Basics Of Statistical Physics 3rd Edition Harald J W Muller-Kirsten - Solutions

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When the Yankelovich polling organization asked,6“Should laws be passed to eliminate all possibilities of special interests giving huge sums of money to candidates?”80% of the sample answered yes. When they posed the question “Should laws be passed to prohibit interest groups from
With quota sampling a researcher stands at a street corner and conducts interviews until obtaining a quota representing the relative sizes of various groups in the population. For instance, the quota might be 50 factory workers, 100 housewives, 60 elderly people, and 30 blacks. Is this a
You are directing a study to determine the factors that relate to good academic performance at your school.(a) Describe how you might select a sample of 100 students for the study.(b) List some variables that you would measure. For each, provide the scale you would use to measure it, and indicate
Repeat the previous exercise for the data file created in Exercise 1.12 (page 10).
Refer to the Students data file introduced in Exercise 1.11 (page 9). For each variable in the data set, indicate whether it is(a) Categorical or quantitative.(b) Nominal, ordinal, or interval.
Clusters versus strata:(a) With a (one-stage) cluster random sample, do you take a sample of (i) the clusters? (ii) the subjects within every cluster?(b) With a stratified random sample, do you take a sample of (i) the strata? (ii) the subjects within every stratum?(c) Summarize the main
You plan to sample from the 32,008 undergraduate students enrolled at the University of Florida, to compare the proportions of black and white students who believe that racism is still a serious problem in the United States.(a) Suppose that you use random numbers to select students, but you stop
You plan to sample from the 5000 students at a college, to compare the proportions of men and women who believe that the legal age for drinking alcohol should be changed to 18. Explain how you would proceed, if you want a systematic random sample of 100 students.
Refer to Exercise 2.12 about selecting 5 of 52,000 names on 400 pages of a directory.(a) Select five numbers to identify subjects for a systematic random sample of five names from the directory.(b) Is cluster sampling applicable? How could it be carried out, and what would be the advantages and
A Roper Poll was designed to determine the percentage of Americans who express some doubt that the Nazi Holocaust occurred. In response to the question“Does it seem possible or does it seem impossible to you that the Nazi extermination of the Jews never happened?”22% said it was possible the
Explain how the following had sampling bias, and explain what it means to call their samples “volunteer samples.”(a) The BBC in Britain requested viewers to call the network and indicate their favorite poem. Of more than 7500 callers, more than twice as many voted for Rudyard Kipling’s
Table 2.2 shows the result of the 2012 Presidential election and the predictions of several organizations in the days before the election. The sample sizes were typically about 2000. (The percentages for each poll do not sum to 100 because of voters reporting as undecided or favoring another
A study is planned about whether passive smoking(being exposed to secondhand cigarette smoke on a regular basis) leads to higher rates of lung cancer.(a) One possible study would take a sample of children, randomly select half of them for placement in an environment where they are passive smokers,
Explain whether an experiment or an observational study would be more appropriate to investigate the following:(a) Whether or not cities with higher unemployment rates tend to have higher crime rates.(b) Whether a HondaAccord or a Toyota Camry gets better gas mileage.(c) Whether or not higher
A local telephone directory has 400 pages with 130 names per page, a total of 52,000 names. Explain how you could choose a simple random sample of five names. Show how to select five random numbers to identify subjects for the sample.
A class has 50 students. Use software to select a simple random sample of three students. If the students are numbered 01 to 50, what are the numbers of the three students selected? Show how you used software to do this.
Which of the following variables are continuous when the measurements are as fine as possible?(a) Age of mother(b) number of children in family(c) income of spouse(d) population of cities(e) latitude and longitude of cities(f) distance of home from place of employment(g) number of foreign languages
Which of the following variables could theoretically be measured on a continuous scale?(a) Method of contraception used(b) length of time of residence in a state(c) task completion time(d) intelligence(e) authoritarianism(f) alienation(g) county of residence.
