Equations, Differential

Equation (16) is a differential equation and applies equally to activity coefficients normalized by the symmetric or unsymme-tric convention. It is only in the integrated form of the Gibbs-Duhem equation that the type of normalization enters as a boundary condition. 

Consider the reflection of a normally incident time-harmonic electromagnetic wave from an inhomogeneous layered medium of unknown refractive index n(x). The complex reflection coefficient r(k,x) satisfies the Riccati nonlinear differential equation [2] ... 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

Equations II-12 and 11-13 illustrate that the shape of a liquid surface obeying the Young-Laplace equation with a body force is governed by differential equations requiring boundary conditions. It is through these boundary conditions describing the interaction between the liquid and solid wall that the contact angle enters. 

The time evolution of the wavefiinction is described by the differential equation... 

A differential equation for the time evolution of the density operator may be derived by taking the time derivative of equation (Al.6.49) and using the TDSE to replace the time derivative of the wavefiinction with the Hamiltonian operating on the wavefiinction. The result is called the Liouville equation, that is. 

The strategy for representing this differential equation geometrically is to expand both H and p in tenns of the tln-ee Pauli spin matrices, 02 and and then view the coefficients of these matrices as time-dependent vectors in three-dimensional space. We begin by writing die the two-level system Hamiltonian in the following general fomi. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

Note that the differential equation obtained from this approaeh will never agree perfeetly with the results of a simulation. The above fomuilation is essentially an adiabatie fomuilation of die proeess the spontaneous emission is eonsidered to be slow eompared with the time seale for the purity-preserving transformations generated by the external field, whieh is what allows us to assume m the theory that the external field... 

One must now examine the integrability of the differentials hi equation (A2.1.121 and equation (A2.1.13), which are examples of what matliematicians call Pfoff differential equations. If the equation is integrable, one can find an integrating denominator k, a fiiiictioii of the variables of state, such that = d( ) where d( ) is... 

The molar entropy and the molar enthalpy, also with constants of integration, can be obtained, either by differentiating equation (A2.1.56) or by integrating equation (A2.T42) or equation (A2.1.50) ... 

The free energy minimum is found by differentiating equation (A2.5.18) with respect to s at constant Tand setting the derivative equal to zero. In its simplest fonn the resultant equation is... 

Arnold L 1974 Stoohastio Differential Equations (New York Wiley-Interscience)... 

The solutions of such partial differential equations require infomiation on the spatial boundary conditions and initial conditions. Suppose we have an infinite system in which the concentration flucPiations vanish at the infinite boundary. If, at t = 0 we have a flucPiation at origin 5C(f,0) = AC (f), then the diflfiision equation... 

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. 

A general fonn of the rate law , i.e. the differential equation for the concentrations is given by... 

Integration of the differential equation with time-mdependent/r leads to the familiar exponential decay ... 

One may justify the differential equation (A3.4.371 and equation (A3.4.401 again by a probability argument. The number of reacting particles VAc oc dc is proportional to the frequency of encounters between two particles and to the time interval dt. Since not every encounter leads to reaction, an additional reaction probability has to be introduced. The frequency of encounters is obtained by the following simple argument. Assuming a statistical distribution of particles, the probability for a given particle to occupy a... 

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

Written in matrix notation, the system of first-order differential equations, (A3.4.139) takes the fomi... 

As an example we take again the Lindemaim mechanism of imimolecular reactions. The system of differential equations is given by equation (A3.4.127T equation (A3.4.128 ) and equation (A3.4.129T The rate coefficient matrix is... 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

The importance of numerical treatments, however, caimot be overemphasized in this context. Over the decades enonnous progress has been made in the numerical treatment of differential equations of complex gas-phase reactions [8, 70, 71], Complex reaction systems can also be seen in the context of nonlinear and self-organizing reactions, which are separate subjects in this encyclopedia (see chapter A3,14. chapter C3.6). 

Gear C W 1971 Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ Prentice-Hall)... 

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... 

One way to solve this is to invert the operator on the left hand side, thereby converting this differential equation into an integral equation. The general result is... 

---