Macroscopic equations

Macroscopic properties of nematic elastomers have been discussed [56, 60]. De Gennes focused on the static properties, emphasizing especially the importance of coupling terms associated with relative rotations between the network and the director field [60]. The electrohydrodynamics of nematic elastomers has been considered generalizing earlier work by the same authors [54,55] on the macroscopic properties of nematic sidechain polymers [56]. The static considerations of earlier work [60] were extended to incorporate electric effects in addition a systematic overview of all terms necessary for linear irreversible thermodynamics was given [56]. 

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The passage from microscopic to macroscopic (equation (B1.3.11)) clearly exposes the additivity of the... 

The NAs such as DNA usually used in the experiments consist of 10" -1 o nucleotides. Thus, they should be considered as macrosystems. Moreover, in experiments with wet NA samples macroscopic quantities are measured, so averaging should also be performed over all nucleic acid molecules in the sample. These facts justify the usage of the macroscopic equations like (3) in our case and require the probabilities of finding macromolecular units in the certain conformational state as variables of the model. 

Macroscopic Equations An arbitraiy control volume of finite size is bounded by a surface of area with an outwardly directed unit normal vector n. The control volume is not necessarily fixed in space. Its boundary moves with velocity w. The fluid velocity is v. Figure 6-3 shows the arbitraiy control volume. 

Actually, this is not really diffusion-XimiiQd, but rather Laplacian growth, since the macroscopic equation describing the process, apart from fluctuations, is not a diffusion equation but a Laplacian equation. There are some crucial differences, which will become clearer below. In some sense DLA is diffusion-limited aggregation in the limit of zero concentration of the concentration field at infinity. 

These conditions show us immediately that in the case of the four-neighbor HPP lattice (V = 4) f is noni.sotropic, and the macroscopic equations therefore cannot yield a Navier-Stokes equation. For the hexagonal FHP lattice, on the other hand, we have V = 6 and P[. is isotropic through order Wolfram [wolf86c] predicts what models are conducive to f lavier-Stokes-like dynamics by using group theory to analyze the symmetry of tensor structures for polygons and polyhedra in d-dimensions. 

A typical elongation that has already reached x=h l/K, exceeds h by about Mk. We estimate Xo as the time required for the elongation to recede from h+l/K to h, using macroscopic equations for the motion of the step. It may be shown from the statistics of the step that the typical base X at x=0 of an elongation of amplitude x=A is given by... 

The expansion method described above enables one to compute the spectral density of fluctuations in successive orders of O 1, provided the Master Equation is known.14 In the linear case, however, it was sufficient to know the macroscopic equation and the equilibrium distribution, as... 

The basic remark is that linearity of the macroscopic law is not at all the same as linearity of the microscopic equations of motion. In most substances Ohm s law is valid up to a fairly strong field but if one visualizes the motion of an individual electron and the effect of an external field E on it, it becomes clear that microscopic linearity is restricted to only extremely small field strengths.23 Macroscopic linearity, therefore, is not due to microscopic linearity, but to a cancellation of nonlinear terms when averaging over all particles. It follows that the nonlinear terms proportional to E2, E3,... in the macroscopic equation do not correspond respectively to the terms proportional to E2, E3,... in the microscopic equations, but rather constitute a net effect after averaging all terms in the microscopic motion. This is exactly what the Master Equation approach purports to do. For this reason, I have more faith in the results obtained by means of the Master Equation than in the paradoxical result of the microscopic approach. 

Of course, the macroscopic equations cannot actually be derived from the microscopic ones. In practice they are pieced together from general principles and experience. The stochastic mesoscopic description must be obtained in the same way. This semi-phenomenological approach is remarkably successful in the range where the macroscopic equations are linear, see chapter VIII. In the nonlinear case, however, difficulties appear, which can only be resolved by the improved, but still mesoscopic, method of chapter X. 

As t increases the peak slides bodily along the y-axis from its initial location y(0) = y0 to its final location y(oo) = < 7>e. (The width only grows from its initial value zero to its final value, being the width of Pe.) This motion determines the evolution of y t) and hence the macroscopic equation. Bearing this picture in mind we can now readily derive the macroscopic equation. 

This is no longer a closed equation for , but higher moments enter as well. The evolution of < Y> in the course of time is therefore not determined by itself, but is influenced by the fluctuations around this average. The macroscopic approximation consists in ignoring these fluctuations, and keeping only the first term in the expansion (8.5). With this approximation therefore (8.4) is valid even when a y) is nonlinear. Thus one obtains as macroscopic equation... 

