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modified: ../../env/.aspell.en.pws

modified:   basicStatMechProblemSet6Problem1.tex
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1 parent 1ca6259 commit 144befb5e4ee5eb330d09c150e9a2858896ee71a @peeterjoot committed Mar 31, 2013
Showing with 99 additions and 13 deletions.
  1. +12 −11 env/.aspell.en.pws
  2. +87 −2 notes/blogit/basicStatMechProblemSet6Problem1.tex
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@@ -1,4 +1,4 @@
-personal_ws-1.1 en 469
+personal_ws-1.1 en 470
CDF
Google
tuples
@@ -8,8 +8,8 @@ ParametericPlot
Indistinguishability
Exponentiating
extremum
-Ficks
Fick's
+Ficks
cgs
phasor
linestyle
@@ -88,8 +88,8 @@ Eikonal
Goldstein's
GPS
colinear
-hoc
Hmm
+hoc
Peeter's
Benard
ijk
@@ -162,8 +162,8 @@ Hestenes's
parametrizations
imaginaries
reparametrize
-n'l'm
nlm
+n'l'm
PDE
Lut
quantized
@@ -190,8 +190,8 @@ arctan
entropic
invertible
pion
-outermorphism
OuterMorphism
+outermorphism
QFT
rescaling
spinors
@@ -313,8 +313,8 @@ df
LIGO
spacetime
dH
-BT
dj
+BT
Routhian
dk
dL
@@ -342,9 +342,9 @@ Prandtl
ia
isync
inferometer
-eV
ib
iB
+eV
orthonormalization
ic
ie
@@ -359,16 +359,16 @@ im
Dekker's
ji
unnormalized
-jj
KE
+jj
elastostatics
amino
-ip
jk
+ip
indices
kj
-iu
kk
+iu
mc
mE
iz
@@ -381,8 +381,8 @@ Strang's
xyz
mk
resistive
-mn
kx
+mn
Eulerian
anticommutes
n'l
@@ -461,6 +461,7 @@ Sipe's
NFCM
Rindler
irrotational
+microcanonical
eigenvectors
lightlike
Hecht
@@ -149,7 +149,7 @@
\makeSubAnswer{}{basicStatMech:problemSet6:1b}
-For the ``action'' that we want to minimize, let's write
+For the ``action'' like quantity that we want to minimize, let's write
\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:380}
f = S - \alpha \sum_j p_j - \beta \sum_j E_j p_j,
@@ -293,5 +293,90 @@
\makeSubAnswer{}{basicStatMech:problemSet6:1c}
-TODO.
+Again write
+
+\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:580}
+f = S - \alpha \sum_j p_j - \beta \sum_j E_j p_j - \gamma \sum_j N_j p_j.
+\end{dmath}
+
+The unconstrained minimization requires
+
+\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:600}
+0
+= \PD{p_i}{f}
+= -\PD{p_i}{}
+\lr{
+\kB (\ln p_i + 1) + \alpha + \beta E_i + \gamma N_i
+},
+\end{dmath}
+
+or
+
+\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:620}
+p_i
+=
+\exp\lr{ -\alpha/\kB - 1 }
+\exp\lr{ -(\beta E_i + \gamma N_i)/\kB }.
+\end{dmath}
+
+The unit probability constraint requires
+
+\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:640}
+1
+= \sum_j p_j
+=
+\exp\lr{ -\alpha/\kB - 1 }
+\sum_j
+\exp\lr{ -(\beta E_j + \gamma N_j)/\kB },
+\end{dmath}
+
+or
+\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:660}
+\exp\lr{ -\alpha/\kB - 1 } =
+\inv{
+\sum_j
+\exp\lr{ -(\beta E_j + \gamma N_j)/\kB }
+}.
+\end{dmath}
+
+Our probability is then
+
+\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:680}
+\myBoxed{
+p_i =
+\frac{
+\exp\lr{ -(\beta E_i + \gamma N_i)/\kB }
+}{
+\sum_j
+\exp\lr{ -(\beta E_j + \gamma N_j)/\kB }
+}.
+}
+\end{dmath}
+
+The average energy $\expectation{E} = \sum_j p_j E_j$ and average number of particles $\expectation{N} = \sum_j p_j N_j$ are given by
+
+\begin{subequations}
+\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:700}
+\expectation{E} =
+\frac{
+E_i
+\exp\lr{ -(\beta E_i + \gamma N_i)/\kB }
+}{
+\sum_j
+\exp\lr{ -(\beta E_j + \gamma N_j)/\kB }
+}
+\end{dmath}
+\begin{dmath}\label{eqn:basicStatMechProblemSet6Problem2:720}
+\expectation{N} =
+\frac{
+N_i
+\exp\lr{ -(\beta E_i + \gamma N_i)/\kB }
+}{
+\sum_j
+\exp\lr{ -(\beta E_j + \gamma N_j)/\kB }
+}.
+\end{dmath}
+\end{subequations}
+
+The values $\beta$ and $\gamma$ are fixed implicitly by requiring simultaneous solutions of these equations.
}

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