Evolution equation for

For the two-level system, the evolution equation for p may also be expressed, as before, in tenns of the tliree-vector r ... 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Prager s rule of kinematic hardening is expressed by a = ce where c is a constant. Generalizing these concepts, the evolution equations for the internal state variables will be taken in the form... 

Using (5.77), the evolution equations for the hardening parameter (5.76) becomes... 

The choice (5.77) for the evolution equation for the plastic strain sets the evolution equations for the internal state variables (5.78) into the form (5.11) required for continuity. The consistency condition in the stress space description may be obtained by differentiating (5.73), or directly by expanding (5.29)... 

The energy conservation equation is not normally solved as given in (9.4). Instead, an evolution equation for internal energy is used [9]. First an evolution equation for the kinetic energy is derived by taking the dot product of the momentum balance equation with the velocity and integrating the resulting differential equation. The differential equation is... 

The first part of Eq. (89), proportional to the inverse viscosity r] of the liquid film, describes a creeping motion of a thin film flow on the surface. In the (almost) dry area the contributions of both terms to the total flow and evaporation of material can basically be neglected. Inside the wet area we can, to lowest order, linearize h = hoo[ + u x,y)], where u is now a small deviation from the asymptotic equilibrium value for h p) in the liquid. Since Vh (p) = 0 the only surviving terms are linear in u and its spatial derivatives Vw and Au. Therefore, inside the wet area, the evolution equation for the variable part u of the height variable h becomes... 

O. Pierre-Louis, C. Misbah, Y. Saito, J. Krug, P. Politi. New nonlinear evolution equation for steps during molecular beam epitaxy on vicinal surfaces. Phys Rev Lett 50 4221, 1998. 

The goal of kinetic theory is to find the time-evolution equation for f x,v,t), with the ultimate aim being to obtain the equilibrium properties of the system through the limiting form of f x, v,t) as t cx>. 

Consider next the case of binary rupture when A = 0, s = 1, and br(xly) = 2. Assuming that fragments once eroded do not fragment further, an approximate form of the evolution equation for clusters larger than size e is... 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

At the same time, Prigogine and his co-workers14 15,17 developed a general theory of non-equilibrium statistical mechanics. They derived a non-Markovian evolution equation for the velocity distribution function. Their results contain a generalization of the Boltzmann equation for arbitrary concentration and coupling parameter. This generalization is the long-time limit of their evolution equation. 

In this section we shall explain somewhat the results which we have just presented. We are interested this time in the evolution equation for the one-particle distribution function. We write down the virial series expansion of the transport equation and we recall that every contribution to this equation is proportional to V n+d, where n is the number of particles which are involved... 

The time-evolution equation for the spherically integrated scalar-variance spectrum Eaa(K, t) obtained from (A.l) can be written as... 

Starting from the Liouville equation as the fundamental microscopic evolution equation for the dynamics of all phase-space variables, MCT uses... 

Because of differences in chemical properties of U, Th, and Pa, the elements are fractionated in many geochemical processes, such as sedimentation, mantle partial melting, and coral precipitation from water. With fractionation, the nuclide activities of Th, and Pa do not equal one another. Define the time of disturbance to be time zero. Use hi, A2, and A3 to denote the decay activity of Th, and Pa, respectively, and Xy X2, and X3 to denote the decay constants of Th, and Pa. Start from the full evolution equation for Pa in Box 2-6,... 

To solve the preceding set of equations, Equation 5.62 is plugged into Equation 5.60. By separately determining the compaction properties of the fiber bed [32] an evolution equation for the pressure can be obtained. Because this is a moving boundary problem the derivative in the thickness direction can be rewritten [32] in terms of an instantaneous thickness. The pressure field can then be solved for by finite difference or finite element techniques. Once the pressure is obtained and the velocity computed, the energy and cured species conservation equations can be solved using the methodology outlined in Section 5.4.1. 

The evolution equation for the two layers in the thin soot deposit limit (used here for analytical convenience) are ... 

The first one is that this particular form of H can also be used to prove the approach to equilibrium in the case of Boltzmann s kinetic equation for dilute gases. The Boltzmann equation is nonlinear and a different technique is needed to prove that all solutions tend to equilibrium. This technique is based on (5.6) other convex functions cannot be used. Incidentally, the Boltzmann equation is not a master equation for a probability density, but an evolution equation for the particle density in the six-dimensional one-particle phase space ( /i-space ). The linearized Boltzmann equation, however, has the same structure as a master equation (compare XIV.5). 

Internal or intrinsic noise is caused by the fact that the system itself consists of discrete particles it is inherent in the very mechanism by which the system evolves, as described in III.2. All our examples concerning chemical reactions, emission and absorption of light, growth of populations, etc., were of this type. Internal noise cannot be switched off and it is therefore impossible to identify A(y) as the evolution equation for the system in isolation. One usually identifies it... 

If we take time average over shorter period than an evolutional time scale and over spherical angular average, we obtain the evolutional equation for the averaged mean molecular weight U ... 

EVOLUTION EQUATIONS FOR CONSERVED AND NON-CONSERVED ORDER PARAMETERS... 

A. Fick, Ann. Phys. (Leipzig) 170, 50 (1855). He actually set up his two laws for the temporal spreading of the concentration of a tracer substance, not for the probability. The first evolution equation for a probability was the Boltzmann equation [L. Boltzmann Vorlesungen tiber Gastheorie I (J. A. Barth, Leipzig, 1896)], following Maxwell s theory of gas kinetics. 

Schut, J. H. Novel Low-VOC Paint Technology. Plast. Technol. 1991, 37, 29-37. Scriven, L. E. Higgins, B. G. Interfacial Shape and Evolution Equations for Liquid Films and Other Viscocapillary Flows. Ind. Eng. Chem. Fundam. 1979, 18, 208-215. 

The evolution equations for the quantities entering the right side of this equation are obtained by substitution into the quantum-classical Liouville equation. For a variety of one- and two-dimensional systems for which exact results are known, excellent agreement was found. 

The quantum-classical limit of the transport coefficient is obtained by evaluating the evolution equation for the matrix elements of W in the quantum-classical limit. This limit was taken in Ref. [68] and the result is... 

In the present and in the following section we discuss the application of the group-theoretical formalism to the formulation of quantum-classical mechanics. Our purpose is to determine evolution equations for two coupled subsystems, with two different degrees of quantization. We have shown in the previous sections that the classical behaviour of a system is formally obtained as a limiting case of the quantum behaviour, when the Planck constant h tends to zero. In this section we will associate two different values of the Planck constant, say hi and /12, to the two subsystems and introduce suitable Lie brackets to determine the evolution of the two subsystems [15]. The consistency, e.g., with respect to Jacobi identity, is guaranteed by the very definition of the... 

The evolution equation for the degree of cement hydration T. appearing in (2), has the following form [10, 15],... 

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