Intermediate regime

Concerning the chain swelling, a very sensitive test exists for the agreement between experiment and theory it consists in determining the effective exponents associated with the increase in size of polymers, and in comparing them with the predictions of the standard continuous model. 

Furthermore, by examining how an effective exponent varies with molecular mass and temperature, it is possible to determine whether one has to do with the intermediate or with the asymptotic regime of the polymer state in good solvent. 

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The Intermediate Regime (Best known as the Transition Regime Law, for 2 [Pg.273]

The authors introduced a new criterion to delineate the two intermediate regimes = 1, which is based on... 

Premixed turbulent combustion regime diagram proposed by Chen and Bilger. Two intermediate regimes are delineated between distributed flame front and wrinkled laminar flamelets. (Reprinted from Chen, Y.C. and Bilger, R., Combust. Flame, 131, 400, 2002. With permission. Figure 9, p. 411, copyright Elsevier editions.)... 

This model breaks down when hv becomes small in comparison to k T, as the potential becomes so soft that the atoms will no longer be located at a specific site but diffuse over the surface. In this intermediate regime the atoms move around in a potential... 

In practice, the process regime will often be less transparent than suggested by Table 1.4. As an example, a process may neither be diffusion nor reaction-rate limited, rather some intermediate regime may prevail. In addition, solid heat transfer, entrance flow or axial dispersion effects, which were neglected in the present study, may be superposed. In the analysis presented here only the leading-order effects were taken into account. As a result, the dependence of the characteristic quantities listed in Table 1.5 on the channel diameter will be more complex. For a detailed study of such more complex scenarios, computational fluid dynamics, to be discussed in Section 2.3, offers powerful tools and methods. However, the present analysis serves the purpose to differentiate the potential inherent in decreasing the characteristic dimensions of process equipment and to identify some cornerstones to be considered when attempting process intensification via size reduction. 

The last two results are rather similar to the quadratic forms given by Fox and Uhlenbeck for the transition probability for a stationary Gaussian-Markov process, their Eqs. (20) and (22) [82]. Although they did not identify the parity relationships of the matrices or obtain their time dependence explicitly, the Langevin equation that emerges from their analysis and the Doob formula, their Eq. (25), is essentially equivalent to the most likely terminal position in the intermediate regime obtained next. 

Higher-order terms have been neglected in this small-x expansion that is valid in the intermediate regime. This expression obeys exactly the symmetry relationships, and it obeys the reduction condition to leading order. (See Eq. (68) for a more complete expression that obeys the reduction condition fully.)... 

From this the most likely terminal position in the intermediate regime is... 

In the intermediate regime, xShort < x < xiong, the transport matrix is linear in x and it follows that... 

That the time correlation function is the same using the terminal velocity or the coarse velocity in the intermediate regime is consistent with Eqs (53) and (54). 

The matrix is readily shown to be antisymmetric, as it must be. In the intermediate regime, the transport matrix must be independent of x, which means that for nonzero x,... 

That the terminal acceleration should most likely vanish is true almost by definition of the steady state the system returns to equilibrium with a constant velocity that is proportional to the initial displacement, and hence the acceleration must be zero. It is stressed that this result only holds in the intermediate regime, for x not too large. Hence and in particular, this constant velocity (linear decrease in displacement with time) is not inconsistent with the exponential return to equilibrium that is conventionally predicted by the Langevin equation, since the present analysis cannot be extrapolated directly beyond the small time regime where the exponential can be approximated by a linear function. 

Beyond the intermediate regime, in the long time limit the correlation function vanishes, <2(x) —> 0. In this regime the second entropy is just the sum of the two first entropies, as is expected,... 

Most of the previous analysis has concentrated on the intermediate regime, Thort < "t < Uong- It is worth discussing the reasons for this in more detail, and to address the related question of how one chooses a unique transport coefficient since in general this is a function of x. 

These scaling relations indicate that in the intermediate regime the second entropy, Eq. (86), may be written... 

Clearly, the first two bracketed terms have to individually vanish. Since the first bracket contains two symmetric matrices, this implies that g (x) = — g (x), and since the second bracket contains a symmetric matrix and an antisymmetric matrix, this also implies that g (x) = g (x) = 0. Furthermore, since g (x) = 5(x)/2 = Sfo(x)/SxT, it is also-concluded that g (x) = 0. Explicidy then, in the intermediate regime it follows that -... 

Hence to linear order in the differences, in the intermediate regime this is... 

In the intermediate regime, this may be recognized as the Green-Kubo expression for the thermal conductivity [84], which in turn is equivalent to the Onsager expression for the transport coefficients [2]. 

In the final equality the asymptote has been used. Although this result has been derived for infinitesimal A(, it is also valid in the intermediate regime, A, Ox. 

This is reasonable, since the arrow of time is provided by the adiabatic evolution of the subsystem in the intermediate regime. The perturbative influence of the... 

Now consider the transition E —> E in time x > 0. Most interest lies in the linear regime, and so for an isolated system the results of Section IID apply. From Eq. (50), the second entropy in the intermediate regime is... 

Terminal velocity, linear thermodynamics intermediate regimes and maximum flux, 25-27 regression theorem, 18-20 Test particle density, multiparticle collision dynamics, macroscopic laws and transport coefficients, 100-104 Thermodynamic variables heat flow, 58-60... 

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