Statistical Methods 3rd Edition Donna L. Mohr - Solutions

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7.4. ϕ3 Lagrangian theory Calculate the first non-vanishing perturbative order of the partition function for the Lagrangian theory with interaction g 3! ϕ3. Determine the upper critical dimension ds and discuss the renormalization of this theory.
7.3. Correlation functions and Feynman graphs Draw the Feynman diagrams relative to the g2 correction of the 4-point correlation function G(x1, . . . ,x4) for the ϕ4 theory. Discuss the convergence of the integrals as functions of the dimensionality d of the system.
7.2. Perturbative series Consider the one-dimensional integral I(λ) = +∞−∞dxe−αx2+λx4 .Write the perturbative series of this expression expanding the term e−λx4 in power series of λ. Compute the perturbative coefficients and show that the series has zero radius of convergence. Give
7.1. Lagrangian theory with Z3 symmetry Consider a Lagrangian of a complex field (x) and its conjugate †(x) that under a Z3 transformation transform as(x) → e2πi/3 (x), †(x) → e−2πi/3†(x).Write down the most general Lagrangian that is invariant under these transfomations.
6.4. Bethe ansatz equation The solution of the six-vertex model consists in finding the eigenvalues of the transfer matrix T(p)ψ = (A(p)+D(p))ψ = λψ.This problem can be solved by the algebraic Bethe ansatz, whose main steps are as follows: define the pseudo-vacuum φ, as the state annihilated
6.3. Free energy Analyse the expression of the free energy of the Ising model, given in eqn. (6.3.63), as a function of the parameter h. Show that, with t = T−Tc Tc= h−1, for t→0 we have F t2 log |t|.
6.2. Inverse of the matrix V Consider the operator R defined in eqn. (6.1.31). Using the property R2 = I, prove that the inverse of the operator A = (2i sinh2L)n I+(−2i sinh2K)nR is given by A−1 = 1(2sinh2K)2n −(2sinh2L)2n (2i sinh2L)n I−(−2i sinh2L)nR.Use this expression and the
6.1. Perron–Frobenius Theorem Consider a finite dimensional positive matrix M, i.e. with all its matrix elements positive Mij > 0. Assume, for simplicity, thatM is also a symmetrical matrix. Prove that its maximum eigenvalue is positive and non-degenerate. Moreover, prove that the corresponding
5.4. Generating function of dimers on a square lattice Consider the function F(x,y) = 1(2π)2 2π0 2π0 dβ1 dβ2 log [x+y−xcosβ1 −ycosβ2] .Its value at x = y =, i.e.F(2,2), provides the solution to the problem of the dimer covering of a square lattice, eqn. (5.2.14).a. Prove that F(x,0) =
5.3. Number of dimersa. Give an argument to justify the exponential growth of eqn. (5.2.16) for the dimer coverings in N, where N is the number of sites of a lattice.b. Use eqn. (5.2.16) to estimate the number of dimer coverings of a 4×4 square lattice.
5.2. Pfaffian and Determinant Prove that for a 2N ×2N anti-symmetric matrix A it holds the identity(PfA)2 = detA.
5.1. High-temperature series Determine the first three terms of the high-temperature expansion of the correlation function σi+2,j σi,j of two spins separated by two lattice sites.
2.8. One-dimensional Tonks gas Consider a gas made of molecules of sizea, that can move freely along a line of length L. Their interaction potential is given by U(xi −xj) = +∞, if | xi −xj |< a 0, if | xi −xj |> a .Compute the exact expression of the partition function of the system and
4.7. Quadratic model LetβH = −J 2r,r(r −r )2 + 1 4lnJ be theHamiltonian of a two-dimensional system where its variables assume all real values.Show that the model is self-dual under the transformation J ←→ 1J
4.6. Critical temperature of the 3-state Model Using the self-duality of the 3-state model to determine its critical temperature.
