- #1
Kashmir
- 439
- 73
Reif,pg 14. ##n_1## is the number of steps to the right in a 1D random walk. ##N## are the total number of steps
"When ##N## is large, the binomial probability distribution ##W\left(n_{1}\right)##
##W\left(n_{1}\right)=\frac{N !}{n_{1} !\left(N-n_{1}\right) !} p^{n_{1}} q^{N-n_{1}}##
tends to exhibit a pronounced maximum at some value ##n_{1}=n_{1}##, and to decrease rapidly as one goes away from ##\tilde{n}_{1}##. Let us exploit this fact to find an approximate expression for ##W\left(n_{1}\right)## valid when ##N## is sufficiently large.
If ##N## is large and we consider regions near the maximum of ##W## where ##n_{1}## is also large, the fractional change in ##W## when ##n_{1}## changes by unity is relatively quite small, i.e.,
##\left|W\left(n_{1}+1\right)-W\left(n_{1}\right)\right| \ll W\left(n_{1}\right)##
Thus ##W## can, to good approximation, **be considered as a continuous function of the continuous variable ##n_{1}##**, although only integral values of ##n_{1}## are of physical relevance. The location ##n_{1}=\tilde{n}## of the maximum of ##W## is then approximately determined by the condition ##\frac{d W}{d n_{1}}=0 \quad##"
* I'm not able to understand why
##
\left|W\left(n_{1}+1\right)-W\left(n_{1}\right)\right| \ll W\left(n_{1}\right)
##means we can use a continuous approximation.
* How do we approximate a discrete function by a continuous one. Since ##W## has values defined only at integral values, what values do we assign to the continuous function between any two consecutive integers i.e what value does ##W(0.5)## have in the continuous approximation?
"When ##N## is large, the binomial probability distribution ##W\left(n_{1}\right)##
##W\left(n_{1}\right)=\frac{N !}{n_{1} !\left(N-n_{1}\right) !} p^{n_{1}} q^{N-n_{1}}##
tends to exhibit a pronounced maximum at some value ##n_{1}=n_{1}##, and to decrease rapidly as one goes away from ##\tilde{n}_{1}##. Let us exploit this fact to find an approximate expression for ##W\left(n_{1}\right)## valid when ##N## is sufficiently large.
If ##N## is large and we consider regions near the maximum of ##W## where ##n_{1}## is also large, the fractional change in ##W## when ##n_{1}## changes by unity is relatively quite small, i.e.,
##\left|W\left(n_{1}+1\right)-W\left(n_{1}\right)\right| \ll W\left(n_{1}\right)##
Thus ##W## can, to good approximation, **be considered as a continuous function of the continuous variable ##n_{1}##**, although only integral values of ##n_{1}## are of physical relevance. The location ##n_{1}=\tilde{n}## of the maximum of ##W## is then approximately determined by the condition ##\frac{d W}{d n_{1}}=0 \quad##"
* I'm not able to understand why
##
\left|W\left(n_{1}+1\right)-W\left(n_{1}\right)\right| \ll W\left(n_{1}\right)
##means we can use a continuous approximation.
* How do we approximate a discrete function by a continuous one. Since ##W## has values defined only at integral values, what values do we assign to the continuous function between any two consecutive integers i.e what value does ##W(0.5)## have in the continuous approximation?