Questions and Answers of Statistical Reasoning For Everyday Life

(a) Show that, to the first order in the scattering length $a$, the discontinuity in the specific heat $C_{V}$ of an imperfect Bose gas at the transition temperature $T_{c}$ is given
(a) Complete the mathematical steps leading to equations (11.3.15) and (11.3.16).(b) Complete the mathematical steps leading to equations (11.3.23) and (11.3.24).
The ground-state pressure of an interacting Bose gas (see Lee and Yang, 1960a) turns out to be\[P_{0}=\frac{\mu_{0}^{2} m}{8 \pi a \hbar^{2}}\left[1-\frac{64}{15 \pi} \frac{\mu_{0}^{1 / 2} m^{1 / 2}
Show that in an interacting Bose gas the mean occupation number $\bar{n}_{\boldsymbol{p}}$ of the real particles and the mean occupation number $\bar{N}_{p}$ of the quasiparticles are connected
Using Fetter's analytical approximation,\[f_{1}\left(ho^{\prime}\right)=\frac{ho^{\prime}}{\sqrt{\left(1+ho^{\prime 2}\right)}}\]for the solution of equation (11.5.23) with $s=1$, calculate the
(a) Study the nature of the velocity field arising from a pair of parallel vortex lines, with $s_{1}=$ +1 and $s_{2}=-1$, separated by a distance $d$. Derive and discuss the general equation of
In the ground state of a Fermi system, the chemical potential is identical to the Fermi energy: $(\mu)_{T=0}=\varepsilon\left(p_{F}\right)$. Making use of the energy spectrum $\varepsilon(p)$ of
The energy levels of an imperfect Fermi gas in the presence of an external magnetic field $\boldsymbol{H}$, to the first order in $a$, may be written
Rewrite the Gross-Pitaevskii equation and the mean field energy, see equations (11.2.21) and (11.2.23), for an isotropic harmonic oscillator trap with frequency $\omega_{0}$ in a dimensionless form
Solve the Gross-Pitaevskii equation and evaluate the mean field energy, see equations (11.2.21) and (11.2.23), for a uniform Bose gas to show that this method yields precisely equation (11.2.6).
Solve the Gross-Pitaevskii equation (11.2.23) in a harmonic trap for the case when the scattering length $a$ is zero. Show that this reproduces the properties of the ground state of the
Solve the Gross-Pitaevskii equation and evaluate the mean field energy, see equations (11.2.21) and (11.2.23), for an isotropic harmonic oscillator trap with frequency $\omega_{0}$ for the case \(N
Assume that in the virial expansion\[\begin{equation*}\frac{P v}{k T}=1-\sum_{j=1}^{\infty} \frac{j}{j+1} \beta_{j}\left(\frac{\lambda^{3}}{v}\right)^{j} \tag{10.4.22}\end{equation*}\]where
Assuming the Dieterici equation of state P(v−b)=kTexp(−a/kTv)P(v−b)=kTexp⁡(−a/kTv) evaluate the critical constants Pc,vcPc,vc, and TcTc of the given system in terms of the parameters aa
Consider a nonideal gas obeying a modified van der Waals equation of state\[\left(P+a / v^{n}\right)(v-b)=R T \quad(n>1) .\]Examine how the critical constants $P_{c}, v_{\mathrm{c}}$, and $T_{c}$
Following expressions (12.5.2), define\[\begin{equation*}p=(1+L) / 2 \quad \text { and } \quad q=(1-L) / 2 \quad(-1 \leq L \leq 1) \tag{1}\end{equation*}\]as the probabilities that a spin chosen at
Using the correspondence established in Section 12.4, apply the results of the preceding problem to the case of a lattice gas. Show, in particular, that the pressure, $P$, and the volume per
Consider an Ising model with an infinite-range interaction such that each spin interacts equally strongly with all other
Study the Heisenberg model of a ferromagnet, based on the interaction (12.3.6), in the mean field approximation and show that this also leads to a phase transition of the kind met in the Ising model.