7.6. Invariant functions Consider the functions(±)(x) = i(2π)d) dd−1kC(±)dk0 eik·x k2 −m2 , where the contours of integration are shown in Figure 7.21:a. Show that the correlation function of the commutator of the field is given by 0| [ϕ(x),ϕ(y) ] |0 =(x−y),where (x−y) =
7.8. Anti-particles Consider the free theory of a complex field φ(x). In the Minkowski space the action is S = ddx∂μφ∗∂μφ −m2φ∗φ.a. Show that the Hamiltonian is given by H = dd−1x(π∗π +∇φ∗ · ∇φ +m2φ∗φ).b. Prove that the system is invariant under the
7.9. Conserved currents.Consider a multiplet of n scalar fields, = (φ1, . . . ,φn).a. Write the most general Lagrangian that is invariant under a rotation of the vectork→(R)kll .b. Use the Noether theorem to derive the conserved currents associates to this symmetry.
10.8. Central extension Consider commutation relations given by[Ln,Lm] = (n−m)Ln+m +f (n,m)C, where f (n,m)=−f (m, n) and [C,Ln] = 0.a. Show that these commutation relations satisfy the Jacobi identities if and only if f (n,m) = ηn(n2 −1)δn+m,0 +(n−m)λ(n+m)where η is a constant and
10.7. Casimir effect Consider two parallel horizontal planes, separated by a distance a along the axis z.Suppose that a massless field theory is defined between the two planes, with boundary conditions that ensure a non-zero value of the expectation value of the stress-energy tensor Tμν, tμν(t,
10.6. Operator product expansion in the channel of the identity operator Let φi(z) a primary field of a CFT with central chargec. Let i its conformal weight.Prove that the Ward identity uniquely fixes the first terms of the operator expansion in the channel of the identity operator,
10.5. Moebius transformations Show that the transformation w = (z−a)/(z+a), a = c2 −ρ2 with c and ρ real and 0 < ρ
10.4. Hydrodynamics of an ideal fluid in two dimensions Consider the stationary motion of an incompressible and irrotational fluid in two dimensions. Denoting by v(x, y) = (v1, v2) the vector field of its velocity at the point(x, y) of the plane, it satisfies∇. v = 0, ∇ ∧ v = 0a. Show
10.3. Laplace equation and conjugate harmonic functionsa. Show that the real and imaginary parts of an analytic function of a complex variable z f (z) = (x, y)+i(x, y)are both harmonic functions, i.e. they satisfy the Laplace equation ∇2 = ∇2 = 0.b. Vice-versa, use the Cauchy–Riemann
10.2. Correlation functions Assuming the validity of the OPE, show that all the correlation functions of a massless field theory can be expressed in terms of the propagators and the structure constants.
10.1. Operatorial identities There is a simple example that shows the necessity of considering the operatorial identities only in a weak sense, i.e. true only for the matrix elements. Consider an interacting scalar field φ(x) and suppose that for x0→−∞ its interactions vanish. In this case
9.5. XXZ model Consider the quantum spin chain of the XXZ model with Hamiltonian H = J1 Nk=1(Sx kSx k+1+Sy kSyk+1)+J2k Sz kSz k+1.Assume periodic boundary condition. The spin operator S is in the 1/2 representation and given by S = 1/2σ, where σ is the set of Pauli matrices.a. Show that
9.4. Correlation functions Use the mode expansion of the fermion field, given in eqn. (9.7.19), and the anticommutation relations of the operators A(θ) and A†(θ) to compute the correlation functions G1(x, t) = 0|(x, t)(0,0)|0G2(x, t) = 0| ¯ (x, t)(0,0)|0.
9.3. Duality Show that the Dirac equation (9.7.12) with m > 0 is linked by a unitary transformation to the one m < 0. The map between the two Hamiltonians establishes the duality relation of the Ising model in its fermionic formulation.
9.2. Fermion identities Prove that c(a) c†(a+1) = −˜σ−(a) ˜σ+(a+1)c†(a) c†(a+1) = ˜σ+(a) ˜σ+(a+1)c(a) c(a+1) = −˜σ−(a) ˜σ−(a+1).Moreover, show that the order operators of the Ising model can be expressed in terms of the fermion operators c(a) and c†(a) asσ3(a) =
9.1. Anti-commutation relations Prove that the operators c(a) and c†(b) satisfy the anti-commutation relationsc(a), c† = δa,b.
