# peeterjoot/physicsplay

questions for problem set 6.

	modified:   basicStatMechProblemSet6Problem1.tex
modified:   basicStatMechProblemSet6Problem2.tex
modified:   basicStatMechProblemSet6Problem3.tex
modified:   basicStatMechProblemSet6Problem4.tex
modified:   basicStatMechProblemSet6Problem5.tex
modified:   basicStatMechProblemSet6Problem6.tex
modified:   basicStatMechProblemSet6Problem7.tex
modified:   renumber
modified:   ../phy452/renumber
1 parent e5668ed commit 84d8307b60d30db8562128219ab3f8fb4f1cefad committed Mar 25, 2013
 @@ -2,44 +2,89 @@ % Copyright © 2013 Peeter Joot. All Rights Reserved. % Licenced as described in the file LICENSE under the root directory of this GIT repository. % -\makeproblem{Maximum entropy principle}{basicStatMech:problemSet6:1}{ +\makeproblem{Maximum entropy principle}{basicStatMech:problemSet6:1}{ +%{2013 problem set 6, problem 1} %(3 points) -%\makesubproblem{}{basicStatMech:problemSet6:1a} -Consider the “Gibbs entropy” S = -kB -P -ipilnpiwhere piis the equilibrium probability of occurrence of a microstate -i in the ensemble. -(i) For a microcanonical ensemble with ? configurations (each having the same energy), assigning an equal probability -pi= 1/? to each microstate leads to S = kBln?. Show that this result follows from maximizing the Gibbs entropy -with respect to the parameters pisubject to the constraint ofP -ipi= 1 (for pito be meaningful as probabilities). In -order to do the minimization with this constraint, use the method of Lagrange multipliers - first, do an unconstrained -minimization of the function S - aP -ipi, then fix a by demanding that the constraint be satisfied. -(ii) For a canonical ensemble (no constraint on total energy, but all microstates having the same number of particles -N), maximize the Gibbs entropy with respect to the parameters pi subject to the constraint ofP -ipi = 1 (for pi -to be meaningful as probabilities) and with a given fixed average energy hEi =P -iEipi, where Ei is the energy -of microstate i. -Use the method of Lagrange multipliers, doing an unconstrained minimization of the function -S - aP -ipi- ßP -iEipi, then fix a,ß by demanding that the constraint be satisfied. What is the resulting pi? -(iii) For a grand canonical ensemble (no constraint on total energy, or the number of particles), maximize the Gibbs -entropy with respect to the parameters pisubject to the constraint ofP -ipi= 1 (for pito be meaningful as probabili- -ties) and with a given fixed average energy hEi =P -iEipi, and a given fixed average particle number hNi =P -iNipi. -Here Ei,Nirepresent the energy and number of particles in microstate i. Use the method of Lagrange multipliers, -doing an unconstrained minimization of the function S-aP +Consider the Gibbs entropy'' +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem1:20} +S = - \kB \sum_i p_i \ln p_i +\end{dmath} + +where $p_i$ is the equilibrium probability of occurrence of a microstate $i$ in the ensemble. + +\makesubproblem{}{basicStatMech:problemSet6:1a} +For a microcanonical ensemble with $\Omega$ configurations (each having the same energy), assigning an equal probability $p_i= 1/\Omega$ to each microstate leads to $S = \kB \ln \Omega$. Show that this result follows from maximizing the Gibbs entropy with respect to the parameters $p_i$ subject to the constraint of + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem1:40} +\sum_i p_i = 1 +\end{dmath} + +(for $p_i$ to be meaningful as probabilities). In order to do the minimization with this constraint, use the method of Lagrange multipliers - first, do an unconstrained minimization of the function + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem1:60} +S - \alpha \sum_i p_i, +\end{dmath} + +then fix $\alpha$ by demanding that the constraint be satisfied. + +\makesubproblem{}{basicStatMech:problemSet6:1b} +For a canonical ensemble (no constraint on total energy, but all microstates having the same number of particles $N$), maximize the Gibbs entropy with respect to the parameters $p_i$ subject to the constraint of + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem1:200} +\sum_i p_i = 1, +\end{dmath} + +(for $p_i$ to be meaningful as probabilities) and with a given fixed average energy + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem1:80} +\expectation{E} = \sum_i E_i p_i, +\end{dmath} + +where $E_i$ is the energy of microstate $i$. Use the method of Lagrange multipliers, doing an unconstrained minimization of the function + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem1:100} +S - \alpha \sum_i p-I - \beta \sum_i E_p p_i, +\end{dmath} + +then fix $\alpha, \beta$ by demanding that the constraint be satisfied. What is the resulting $p_i$? + +\makesubproblem{}{basicStatMech:problemSet6:1c} +For a grand canonical ensemble (no constraint on total energy, or the number of particles), maximize the Gibbs entropy with respect to the parameters $p_i$ subject to the constraint of + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem1:120} +\sum_i p_i = 1, +\end{dmath} + +(for $p_i$ to be meaningful as probabilities) and with a given fixed average energy + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem1:140} +\expectation{E} = \sum_i E_i p_i, +\end{dmath} + +and a given fixed average particle number + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem1:160} +\expectation{N} = \sum_i N_i p_i. +\end{dmath} + +Here $E_i, N_i$ represent the energy and number of particles in microstate $i$. Use the method of Lagrange multipliers, doing an unconstrained minimization of the function + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem1:180} +S - \alpha \sum_i p_i - \beta \sum_i E_i p_i - \gamma \sum_i N_i p_i, +\end{dmath} + +then fix $\alpha, \beta, \gamma$ by demanding that the constrains be satisfied. What is the resulting $p_i$? } % makeproblem -\makeanswer{basicStatMech:problemSet6:1}{ -%\makeSubAnswer{XXX}{basicStatMech:problemSet6:1a} +\makeanswer{basicStatMech:problemSet6:1}{ +\makeSubAnswer{}{basicStatMech:problemSet6:1a} TODO. -} +\makeSubAnswer{}{basicStatMech:problemSet6:1b} + +TODO. +\makeSubAnswer{}{basicStatMech:problemSet6:1c} +TODO. +}
 @@ -2,32 +2,40 @@ % Copyright © 2013 Peeter Joot. All Rights Reserved. % Licenced as described in the file LICENSE under the root directory of this GIT repository. % -\makeoproblem{Fugacity expansion}{basicStatMech:problemSet6:2}{\citep{pathriastatistical} Pathria, Appendix D, E}{ -%\makesubproblem{}{basicStatMech:problemSet6:2a} +\makeoproblem{Fugacity expansion}{basicStatMech:problemSet6:2}{\citep{pathriastatistical} Pathria, Appendix D, E}{ %(3 points) The theory of the ideal Fermi or Bose gases often involves integrals of the form -f± -?(z) = -1 -G(?) -Z8 -0 -dx -x?-1 -z-1ex± 1 -where G(?) =R8 -0dyy?-1e-ydenotes the gamma function. Obtain the behavior of f± -?(z) for z ? 0 keeping the two -leading terms in the expansion. For fermions, obtain the behavior of f+ -?(z) for z ? 8 again keeping the two leading -terms. For bosons, we must have z = 1 (why?), obtain the leading term of f- -?(z) for z ? 