Confusing velocity distribution result

[zn5252]

hello,
Please see attached excerpt of a solution to a problem. I do not understand why the result obtained is to second order ? (this is a diffusion equation they say)
Please let me know if you would need more info.
Thanks,

 

[HallsofIvy]

What does "at n- 1" mean?

 

[zn5252]

n-1 means at the step which is prior to step n where n is the number of boosts . I have attached the first paragraph for more details.

 

[haruspex]

It's because the first order differences will cancel out: f(x+dx) + f(x-dx) ≈ f(x) + f'(x)dx + f''(x)dx²/2 + f(x) - f'(x)dx + f''(x)dx²/2 = 2f(x) + f''(x)dx²

 

[zn5252]

I see. Thanks a lot !