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PHYS 420 The Poisson distribution

创建日期：2024-06-03 15:41:18

所属分类：物理Physical

标签： PHYS 420

The Poisson distribution

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PHYS 420

1 The Poisson distribution (15 points)
The classical example of a Poisson process is radioactive decay. Observing a piece of ra-
dioactive material over a time interval of T shows that:
• The probability of one and only one decay event in a time interval [t, t+ dt] is γdt.
• The probabilities of events at different intervals are independent of each other.
Poisson distribution describes the probability of observing exactly n decay events in the
time interval T . In this problem, we will derive this important distribution function.
The Poisson distribution can be viewed as a limiting case of the binomial distribution
that we discussed in class. Consider the an radioactive experiment has been running for a
time interval T . We can divide the time interval into N small intervals each lasts ∆t = T/N .
Assume ∆t is small enough such that the probability of having 2 decay events in interval ∆t
is negligible. The probability of having one and only one decay in time ∆t is p = γ∆t.
(a) Use the binomial distribution to write down the probability, P , of having n radioactive
decays in the experiment. Express this probability P as function of p, n,N .
(b) Let us define λ = Np. (Notice λ is also equal to γT .) First, write the probability
function P above in terms of λ, n,N . Now let us consider the limit where λ and n are kept
fixed, but N is sent to infinity. Show that now the probability distribution becomes
P (n, λ) =
λn
n!
e−λ, (1)
which is the Poisson distribution.
(c) Show that the Poisson distribution above is properly normalized.
(d) Calculate 〈n〉, 〈n2〉, and the standard deviation √〈n2〉 − 〈n〉2.
(e) An application for the Poisson distribution. Assuming that the stars in the
galaxy is randomly distributed with an average density ns, sitting at a given point, what is
1
the probability that the nearest star is in a distance range between R and R + dR? You
can assume dR is sufficiently small, therefore, only keep the leading term in dR. (Hint: the
probability should be a product of the probability of finding no stars in the ball of radius R
and the probability of finding precisely one star in the thin spherical shell of radius R.)
2 Textbook chapter 2 problem 1 (5 points)
Suppose g(U) = CU3N/2, where C is a constant and N is the number of particles. This form
of g(U) actually applies to an ideal gas.
(a) Show that U = 3
2
Nτ .
(b) Show that
(
∂2σ
∂U2
)
N
is negative.
3 Negative temperature (10 points)
In this exercise we will learn that the temperature of a system can actually be negative. Sur-
prisingly, the negative temperature state is actually “hotter” than the positive temperature
states.
Consider a closed system of N spins. Each spin has two states, namely ↑ and ↓. The
energy associated with the two states are + and − respectively.
(a) Consider in a macrostate with n+ spin up and n− spin down. Define s = n+ − n−.
The total energy for such states are E = s. What is the multiplicity for this macrostates?
(b) Write down the entropy σ as a function of the energy E. Use the definition of
temperature, namely 1
τ
= ∂σ
∂E
, to show that the τ as a function of the energy E is given by
τ(E) =
2
log(N− E)− log(N+ E) (2)
(Remember to use the Stirling approximation logN ! ∼= N logN −N).
(c) Find the approximate form of τ at small E. Plot the τ(E) near E = 0.
(d) Consider two copies of the system. One of them is prepared such that the temperature
is at τ0 > 0. The other one is at temperature −τ0 < 0. What is the equilibrium temperature
when we put the two systems in thermal contact? What is the direction of the energy flow
during this process? Based on this, which system is “hotter” by our conventional definition?

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