A survey asks subjects to rate three issues according to their importance in determining voting intention for a U.S. senator, using the scale (very important, somewhat important, unimportant). The issues are foreign policy, the economy, and the environment. The evaluations can be treated as three
A poll conducted by YouGov (yougov.com) for the British newspaper The Daily Telegraph in 2006 asked a random sample of 1962 British adults, “How would you rate George W. Bush as a world leader?” The possible choices were (great, reasonably satisfactory, pretty poor, terrible).(a) Is this
Give an example of a variable that is(a) categorical(b) quantitative(c) ordinal scale(d) nominal scale(e) discrete(f) continuous(g) quantitative and discrete.
Which scale of measurement is most appropriate for“attained education” measured as(a) Number of years (0, 1, 2, 3, . . .)?(b) Grade level (elementary school, middle school, high school, college, graduate school)?(c) School type (public school, private school, home schooling)?
Which scale of measurement is most appropriate for(a) Occupation (plumber, teacher, secretary, . . .)?(b) Occupational status (blue collar, white collar)?(c) Social status (lower, middle, upper class)?(d) Statewide murder rate (number of murders per 1000 population)?(e) County population size
Which scale of measurement (nominal, ordinal, or interval)is most appropriate for(a) Attitude toward legalization of marijuana (favor, neutral, oppose)?(b) Gender (male, female)?(c) Number of children in family (0, 1, 2, . . .)?(d) Political party affiliation (Democrat, Republican, Independent)?(e)
Identify each variable as categorical or quantitative:(a) Number of pets in family.(b) County of residence.(c) Choice of auto (domestic or import).(d) Distance (in miles) commuted to work.(e) Choice of diet (vegetarian, nonvegetarian).(f) Time spent in previous month browsing the World Wide Web.(g)
Explain the difference between(a) Discrete and continuous variables.(b) Categorical and quantitative variables.(c) Nominal and ordinal variables.Why do these distinctions matter for statistical analysis?
Your school library probably has electronic access to many research journals. Select a journal from an area of interest to you, such as American Sociological Review, Political Analysis, Annual Review of Psychology, Annual Review of Anthropology, Journal of Sports Sciences, or New England Journal of
At the homepage www.gallup.com for the Gallup Poll, from information listed or linked, give an example of(a) a descriptive statistical analysis, (b) an inferential statistical analysis.
In a recent survey by Eurobarometer of Europeans about energy issues and global warming,3 the percentage who agreed that environmental issues have a direct effect on their daily life varied among countries between 56%(in Denmark) and 95% (in Cyprus). Of the 1306 subjects interviewed in the United
A sociologist wants to estimate the average age at marriage for women in New England in the early eighteenth century. She finds within her state archives marriage records for a large Puritan village for the years 1700–1730. She then takes a sample of those records, noting the age of the bride for
You have data for a population, from a census. Explain why descriptive statistics are helpful but inferential statistics are not needed.
Illustrating with an example, explain the difference between(a) a statistic and a parameter.(b) description and inference as two purposes for using statistical methods.
For the statistical software your instructor uses for your course, find out how to access the software, enter data, and print any data files that you create. Create a data file using the data in Figure 1.1 on page 7, and print it.
