Dynamical variables

Dynamical variable A Classical quantity Quantum-mechanical operator A... 

Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore... 

If now the nuclear coordinates are regarded as dynamical variables, rather than parameters, then in the vicinity of the intersection point, the energy eigenfunction, which is a combined electronic-nuclear wave function, will contain a superposition of the two adiabatic, superposition states, with nuclear... 

Election nuclear dynamics theory is a direct nonadiababc dynamics approach to molecular processes and uses an electi onic basis of atomic orbitals attached to dynamical centers, whose positions and momenta are dynamical variables. Although computationally intensive, this approach is general and has a systematic hierarchy of approximations when applied in an ab initio fashion. It can also be applied with semiempirical treatment of electronic degrees of freedom [4]. It is important to recognize that the reactants in this approach are not forced to follow a certain reaction path but for a given set of initial conditions the entire system evolves in time in a completely dynamical manner dictated by the inteiparbcle interactions. 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

A powerful analytical tool is the time correlation function. For any dynamic variable A (it), such as bond lengths or dihedral angles, the time autocorrelation function Cy) is defined... 

A time cross-correlation function between dynamic variables A (it) and B(t) is defined in a similar way ... 

The potential of mean force is a useful analytical tool that results in an effective potential that reflects the average effect of all the other degrees of freedom on the dynamic variable of interest. Equation (2) indicates that given a potential function it is possible to calculate the probabihty for all states of the system (the Boltzmann relationship). The potential of mean force procedure works in the reverse direction. Given an observed distribution of values (from the trajectory), the corresponding effective potential function can be derived. The first step in this procedure is to organize the observed values of the dynamic variable, A, into a distribution function p(A). From this distribution the effective potential or potential of mean force, W(A), is calculated from the Boltzmann relation ... 

The advantage of a potential of mean force is that it reflects the effect of environmental factors on the behavior of the dynamic variable of interest. Such an effect may be... 

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... 

The flow field in front of an expanding piston is characterized by a leading gas-dynamic discontinuity, namely, a shock followed by a monotonic increase in gas-dynamic variables toward the piston. If both shock and piston are regarded as boundary conditions, the intermediate flow field may be treated as isentropic. Therefore, the gas dynamics can be described by only two dependent variables. Moreover, the assumption of similarity reduces the number of independent variables to one, which makes it possible to recast the conservation equations for mass and momentum into a set of two simultaneous ordinary differential equations ... 

Now the distribution of the gas-dynamic variables can be computed from the isen-tropic relations ... 

If the values of the gas dynamic variables are known, these expressions may be evaluated for any position throughout the flow field. The location of the flame front is found where Q matches the heat of combustion of the fuel-air mixture in question. If the coordinate of the front X, is known, the burning velocity Mach number can be computed from... 

Most vibration programs that use microprocessor-based analyzers are limited to steady-state data. Steady-state vibration data assumes the machine-train or process system operates in a constant, or steady-state, condition. In other words, the machine is free of dynamic variables such as load, flow, etc. This approach further assumes that all vibration frequencies are repeatable and maintain a constant relationship to the rotating speed of the machine s shaft. 

In the cast of the discrete theory, thfa e are two set of equations, since both and t,i are eonsidered dynamical variables ... 

The first equation gives the diserete version of Newton s equation the second equation gives energy c onservation. We make two comments (1) Notice that while energy eouseivation is a natural consequence of Newton s equation in continuum mechanics, it becomes an independent property of the system in Lee s discrete mechanics (2) If time is treated as a conventional parameter and not as a dynamical variable, the discretized system is not tiine-translationally invariant and energy is not conserved. Making both and t , dynamical variables is therefore one way to sidestep this problem. 

In continuum field theory, the field dynamical variable and the coordinates X and t are only parameters. With the conjugate momentum f (T, t), and Hamiltonian operator... 

Jourjine [jour85] generalizes Euclidean lattice field theory on a d-dimensional lattice to a cell complex. He uses homology theory to replace points by cells of various dimensions and fields by functions on cells, the cochains, in hopes of developing a formalism that treats space-time as a dynamical variable and describes the change in the dimension of space-time as a phase transition (see figure 12.19). 

Here, the second term describes the change of the hopping amplitudes due to the displacement of the atoms parallel to the chain [cf. Eq. (3.5)] and the third term is a random contribution resulting from the conformational disorder (chain twists). While the lattice displacements u are dynamic variables, the fluctuations dl +t due to disorder are assumed to be frozen ( quenched disorder). 

In practice, even though with the developments that has occurred in the past and continues this perfect stable situation is never achieved and there are variables that affects the output. If the process is analyzed one can decide that two types of variables affect the quality and output rate. They can be identified as (1) the variables of the machine s design and manufacture and (2) the operating or dynamic variables that control how the machine is run. 

Stated more abstractly, in quantum mechanics, a particle is characterized by a set of dynamical variables, p,q, which are represented by operators that obey the fundamental commutation rules... 

The dynamical variables Ll that correspond to the three components of the orbital angular momentum pseudovector are related to the dynamical variables q and p corresponding to position and momentum by... 

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