text:
\section{Introduction}
\label{sec_intro}
It has been predicted that ultracold two-component Fermi gases with
mismatching Fermi surfaces support novel phases such as the
Fulde-Ferrell-Larkin-Ovchinnikov~\cite{FuldeFerrell1964,LarkinOvchinnikov1964,LarkinOvchinnikov1965} state
or other interesting phase separations, both in the
homogeneous and inhomogeneous systems~\cite{RevModPhys2008,CLoboPRL2006,DESheehyPRL2006,ABulgacPRA2007,ARecatiPRA2008,PilatiGiorginiPRL2008}.
In equal-mass systems, the two Fermi surfaces differ if the number of
fermions of species 1 and 2 differ, i.e., if the system exhibits a
spin imbalance~\cite{YShinPRL2006,GBPartridgePRL2006}.
Alternatively, mismatching Fermi surfaces arise if
species 1 and 2 have different masses~\cite{MIskinPRL2006,STWuPRB2006,CHPaoPRA2007,MMParishPRL2007,MIskinArxiv2008,CarlsonPRL2009,IBausmerthPRA2009,JStecherPRA2007}.
Experimentally, mass-imbalanced systems can be realized by simultaneously
trapping, e.g., $\!^6$Li and $\!^{40}$K~\cite{EWillePRL2008,MTaglieberPRL2008,FMSpiegelhalderPRA2010,LCostaPRL2010,ARidingerEPJD2011,DNaikEPJD2011,ATrenkwalderPRL2011}.
The application of an external magnetic field in the vicinity of a
Fano-Feshbach resonance allows for the realization of strongly-interacting
gases, thus motivating our studies of the unequal-mass Fermi gas at unitarity.

At the few-body level, two-component
unequal mass systems with short-range interspecies interactions and
mass ratio $\kappa$
are interesting because they support a variety of intriguing states~\cite{DBlumeReview2011}.
At unitarity, the three-body system consisting of two heavy and one light
particle  [the $(n_1,n_2)=(2,1)$ system]
with zero-range interspecies interactions supports no three-body
bound state in free space
for $\kappa \le 8.62$.  For $8.62 \le \kappa \le 13.61$, however,
the $(n_1,n_2)=(2,1)$
system can support a bound state whose binding energy is determined
by a three-body parameter~\cite{Petrov2003,WernerCastinPRA2006,NishidaSonTan2008,DBlumePRL2010,DBlumePRA2010}.
For yet larger mass ratios ($\kappa > 13.61$),
the three-body system in free space supports an infinite number of three-body
Efimov states, which are geometrically spaced~\cite{VNEfimov1970,EfimovLetter1972,Efimov1973}.
The spacing depends on the mass ratio $\kappa$ between the heavy and light
particles with the energy of the most deeply bound state being determined
by the so-called three-body Efimov parameter~\cite{EBraatenPhysRep2006}.

The objective of this paper is to investigate the thermodynamics
of inhomogeneous and homogeneous two-component Fermi gases
as a function of the mass ratio $\kappa$.
Throughout, we consider the unitary regime where the interspecies
$s$-wave scattering length diverges and determine the system properties
as functions of the polarization $P$, $P=(N_1-N_2)/(N_1+N_2)$, and the
temperature $T$. We are limiting ourselves to the so-called high temperature
regime, where the virial equation of state up to the third order is expected
to provide a valid description~\cite{HoPRL2004,HoMuellerPRL2004}.
For the trapped equal mass system,
the virial equation of state up to the third order
has been shown to be applicable over the temperature range of well above
the Fermi temperature $T_F$ down to about $T_F/2$~\cite{LiuPRL2009,HuNewJPhys2010,LiuPRA2010,TokyoScience2010,SalomonNature2010}.

Our key findings are: (i) The second order virial equation of state
in a harmonic trap is independent of the mass ratio $\kappa$.  The
mass ratio first enters at the third order in the virial expansion.
(ii) The equilibrium state of the trapped system at unitarity
is stable and favors a majority of heavy particles for $\kappa \ne 1$.
(iii) The equation of state of the homogeneous system
depends strongly on the mass ratio. The mass
ratio first enters at the zeroth order in the virial expansion.
(iv) The equilibrium state of the homogeneous system at unitarity
is stable and favors a
majority of heavy particles.
The equilibrium polarization exhibits an intricate dependence on the
third-order virial coefficients.

