Functional evolution equations, direct

Instead of solving evolution equations for the moments, the evolution of the weights and nodes in the quadrature approximation can be directly tracked (Marchisio Fox, 2005 McGraw Wright, 2003). The evolution equations for weights and nodes can be derived by formally substituting the delta-function representation of the NDF into Eq. (7.3). If the weights and nodes are continuous functions of time, this procedure yields... 

The level set method [1,2] is an interface capturing scheme in which the evolution of an interface is tracked by evolving a level set function (f>(x, t) throughout space but focusing on the location of a specific level surface (or curve in 2D) of (f> to capture the position of the desired interface. In contexts other than two-phase fluid flow, the evolution equation for the level set function can account for advection by a specified external velocity field v(x, t), as well as propagation of the interface in the direction of the local normal vector at a constant speed a, and the motion of the interface in the normal direction at a speed proportional to the local curvature k with proportionality constant —b. The resulting evolution equation for cj) then takes the form... 

An approximate method for the response variability calculation of dynamical systems with uncertain stiffness and damping ratio can be found in Papadimitriou et al. (1995). This approach is based on complex mode analysis where the variability of each mode is analyzed separately and can efficiently treat a variety of probability distributions assumed for the system parameters. A probability density evolution method (PDEM) has also been developed for the dynamic response analysis of linear stochastic structures (Li and Chen 2004). In this method, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation from which the instantaneous probability density function (PDF) of the response and its evolution are obtained. Finally, variability response functions have been recently proposed as an alternative to direct MCS for the accurate and efficient computation of the dynamic response of linear structural systems with rmcertain Young modulus (Papadopoulos and Kokkinos 2012). 

Comparison with (X.2.16) shows the following drastic differences. The dominant term of (X.2.16) is absent and therefore no equation for the macroscopic part of X can be extracted. In other words, on the macroscopic scale the system does not evolve in one direction rather than the other. The remaining evolution of P is merely the net outcome of the fluctuations. Accordingly the time scale of the change is a factor slower than in the preceding case, compare (X.2.14). Since P is not sharply peaked the coefficients a(x) cannot be expanded around some central value but they remain as nonlinear functions in the equation. The first line of (1.4) contains the main terms and is called the diffusion approximation... 

The diffusion equation is the partial-differential equation that governs the evolution of the concentration field produced by a given flux. With appropriate boundary and initial conditions, the solution to this equation gives the time- and spatial-dependence of the concentration. In this chapter we examine various forms assumed by the diffusion equation when Fick s law is obeyed for the flux. Cases where the diffusivity is constant, a function of concentration, a function of time, or a function of direction are included. In Chapter 5 we discuss mathematical methods of obtaining solutions to the diffusion equation for various boundary-value problems. 

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

Except for a number of cases mentioned below a large dimensionality of system (24), which is equal to sN does not allow to use it for direct integration. In the theory of kinetic equations for describing the evolution of an object, instead of a full function of the distribution one uses a... 

The next steps are the following Step 1 Passage to the entropy representation and specification of the dissipative thermodynamic forces and the dissipative potential E. Step 2 Specification of the thermodynamic potential o. Step 3 Recasting of the equation governing the time evolution of the np-particle distribution function/ p into a Liouville equation corresponding to the time evolution of np particles (or p quasi-particles, Up > iip —see the point 4 below) that then represent the governing equations of direct molecular simulations. 

E>vib and Our also show up in the theory of spontaneous Raman spectroscopy describing fluctuations of the molecular system. The functions enter the CARS interaction involving vibrational excitation with subsequent dissipation as a consequence of the dissipation-fluctuation theorem and further approximations (21). Equations (2)-(5) refer to a simplified picture a collective, delocalized character of the vibrational mode is not included in the theoretical treatment. It is also assumed that vibrational and reori-entational relaxation are statistically independent. On the other hand, any specific assumption as to the time evolution of vib (or or), e.g., if exponential or nonexponential, is made unnecessary by the present approach. Homogeneous or inhomogeneous dephasing are included as special cases. It is the primary goal of time-domain CARS to determine the autocorrelation functions directly from experimental data. 

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