1. - - -} % makeproblem - -\makeanswer{basicStatMech:problemSet6:2}{ -%\makeSubAnswer{XXX}{basicStatMech:problemSet6:2a} + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:20} +f_\nu^\pm(z) = \inv{\Gamma(\nu)} \int_0^\infty dx \frac{x^{\nu - 1}}{z^{-1} e^x \pm 1} +\end{dmath} + +where + +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:40} +\Gamma(\nu) = \int_0^\infty dy y^{\nu-1} e^{-y} +\end{dmath} + +denotes the gamma function. + +\makesubproblem{}{basicStatMech:problemSet6:2a} +Obtain the behavior of $f_\nu^\pm(z)$ for $z \rightarrow 0$ keeping the two leading terms in the expansion. + +\makesubproblem{}{basicStatMech:problemSet6:2b} +For fermions, obtain the behavior of $f_\nu^\pm(z)$ for $z \rightarrow \infty$ again keeping the two leading terms. + +\makesubproblem{}{basicStatMech:problemSet6:2c} +For bosons, we must have $z \le 1$ (why?), obtain the leading term of $f_\nu^\pm(z)$ for $z \rightarrow 1$. +} % makeoproblem + +\makeanswer{basicStatMech:problemSet6:2}{ +\makeSubAnswer{}{basicStatMech:problemSet6:2a} + +TODO. +\makeSubAnswer{}{basicStatMech:problemSet6:2b} + +TODO. +\makeSubAnswer{}{basicStatMech:problemSet6:2c} TODO. }
 @@ -2,19 +2,13 @@ % Copyright © 2013 Peeter Joot. All Rights Reserved. % Licenced as described in the file LICENSE under the root directory of this GIT repository. % -\makeproblem{Nuclear matter}{basicStatMech:problemSet6:3}{ +\makeoproblem{Nuclear matter}{basicStatMech:problemSet6:3}{K. Huang, prob 16.2}{ %\makesubproblem{}{basicStatMech:problemSet6:3a} %(3 points) -Consider a heavy nucleus of mass number A. i.e., having A total nucleons including neutrons and protons. Assume -that the number of neutrons and protons is equal, and recall that each of them has spin-1/2 (so possessing two spin -states). Treating these nucleons as a simply free ideal Fermi gas of uniform density contained in a radius R = r0A1/3, -where r0= 1.4×10-13cm, calculate the Fermi energy and the average energy per nucleon in MeV. (From: K. Huang, -prob 16.2). - -} % makeproblem +Consider a heavy nucleus of mass number $A$. i.e., having $A$ total nucleons including neutrons and protons. Assume that the number of neutrons and protons is equal, and recall that each of them has spin-$1/2$ (so possessing two spin states). Treating these nucleons as a free ideal Fermi gas of uniform density contained in a radius $R = r_0 A^{1/3}$, where $r_0 = 1.4 \times 10^{-13} \text{cm}$, calculate the Fermi energy and the average energy per nucleon in MeV. +} % makeoproblem \makeanswer{basicStatMech:problemSet6:3}{ -%\makeSubAnswer{XXX}{basicStatMech:problemSet6:3a} TODO. }
 @@ -4,16 +4,16 @@ % \makeoproblem{Neutron star}{basicStatMech:problemSet6:4}{K. Huang, prob 16.5}{ %(3 points) -Model a neutron star as an ideal Fermi gas of neutrons at T = 0 moving in the gravitational field of a heavy point mass -M at the center. Show that the pressure P obeys the equation dP/dr = -?M?(r)/r2where ? is the gravitational -constant, r is the distance from the center, and ?(r) is the density which only depends on distance from the center. +Model a neutron star as an ideal Fermi gas of neutrons at $T = 0$ moving in the gravitational field of a heavy point mass $M$ at the center. Show that the pressure $P$ obeys the equation -%\makesubproblem{}{basicStatMech:problemSet6:4a} +\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem4:20} +\frac{dP}{dr} = - \gamma M \frac{\rho(r)}{r^2}, +\end{dmath} + +where $\gamma$ is the gravitational constant, $r$ is the distance from the center, and $\rho(r)$ is the density which only depends on distance from the center. } % makeoproblem -\makeanswer{basicStatMech:problemSet6:4}{ -%\makeSubAnswer{XXX}{basicStatMech:problemSet6:4a} +\makeanswer{basicStatMech:problemSet6:4}{ TODO. } -
 @@ -4,7 +4,7 @@ % \makeoproblem{description}{basicStatMech:problemSet6:5}{Huang 16.3}{ %\makesubproblem{}{basicStatMech:problemSet6:5a} -} % makeproblem +} % makeoproblem \makeanswer{basicStatMech:problemSet6:5}{ %\makeSubAnswer{XXX}{basicStatMech:problemSet6:5a}