# 固体物理

1. Poisson Distribution

In the Drude model the probability of an electron suffering a collision in any infinitesimal interval dt is just dt/τ.

(a) Show that an electron picked at random at a given moment had no collision during the

preceding t seconds with probability e (-t /τ). Show that it will have no collision during the next t seconds with the same probability.

The probability of an electron suffering a collision during the preceding t second is

/t t

p dt ττ

==?,

The probability of an electron picked at random at a given moment having no collision during the preceding t seconds, given by the Poisson distribution

()!

n e P n n λ

λ-=

,

Where n = 0,λ=p, as there is no collision happen;thus, the probability is

()0/00!

t e P e λ

τλ--=

=,

In the next t second

2/t t

t p dt ττ

'==?.

So it will have no collision during the next t seconds with the same probability.

(b) Show that the probability that the time interval between two successive collisions of an

electron falls in the range between t and t + dt is (dt/τ)e (-t/τ).

As we known, the probability of an electron suffering a collision in any infinitesimal interval dt is just dt/τ, and the probability that it has no collision during the preceding t seconds is e (-t /τ), so the probability that the time interval between two successive collisions of an electron falls in (t, t+dt ) is

()////t t P e dt dt e ττττ--=?=.

(c) Show as a consequence of (a) that at any moment the mean time back to the last collision

(or up to the next collision) averaged over all electrons is τ.

The probability of a collision in the time interval t is t/τ, so

/t λτ=,

And there is only one collision in the time interval t,

1λ=,

Hence

t τ=.

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