Generalized equations of motion

In this section we present the concept of a generalized set of equations of motion that will describe a system coupled to an external field (e.g., shear flow, electric field, etc.). This general set of equations of motion will also describe coupling to the extended phase space (e.g., thermostats, barostats) and is of the form  

The notation reminds us that that the phase variable B has no explicit time dependence. However, its average in the nonequilibrium steady state explicitly depends on time through the distribution function /(F, t) generated by Eqs. [91]. 

we have a set of equations of motion given by Eqs. [91] that describe a general coupling to an external field. Our objective is to compute averages of functions of the phase space when the system coupled to the external field has reached a steady state. This is the procedure of nonequilibrium molecular dynamics (NEMD) simulations. An illustrative example to consider is the computation of the shear viscosity from Newton s law of viscosity, which reads  

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General Equation of Motion. Neglecting relativistic effects, the rate of accumulation of mass within a Cartesian volume element dx-dy-dz must equal the sum of the rates of inflow minus outflow. This is expressed by the equation of continuity ... 

The time derivative of the current density can easily be obtained from the general equation of motion for an arbitrary operator given by... 

We shall shortly consider such fundamental concepts as density matrices and the superoperator formalism which are convenient to use in a formulation of the lineshape theory of NMR spectra. The general equation of motion for the density matrix of a non-exchanging spin system is formulated in the laboratory (non-rotating) reference frame. The lineshape of a steady-state, unsaturated spectrum is given as the Fourier transform of the free induction decay after a strong radiofrequency pulse. The equations provide a starting point for the formulation of the theory of dynamic NMR spectra presented in Section III. The reader who may be interested in a more detailed consideration of the problems is referred to the fundamental works of Abragam and... 

The general equation of motion for a thin slice of thickness dz for a moving bed, shown schematically in Fig. 29, can be written as... 

The fully general situation of a particle diffusing in an out-of-equilibrium environment is much more difficult to describe. Except for the particular case of a stationary environment, the motion of the diffusing particle cannot be described by the generalized Langevin equation (22). A more general equation of motion has to be used. The fluctuation-dissipation theorems are a fortiori not valid. However, one can try to extend these relations with the help of an age- and frequency-dependent effective temperature, such as proposed and discussed, for instance, in Refs. 5 and 6. 

If we consider that the flow is steady, two-dimensional, and fully developed, and there is no pressure gradient in the microchannel, the general equation of motion is given by a balance between the viscous or shear stresses in the fluid and the externally imposed electrical field force ... 

As stated in the earlier section on Generalized Equations of Motion, we would ultimately like to find a set of equations of motion in the form of Eq. [91] to compute transport coefficients such as the shear viscosity and self diffusion... 

In describing polarization propagator methods it is instructive to start out with the simplest consistent method of the kind, namely the random-phase approximation (RPA). Within the framework we use here, RPA is described as the approximation to the general equation of motion (Eq. (58)) in which we set h = hj and assume 0> = HF>, that is, use the simplest truncation in both Eqs (64) and (89). It is convenient to split hj up into p-h and h-p excitation operators... 

The equation of motion for the MCRPA propagator is derived from the general equation of motion (Eq. (58)) using i0> in Eq. (113) as the reference state and a projection manifold h consisting of... 

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