Using statistical software or a spreadsheet program(such as Microsoft Excel), your instructor will help the class create a data file consisting of the values for class members of characteristics such as those in the previous exercise. One exercise in each chapter will use this data file.(a) Copy
The Students data file at the text websites www.pearsonhighered.com/mathstatresources and www.stat.ufl.edu/~aa/smss/data shows responses of a class of social science graduate students at the University of Florida to a questionnaire that asked about GENDER (1 = female, 0 = male), AGE, HSGPA= high
With the software used in your course, construct a data file for the criminal behavior of five inmates in a local prison. The characteristics measured (with observations for the five subjects) were race (white, black, white, Hispanic, white), age (19, 23, 38, 20, 41), length of sentence in years
In October 2012, a poll (see www.worldpublicopinion.org) in 21 countries that asked whether people favored Barack Obama or Mitt Romney in the U.S. Presidential election stated that Obama was clearly preferred. Of the sample from France, 72% preferred Obama, 2% preferred Romney, with the rest
The Current Population Survey (CPS) is a monthly survey of households conducted by the U.S. Census Bureau.A CPS of 68,000 households in 2013 indicated that of those households, 9.6% of the whites, 27.2% of the blacks, 23.5% of the Hispanics, and 10.5% of the Asians had annual income below the
At the General Social Survey website, sda.berkeley.edu/GSS, by entering HEAVEN in the Row box, you can find the percentages of people who said yes, definitely; yes, probably; no, probably not; and no, definitely not when asked whether they believed in heaven.(a) Report the percentage who gave one
Go to the General Social Survey website, sda.berkeley.edu/GSS. By entering TVHOURS in the Row box, find a summary of responses to the question“On a typical day, about how many hours do you personally watch television?”(a) What was the most common response?(b) Is your answer in (a) a descriptive
A General Social Survey asked subjects whether astrology—the study of star signs—has some scientific truth. Of 1245 sampled subjects, 651 responded definitely or probably true, and 594 responded definitely or probably not true. The proportion responding definitely or probably true was 651/1245
For several years, the Gallup Poll has asked, “Do you think abortions should be legal under any circumstances, legal only under certain circumstances, or illegal in all circumstances?”The poll reported in May 2014, based on a sample of 1028 adults, that the percentages for these three responses
The student government at the University of Wisconsin in Madison conducts a study about alcohol abuse among students. One hundred of the 43,193 members of the student body are sampled and asked to complete a questionnaire. One question asked is, “On how many days in the past week did you consume
In the 2014 gubernatorial election in California, a CBS News exit poll of 1824 voters stated that 60.5% voted for the Democratic candidate, Jerry Brown. Of all 7.3 million voters, 60.0% voted for Brown.(a) What was the population and what was the sample?(b) Identify a statistic and a parameter.
The Environmental Protection Agency (EPA) uses a few new automobiles of each brand every year to collect data on pollution emission and gasoline mileage performance.For the Toyota Prius brand, identify the (a) subject,(b) sample, (c) population.
Example 7.7: Fermi–Dirac distribution from detailed balancing Assuming Pauli’s principle, energy conservation and the principle of microscopic reversibility, also known as the principle of detailed balancing, (i.e. that the number of reactions 1 + 2 → 3 + 4 is the same as the number of
Example 7.8: Bose–Einstein distribution from detailed balancing Obtain the Bose–Einstein distribution function by repeating the calculations of Example 7.7 but without the Pauli principle.
