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PHYS 420 The Poisson distribution

Creation date：2024-06-03 15:41:18

Belonging category：物理Physical

Tag 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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LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 FINC5090 MATH1052 SciComp Elec4622 FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 MAT 237 BISM2201 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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum

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