Permalink
Browse files

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
  • Loading branch information...
1 parent e5668ed commit 84d8307b60d30db8562128219ab3f8fb4f1cefad @peeterjoot 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}
@@ -2,9 +2,9 @@
% 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}{
+\makeoproblem{description}{basicStatMech:problemSet6:6}{Huang 16.4}{
%\makesubproblem{}{basicStatMech:problemSet6:6a}
-} % makeproblem
+} % makeoproblem
\makeanswer{basicStatMech:problemSet6:6}{
%\makeSubAnswer{XXX}{basicStatMech:problemSet6:6a}
@@ -2,9 +2,9 @@
% 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}{
+\makeoproblem{description}{basicStatMech:problemSet6:7}{Huang 16.6}{
%\makesubproblem{}{basicStatMech:problemSet6:7a}
-} % makeproblem
+} % makeoproblem
\makeanswer{basicStatMech:problemSet6:7}{
%\makeSubAnswer{XXX}{basicStatMech:problemSet6:7a}
View
@@ -9,13 +9,10 @@
#perl -p -i ./p energyProbabilityPathriaQuestion.tex
#~/bin/lgrep basicStatMechLecture17.tex | tee o ; . ./o
#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
+#~/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
View
@@ -52,6 +52,7 @@
#~/bin/lgrep phaseSpaceChangeOfVars.tex | tee o ; . ./o
#~/bin/lgrep phaseSpaceChangeOfVarsSpherical.tex | tee o ; . ./o
#~/bin/lgrep poissonStirling.tex | tee o ; . ./o
+#~/bin/lgrep relStatMechExploration.tex | tee o ; . ./o
#~/bin/lgrep spinOneHalfAdditionReview.tex | tee o ; . ./o
#~/bin/lgrep stirlingLike.tex | tee o ; . ./o
#~/bin/lgrep thermodynamicIdentity.tex | tee o ; . ./o
@@ -68,7 +69,8 @@
#perl -p -i ./p basicStatMechProblemSet5Problem1.tex
#perl -p -i ./p basicStatMechProblemSet5Problem2.tex
#perl -p -i ./p basicStatMechProblemSet5Problem3.tex
+#perl -p -i ./p kittelCh3Zipper.tex
#perl -p -i ./p kittelRotationalPartition.tex
#perl -p -i ./p midterm2reflection.tex
#perl -p -i ./p pathriaHarmonicOscPertubation.tex
-perl -p -i ./p kittelCh3Zipper.tex
+#perl -p -i ./p relStatMechExploration.tex

0 comments on commit 84d8307

Please sign in to comment.