Example 7.9: Energy of an ideal gas in a spherical volume V
What is the relation corresponding to Eq. (7.63), i.e. PV = 2E/3, in the case of an ideal gas enclosed in a spherical volume of radius R? [Hint: The radial part of the Schr¨odinger equation is solved in terms of spherical Bessel functions. The zeros of these functions are known and can be looked
Example 9.6: Estimate of Einstein temperature Consider a solid consisting of a cubic lattice with lattice constant a (shortest distance between neighbouring lattice points), and assume a spring-like interaction between neighbouring atoms at lattice points. Show that the compressibility κ of the
Example 9.7: Electron contribution to capacity of heat In the case of some substances like O2 and NO there are additional contributions to thermodynamical properties resulting from two closely lying electronic levels, say 1, 2 with 2 − 1 kT, so that thermal excitation suffices in order to
Example 9.8: Chemical potential derived from free energy What is the free energy F of an Einstein solid of 3N one-dimensional oscillators, each of natural frequency ν, if the interaction between them lowers F by the amount AN,A = const. From this derive the chemical potential 0 of the solid. What
Example 9.9: Bose–Einstein integrals and Riemann zeta functions ζ(x)Show that¶¶∞0 x2e−x2 1 − e−x2 dx =1 4√πζ3 2, ζ3 2= 2.612. (9.43)More generally ∞0 xn ex − 1 dx = n! ζ(n + 1). (9.44)Thus Bose–Einstein integrals can be related to Riemann zeta functions.[Hint: Use the
Example 9.10: Entropy of free massless scalar fields Show that the entropy S of a free scalar field in (a) 1+1 dimensions (length L), and in (b) 1+3 dimensions (volume V ) is given by(a) S =1 3πLT, and (b) S =2π2 Vζ(4)T3, (9.45)‡‡F. Reif [59], Ex. 7.12.§§From P.T. Landsberg [38], problem
Example 9.11: Debye frequency and velocity of sound Assuming one ion in a cube of volume (d/2)3 (d being the distance between neighbouring ions)and ut = ul = u, show thatωD 8u d. (9.47)
Example 9.12: Debye frequency in a superconductor Considering the cubic metal lattice of a superconductor with neighbouring ion separation d at temperature T = 0, show that the Debye frequency is given byωD =2√2(meMaBd3)1/2 2√2√meM√nd2, (9.48)where me is the mass of the electron, M that
Example 9.13: Debye frequency evaluation Evaluate the Debye frequency of Example 9.12 for mercury (atomic mass 0.331 × 10−24 kg) with d = n Bohr radii = n × 0.529 × 10−10 meters.(Answer: 1.95 × 1014/n3/2 sec−1).
What is the equation of state of a gas of noninteracting bosonic particles?
Example 10.11: Mean energy of electrons How does the energy of electrons in a metal vary with volume at constant density N/V , and how with N at constant volume? Can one obtain a sensible value of a mean energy of an electron as in the case of an ideal gas? What is the explanation? [Hint: Recall
Example 10.12: Temperature dependence of entropy Without detailed calculations explain: (a) How does the entropy of electrons in a metal vary with temperature, and (b) how does the entropy of the radiation in a cavity vary at high temperatures?
Example 10.13: Fermi gas in a magnetic field Consider a perfect Fermi gas consisting of spin-1/2 particles in a constant magnetic field B in a volume V . The energy of a particle is=p2 2m − 2μBBsz, sz = ±1 2. (10.100)Calculate at least to leading order the energy E of the gas and evaluate N =i
Example 10.14: Pressure of a Fermi gas
In Example 7.3 it was shown that in all 3 statistics PV = 2E/3 for a perfect gas of massive particles with energy = p2/2m. What is in the limit T → 0 the value of E/N? Hence show that in the case of a nonrelativistic perfect Fermi gas the pressure P in the limit T → 0 is P =2 50. (10.101)
How does this differ from the case of a Bose–Einstein gas? What are the implications of this result for solids and fluids?
Example 10.15: Fermi momentum Show that at temperature T 0 the momentum of the fastest electrons on top of the Fermi distribution is pFdπ, vFmdπ, (10.102)where d is the lattice distance.¶¶ Assuming d = n Bohr radii, show that the corresponding velocity is vF = 7× 106/n meters/second.Example
Example 10.17: An electron pair at the Fermi surface The wave function of the relative motion of a pair of free electrons of mass m and moving against each other with relative momentum p as for a Cooper pair on or very near the Fermi surface in a large sphere of radius R is given byψ(p) =2
Show that the number of S-states in an energy interval d near the Fermi surface is dn =mR 2πpFd . (10.105)
Example 10.18: The electron motion in a superconductor In superconductivity the motion of an electron on top of the Fermi distribution, i.e. at T = 0, attracts the positive ions of the lattice (here taken with charge +1 and lattice separationd) towards its path over a distance l and breadth δ
d, (10.106)where M is the mass of the ion, and me that of the electron. [84] (Note the difference between this displacement of the ions and their vibration).[Hint: Start from p = M(δωD), p = ion momentum e2/ved, ve = electron velocity, and l =ve/ωD].