The paper is organized as follows.  Section~\ref{sec_virials} defines the
virial expansion and the first few virial expansion coefficients
for a two-component Fermi gas in a harmonic trap where both species feel the
same angular trapping frequency $\omega$.
The third order virial expansion coefficients at unitarity are determined as
functions of the temperature and the mass ratio.
Sections~\ref{sec_thermo2} and~\ref{sec_thermo3}
explore the virial equation of state of the
harmonically trapped and homogeneous two-component Fermi gas at unitarity.
Finally, Sec.~\ref{sec_conclusion} concludes.
Appendix~\ref{sec_thermo1} summarizes the virial expansion for
the harmonically trapped single-component Fermi gas
and connects the results with those for the two-component Fermi gas.

\section{Determination of the virial coefficients}
\label{sec_virials}
The model Hamiltonian $H(n_1,n_2)$
that describes unequal mass two-component Fermi gases
with $n_1$ heavy and $n_2$ light particles in a harmonic trap
with angular trapping frequency $\omega$ reads
\begin{align}
\label{eq_Hamiltonian}
& H(n_1,n_2) = \sum_{j=1}^{n_1} \left( \frac{-\hbar^2}{2 m_1}
\vec{\nabla}_{\vec{r}_j}^2 +
\frac{1}{2} m_1 \omega^2 \vec{r}_j^2 \right) \\
& + \sum_{j=n_1+1}^{n_1+n_2} \left( \frac{-\hbar^2}{2 m_2}
\vec{\nabla}_{\vec{r}_j}^2 +
\frac{1}{2} m_2 \omega^2 \vec{r}_j^2 \right) \nonumber
+ \sum_{i=1}^{n_1} \sum_{j=n_1+1}^{n_1+n_2} V_{tb}(r_{ij}),
\end{align}
where $m_1$ denotes the mass of the heavy species and $m_2$ denotes
the mass of the light species.
We define the mass ratio $\kappa$ as $\kappa=m_1/m_2$ ($\kappa \ge 1$).
In Eq.~(\ref{eq_Hamiltonian}), $\vec{r}_j$ denotes the
position vector of the $j^{th}\!$ atom measured with respect
to the trap center.
The interspecies interaction potential $V_{tb}$ is parameterized by a
zero-range potential,
$V_{tb}(r_{ij})=(2\pi\hbar^2 a_s/ \mu_{red})
\delta(\vec{r}_{ij})(\partial/\partial r_{ij}) r_{ij}$,
where $\mu_{red}$ denotes the reduced mass, $\mu_{red}=m_1m_2/(m_1+m_2)$,
$a_s$ the interspecies $s$-wave scattering length, and
$r_{ij}$ the interparticle distance, $r_{ij}=|\vec{r}_i-\vec{r}_j|$.

If the complete energy spectrum of the Hamiltonian given in
Eq.~(\ref{eq_Hamiltonian}) were known,
one could calculate thermodynamic quantities like the
average energy $U$ or the entropy $S$ from the
free energy $F$, $F=-k_B T \ln{Q_{n_1,n_2}}$,
using the canonical ensemble.  Here,
$k_B$ is Boltzmann's constant and $Q_{n_1,n_2}$ is
the canonical partition function~\cite{McQuarrie,HuangStatMech},
\begin{align}
\label{eq_canonicalPartFcta}
Q_{n_1,n_2} = \; \mbox{Tr} \; \exp{[-H(n_1,n_2)/(k_B T)]},
\end{align}
where Tr is the trace operator. To evaluate the trace, we insert
a complete set of eigenstates, yielding
\begin{align}
\label{eq_canonicalPartFctb}
Q_{n_1,n_2} = \sum_j \exp[-E_j^{(n_1,n_2)}/(k_B T)].
\end{align}
The summation index $j$
collectively denotes the complete set of quantum numbers allowed by symmetry.

