Dynamical nonlinear terms

The nonlinear Smoluchowski-Vlasov equation is calculated to investigate nonlinear effects on solvation dynamics. While a linear response has been assumed for free energy in equilibrium solvent, the equation includes dynamical nonlinear terms. The solvent density function is expanded in terms of spherical harmonics for orientation of solvent molecules, and then only terms for =0 and 1, and m=0 are taken. The calculated results agree qualitatively with that obtained by many molecular dynamics simulations. In the long-term region, solvent relaxation for a change from a neutral solute to a charged one is slower than that obtained by the linearized equation. Further, in the model, the nonlinear terms lessen effects of acceleration by the translational diffusion on solvent relaxation. 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

Detonation, Nonlinear Theory of Unstable One-Dimensional. J.J. Erpenbeck describes in PhysFluids 10(2), 274-89(1969) CA 66, 8180-R(1967) a method for calcg the behavior of 1-dimensional detonations whose steady solns are hydrodynamic ally unstable. This method is based on a perturbation technique that treats the nonlinear terms in the hydro-dynamic equations as perturbations to the linear equations of hydrodynamic-stability theory. Detailed calcns are presented for several ideal-gas unimol-reaction cases for which the predicted oscillations agree reasonably well with those obtd by numerical integration of the hydrodynamic equations, as reported by W. Fickett W.W. Wood, PhysFluids 9(5), 903-16(1966) CA 65,... 

It should be stressed that the above calculations refer to a perhaps overly simplified model. When anharmonicities in the oscillator potential and processes like (n + m) + (n — m) 2n with m / 1 are taken into account, both the linear and the nonlinear term in the equation changes drastically. For this case one can, however, still obtain a simple analytical expression for the steady-state distribution, as discussed by Treanor9 and Fisher and Kummler.8 Our main concern here has been the dynamic... 

When motion of the fluid consists of only small fluctuations about a state of near-rest, Lhe continuum equations are linearized by neglecting nonlinear terms and they become lhe equalions of acoustics. A large variety of fluid motions are described as sound waves when the small-motion or acoustic description can be used, the principle of superposition is valid. This powerful principle allows addition of simple simultaneous motions to represent a more complex motion, such as the sound reaching lhe audience from the instruments of a symphony orchestra. The superposition principle does not apply to large-scale (nonacoustical) motions, and the subject of fluid dynamics (in distinction from acoustics) treats nonlinear flows. i.e.. those that cannot be described as superpositions of other flows. 

So, apart from the regular behavior, which is either steady-state, periodic, or quasi-periodic behavior (trajectory on a torus, Figure 3.2), some dynamic systems exhibit chaotic behavior, i.e., trajectories follow complicated aperiodic patterns that resemble randomness. Necessary but not sufficient conditions in order for chaotic behavior to take place in a system described by differential equations are that it must have dimension at least 3, and it must contain nonlinear terms. However, a system of three nonlinear differential equations need not exhibit chaotic behavior. This kind of behavior may not take place at all, and when it does, it usually occurs only for a specific range of the system s control parameters 9. 

If a given her is of higher than first order, nonlinear terms arise in the dynamic equation(s). With terms, for example, in squared concentrations (see below), there is the danger, due to computational errors, that a concentration becomes negative, after which it can never be corrected. The technique CN is especially prone to this, because of the oscillations it engenders as a response to sharp transients such as a potential jump. This is one reason some workers prefer the Laasonen method or its improved offshoots, which have a smooth error response without any oscillations. With a Pearson start, however, CN can be used safely, without the appearance of negative concentrations. 

We can also choose not to linearise the nonlinear term by an approximation, in which case we do not run the (minimal) risk of adding errors to the simulation by the linearising approximation. The same example as used above (8.57) and again choosing CN as the method, the dynamic (8.58) is discretised as... 

Molecular electric properties give the response of a molecule to the presence of an applied field E. Dynamic properties are defined for time-oscillating fields, whereas static properties are obtained if the electric field is time-independent. The electronic contribution to the response properties can be calculated using finite field calculations , which are based upon the expansion of the energy in a Taylor series in powers of the field strength. If the molecular properties are defined from Taylor series of the dipole moment /x, the linear response is given by the polarizability a, and the nonlinear terms of the series are given by the nth-order hyperpolarizabilities ()6 and y). 

It is important to clarify the mathematical and physical significance of the hydro-dynamic transport terms V(vu)/m) in eqs. (12.141). From the mathematical point of view the terms V(vuKm) emerge as a result of a nonlinear transformation of the state variables, from species densities to species fractions. The physical interpretation of the transport terms V(VuK ) depends on the direction and orientation of the speed vectors for expanding populations v are generally oriented toward the direction of expansion of the population cloud, resulting in enhanced transport. For shrinking population clouds the terms V(vu Km) lead to the opposite effect, that of the transport process slowing down. 

The terms Pmax and Pmax denote the limiting capacities of the inspiratory muscles, and n is an efficiency index. The optimal Pmus(t) output is found by minimization of / subjects to the constraints set by the chemical and mechanical plants. Equation 11.1 and Equation 11.9. Because Pmusif) is generally a continuous time function with sharp phase transitions, this amounts to solving a difficult dynamic nonlinear optimal control problem with piecewise smooth trajectories. An alternative approach adopted by Poon and coworkers [1992] is to model Pmus t) as a biphasic function... 

In conclusion from the above formalism we can deduce that the Davydov soliton is the macroscopic envelope of the localized boson condensation of the excitation quanta /) induced by the j8 transformation (3.19). The vanishing of the coefficient G of the nonlinear term of Eq. (3.29), via Eq. (3.13), implies the disappearance of the soliton, which then looks like a necessary consequence of the nonlinearities of the dynamics. 

The solution of the system produces a time series of the value of x in time t with the initial value of X given at time t = 0 or x(f = 0) = xq. The space x R") with t as a parameter is termed a phase space. The function/determines the time evolution of the system. The dynamic system is nonlinear with nonlinear term in the function/. One example in fluid dynamics is the Navier-Stokes equations for Newtonian fluids. The nonlinearity of the system comes mainly from the convective motion of the fluid flow. The fluid motion can develop unpredictable, yet observable, behavior with the proper choice of the initial conditions. 

The methods of Nonlinear Dynamics(8) can be applied to gain new theoretical insight into the underlying dynamics in terms of the molecular phase space structure. In particular, the existence of low order Fermi resonances between vibrationally anharmonic local (or normal) modes or between bending vibrations and rotations, can cause dramatic changes in phase space structure, manifest in the breakdown... 

Using the Oldroyd determination of derivatives In the law (1.1) and keeping nonlinear terms In the balance equations of mass and momentum, one can get the system of dynamic equations for the medium with Internal oscillators. 

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