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modified: ../../bin/cbp

modified:   GNUmakefile
new file:   basicStatMechProblemSet6.tex
new file:   basicStatMechProblemSet6Problem1.tex
new file:   basicStatMechProblemSet6Problem2.tex
new file:   basicStatMechProblemSet6Problem3.tex
new file:   basicStatMechProblemSet6Problem4.tex
new file:   basicStatMechProblemSet6Problem5.tex
new file:   basicStatMechProblemSet6Problem6.tex
new file:   basicStatMechProblemSet6Problem7.tex
modified:   renumber
modified:   ../peeter_prologue_print2.tex
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1 parent b7c229a commit e5668edce874242aec2035e2f203119352fbfacf @peeterjoot committed Mar 24, 2013
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@@ -4,7 +4,7 @@ n=${1?}
~/bin/ct basicStatMechProblemSet${n} basicStatMechProblemSetN tp$n
perl -p -i -e "s/{N}/$n/g" basicStatMechProblemSet${n}.tex
-for i in 1 2 3 ; do
+for i in 1 2 3 4 5 6 7 ; do
~/bin/ct basicStatMechProblemSet${n}Problem${i} basicStatMechProblemSetNProblemM tp${n}$i
perl -p -i -e "s/{N}/$n/g;s/{M}/$i/g;" basicStatMechProblemSet${n}Problem${i}.tex
done
View
@@ -1,9 +1,13 @@
THISDIR := blogit
RAWTEXFILES := $(shell ls *.tex)
-#INCOMPLETE += basicStatMechProblemSet6Problem1.tex
-#INCOMPLETE += basicStatMechProblemSet6Problem2.tex
-#INCOMPLETE += basicStatMechProblemSet6Problem3.tex
+INCOMPLETE += basicStatMechProblemSet6Problem1.tex
+INCOMPLETE += basicStatMechProblemSet6Problem2.tex
+INCOMPLETE += basicStatMechProblemSet6Problem3.tex
+INCOMPLETE += basicStatMechProblemSet6Problem4.tex
+INCOMPLETE += basicStatMechProblemSet6Problem5.tex
+INCOMPLETE += basicStatMechProblemSet6Problem6.tex
+INCOMPLETE += basicStatMechProblemSet6Problem7.tex
# buildable:
#INCOMPLETE += dotBlade.tex
@@ -51,9 +55,13 @@ all :: peeters_layout.sty
all :: $(LOCAL_FILES) $(PDFS) .gitignore
#all :: ftMaxVacuum.pdf
-#basicStatMechProblemSet6.pdf : basicStatMechProblemSet6Problem1.tex
-#basicStatMechProblemSet6.pdf : basicStatMechProblemSet6Problem2.tex
-#basicStatMechProblemSet6.pdf : basicStatMechProblemSet6Problem3.tex
+basicStatMechProblemSet6.pdf : basicStatMechProblemSet6Problem1.tex
+basicStatMechProblemSet6.pdf : basicStatMechProblemSet6Problem2.tex
+basicStatMechProblemSet6.pdf : basicStatMechProblemSet6Problem3.tex
+basicStatMechProblemSet6.pdf : basicStatMechProblemSet6Problem4.tex
+basicStatMechProblemSet6.pdf : basicStatMechProblemSet6Problem5.tex
+basicStatMechProblemSet6.pdf : basicStatMechProblemSet6Problem6.tex
+basicStatMechProblemSet6.pdf : basicStatMechProblemSet6Problem7.tex
.gitignore : GNUmakefile
rm -f .gitignore
@@ -0,0 +1,29 @@
+%
+% Copyright © 2013 Peeter Joot. All Rights Reserved.
+% Licenced as described in the file LICENSE under the root directory of this GIT repository.
+%
+\input{../assignment.tex}
+\renewcommand{\basename}{basicStatMechProblem Set6}
+\renewcommand{\dirname}{notes/phy452/}
+\newcommand{\keywords}{Statistics mechanics, PHY452H1S}
+\newcommand{\dateintitle}{}
+\input{../peeter_prologue_print2.tex}
+
+\beginArtNoToc
+\generatetitle{PHY452H1S Basic Statistical Mechanics. Problem Set 6: Max entropy, fugacity, and Fermi gas}
+%\chapter{Max entropy, fugacity, and Fermi gas}
+\label{chap:basicStatMechProblemSet6}
+
+%\section{Disclaimer}
+%
+%This is an ungraded set of answers to the problems posed.