8.4. β-functions Consider a statistical system with the space of the coupling constants described by the variables (x,y2). The fixed point is identified by the origin (0,0). Suppose that the βfunctions of these coupling constants are given by dx db= −y2, dy2 db= −2xy2.Study the RG flows, with
8.3. Approximated values of the critical exponents Use the formulae (8.10.7) to obtain an approximate values of the three-dimensional Ising model and the spherical model. Compare with the numerical values obtained for the three-dimensional Ising model and with the exact expressions of the spherical
8.2. Universal Ratios Consider the mean field solution of the Ising model and, with the notations used in Chapter 1, compute the universal ratios C+ χ+/M2 0, χ+Mδ−1 0M−δ0 .
8.1. Logistic map Consider the map xn+1 = fr(xn) = r xn(1−xn).Introduced by Verhulst in 1845, the map models the growth of a population in a region of a finite area: the population at time n+1, expressed by xn+1, is proportional to the population xn at time n, but at the same time is also
4.5. Self-duality Consider the function Z(K) = K1/4∞n=−∞e−πKn2 that satisfies Z(K) = Z 1 K .a. Show that in this case the duality relation does not imply a phase transition at K = 1.b. How many terms are necessary in the original expression to compute Z(K) with a precision 10−4 for K
4.4. Poisson sum rulea. Generalize the Poisson sum rule to the d-dimensional case.b. Using the Poisson sum rule show∞n=0 1x2 +n2= 1 2x2− π2x+ πx 11−e−2Pix
4.3. Free energy Consider the high-temperature series expansion of a homegeneous Ising model on a two dimensional square lattice ZN = (2coshβJ)2N× 1+Nv4 +2Nv6 +2Nv8 + 1 2N(N −9)v8+· · ·(v = tanhβJ). In the thermodynamic limit, we should have ZN (Z1)N = e−Nβf, where f is the
2.2. Multiple couplings Consider the one-dimensional Ising model with N spins σi =±1, with the Hamiltonian given by H = −J1 N−1i=1σi σi+1 −J2 N−2i=1σiσi+2 Compute the free energy of such a system and analyse its behaviour by varying the parameters J1 and J2.Hint. It is convenient to
2.1. Ising model with vacancies Consider the one-dimensional Ising model with the familiar Hamiltonian H = −J Ni=1 Si Si+1 but with the values of the spins Si = {0,±1}. The null value may be interpreted as a vacancy of the corresponding site. Use the transfer matrix method to compute the
1.15. Braid group The braid group on n strands, denoted by Bn, is a set of operations has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number. Braid groups find applications in knot theory, since any knot may be represented as the
1.14. Dimensional regularization Let d the dimension of the space. Discuss the convergence of the integral I(d) = ∞0 rd−1 r2 +1 dr by varying d.a. Determine, in its convergent domain, the exact expression of the integral as a function of d and identify the position of its poles.b. Analytically
1.13. Bose–Einstein Condensation Appendix 1.B showed that, in three dimensions, an ideal gas with a bosonic statistics presents a Bose–Einstein condensation for sufficiently low temperature. Discuss if the same phenomenon can take place in one and two dimensions. Explain if a Bose–Einstein
1.12. Riemann function The Riemann function ζ(β) is defined byζ(β) =∞n=1 1nβa. Interpret this expression as the partition function in the canonical ensemble of a quantum system and identify the discrete spectrum of the energies.b. Compute the density of states and the entropy of the quantum
1.11. Quantum harmonic oscillator The one-dimensional quantum oscillator has the energy spectrum given by En = ¯hω(n+1/2), n = 0, 1, 2, . . . .a. Compute the partition function in the canonical ensemble.b. Compute the specific heat as a function of the temperature and discuss how this quantity
1.10. Density of states Determine the number of quantum states with energy less than E for a free particle in a cubic box of length L. Compare this quantity with the volume of the classical phase space and find the corresponding density of states of the system.
1.9. Ergodic system Consider a classical dynamical system with a phase space (0 < q < 1; 0 < p < 1) and equation of motion given by q(t) = q0 +t ; p(t) = p0 +α t.a. Discuss the trajectories in the phase space when α is a rational and irrational number.b. Show that the system is ergodic when α is
1.8. First-order phase transitions In the second-order phase transitions, the state with the lowest value of the free energy changes continuously when the system crosses its critical point. On the contrary, in the first-order phase transition, the order parameter changes discontinuously.a. Study
1.7. Scaling laws Given the equation of state of a magnetic system in the form B = MδQ tM1/β ,a. Prove that the parameters β and δ in the expression above are the critical exponents of the system, as defined in the chapter;b. Show the identity γ = β(δ −1).