Study the spontaneous magnetization of the Heisenberg model in the mean field approximation and examine the dependence of $\bar{L}_{0}$ on $T$ (i) in the neighborhood of the critical temperature
An antiferromagnet is characterized by the fact that the exchange integral $J$ is negative, which tends to align neighboring spins antiparallel to one another. Assume that a given lattice structure
The Néel temperature $T_{N}$ of an antiferromagnet is defined as that temperature below which the sublattices $a$ and $b$ possess nonzero spontaneous magnetizations $M_{a}$ and $M_{b}$,
Suppose that each atom of a crystal lattice can be in one of several internal states (which may be denoted by the symbol $\sigma$ ) and the interaction energy between an atom in state
Consider a binary alloy containing $N_{A}$ atoms of type $A$ and $N_{B}$ atoms of type $B$ so that the relative concentrations of the two components are \(x_{A}=N_{A}
Modify the Bragg-Williams approximation (12.5.29) to include a short-range order parameter $s$, such that\[\begin{aligned}& \bar{N}_{++}=\frac{1}{2} q N
Mean field theories can be derived using a variational approach in which the exact free energy is a lower bound of an approximate free energy; see Chaikin and Lubensky (1995). First prove the
Following the results of Problem 12.28, one can form a mean field theory by choosing a special form for the density function $ho$ and solve for $ho$. Suppose that the Hamiltonian describes a
How small must the volume, $V_{A}$, of a gaseous subsystem (at normal temperature and pressure) be, so that the root-mean-square deviation in the number, $N_{A}$, of particles occupying this
Derive the Jarzynski equality for the case in which the system of interest is described by classical Hamiltonian $H_{\lambda}(\boldsymbol{q}, \boldsymbol{p})$ that is coupled to a thermal reservoir
Write an MC code for a fluid of $N$ hard spheres in a two-dimensional $L \times L$ square box and include a one-body gravity term $\sum_{i=1}^{N} m g y_{i}$ in the algorithm, that is, accept
Write an MD code for $N$ Lennard-Jones particles in a two-dimensional $L \times L$ square box, and include a one-body gravity term in the energy: $\sum_{i=1}^{N} m g y_{i}$. Apply periodic
Write an MC code to simulate the two-dimensional nearest-neighbor Ising model on a periodic $L \times L$ lattice in zero field. Calculate the energy distribution $P(E)$ over a range of
For imperfect-gas calculations, one sometimes employs the Sutherland potential\[u(r)= \begin{cases}\infty & \text { for } rD\end{cases}\]Using this potential, determine the second virial coefficient
According to Lennard-Jones, the physical behavior of most real gases can be well understood if the intermolecular potential is assumed to be of the form\[u(r)=\frac{A}{r^{m}}-\frac{B}{r^{n}}\]where
(a) Show that for a gas obeying the van der Waals equation of state (10.3.9),\[C_{P}-C_{V}=N k\left\{1-\frac{2 a}{k T v^{3}}(v-b)^{2}\right\}^{-1}\](b) Also show that, for a van der Waals gas with
The coefficient of volume expansion $\alpha$ and the isothermal bulk modulus $B$ of a gas are given by the empirical expressions\[\alpha=\frac{1}{T}\left(1+\frac{3 a^{\prime}}{v T^{2}}\right)
Show that the first-order Joule-Thomson coefficient of a gas is given by the formula\[\left(\frac{\partial T}{\partial P}\right)_{H}=\frac{N}{C_{P}}\left(T \frac{\partial\left(a_{2}
Determine the lowest-order corrections to the ideal-gas values of the Helmholtz free energy, the Gibbs free energy, the entropy, the internal energy, the enthalpy, and the (constantvolume and
The molecules of a solid attract one another with a force $F(r)=\alpha(l / r)^{5}$. Two semiinfinite solids composed of $n$ molecules per unit volume are separated by a distance $d$, that is,
Using the wavefunctions\[u_{p}(\boldsymbol{r})=\frac{1}{\sqrt{ } V} e^{i(\boldsymbol{p} \cdot \boldsymbol{r}) \hbar}\]to describe the motion of a free particle, write down the symmetrized
Show that for a gas composed of particles with spin $J$\[\hbar_{2}^{S}(J)=(J+1)(2 J+1) \hbar_{2}^{S}(0)+J(2 J+1){\varpi_{2}^{A}}^{A}(0)\]and\[\bar{b}_{2}^{A}(J)=J(2 J+1) \hbar_{2}^{S}(0)+(J+1)(2
Show that the coefficient $\bar{b}_{2}$ for a quantum-mechanical Boltzmannian gas composed of "spinless" particles satisfies the following relations:\[\begin{aligned}\bar{b}_{2} & =\lim _{J
Use a virial expansion approach to determine the first few nontrivial order contributions to the pair correlation function $g(r)$ in $d$ dimensions. Show that the pair correlation function is of
