# Changing Summation to Integral

This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However, I don't see how. Can anyone justify this change?

• What is the context? What does $W$, $n_1$ means and why do we sum? I guess that in general, if you approximate a summation variable as being quasicontinuous, you can always approximate by an integral. – fgoudra Mar 22 at 9:59
• Here W(n) refers to probability of given problem(The random walk by drunken man). My question is how can you approximate the quasicontinuous variables which is W above by integrals? Can you justify it? – Abhi7731756 Mar 22 at 10:07
• Well usually one does this by saying that the $n_1$ are "really close to each other" such that changing the summation by an integral introduces a negligible error. It's an approximation, thus it's justified depending on how large would you allow the error to be. – fgoudra Mar 22 at 12:33
• I will apply mean value theorem. If we apply this I get $W_{n}\delta{n}$. Now,how do I tell that this and above sum yield approximately same result? – Abhi7731756 Mar 22 at 13:58

Say we're talking about the normalization of a probability mass function: $$1= \sum_{n=0}^N W(n) \,.$$ You first note that $$n$$ is a counting number, so its increments are by 1. Namely, $${\rm d}n = 1$$. So you can just insert one: $$1 = \sum_{n=0}^N {\rm d}n W(n) \,.$$ Then for some reason, you might be interested in the variable $$r=n/N$$. The ratio of rightward steps or something. You rewrite: $$1 = \sum_{n=0}^N \frac{{\rm d}n} {N} N W(n) \,.$$ Now you realize that as $$N\to\infty$$, the quantity $${\rm d}n/N$$ approaches an infinitesimal $${\rm d}r$$ . Now, thanks to the infinitesimal element, the summation looks like a Riemann sum, allowing the passage to an integral $$1= \sum_{n=0}^N \frac{{\rm d}n} {N} N W(n) \to \int_0^\infty {\rm d} r \, N W(Nr) =1 \,.$$ Then, one proclaims that a new function, say, $$P(r)$$ has to be the probability density for the continuous variable $$r=n/N$$ such that $$P(r) = N W(N r) \,.$$
From what I can see, Reif's treatment is not emphasising that the new variable is proportional to $$1/N$$, and also failing to identify that the probability density of the new variable is not the same $$(W)$$ as that of the old variable, even though they are very closely connected (equation above).