Random-flight Chains Are Gaussian

Random-flight Chain Conformations Are Described by the Gaussian Distribution Function 

To be more quantitative, we first consider a one-dimensional random w alk, then we generalize to three dimensions. Each bond vector, because it is oriented randomly, can have a different projection onto the x-axis. We first compute the distribution of the number of forw ard and reverse steps, then we compute the average x-axis distance travelled per step. 

We want the probability, P(m,N) that the chain takes m steps in the +x direction out of N total steps, giving N -m steps in the -x direction. Just like the probability of getting m heads out of N coin flips, this probability is given approximately by the Gaussian distribution function. Equation (4.34), 

Now v e convert from the number of steps m to coordinate position x. The net forward progress x is a product of two factors (1) the number of forward minus reverse steps, which equals m- (N-m) = 2m-N (2) the average x-axis distance travelled per step. The average x-axis projection of the bond vectors is (r ) = 0 (see Equation (32.5)) because of the symmetry between forward and 

The product of these two factors is x = (2m - N)b/v 3. Since the average number offorward steps ism = N/2, you havex = 2(m-m )h/v3- Squaring both sides and rearranging gives 

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