For the particular case of hard spheres, the pressure in the virial equation of state is determined by evaluating the pair correlation function at contact. Write the pair correlation function as
Derive the probability distribution $w(r)$ for the distance to the closest neighboring particle using the pair correlation function $g(r)$ and the number density $n$. Show that in three
Show that for a gas of noninteracting bosons, or fermions, the pair correlation function $g(r)$ is given by the expression\[g(r)=1 \pm \frac{g_{s}}{n^{2} h^{6}}\left|\int_{-\infty}^{\infty}
Show that, in the case of a degenerate gas of fermions $\left(T \ll T_{F}\right)$, the correlation function $g(r)$, for $r \gg \hbar / p_{F}$, reduces to the expression\[g(r)-1=-\frac{3(m k
(a) For a dilute gas, the pair correlation function $g(r)$ may be approximated as\[g(r) \simeq \exp \{-u(r) / k T\}\]Show that, under this approximation, the virial equation of state (10.7.11)
Show that the pressure and Helmholtz free energy of a fluid at temperature $T$ can be determined by performing a thermodynamic integration of the inverse of the isothermal compressibility from the
Show that, for a general Gaussian distribution of variables $u_{j}$, the average of the exponential of a linear combination of the variables obeys the relation\[\left.\left\langle\exp
Calculate the isothermal compressibility and Helmholtz free energy for the CarnahanStarling equation of state (10.3.25) and show that the Helmholtz free energy is given by\[\frac{\beta
Use the Hubble expansion relation (9.1.1), the temperature scaling relation (9.1.3), and the energy density relation before the electron-positron annihilation (9.3.6b) to show that the temperature as
While the electromagnetic interaction between the photons and the charged electrons and positrons kept them in equilibrium with each other during the early universe, show that the direct
Show that during the early part of the electron-positron annihilation era, the ratio of the electron number density to the photon number density scaled with temperature as\[\frac{n_{-}}{n_{\gamma}}
Show that after nearly all of the positrons were annihilated and the electron number density had nearly leveled off at the proton density, the ratio of the positron number density to the photon
After the positrons were annihilated, the energy density of the universe was dominated by the photons and the neutrinos. Show that the energy density in that era was given by \(u_{\text {total
How would the primordial helium content of the universe have been affected if the present cosmic background radiation temperature was $27 \mathrm{~K}$ instead of $2.7 \mathrm{~K}$ ? What about
Gold-on-gold nuclear collisions at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory create a quark-gluon plasma with an energy density of about \(4 \mathrm{GeV} /
Calculate the energy density versus temperature very early in the universe when the temperatures were above $k T=300 \mathrm{MeV}$. At those temperatures, quarks and gluons were released from
(a) Assuming that the total number of microstates accessible to a given statistical system is $\Omega$, show that the entropy of the system, as given by equation (3.3.13), is maximum when all
Sketch the $P-V$ phase diagram for helium-4 using the sketch of the $P-T$ phase diagram in Figure 4.3. Ps P S superfluid Pe T To T FIGURE 4.3 Sketch of the P-T phase diagram for helium-4. The
Near room temperature, the classical Helmholtz free energy of $N$ diatomic gas molecules in a container of volume $V$ is given by\[A(N, V, T)=-N k T \ln \left(\frac{V}{N \lambda^{3}}\left(\frac{2
Derive the density matrix $ho$ for (i) a free particle and (ii) a linear harmonic oscillator in the momentum representation and study its main properties along the lines of Section 5.3. 5.3.A An
Show that the density matrix $\hat{ho}(t)$ has the properties $\operatorname{Tr} \hat{ho}(t)=1$ and $\operatorname{Tr} \hat{ho}^{2}(t) \leq 1$, and that the equality is manifested only for the
Consider a composite system $A B$ composed of two noninteracting subsystems $A$ and $B$, with the principle of independence of noninteracting subsystems requiring that \(\hat{ho}_{A
Consider the properties of a $2 \times 2$ real symmetric GOE random matrix of the form\[\boldsymbol{R}=\left[\begin{array}{ll}x & z \\z & y\end{array}\right]\]where the symmetric off-diagonal
Numerically calculate a histogram of the distribution of eigenvalues of $2 \times 2$ GOE random matrices, and show that the distribution $P(\lambda)$ agrees with the result in Problem 5.11(e).
