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Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬
Exercise 1 – Mathematics and the Model Spin System
1.
The birthday problem
a. In a room containing N people, calculate the probability that at least 2 people were
born on the same date (day and month, not year). Hint: calculate the probability
that no pair of people was born on the same day. What is the minimal number in
order to get a probability of 50%? Of 90%?
b. What is the average value of a throw-of-a-dice experiment? What is the variance?
And the standard deviation?
2.
Weather statistics
Visit the website of the Israeli meteorological service at the following address
http://www.ims.gov.il/IMS/tazpiot/HourObservations/. Click on the link for hourly
observations, where you can get the hourly temperature at a number of locations (The hour
can be changed above the map). You are requested to perform the following calculations
over a single day (please state the date) between 07:00 and 16:00.
a. What is the average temperature in the following cities: Tel Aviv, Eilat and Haifa?
b. What is the variance and standard deviation for each of the cities?
c. Given the correlation function between two cities G (a, b) 
 Ta Tb
, calculate
 a b
the correlations between Eilat and TA, Eilat and Haifa, TA and Haifa. Describe the
meaning of the correlation function
d. Now go to the stock-market site in http://www.tase.co.il/TASEEng/Homepage.htm
and click on the TA25 index. Go to "Chart Data" and obtain the values of the index
that relate (or are closest) to the same times as your weather data. Calculate the
correlation between the weather in Tel-Aviv and the value of the TA25 index. What
can you conclude?
Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
3.
‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬
Estimating Pi
In the following problem we will develop a probabilistic way to calculate the value of π, and
conduct an experiment to verify it. Consider a sheet of paper with many horizontal, parallel
lines drawn on it. The lines are at distances D from each other. Consider also a needle of
length l, with l < D.
a. For a given angle  , where    0,   , at what distance must the central point of
b.
c.
d.
e.
4.
the needle be from one of the lines in order to cross it?
What, then, is the probability that the needle that was placed at the given angle
crosses a line?
Assuming needles are thrown randomly on the sheet of paper, find the probability
that a needle will cross a line
Now, use the result obtained in (c) to evaluate π experimentally. Please attach your
results in an organized table and state N, the number of times the experiment was
conducted
Bonus: estimate theoretically how many experiments (N) have to be performed to
reach an accuracy ε in the determination of π. Is the deviation of your results from
real value of π consistent with this calculation?
Gaussian approximation of the multiplicity function
a. In order to get a better "feel" of the Stirling approximation, plot on the same graph
(using different colors or patterns) the following functions: ln(n!) , n ln(n)  n ,
(n  12 ) ln(n)  n  ln(2 ) / 2 for n between 0 and 5, 0 and 20, 0 and 100 (3 graphs
altogether)
b. For N=50, Plot on the same graph the function g (m) 
(
N m
2
N!
and its
)!( N 2 m )!
Gaussian approximation
c. What is the probability to get m=8 according to the precise and approximated
distribution functions (for the approximated choose a range close to 8)? What is the
probability to get 0<m<8 according to each distribution?