Example 11.6: Bose–Einstein distribution Find by the method of maximum probability (i.e. parallel to the method of Example 11.3) the distribution function for a Bose–Einstein system placed in a heat bath.
Example 11.7: The ergodic hypothesis and the linear harmonic oscillator The classical ergodic hypothesis claims that the phase space trajectory (traced by (q(t), p(t))) of a system passes through each point of the hyperplane E = H(q, p). Show that the classical linear harmonic oscillator with H(q,
Example 11.8: Mean pressure in extreme relativistic case Consider a perfect quantum gas in a volume V and determine the relation between its mean energy E and the mean pressure P in both the nonrelativistic and extreme relativistic cases using P = kT∂ ln Z/∂V,Z =i exp(−Ei/kT) of Eq. (11.40).
Example 11.9: Difference between entropies Boxes A and B are separated by a common partition with a tiny hole. With an explosive device A is filled with photons which then gradually escape through the hole into B. Sketch roughly the behaviour of the difference of coarse grained and fine grained
Example 11.10: Entropy as measure of entanglement The von Neumann entropy S is given by the relation S = −kTrρ ln ρ = −kΣiρi ln ρi, (11.78)and provides a measure of the departure from a pure state (in which case S = 0). An eigenvalue(i.e. diagonalized element) ρi describes the probability
11.8 Problems without Worked Solutions 175 matrix is in state i. What is (a) the von Neumann entropy in the case of the diagonalized density matrixρ =1 21 0 0 1
(11.79)and (b) the number of states in the ensemble? (Answers: (a) k ln 2 = 0.69, (b) 2).
Example 11.11: Density matrix of a pure state Show by diagonalization¶¶ that the matrixρ =1 21 1 1 1(11.80)represents a pure state (i.e. has eigenvalues 1, 0).
Example 13.2: Screening effect of the electron cloud Consider the scattering of an electron by an ionized impurity atom in a semiconductor (see Sec. 13.3.4, case (ii)). Assume an exponential screening (due to the electron cloud) of the Coulomb potential and calculate the relaxation time τ.[Hint:
13.5 Problems without Worked Solutions 229 Example 13.3: Maxwell–Boltzmann distribution Show that in the presence of an external potential U(r) the stationary solution of the Boltzmann equation is the Maxwell–Boltzmann distribution.§§
Example 13.4: The insulator versus superconductor Why is an insulator not a perfect superconductor?[Hint: Recall Eq. (13.23). If the conduction band is full, the velocity of the electrons at the upper edge is zero. No current is induced by a magnetic field in a completely filled conduction band.
Example 2.9: Maxwell averaging of 1/c Using the Maxwell distribution for a perfect monatomic gas in thermal equilibrium obtain the average of 1/c, i.e. (1/c), where c is the speed of a molecule. How does the result compare with 1/c?
Example 2.10: Probability of one particle picked out from among N Show that the probability for one particle of a total of N noninteracting particles in a volume V to have a velocity component u1 in the interval (u1, u1 +du1) is given by the Maxwell distribution law Πu1du1. Start from the
Example 2.11: Standard deviation σConsider the probability density Πu of Eq. (2.50a) and the average |u| of Eq. (2.50b). Evaluate correspondingly(u − |u|)2 =∞−∞(u − |u|)2Πudu = σ2 (2.69)(the quantity σ being called standard deviation). Show that generally:(u − |u|)2i =
Example 2.12: Mean energy of effusing molecules A vapour of molecules of mass m is held at temperature T in some vessel and is allowed to escape through a small hole. What is the mean energy of the molecules emitted from the hole (or through unit area)? (Answer: kT/2 multiplied by the particle
Example 3.8: System of linear oscillators Show that for a system of one-dimensional linear harmonic oscillators with coordinate q given by q = Acos(ωt + θ), the phase θ in the range 0 ≤ θ ≤ 2π, the a priori probability P(q)dq is proportional to dq/A2 − q2, and when normalized to 1 as a
Example 3.9: Throwing a die How many times must a die be thrown on the average to produce the number 6? [Hint: The probability that in n trials the six appears once and no six n − 1 times is P(n) = (1/6)(5/6)n−1].The answer is ˜n = 6.