Consider a $\Gamma \times \Gamma$ GOE random matrix $\boldsymbol{R}$ whose off-diagonal elements are selected from a normal distribution with zero mean and unit variance, and diagonal elements
Numerically calculate a histogram of the distribution of eigenvalues of $\Gamma \times \Gamma$ GOE random matrices for moderate values of $\Gamma$, and show that the distribution $P(\lambda)$
Derive equation (5.6.21). Show that if equation (5.6.15) applies, then you get the result in equation (5.6.22).
For systems that obey the ETH, equation (5.6.22) implies that $\delta\langle Aangle_{t} \sim 1 / \sqrt{\Gamma}$ for almost all times. Show that if the set of coefficients $\left\{c_{n}\right\}$
Consider a two-level quantum system with energy eigenstates $|0angle$ and $|1angle$, with energies 0 and $\varepsilon=\hbar \omega$, respectively, and two different pure states:
Use particle in a box wavefunctions to construct the $N$-body quantum canonical partition function $Q_{N}(V, T)$ for indistinguishable noninteracting particles with mass $m$ in a rectangular
Analyze the combustion reaction\[\begin{equation*}\mathrm{CH}_{4}+2 \mathrm{O}_{2} \rightleftarrows \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O}, \tag{6.6.8}\end{equation*}\]assuming that at
Diatomic nitrogen and oxygen molecules can stick to the surface of a container because of van der Waals forces. Assume that these molecules are stuck to the surface with binding energy
Polyatomic molecules in which the three moments of inertia are all approximately $I$ have a rotational energy spectrum given by\[E_{\mathrm{rot}}=\frac{\hbar^{2} l(l+1)}{2 I}\]with level degeneracy
The classical Lagrangian for a diatomic molecule with particle masses $m_{1}$ and $m_{2}$ is\[\mathcal{L}=\frac{1}{2} M\left(\dot{X}^{2}+\dot{Y}^{2}+\dot{Z}^{2}\right)+\frac{1}{2} \mu
Show that for an ideal Bose gas\[\frac{1}{z}\left(\frac{\partial z}{\partial T}\right)_{P}=-\frac{5}{2 T} \frac{g_{5 / 2}(z)}{g_{3 / 2}(z)}\]compare this result with equation (7.1.36). Thence, show
Calculate the occupancy of the first excited state of a Bose-condensed system in a threedimensional harmonic trap. Show that this state is not macroscopically occupied for any temperature, i.e., that
Calculate the Bose-Einstein condensation temperature for a system with energy density of states given
Let the Fermi distribution at low temperatures be represented by a broken line, as shown in Figure 8.13, the line being tangential to the actual curve at $\varepsilon=\mu$. Show that this
For a Fermi-Dirac gas, we may define a temperature $T_{0}$ at which the chemical potential of the gas is zero $(z=1)$. Express $T_{0}$ in terms of the Fermi temperature $T_{F}$ of the gas.