1.6. Two-state systems Consider a system of N free classical particles. The energy of each particle can take only two values: 0 and E(E > 0). Let n0 and n1 the occupation numbers of the two energy levels and U the total energy of the system.a. Determine the entropy of the system.b. Determine n0,
1.5. Zeros of the partition function Consider a classical system with only two states of magnetization, both proportional to the volume V of the system M =±αV. In the presence of an external magnetic field B, the Hamiltonian is given by H = BM.a. Compute the partition function in the canonical
1.4. Equation of state for homogeneous potentials Consider a system of classical particles whose interaction potential is given by a homogeneous function of degree ηU(λr1,λ r2, . . . ,λrN) = ληU(r1, r2, . . . , rN).Show that the equation of state of such a system assumes the form
1.3. Theorem of equi-partition Consider a classical one-dimensional harmonic oscillator, with Hamiltonian H = p2 2m+ mω2x2 2,a. Determine the surface E = constant in the phase space and derive the thermodynamics of the system by using the microcanonical ensemble.b. Put the system in contact with a
1.2. Potts model In the Potts model, the spin variable σi assume q value, as {0, 1, . . . ,q−1}. The energy of the configurations is given by H = −Jijδσi ,σj ,where δa,b = 1 if a = b 0 if a = b.a. Identify the symmetry transformations of the spins that leave the Hamiltonian
1.1. Lattice Gas Consider a lattice gas in which the particles occupy the sites of a d-dimensional lattice, with the constraint that each site cannot be occupied by more than one particle. Let ei equal a variable that takes values {0,1}: 0 when is the site is vacant and 1 when it is occupied. The
2.3. Grand canonical partition function Consider a one-dimensional lattice gas on M sites. The variable ti has values 0 and 1, according whether the relative site is occupied or not. The Hamiltonian is given by H = −J Mi=1 ti ti+1 with ti+M = ti . Moreover, we assume that Mi=1 ti = Na. Compute
2.4. Ising model in a magnetic field Let us consider the one-dimensional Ising model with nearest-neighbour interactions in the presence of a magnetic field. Show that the partition function ZN(T,B) is an even function of B and compute its first non-vanishing term of the series expansion with
4.2. Low-temperature series in the presence of a magnetic field Consider the two-dimensional Ising model on a square lattice with equal coupling along the horizontal and vertical links and in presence of an external magnetic field B.Generalize the discussion on the series expansion of the free
4.1. Three-dimensional lattices Generalize Peierls’s argument to the Ising model on three-dimensional lattices and prove that the model admits a phase transition.
3.10. Langevin equation and Brownian motion Consider a particle of mass m in motion in a fluid (for simplicity we consider a onedimensional motion), subjected to a friction force proportional to the velocity and a random force η(t) due to the randomfluctuations of the fluid density. Denoting by
3.9. Brownian motion on a ring Consider a Brownian motion on a ring of N sites, with a transition rate to next-neighbour sites equal to 1/2. Let Pn(s) be the probability to find the walker at the site s at the time n(s = 1, 2, . . . ,N). Show that ifN is an odd number, there is a unique stationary
3.8. Markov processes Brownian motion is a particular example of a general class of stochastic processes known as Markov processes, characterized by a transition probability w(i→j) = wij between the discrete states {A} = {a1,a2,a3, . . . ,an} of a stochastic variable A (wij ≥ 0 andnj=1wij =1).
3.7. Distinct points visited in the Brownian motion Give a physical argument for the mean value of the number of distinct sites visited in the Brownian motion and show that this mean value depends on the dimensionality of the lattice as predicted by eqn. (3.B.28).
3.6. Brownian motion with boundary conditions Let P(s, t) = 1 √4Dt exp[−s2/(4Dt)]be the probability distribution in the continuum limit of the one-dimensional Brownian motion.a. Find the probability distribution Pa(s, t) when the origin is an absorbent point, i.e.when Pa(0, t) = 0 for all t.b.