+
+\input{basicStatMechProblemSet6Problem1.tex}
+\input{basicStatMechProblemSet6Problem2.tex}
+\input{basicStatMechProblemSet6Problem3.tex}
+\input{basicStatMechProblemSet6Problem4.tex}
+%\input{basicStatMechProblemSet6Problem5.tex}
+%\input{basicStatMechProblemSet6Problem6.tex}
+%\input{basicStatMechProblemSet6Problem7.tex}
+
+\EndArticle
@@ -0,0 +1,45 @@
+%
+% 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}{
+%(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
+
+} % makeproblem
+
+\makeanswer{basicStatMech:problemSet6:1}{
+%\makeSubAnswer{XXX}{basicStatMech:problemSet6:1a}
+
+TODO.
+}
+
@@ -0,0 +1,34 @@
+%
+% 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}
+%(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}
+
+TODO.
+}
+
@@ -0,0 +1,21 @@
+%
+% 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}{
+%\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
+
+\makeanswer{basicStatMech:problemSet6:3}{
+%\makeSubAnswer{XXX}{basicStatMech:problemSet6:3a}
+
+TODO.
+}
+
@@ -0,0 +1,19 @@
+%
+% Copyright © 2013 Peeter Joot. All Rights Reserved.
+% Licenced as described in the file LICENSE under the root directory of this GIT repository.
+%
+\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.
+
+%\makesubproblem{}{basicStatMech:problemSet6:4a}
+} % makeoproblem
+
+\makeanswer{basicStatMech:problemSet6:4}{
+%\makeSubAnswer{XXX}{basicStatMech:problemSet6:4a}
+
+TODO.
+}
+
@@ -0,0 +1,14 @@
+%
+% Copyright © 2013 Peeter Joot. All Rights Reserved.
+% Licenced as described in the file LICENSE under the root directory of this GIT repository.
+%
+\makeoproblem{description}{basicStatMech:problemSet6:5}{Huang 16.3}{
+%\makesubproblem{}{basicStatMech:problemSet6:5a}
+} % makeproblem
+
+\makeanswer{basicStatMech:problemSet6:5}{
+%\makeSubAnswer{XXX}{basicStatMech:problemSet6:5a}
+
+TODO.
+}
+
@@ -0,0 +1,14 @@
+%
+% Copyright © 2013 Peeter Joot. All Rights Reserved.
+% Licenced as described in the file LICENSE under the root directory of this GIT repository.
+%
+\makeproblem{description}{basicStatMech:problemSet6:6}{Huang 16.4}{
+%\makesubproblem{}{basicStatMech:problemSet6:6a}
+} % makeproblem
+
+\makeanswer{basicStatMech:problemSet6:6}{
+%\makeSubAnswer{XXX}{basicStatMech:problemSet6:6a}
+
+TODO.
+}
+
@@ -0,0 +1,14 @@
+%
+% Copyright © 2013 Peeter Joot. All Rights Reserved.
+% Licenced as described in the file LICENSE under the root directory of this GIT repository.
+%
+\makeproblem{description}{basicStatMech:problemSet6:7}{Huang 16.6}{
+%\makesubproblem{}{basicStatMech:problemSet6:7a}
+} % makeproblem
+
+\makeanswer{basicStatMech:problemSet6:7}{
+%\makeSubAnswer{XXX}{basicStatMech:problemSet6:7a}
+
+TODO.
+}
+
View
@@ -11,3 +11,11 @@
#perl -p -i ./p basicStatMechLecture17.tex
~/bin/lgrep relStatMechExploration.tex | tee o ; . ./o
perl -p -i ./p relStatMechExploration.tex
+~/bin/lgrep basicStatMechProblemSet6.tex | tee o ; . ./o
+~/bin/lgrep basicStatMechProblemSet6Problem1.tex | tee o ; . ./o
+~/bin/lgrep basicStatMechProblemSet6Problem2.tex | tee o ; . ./o
+~/bin/lgrep basicStatMechProblemSet6Problem3.tex | tee o ; . ./o
+~/bin/lgrep basicStatMechProblemSet6Problem4.tex | tee o ; . ./o
+~/bin/lgrep basicStatMechProblemSet6Problem5.tex | tee o ; . ./o
+~/bin/lgrep basicStatMechProblemSet6Problem6.tex | tee o ; . ./o
+~/bin/lgrep basicStatMechProblemSet6Problem7.tex | tee o ; . ./o
@@ -83,8 +83,8 @@ \section*{}
\colorSectionsForArticle
% (a), (b), (c), ... numbering in ex (exercise pulled in by peeters_layout)
-\renewcommand{\QuestionNB}{\alph{Question}.\ }
-\renewcommand{\theQuestion}{\alph{Question}}
+%\renewcommand{\QuestionNB}{\alph{Question}.\ }
+%\renewcommand{\theQuestion}{\alph{Question}}
% have \cref fixup in book_layout.sty. Use this macro instead:
\newcommand{\eqnref}[1]{eq. (\ref{#1})}

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