The continuous limit

We start with a discrete link model and we try to evaluate a correlation function or, in other words, the mean value of an observable. The grand 

We approach the continuous limit by increasing the number of links, but in such a way as to ensure that the size of non-interacting chains remains constant. However, as the number of degrees of freedom increases, the value of the partition function becomes infinite even in the absence of interaction. For periodical boundary conditions and a large volume, we have 

on condition that regularizations exist (they are recalled to mind by the symbol + ), the new partition functions have continuous limits. In the same way, the fugacities will be renormalized by setting 

The grand partition functions remain unchanged, but they must be re-expressed 

Moreover, in the continuous case, we shall introduce the following connected partition functions 

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By inserting the solutions proposed in Eq. (6.189) and condition (6.175) in Eq. (6.185), recurrent expressions for coefficients 8lp) and are deduced [68] and by inserting these expressions into (6.191) the current is calculated. These expressions allow us to obtain limiting cases like the reversible and irreversible ones which have a discrete character which makes them applicable to any multipulse technique by simply changing the potential time waveform, including the continuous limit of Cyclic Voltammetry. Moreover, they are independent of the kinetic formalist considered for the process. 

Using the standard master equation technique, we consider the continuous limit, supposing that w(x) and v(x) vary slightly from one bond to another. Then,... 

In the general case, a numerical solution of (4.10) is required. Here, we shall focus on two cases N — 2, and the continuous limit N - oo. 

For An> T we have two states with the molecular decay rate rj2. For An< r we have two states with the same real energy (Rez1 0), but with different decay rates (superradiant y > r j2, subradiant y < r/2). We find a sudden qualitative change in behavior for the system for A = T the time decay passes from biexponential for An> T to a decrease with oscillating beats for A < T.153 This transition is not a special feature of the N = 2 case, but even survives in the continuous limit, as we shall see now. 

The continuous limit of a simple random walk model leads to a stochastic dynamic equation, first discussed in physics in the context of diffusion by Paul Langevin. The random force in the Langevin equation [44], for a simple dichotomous process with memory, leads to a diffusion variable that scales in time and has a Gaussian probability density. A long-time memory in such a random force is shown to produce a non-Gaussian probability density for the system response, but one that still scales. 

To derive the macroscopic transport equations, the conservation Relation [10] and [11] must be converted to differential equations. The main assumption needed is that the mean density and the mean velocity vary slowly in space and in time. Starting from Eq. [10] and [11], the macro dynamic equations describing the large-scale behavior of the lattice gas are obtained by multiple-scale perturbation expansion technique (Frisch et al., 1987). We shall not derive this formalism here. In the continuous limit, Eq. [10] leads to the macro dynamical conservation of mass or Euler equation... 

We may now go to the continuous limit, which is called a Brownian chain. This limit is obtained by reducing the size of the links and by increasing their number so as to keep %,r2y fixed. 

The area s can thus be used as a curvilinear coordinate to locate a point on the continuous curve. Therefore, the configuration of a Brownian chain is defined by the vectorial function r(s) (where r(s) is the continuous limit of r ). 

Let us note that, here, we introduce a weight and not a probability distribution, in order to move more easily to the continuous limit. The weight defined above has a continuous limit on the contrary the normalization constant has no limit when the number of variables becomes infinite. Thus, the probability distribution corresponding to the weight has no limit. 

As we did for the chain with independent links, we can eliminate the chemical microstructure by going to the continuous limit. The generalization is trivial. Let us set... 

Let us now go to the continuous limit by letting the number of links become infinite. In this case, again, SN becomes infinite. However, by choosing a proper length scale (independent of N), it is always possible so to arrange that the coefficient of N in (2.3.17) vanishes it is sufficient for that to put k = jjl in the above transformation. It is then possible to compare the entropy of two chains characterized respectively by the courses S and S0. We find... 

The passage to the continuous limit is also trivial and gives... 

However, it is more interesting to go directly to the continuous limit. Setting... 

We can now consider the continuous limit. In this case, the length / which appears in (8.1.70) becomes very small. Consequently, one must look at large values of x, i.e. small values of k. 

The non-realistic character of the model becomes more conspicuous in the continuous limit. Then, to any chain configuration defined by the vector r(s) (0 < s < S), we attribute the weight... 

Therefore, we obtain the so-called Rouse equation in the continuous limit. 

The formula in the functional integral is obtained by taking the continuous limit of the discrete variables. For oample, as a generalization of eqn (2.1.16), we have... 

As in the case of the Gaussian chain, the suffix n in the Rouse model can be regarded as a continuous variable. In the continuous limit, eqn (4.6) is rewritten as (see the transformation rule given in Table 2.1, Section 2.2)... 

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