3.5. Critical exponents of the spherical model Use the equation of state and the other relations discussed in the text to derive the critical exponents of the spherical model.
3.4. Quantum magnets Consider the Hamiltonian H = −1 2R,R J(R−R) S(R) · S(R)where J(R−R) > 0 and S(R) is the quantum operator of spin S.a. Prove initially the following result: the largest (smallest) diagonal element that a Hermitian operator can have is equal to its largest (smallest)
3.3. Spontaneous magnetization in low temperature Show that, when T is much less than Tc, the mean field theory of a ferromagnet predicts a spontaneous magnetization that differs from its saturation value for terms that are exponentials in −1/T.
3.2. Mean field theory for coupled lattices Consider a system of Ising spins made of two-dimensional square lattices A and B, coupled together. Let Ja and Jb be the coupling constants between next-neighbour spins in the lattices A and B respectively, and J the coupling constant between the
3.1. Mean field theory for anti-ferromagnetic systemsa. Prove that the mean field theory can be applied to anti-ferromagnetic systems defined on a square lattice.b. Generalize this result to all lattices that admit two sub-lattices such that the nextneighbour sites of one are sites of the other,
2.10. Thermodynamics of one-dimensional oscillators Consider a set of N particles of mass m placed along a line of length L. Let pi and qi be their momentum and coordinate respecively. Their Hamiltonian is H =Ni=1 p2i 2m+ 1 2k N−1i=1(qi −qi+1)2.a. Determine the frequencies of the normal modes
2.9. One-dimensional gas Considerar a one-dimensional gas of N particles, with potential interaction V(x1,x2, . . . ,xN) = −i
2.7. Potts Model Consider the one-dimensional q-state Potts model, whose Hamiltonian is given by H = −J Ni=1δ(σi ,σi+1).a. Compute the expectation value δ(σk,σk+1).b. Study this quantity for negative values of q.
2.6. Correlation functions For the one-dimensional Ising model with next-neighbour interaction:a. Compute the correlation function σi σjσkσl, with i > j > k > l.b. Use the expression of the quantity above to compute the derivative of the correlation function σi σjσk with respect to the
2.5. Ising model in a purely imaginary magnetic field Consider the one-dimensional Ising model with next-neighbour interaction in the presence of a purely imaginary magnetic field B = ih. Study the correlation length of the model, expressed by eqn. (2.2.28) and show that there exist an infinite
7.7. Field theories with soliton solutions Consider the Lagrangian field theory in 1+1 dimensions˜L= 1 2(∂μϕ)2 + m2β2 [cos(βϕ)−1].a. Expand in powers of β and show that this model corresponds to a Landau–Ginzburg theory with an infinite number of couplings.b. Write the equation of
Twelve patients undergoing recuperation from cardiothoracic surgery are randomly divided into two groups. Because the treatment that one group is about to receive affects blood glucose levels, the researchers first compare values of A1C(a measure of blood glucose control over the last 120 days) in
The number of birth defects in a region is commonly modeled as having a binomial distribution,with a rate of three birth defects per 100 births considered a typical rate in the United States.(a) What is the probability a county that had 50 independent births during the year would have more than
The Kaufman Assessment Battery for Children is designed to measure achievement and intelligence with a special emphasis on nonverbal intelligence. Its global measures, such as its Sequential Processing score, are scaled to have a mean of 100 and a standard deviation of 15. Assume that the
Based on data from the 2007 National Health Interview Survey, it is estimated that “10% of adults experienced feelings of sadness for all, most, or some of the time” during the 30 days prior to the interview. You interview a random sample of 68 people who have recently filed for unemployment
An insurance company wishes to keep the error rates in medical claims at or below 5%. If there is evidence of an error rate greater than this, they will need to introduce new quality procedures. The company has two possible decision plans:Plan A. Randomly select 30 independent claims and audit them
A manufacturer of auto windows uses a thin layer of plastic material between two layers of glass to make safety glass for windshields. The thickness of the layer of this material is important to the quality of the vision through the glass.A constant quality control monitoring scheme is employed by