Show that for an ideal Fermi gas\[\frac{1}{z}\left(\frac{\partial z}{\partial T}\right)_{P}=-\frac{5}{2 T} \frac{f_{5 / 2}(z)}{f_{3 / 2}(z)}\]compare with equation (8.1.9). Hence show that\[\gamma
Show that the velocity of sound $w$ in an ideal Fermi gas is given by\[w^{2}=\frac{5 k T}{3 m} \frac{f_{5 / 2}(z)}{f_{3 / 2}(z)}=\frac{5}{9}\left\langle u^{2}\rightangle\]where \(\left\langle
Making use of another term of the Sommerfeld lemma (E.17), show that in the second approximation the chemical potential of a Fermi gas at low temperatures is given by\[\begin{equation*}\mu \simeq
Consider an ideal Fermi gas, with energy spectrum $\varepsilon \propto p^{s}$, contained in a box of "volume" $V$ in a space of $n$ dimensions. Show that, for this system,(a) \(P V=\frac{s}{n}
Examine results (b) and (c) of the preceding problem in the high temperature limit ( $T \gg T_{F}$ ) as well as in the low temperature limit ( $T \ll T_{F}$ ), and compare the resulting
Show that, in two dimensions, the specific heat $C_{V}(N, T)$ of an ideal Fermi gas is identical to the specific heat of an ideal Bose gas, for all $N$ and $T$.[It will suffice to show that,
Show that, quite generally, the low-temperature behavior of the chemical potential, the specific heat, and the entropy of an ideal Fermi gas are given by\[\mu \simeq
Investigate the Pauli paramagnetism of an ideal gas of fermions with intrinsic magnetic moment $\mu^{*}$ and spin $J \hbar\left(J=\frac{1}{2}, \frac{3}{2}, \ldots\right)$, and derive expressions
Show that expression (8.2.20) for the paramagnetic susceptibility of an ideal Fermi gas can be written in the form\[\frac{\chi}{V}=\frac{N}{V} \frac{\mu^{* 2} \mu_{0}}{k T} \frac{f_{1 / 2}(z)}{f_{3 /
The observed value of $\gamma$, see equation (8.3.6), for sodium is $4.3 \times 10^{-4} \mathrm{cal} \mathrm{mole}^{-1} \mathrm{~K}^{-2}$. Evaluate the Fermi energy $\varepsilon_{F}$ and the
Show that the ground-state energy $E_{0}$ of a relativistic gas of electrons is given by\[E_{0}=\frac{\pi V m^{4} c^{5}}{3 h^{3}} B(x)\]where\[B(x)=8 x^{3}\left\{\left(x^{2}+1\right)^{1 /
Show that the low-temperature specific heat of the relativistic Fermi gas, studied in Section 8.5, is given by\[\frac{C_{V}}{N k}=\pi^{2} \frac{\left(x^{2}+1\right)^{1 / 2}}{x^{2}} \frac{k T}{m
Express the integrals (8.6.19) in terms of the initial slope of the function $\Phi(x)$, and verify equation (8.6.20). and [{$(r)}/rdr, (19)
Derive equations (8.4.3) through (8.4.5) for a Fermi gas in a harmonic trap. Evaluate equations (8.3.4) and (8.3.5) numerically to reproduce the theoretical curves shown in Figures 8.9 and 8.10.
The pressure and number density and energy density for an ideal Fermi gas are given by\[P(\mu, T)=k T \int a(\varepsilon) \ln \left(1+e^{-\beta(\varepsilon-\mu)}\right) d \varepsilon\]where
For the hypothesis test described in Exercise 1, which of the following distributions is most appropriate?a. normal distributionb. t distributionc. chi-square distributiond. uniform
Assuming that we want to use the data in the table below to test for independence between wearing a helmet and receiving facial injuries in a bicycle accident, find the expected frequency for the
A simple random sample of heights of basketball players in the NBA is obtained, and the population has a distribution that is approximately normal. The sample statistics are n = 16, x = 77.9 inches,

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