Lagrangian equations dynamics

The first approach is based on introducing simple velocity or position rescaling into the standard Newtonian MD. The second approach has a dynamic origin and is based on a refonnulation of the Lagrangian equations of motion for the system (so-called extended Lagrangian formulation.) In this section, we discuss several of the most widely used constant-temperature or constant-pressure schemes. 

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]. 

Other methods for performing constant-temperature molecular dynamics calculations have been proposed recently. Evans (72) has introduced an external damping force in addition to the usual intermolecular force in order to keep the temperature constant in the simulation of a dissipative fluid flow. In another method, Haile and Gupta 13) have imposed the constraint of constant kinetic energy on the lagrangian equations of motion to perform calculations al constant temperature. 

We note that the above results are not limited to the case of linear decay, but also apply to any kind of decay-type or stable reaction dynamics in a flow with chaotic advection (Chertkov, 1999 Hernandez-Garcfa et ah, 2002). In such systems where the reaction dynamics is nonlinear, the decay rate b should be replaced by the absolute value of the negative Lyapunov exponent of the Lagrangian chemical dynamics given by the second equation in (6.25), that represents the average decay rate of small perturbations in the chemical concentration along the trajectory of a fluid element. 

The Eulerian equations of motion are more useful for numerical solution of highly distorted fluid flow than are Lagrangian equations of motion. Multicomponent Eulerian calculations require equations of state for mixed cells and methods for moving mass and its associated state values into and out of mixed cells. These complications are avoided by Lagrangian calculations. Harlow s particle-in-cell (PIC) method uses particles for the mass movement. The first reactive Eulerian hydrodynamic code EIC (Explosive-in-cell) used the PIC method, and it is described in reference 2. The discrete nature of the mass movement introduced pressure and temperature variations from cycle to cycle of the calculation that were unacceptable for many reactive fluid dynamic problems. A one-component continuous mass transport Eulerian code developed in 1966 proved useful for solving many one-component problems of interest in reactive fluid dynamics. The need for a multicomponent Eulerian code resulted in a second 2DE code, described in reference 4. Elastic-plastic flow and real viscosity were added in 1976. The technique was extended to three dimensions in the 1970 s and the resulting 3DE code is described in Appendix D. 

The dynamics of a system of N molecules with / holonomic constraints per molecule can be described in Cartesian coordinates via the Lagrangian equations of the following form ... 

The hC ) and the diXP) curves (Figure 34.2a) [9] decomposed from the measured V F) profile of compressed ice [14] and the measured phonon relaxation dynamics (oJP) (Fig. 37.3a) [11-14] provide the input for solving the Lagrangian equation ... 

To obtain the corresponding Lagrangian equations of motion. Eg is initially treated as a constant and later expanded using Eq. (83). It appears that Eq. (78) has exactly the form of Eq. (18), i.e., the Nose-Hoover thermostat has one equation of motion in common with both the Woodcock/Hoover-Evans and the Berendsen thermostats. However, in contrast to these other thermostats where the value of y was uniquely determined by the instantaneous microstate of the system (compare Eq. (79) with Eqs. (45), (50), and (56)), y is here a dynamical variable which derivative (Eq. (79)) is determined by this instantaneous microstate. Accompanying the fluctuations of y, heat transfers occur between the system and a heat bath, which regulate the system temperature. Because y = s s = y = s (Eq. (77)), the variable y in the Nose-Hoover formulation plays the same role as s in the Nose formulation. When y (or s) is negative, heat flows from the heat bath into the system due to Eq. (78) (or Eq. (60)). When the system temperature increases above To, the time derivative of y (or s) becomes positive due to Eq. (79) (or Eq. (63)) and the heat flow is progressively reduced (feedback mechanism). Conversely, when y (or s) is positive, heat is removed from the system until the system temperature decreases below To and the heat transfer is slowed down. 

Arbitrary-Lagrangian-Eulerian (ALE) codes dynamically position the mesh to optimize some feature of the solution. An ALE code has tremendous flexibility. It can treat part of the mesh in a Lagrangian fashion (mesh velocity equation to particle velocity), part of the mesh in an Eulerian fashion (mesh velocity equal to zero), and part in an intermediate fashion (arbitrary mesh velocity). All these techniques can be applied to different parts of the mesh at the same time as shown in Fig. 9.18. In particular, an element can be Lagrangian until the element distortion exceeds some criteria when the nodes are repositioned to minimize the distortion. 

In preparation for this, the equations of gas dynamics will reproduce the conservation laws of impuls, mass and energy that can be written in a number of different ways with respect to Eulerian (x,t) or Lagrangian (s,f) variables, where x is the coordinate of a particle and s is the initial coordinate of a particle or the quantity... 

Equations of gas dynamics with heat conductivity. We are now interested in a complex problem in which the gas flow is moving under the heat conduction condition. In conformity with (l)-(7), the system of differential equations for the ideal gas in Lagrangian variables acquires the form... 

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... 

The Kohn-Sham theory made a dramatic impact in the field of ab initio molecular dynamics. In the 1985, Car and Parrinello38 introduced a new formalism to study dynamics of molecular systems in which the total energy functional defined as in the Kohn-Sham formalism proved to be instrumental for practical applications. In the Car-Parrinello method (CP), the equations of motion are based on a Lagrangian (Lcp) which includes fictitious degrees of freedom associated with the electronic state. It is defined as ... 

We introduced the equations of motion above from the Newtonian point of view using Newton s famous equation F ma. It is useful to realize that this is not the only (or even the best) way to define equations of motion within classical dynamics. Another powerful approach to this task is to define a quantity called the Lagrangian, L, in terms of the kinetic and potential energies,... 

Biot and Daughaday (B6) have improved an earlier application by Citron (C5) of the variational formulation given originally by Biot for the heat conduction problem which is exactly analogous to the classical dynamical scheme. In particular, a thermal potential V, a dissipation function D, and generalized thermal force Qi are defined which satisfy the Lagrangian heat flow equation... 

The Eulerian continuum approach, based on a continuum assumption of phases, provides a field description of the dynamics of each phase. The Lagrangian trajectory approach, from the study of motions of individual particles, is able to yield historical trajectories of the particles. The kinetic theory modeling for interparticle collisions, extended from the kinetic theory of gases, can be applied to dense suspension systems where the transport in the particle phase is dominated by interparticle collisions. The Ergun equation provides important flow relationships, which are useful not only for packed bed systems, but also for some situations in fluidized bed systems. 

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