Slow variable

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

As indicated, the power law approximations to the fS-correlator described above are only valid asymptotically for a —> 0, but corrections to these predictions have been worked out.102,103 More important, however, is the assumption of the idealized MCT that density fluctuations are the only slow variables. This assumption breaks down close to Tc. The MCT has been augmented by coupling to mass currents, which are sometimes termed inclusion of hopping processes, but the extension of the theory to temperatures below Tc or even down to Tg has not yet been successful.101 Also, the theory is often not applied to experimental density fluctuations directly (observed by neutron scattering) but instead to dielectric relaxation or to NMR experiments. These latter techniques probe reorientational motion of anisotropic molecules, whereas the MCT equation describes a scalar quantity. Using MCT results to compare with dielectric or NMR experiments thus forces one to assume a direct coupling of orientational correlations with density fluctuations exists. The different orientational correlation functions and the question to what extent they directly couple to the density fluctuations have been considered in extensions to the standard MCT picture.104-108... 

The adiabatic approximation ignores the action of the slow variable kinetic energy operator, T", on the X /jV) When this approximation is made, then the wavefrmction for the entire system can be written as a product of fast and slow factors, T = and the wavefrmction for the slow variable subsystem satisfies the slow variable Schrodinger equation... 

Transitions can occur between different adiabatic states for the fast variable subsystem, since the adiabatic approximation ignores the action of T on the Yjf. The ignored terms act as the coupling between the different fast variable states. T" involves derivatives with respect to slow variable coordinates. In the discussion below, the coupling between the fast variable states is given by the nonadiabatic coupling vector... 

If the initial state of the system is described by t /ift /os(io), where /oXi o) is the initial wavefunction for the slow variable subsystem at t = 0, then the k-hop contribution to the component of the system in fast variable state //at time t is given by... 

In order to test the small x assumptions in our calculations of condensed phase vibrational transition probabilities and rates, we have performed model calculations, - for a colinear system with one molecule moving between two solvent particles. The positions ofthe solvent particles are held fixed. The center of mass position of the solute molecule is the only slow variable coordinate in the system. This allows for the comparison of surface hopping calculations based on small X approximations with calculations without these approximations. In the model calculations discussed here, and in the calculations from many particle simulations reported in Table II, the approximations made for each trajectory are that the nonadiabatic coupling is constant that the slopes of the initial and final... 

Formulas (99), (100) and (102) give the backgrounds for surgery of cycles with outgoing reactions. The left eigenvector gives the slow variable if there are some incomes to the cycle, then... 

Stiffness occures in a problem if there are two or more very different time scales on which the dependent variables are changing. Since at least one component of the solution is "fast", a small step size must be selected. There is, however, also a "slow" variable, and the time interval of interest is large, requiring to perform a large number of small steps. Such models are common in many areas, e.g., in chemical reaction kinetics, and solving stiff equations is a challenging problem of scientific computing. 

This new timescale T is a slow time and is the timescale on which the slow variable a is changing. 

Higher order terms can be obtained by writing the inner and outer solutions as expansions in powers of e and solving the sets of equations obtained by comparing coefficients. This enzymatic example is treated extensively in [73] and a connection with the theory of materials with memory is made in [82]. The essence of the singular perturbation analysis, as this method is called, is that there are two (or more in some extensions) time (or spatial) scales involved. If the initial point lies in the domain of attraction of steady states of the fast variables and these are unique and stable, the state of the system will rapidly pass to the stable manifold of the slow variables and, one might... 

The hydrodynamic approach to liquid-state dynamics is based on the assumption that many experimental observables (like the intensity in a light scattering experiment) can be rationalized by considering the dynamics of a few slow variables. The natural choice for the slow variables are the densities of the conserved quantities—that is, the number density, p(r, t), the momentum density, g(r, t) and the energy density, e(r,t). The conservation of number, momentum, and energy are expressed locally by the conservation equations... 

When the solvent molecules are explicitly included, one needs to treat a ternary system (two ions and the dipolar solvent molecules). The additional slow variables to be included in the mode coupling theory are the products of the ion charge and solvent densities. This will explicitly introduce terms like Fis(k,t), which is the partial dynamic structure factor involving the ion and the solvent molecules. The calculation of the microscocpic terms of the friction, containing the density terms, does not appears to be difficult, but calculation of the current terms now appears to be formidable. 

A simple theory of the concentration dependence of viscosity has recently been developed by using the mode coupling theory expression of viscosity [197]. The slow variables chosen are the center of mass density and the charge density. The final expressions have essentially the same form as discussed in Section X the structure factors now involve the intermolecular correlations among the polyelectrolyte rods. Numerical calculation shows that the theory can explain the plateau in the concentration dependence of the viscosity, if one takes into account the anisotropy in the motion of the rod-like polymers. The problem, however, is far from complete. We are also not aware of any study of the frequency-dependent properties. Work on this problem is under progress [198]. 

O.I. Tolstikhin, S. Watanabe, M. Matsuzawa, Slow variable discretization a novel approach for Hamiltonians allowing adiabatic separation of variables, J. Phys. B At. Mol. Opt. Phys. 29 (1996) L389. 

For catalytic reactions the fast and slow variables usually considered are the concentrations of surface intermediates on catalysts and gas-phase reactants, respectively. (In the case of high-vacuum conditions, "a vice versa quasi-stationarity is possible, see below.) But in the equations for heterogeneous catalytic reactions (119)... 

Angle brackets in Eqs. (31a) and (31b) denote averaging over the whole spectrum of the states of slow variables. Generally, the states of rapid variables depend on those of slow variables, so that SFj(a) and 8F (r a) cannot be taken out of the angle brackets. An expression in the quasichemical approximation for the free energy that are needed for calculation of the 8Fy(d) and 8F (r d) is given in Appendix A. 

Flere indexes k and m in Eqs. (34a) and (34b) correspond to the rapid and slow variables which correspond to the neighboring species ... 

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