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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?