Example 3.10: Time independence of Jacobi determinant Prove Liouville’s theorem by verification of the time independence of the Jacobi determinant J under infinitesimal contact transformations Q = q+ ˙qδt, P = p+ ˙pδt. [Hint: Start from dQdP =Jdqdp, and prove that dJ/dt = 0].
Example 3.11: Liouville’s theorem and elastic collisions Two identical solid balls collide elastically along a smooth, straight furrow. (a) If their velocities before impact are v1 and v2, and after impact v1 and v2 , show that dv1dv2 = dv1 dv2. (b) Verify Liouville’s theorem for the
Example 3.12: Liouville’s theorem and inelastic collisions Verify Liouville’s theorem in the case of two balls which collide inelastically (i.e. after the collision the dimensionality of the phase space has been reduced).
Example 3.13: Phase space trajectory of a particle What is the (x, px) phase space trajectory of a particle which is free to move along the x-axis between 0 anda, and is reflected at these ends?
Example 3.14: Another evaluation of gShow that the result (3.9c) can also be obtained by evaluation of g =1(2π)3V dqdpδ−1 2m p2
d , (3.28)the delta function representing the number of states per unit interval of . What is the -dependence of g in the case of an N-dimensional configuration space?
Example 3.15: Ratio of statistical weights Consider a single α-particle trapped within a sphere of radius a representing a nucleus. In the Coulomb region r > a the potential barrier falls off as V = +2Ze2/r; inside the sphere V = 0.The α-particle penetrates the barrier from r = a to r =b, b >a,
Example 4.6: The maximum of the binomial distribution Determine the maximum value ˜n of n of the binomial distribution W(n) =N!n!(N − n)!pnqN−n. (4.61)
Example 4.7: The binomial distribution for large N, n Show that for large values of N and n the binomial distribution of Example 4.6 assumes the Gaussian form W(n) = Ne−|V2|(n−˜n)2/2, (4.62)and determine the normalization constant N by evaluating Nn=0 W(n) N 0W(n)dn = 1 (4.63)(Answer: N
Example 4.8: The Poisson distribution Show that with the help of the Stirling approximation (and n N, p 1) the binomial distribution W(n) of Example 4.6 with q = 1 − p reduces to the following expression known as the Poisson distribution, i.e.W(n)λn n!e−λ, λ = Np. (4.64)Show that the
Example 4.9: Probability of sedimentation of atoms N atoms of the same kind in exhaust fumes of some industrial complex are blown by wind into the air and deposit on elements of area a of a larger surrounding region of area A. On the average ˙N = dN/dt atoms are deposited per minute on an
Example 4.10: Decrease of energy fluctuation with increasing N(a) It is a typical characteristic of statistical distributions that, for instance, the percentage deviation of the total energy E of an N-particle system from its mean value E becomes increasingly negligible with rapidly increasing N.
(b) Evaluate D2( ) in classical statistics of noninteracting, nonrelativistic particles in a volume V at absolute temperature T. (The answer follows from i = 3kT/2, 2i= 15(kT)2/4).
Example 4.11: Partition of energy weighting factor of diatomic molecule Consider the energy of a diatomic molecule as the sum of three independent contributions trans, rot and vib arising from translation, rotation and vibration. Consider the relative motion of one of the atoms about the other as

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Basics Of Statistical Physics 3rd Edition Harald J W Muller-Kirsten - Solutions

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