Note Stirlings Approximation

We shall have need for an analytic representation of a factorial (for example, a ) in terms of known functions. For these purposes we can avail 

P(xi)P(x2) dxi dx2 gives the chances that Xi lies between Xi and Xi + dxi and that simultaneously Xi lies between X2 and X2 + dx2. The chances that z lies between Zi and Zi + dzi are then equal to this product integrated over all values of xi( ) when is equal to z — Xi and X2 dx2 — z — Xi + dz (that is, dx2 = dz)  

The relative accuracy of these forms can be seen in Table VI.3. 

For chemical problems in which x is likely to be of the order of magnitude of Avogadro s number, a quite good approximation to ln(x ) is taken as 

While the absolute error in a now increases with increasing Xy the relative error decreases because x increases faster than In (27ro ). 

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It is also instructive to start from the expression for entropy S = log(g(A( m)) for a specific energy partition between the two-state system and the reservoir. Using the result for g N, m) in section A2.2.2. and noting that E = one gets (using the Stirling approximation A (2kN)2N e ). 

When later we replace a< by the Stirling approximation, we will be following the generally accepted procedure of Boltzmann. Requirement (b) of note 119 will be especially important in that case. 

Here it must be noted that we have made an imprecise description of what under more formal terms is known as the Stirling approximation. Within the context of our characterization of the configurational entropy, the Stirling approximation of In N allows us to rewrite the entropy as... 

However, for N of the order 10 , the first approximation is sufficient. It is also useful to note that, within the Stirling approximation, we have... 

For small N we easily find that A 5 7 0. So what is wrong First we note that in a macroscopic system N is large, i.e. V 10 Computing AS for large N requires that we use the Stirling approximation Eq. (4.97). The result is... 

By application of Stirling s approximation for large numbers (note that Nq = 6.022 X 10 3) ... 

Furthermore, suppose that we are interested in using as few bits as possible to represent the collection of subsets used i, 2, The information-theoretic bound on the number of bits needed is [log ( )], which is roughly m log w/m, using Stirling s approximation. (Note that when m /w this represents a factor 2 compression compared to storing ii, 2) m explicitly.) However we are interested in a succinct representation of the collection that allows efficient lookup in this list. It turns out that with an additive factor of 0(m -h log log w) bits it is possible to support an 0(1) lookup, see [10, 38] the results they provide are even slightly better, but this bound is relatively simple to achieve. 

The solution to the fractional diffusion equation is clearly dependent on fluctuations that have occurred in the remote past note the time lag k in the index on the fluctuations and the fact that it can be arbitrarily large. The extent of the influence of these distant fluctuations on the system response is determined by the relative size of the coefficients in the series. Using Stirling s approximation on the gamma functions determines the size of the coefficients in Eq. (25) as the fluctuations recede into the past, that is, as k — oo we obtain... 

The use of Stirling s approximation assumes a sufficiently large value of Njj. This assumption fails at high degrees of crosslinking and high extensions. The failure leads to non-Gaussian behavior. Note the similarity of the last term of this sum and that evaluated in the... 

Note that, in addition to those discussed, the statistical effect of large numbers, of particles TV, or of available energetic (sub) levels g. - it manifested, beyond the thermod5mamic considerations, also analytically by the systematic application of Stirling s approximation (see Appendix A.2)... 

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