Remark. From the linear integro-differential equation for P(y, t) we have derived a nonlinear equation for y(t). Thus the essentially linear master equation may well correspond to a physical process that in the laboratory would be regarded as a nonlinear phenomenon inasmuch as its macroscopic equation is nonlinear. This is not paradoxical provided one bears in mind that the distinction between linear and nonlinear is defined for equations. It is wrong to apply it to a physical phenonemon, unless one has agreed upon a specific mathematical description of it. Newton s equations for the motion of the planets are nonlinear, but the Liouville equation of the solar system is linear. This connection between linear and nonlinear equations is not a matter of approximation the linear Liouville equation is rigorously equivalent with the nonlinear equations of motion of the particles. Generally any linear partial... 

It is also true that near the equilibrium value ye one may approximate the nonlinear macroscopic equation (8.6) by a linear one ... 

Having obtained the equation (8.9) for the variance, one may now include the second term in (8.5) to obtain a correction to the macroscopic equation... 

It will be found in X that this equation together with (8.9) does indeed constitute the first approximation beyond the macroscopic equation (8.6). ... 

Comment. The macroscopic equation (8.6) is a differential equation for y, which determines y(t) uniquely when the initial value /(0) is given. In the next approximation (8.12) the evolution of / depends on the variance of the fluctuations as well. The reason is that y fluctuates around / and thereby feels the value of not merely at / but also in the neighborhood. This effect is proportional to the curvature of al the slope of at is ineffective as the fluctuations are symmetric (in this approximation). The magnitude of the fluctuations, however, is determined by the second equation (8.9), which does contain the slope of ax. 

Exercise. Find the jump moments and the macroscopic equation for the decay process and for the Poisson process. 

Exercise. Prove the following theorem 510. The macroscopic equation is linear if and only if the function Q[y) = y— [Pg.127]

The same assumptions on which the macroscopic equation (2.2) is based lead to a definite form of the master equation for P, apart from a small but essential modification.In (2.1) the probability for a collision involving Sj molecules Xj is taken proportional to ri-j more precisely this factor should be... 

Exercise. Determine the first jump moment a[j)( n ) of rij. Write the macroscopic equation for itj. 

A much larger variety of phenomena can be described as stationary states of open chemical systems, i.e., systems in which molecules can be injected and from which molecules can be extracted. The simplest possibility for injection of molecules X is at a constant rate b. Extraction of X can only be done as long as there are some X present the rate of extraction must therefore vanish when n = 0 the simplest possible choice for it is an. Then the macroscopic equation for the number n of molecules X has the form... 

This equation is identified with the macroscopic equation of motion for the system, which is supposedly known. Thus the function A(y) is obtained from the knowledge of the macroscopic behavior. Subsequently one obtains B(y) by identifying (1.4) with the equilibrium distribution, which at least for closed physical systems is known from ordinary statistical mechanics. Thus the knowledge of the macroscopic law and of equilibrium statistical mechanics suffices to set up the Fokker-Planck equation and therefore to compute the fluctuations. 

Exercise. The rotation of an ellipsoidal particle suspended in a fluid obeys the macroscopic equation of motion... 

Write the deterministic macroscopic equations of motion of the system. 

One sees that, in contrast to the linear case, it is not true that the average obeys the deterministic macroscopic equation from which one started. In fact, (4.2) is... 

Internal noise is described by a master equation. When this equation cannot be solved exactly it is necessary to have a systematic approximation method - rather than the naive Fokker-Planck and Langevin approximations. Such a method will now be developed in the form of a power series expansion in a parameter Q. In lowest order it reproduces the macroscopic equation and thereby demonstrates how a deterministic equation emerges from the stochastic description. 

This is the macroscopic equation (1.2) and it is satisfied since we stipulated that for the function a macroscopic solution should be taken. 

Remark. The equation (3.1) is not quite identical with (V.8.6), which was previously called the macroscopic equation, inasmuch as the latter also includes terms of order Q l. The question which of the two is the correct one is moot, because terms of order Q l (even of order Q l/2) can be transferred from the macroscopic part of (3.2) into the fluctuating part. To put it differently the location of the peak P(X, t) is not defined more precisely than permitted by its width, which is of order Ql/2. Of course, one might agree to define the location as , or as the maximum of the peak, but there is no logical necessity for this and in the case of nonlinear processes it is awkward. 

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