and localized phonon modes (Sec. 3.1) and the construction of an associated diabatic Hamiltonian in a normal-mode representation (Sec. 3.2) [61]. 

The LVC model further allows one to introduce coordinate transformations by which a set of relevant effective, or collective modes are extracted that act as generalized reaction coordinates for the dynamics. As shown in Refs. [54, 55,72], neg = nei(nei + l)/2 such coordinates can be defined for an electronic nei-state system, in such a way that the short time dynamics is completely described in terms of these effective coordinates. Thus, three effective modes are introduced for an electronic two-level system, six effective modes for a three-level system etc., for an arbitrary number of phonon modes that couple to the electronic subsystem according to the LVC Hamiltonian Eq. (7). In order to capture the dynamics on longer time scales, chains of such effective modes can be introduced [50,51,73]. These transformations, which are briefly summarized below, will be shown to yield a unique perspective on the excited-state dynamics of the extended systems under study. 

If the phonon distribution of the model Eq. (8) spans a dense spectrum - as is generally the case for the extended systems under consideration, which are effectively infinite-dimensional - the dynamics induced by the Hamiltonian will eventually exhibit a dissipative character. However, the effective-mode construction demonstrates that the shortest time scales are fully determined by few effective modes, and by the coherent dynamics induced by these modes. The overall picture thus corresponds to a Brownian oscillator type dynamics, and is markedly non-Markovian [81,82],... 

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