In practice, it is easier to work in the grand canonical
ensemble, which can be considered to be a collection of canonical ensembles
with $n_1$ and $n_2$ particles of species 1 and 2
in thermal equilibrium with each other.
In the grand canonical ensemble,
the chemical potentials $\mu_1$ and $\mu_2$ of the two species are fixed.
This implies that the system is characterized by the average numbers
$N_1$ and $N_2$ of atoms of species 1 and 2.
The thermodynamic potential $\Omega^{(2)}$ of the two-component Fermi
gas in the grand canonical ensemble is~\cite{HuangStatMech}
\begin{align}
\label{eq_thermPot1a}
\Omega^{(2)} = & -k_B T \ln \mbox{Tr} \; \exp[-(H(n_1,n_2) \nonumber \\
&- \mu_1 n_1 - \mu_2 n_2)/(k_B T)],
\end{align}
where the trace operator now extends over $n_1$ and $n_2$.
Rewritten in terms of the fugacities $z_i$, $z_i=\exp[\mu_i/(k_B T)]$,
the thermodynamic potential reads
\begin{align}
\label{eq_thermPot1b}
\Omega^{(2)} & = -k_B T \ln \left[ \sum_{n_1=0}^{\infty}
  \sum_{n_2=0}^{\infty} Q_{n_1,n_2} z_1^{n_1} z_2^{n_2} \right].
\end{align}

Throughout, we are interested in the large $T$ limit where $z_i$ is small.
Taylor expanding Eq.~(\ref{eq_thermPot1b}) about $z_i=0$ to third order,
we find~\cite{LiuPRA2010}
\begin{align}
\label{eq_thermPotHighTa}
\Omega^{(2)} = & \; \Omega^{(1)}_1 + \; \Omega^{(1)}_2 + \Delta \Omega,
\end{align}
where
\begin{align}
\label{eq_singCompFGa}
\Omega_1^{(1)} & = -k_B T Q_{1,0} \sum_{n_1=1}^{\infty} b_{n_1,0}
\; z_1^{n_1}, \\
\label{eq_singCompFGb}
\Omega_2^{(1)} & = -k_B T Q_{1,0} \sum_{n_2=1}^{\infty} b_{0,n_2}
\; z_2^{n_2}
\end{align}
and
\begin{align}
\label{eq_thermPotHighTb}
\Delta \Omega = & - k_B T Q_{1,0} [\; b_{1,1} \; z_1 z_2 \; + \nonumber \\
& \; b_{1,2} \; z_1 z_2^2 \; + \; b_{2,1} \; z_1^2 z_2 \; + \; \cdots\; ].
\end{align}
Equations (\ref{eq_thermPotHighTa})-(\ref{eq_thermPotHighTb})
show that the thermodynamic potential of the two-component Fermi gas can
be written as a sum of the thermodynamic potentials $\Omega_1^{(1)}$
and $\Omega_2^{(1)}$ of components 1 and 2 and a correction term
$\Delta\Omega$.
The thermodynamic potentials $\Omega_1^{(1)}$ and $\Omega_2^{(1)}$ of the
single-component species 1 and 2 are written in terms of the expansion
or virial coefficients $b_{n_1,0}$ and $b_{0,n_2}$
(see Appendix~\ref{sec_thermo1})
and the canonical partition function $Q_{1,0}$ of a single heavy
particle of species 1.
The $b_{n_1,0}$ ($b_{0,n_2}$) with $n_1 > 1$ ($n_2>1$)
arise from the fermionic nature of the atoms
and can be viewed as corrections to the Boltzmann gas~\cite{McQuarrie,HuangStatMech}.
The correction term $\Delta\Omega$ contains
the expansion or virial coefficients $b_{n_1,n_2}$, which account
for the interactions between distinguishable fermions~\cite{notation1,notation2},
\begin{align}
\label{eq_v11}
b_{1,1} = & (Q_{1,1}-Q_{1,0}Q_{0,1})/Q_{1,0}, \\
\label{eq_v12}
b_{1,2} = & (Q_{1,2}-Q_{1,0}Q_{0,2} -b_{1,1}Q_{1,0}Q_{0,1})/Q_{1,0},
\end{align}
and
\begin{align}
\label{eq_v21}
b_{2,1} = & (Q_{2,1}-Q_{2,0}Q_{0,1} -b_{1,1}Q_{1,0}Q_{1,0})/Q_{1,0}.
\end{align}
In the non-interacting limit, i.e., for $a_s=0$,
we label the partition functions by a superscript NI. In this case,
$Q_{n_1,n_2}^{\rm{NI}}$ reduces to $Q_{n_1,0}Q_{0,n_2}$,
since $H(n_1,n_2)$ is separable for $a_s=0$.
In this limit,
$b_{1,1}$ vanishes since $Q^{\rm{NI}}_{1,1}$ equals $Q_{1,0}Q_{0,1}$.
Consequently,
$b_{1,2}$ and $b_{2,1}$ as well as all higher order virial coefficients
$b_{n_1,n_2}$ with $n_1+n_2>3$ vanish; thus, $\Delta\Omega$ is zero
for vanishing $a_s$.