The Florida lottery uses a system of numbers ranging in value from 1 to 53. Every week the lottery commission randomly selects six numbers, and every ticket with those numbers wins a share of the grand prize. Individual numbers appear only once (no repeat values), and the order in which they are
To monitor the production of sheet metal screws by a machine in a large manufacturing company, a sample of 100 screws is examined each hour for three shifts of 8 hours each. Each screw is inspected and designated as conforming or nonconforming according to specifications. Management is willing to
Assuming a normal distribution,what values of μ and σ 2 will provide such a curve? LO.1
A teacher wants to curve her grades such that 10% are below 60 and 10% above LO.1
A manufacturer wants to state a specific guarantee for the life of a product with a replacement for failed products. The distribution of lifetimes of the product is normal and has a mean of 1000 days and standard deviation of 150 days. What life length should be stated in the guarantee so that only
What is the probability that the sample mean lies between 138 and 142? LO.1
A random sample of 100 is taken from a population with a mean of 140 and standard deviation of LO.1
The Poisson distribution may also be used to find approximate binomial probabilities when n is large and p is small, by letting μ be np. This method provides for faster calculations of probabilities of rare events such as exotic diseases. For example, assume the incidence rate (proportion in the
As μ increases, say, to values greater than 30, the Poisson distribution begins to be very similar to the normal with both a mean and variance of μ. Using this approximation, determine how many telephone operators are needed to ensure at most 5% busy signals if the mean number of phone calls at
The probabilities for a randomvariable having the Poisson distributionwith μ = 1 is given in the following table.Values of Y 0 1 2 3 4 5 6 Probability 0.368 0.368 0.184 0.061 0.015 0.003 0.001 Note: Probabilities for Y > 6 are very small and may be ignored.(a) Compute the mean and variance of
It is believed that 20% of voters in a certain city favor a tax increase for improved schools. If this percentage is correct, what is the probability that in a sample of 250 voters 60 ormorewill favor the tax increase? (Use the normal approximation.) LO.1
A teacher finds that the scores of a particularly difficult test were approximately normally distributed with a mean of 76 and standard deviation of 14.(a) If a score below 60 represents a grade of F (failure), approximately what percent of students failed the test?(b) If the cutoff for a grade of
Let Y be a normally distributed random variable with mean 10 and variance 25.Find:(a) P(Y > 15)(b) P(8 < Y < 12)(c) the value of C such that P(Y < C) = 0.90 LO.1
Let Z be a standard normal random variable. Use Appendix Table A.1 to find:(a) P(Z > 1)(b) P(Z > −1)(c) P(0 < Z < 1)(d) P(Z < −1.5)(e) P(−2.07 < Z < 0.98)(f) the value A such that P(Z < A) = 0.95(g) the value C such that P(−C < Z < C) = 0.95 LO.1
A system requires two components, A and B, to both work before the system will. Because of the sensitivity of the system, an increased reliability is needed.To obtain this reliability, two duplicate components are to be used. That is, the system will have components A1,A2, B1, and B2. An engineer
A system consists of 10 components all arranged in series, each with a failure probability of 0.001. What is the probability that the system will fail? (Hint: See Section 2.2.) LO.1
The binomial distribution for p = 0.2 and n = 5 is:Value of Y 0 1 2 3 4 5 Probability 0.3277 0.4096 0.2048 0.0512 0.0064 0.0003(a) Compute μ and σ 2 for this distribution.(b) Do these values agree with those obtained as a function of the parameter p and sample size n? (See discussion of random
Assume the random variable y has the continuous uniform distribution defined on the interval a tob, that is, f (y) = 1/(b − a), a ≤ y ≤ b.For this problem let a = 0 and b = 2.(a) Find P(Y < 1). (Hint: Use a picture.)(b) Find μ and σ 2 for the distribution. LO.1
A lottery that sells 150,000 tickets has the following prize structure:(1) first prize of $50,000(2) 5 second prizes of $10,000(3) 25 third prizes of $1000(4) 1000 fourth prizes of $10(a) Let Y be the winning amount of a randomly drawn lottery ticket.Describe the probability distribution of Y.(b)

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Statistical Methods 3rd Edition Donna L. Mohr - Solutions

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