Phenomenological parameters equations

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

To improve the accuracy of the solution, the size of the time step may be decreased. The smaller is the time step, the smaller are the assumed errors in the trajectory. Hence, in contrast (for example) to the Langevin equation that includes the friction as a phenomenological parameter, we have here a systematic way of approaching a microscopic solution. Nevertheless, some problems remain. For a very large time step, it is not clear how relevant is the optimal trajectory to the reality, since the path variance also becomes large. Further-... 

Shortcomings of the above described approach are self-evident the fluctuations entering equation (2.2.2) are independent of the deterministic motion, the passage from the deterministic description given by equation (2.1.1) to the stochastic one needs a large number of additional phenomenological parameters determining Gij. To define them, the fluctuation-dissipative theorem should be used. 

Within the approximation of the effective mass, consideration of the field created by the condensed media is confined to substitution of the real electron mass by the effective mass. Precise calculation of the effective mass is equivalent to solution of the Schrodinger equation with the consideration of the field created by the medium, and, consequently, as noted before, is hardly possible. Thus, as far as the problem of electron tunneling is concerned, the effective mass must be considered as a phenomenological parameter. In the case of tunneling with the energy I of the order of 1-5 eV, the field created by the medium apparently increases considerably the probability of electron tunneling, and the effective mass of electron can be noticeably lower than the real mass. 

As stated in the previous section, the use of a phenomenological kinetic equation derived from Eq. (5.1), for a system that does not verify the required restrictions for its use, may lead to different kinetic expressions when trying to fit experimental results obtained under isothermal and nonisothermal conditions. In particular, it may be observed that different kinetic parameters result by varying the heating rates in nonisothermal experiments. 

The generic equations of balance are statements of truth, which is a priori self-evident and which must apply to all continuum materials regardless of their individual characteristics. Constitutive relations relate diffusive flux vectors to concentration gradients through phenomenological parameters called transport coefficients. They describe the detailed response characteristics of specific materials. There are seven generic principles (1) conservation of mass, (2) balance of linear momentum, (3) balance of ro-... 

The internal viscosity force is defined phenomenologically by equations (2.26) formulated above. Various internal-friction mechanisms, discussed in a number of studies (Adelman and Freed 1977 Dasbach et al. 1992 Gennes 1977 Kuhn and Kuhn 1945 Maclnnes 1977a, 1977b Peterlin 1972 Rabin and Ottinger 1990) are possible. Investigation of various models should lead to the determination of matrices Ca/3 and Ga and the dependence of the internal friction coefficients on the chain length and on the parameters of the macromolecule. 

Dynamics of a single macromolecule in an entangled system is defined by the system of non-linear equations (3.52)-(3.54), containing some phenomenological parameters, which will be identified later. 

Here x is a phenomenological parameter measuring the chirality and / is a size scale factor. Since here the Reynolds number is small ( 10 s), the Stokes equation can be used to get r = DS2. where D is the hydrodynamic drag coefficient and 2 is the rotational speed. The drag coefficient for a cylindrical object rotating about its axis with cross-sectional radius r and length L is D = 4ztT)r2L, where tj is the viscosity of the medium [19]. Therefore, D /3 and the rotational speed 2 of the rotor will scale as... 

In equation (8.60) 7 represent phenomenological parameters which connect the ground state (f"irj and the final state fnir J through the matrix of the square of the unit tensor operators U(X. These quantities are sensitive to the accuracy of the oscillator strength and the nature of transitions used in their computation. 

Judd (13) has shown that the oscillator strength of an induced electric dipole transition may be related to the energy of the transition (v, in cm. ") and the square of the matrix elements of the unit tensor operators connecting the initial and final states via three phenomenological parameters T (A = 2, 4, and 6) according to Equation 2. 

The exact equation (III.3) may be solved if the phenomenological parameters and can be computed. For the special test case of a single advantageous master with homogeneously poorer competitors, s4 may be computed exactly and computed as a function of under the assumption that... 

J 0,1,2) is a one dimensional variable replacing the three dlmen slonal variable tj. Q(E) and Q(E—T) are the scattering probabilities per unit le ht for collisions to be in the forward or backward direction with a loss of energy equal to E and E-T, respectively. e is a phenomenological parameter averaging the angular dependence. The one dimensional version of equation 2, becomes... 

In Eq. (A.17), as before X(f) is the vector of thermodynamic forces while M is a symmetric matrix of phenomenological parameters introduced by Machlup and Onsager [4]. We adopt Eq. (A.17) inasmuch as it is the simplest equation of motion that is consistent with the Machlup-Onsager Eq. (A.24). Notice that Eq. (A.17) is similar in form to Newton s equation of motion for a particle system. Thus, we denote the matrix of phenomenological parameters by M in order to emphasize the analogy to particle masses. The analogy, however, is not perfect because M may be nondiagonal [4]. 

Phenomenological modeling uses a set of partial differential equations that characterize the fines migration process by means of model parameters. The values of these phenomenological parameters are attained through experiments. Phenomenological modeling can also be... 

Where Ptuji, is the dynamic factor of turbulent viscosity, which can be expressed in terms of turbulent pulsation energy K and the specific rate of its dissipation, e. In turn, the corresponding phenomenological defining equations are worked out for these parameters [36, 51]. 

Equations (10.38) and (10.39) give a nonlinear integro-differential equation for W, and its mathematical handling is not easy. A guidance of how to proceed is obtained from the phenomenological theory in nematics. De Gennes showed that the dynamics of nematics is essentially described by the Landau theory of phase transition and proposed a phenomenological nonlinear equation fof the order parameter tensor... 

Equations (10.48) and (10.49) give a special form of the Landau-de Gennes theory equation (10.40) with the phenomenological parameters... 

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