To determine the $b_{n_1,n_2}$, one must know three things:
(i) all $b_{i,j}$ where $i \le n_1$ and $j<n_2$ or $i<n_1$ and $j \le n_2$;
(ii) all $Q_{i,0}$ and $Q_{0,j}$, where $i \le n_1$ and $j \le n_2$; and
(iii) the complete energy spectrum of the ($n_1,n_2$) system.
The $Q_{n_1,0}$ and $Q_{0,n_2}$
characterize the single-component Fermi gas and are
calculated in Appendix~\ref{sec_thermo1}.
In calculating the $Q_{n_1,n_2}$, it is convenient to express the total
energy $E^{(n_1,n_2)}$
as a sum of the center of mass energy $E_{\rm{CM}}$ and the relative energy
$E_{\rm{rel}}^{(n_1,n_2)}$\!,
$E^{(n_1,n_2)} = E_{\rm{CM}} + E^{(n_1,n_2)}_{\rm{rel}}\!\!$.
In Eqs.~(\ref{eq_v11})-(\ref{eq_v21}),
the center of mass contribution cancels
the $Q_{1,0}$ term in the denominator~\cite{LiuPRL2009}.

We now evaluate the $b_{n_1,n_2}$ at unitarity, i.e., for $1/a_s=0$.
The calculation of
$b_{1,1}$ is straightforward~\cite{LiuPRL2009}.
The relative energies of the $(n_1,n_2)=(1,1)$
system in a harmonic trap can be solved
exactly using zero-range $s$-wave interactions~\cite{Busch1998}.
The relative energies of the $(1,1)$ system at unitarity are
$E_{\rm{rel}}^{(1,1)} = (2 q+1/2) \hbar\omega$
for $l=0$ and $E_{\rm{rel}}^{(1,1)} = (2 q+l+3/2) \hbar\omega$
for $l=1,2,\ldots$,
where $q$ is a non-negative integer.
As the relative energy spectrum for $l>0$ is identical to the single particle
energy spectrum,
we have that $Q_{1,1}/Q_{1,0}$ is equal to $Q_{0,1}$ for states with $l>0$.
The remaining sum,
\begin{align}
\label{eq_b11a}
b_{1,1}=\sum_{q=0}^{\infty} \left( e^{-(2q+1/2)\tilde{\omega}} -
e^{-(2q+3/2)\tilde{\omega}} \right),
\end{align}
includes all $s$-wave states and can be calculated analytically~\cite{LiuPRL2009},
\begin{align}
\label{eq_b11b}
b_{1,1}= \frac{e^{\tilde{\omega}/2}}{1+e^{\tilde{\omega}}},
\end{align}
where $\tilde{\omega}$ denotes a dimensionless inverse temperature,
$\tilde{\omega}=\hbar\omega/(k_B T)$.
Equation (\ref{eq_b11b}) shows that $b_{1,1}$ is independent of
the mass ratio.
This is a direct consequence of the fact that
the relative two-body energy spectrum
is independent of the mass ratio.
Furthermore, like the single-component virial coefficients $b_n$
(see Appendix~\ref{sec_thermo1}), $b_{1,1}$ is an even function in
$\tilde{\omega}$.

To calculate $b_{1,2}$ and $b_{2,1}$, we need the complete relative energy
spectrum of the $(1,2)$ system (1 light atom and 2 heavy atoms)
and the $(2,1)$ system (2 light atoms and 1 heavy atom), respectively.
The three-body energies are conveniently expressed in terms of the quantity
$\bar{\kappa}$,
\begin{align}
\label{eq_massratio}
\bar{\kappa} =
\begin{cases}
1/\kappa & \text{for a majority of light species} \\
\kappa & \text{for a majority of heavy species}.
\end{cases}
\end{align}
At unitarity, the relative three-body energies for interspecies $s$-wave
zero-range interaction potentials can be written as
$E_{\rm{rel}}^{(n_1,n_2)}
=(2q+s_{l,\nu}+1)\hbar\omega$~\cite{WernerCastinPRA2006},
where $q=0,1,\ldots$,
and where the $s_{l,\nu}$ denote non-integer values.
The $s_{l,\nu}$ depend on the particle symmetry and $\bar{\kappa}$, and
are obtained by solving the five-dimensional hyperangular Schr\"odinger
equation~\cite{WernerCastinPRL2006,OIKartavtsevJPB2007,RittenhousePRA2010}.
The hyperangular eigenvalues were first obtained for
three identical bosons by Efimov~\cite{VNEfimov1970},
but have since been extended to
any particle symmetry and mass ratio by both Efimov
\cite{Efimov1973,EfimovLetter1972} and
others
\cite{WernerCastinPRL2006,OIKartavtsevJPB2007,RittenhousePRA2010}.
Application of Ref. \cite{RittenhousePRA2010} to the two-component system with
$\bar{\kappa}$, relative angular momentum $l$, and
parity $(-1)^l$ at unitarity yields
\begin{align}
\label{eq_trans}
-& \frac{\sqrt{4 \pi} \; \Gamma(l+3/2)}{\Gamma\!\left(\frac{1+l-s_{l,\nu}}{2}\right)
 \Gamma\!\left(\frac{1+l+s_{l,\nu}}{2}\right)} =
\left(\frac{-\bar{\kappa}}{\bar{\kappa} + 1}\right)^l  \times \\
& _2F_1 \!\!
\left(1+\frac{l-s_{l,\nu}}{2},1+
\frac{l+s_{l,\nu}}{2},l+\tfrac{3}{2},
\frac{\bar{\kappa}^2}{(\bar{\kappa}+1)^2}\right)\!, \nonumber
\end{align}
where $\Gamma$ is the gamma function and
$\!\;_2F_1$ is the hypergeometric function.
Equation~(\ref{eq_trans}) has a spurious
root at $s_{0,\nu}=2$ for all $\bar{\kappa}$;
this root needs to be removed ``by hand".

Solid lines in Fig.~\ref{fig_sln}
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig1.eps}
\vspace*{0.1cm}
\caption{(Color online) Hyperangular eigenvalues $s_{l,\nu}$
as a function of $\bar{\kappa}$ for (a) $l=0$, (b) $l=1$, and
(c) $l=2$ at unitarity. Note that $\bar{\kappa}$ is shown on a log scale.
Solid lines show the $s_{l,\nu}$ obtained by solving Eq.~(\ref{eq_trans}),
dashed lines show the limiting
expressions, Eqs.~(\ref{eq_transPerta})-(\ref{eq_transPertd}),
for $\bar{\kappa} \ll 1$, and thin dotted lines show the non-interacting
$s^{\rm{NI}}_{l,\nu}$.
}\label{fig_sln}
\end{figure}
show the $s_{l,\nu}$ as a function of $\bar{\kappa}$
for the three lowest relative angular momenta, i.e., for $l=0$, 1, and 2.
The hyperangular quantum number $\nu$, $\nu=0,1,\ldots$,
counts the number of times that the slope of $s_{l,\nu}$
changes sign.
The thin dotted lines indicate the non-interacting limits,
$s^{\rm{NI}}_{0,\nu}=2\nu+4$ for $l=0$
and $s^{\rm{NI}}_{l,\nu}=2\nu+l+2$ for $l>0$.
The dashed lines show an expansion of Eq.~(\ref{eq_trans}) for
$\bar{\kappa} \ll 1$,
\begin{align}
\label{eq_transPerta}
s_{0,0} & \approx 2 + \frac{16}{3 \pi^2} \; \bar{\kappa}^2, \\
\label{eq_transPertb}
s_{0,2 j} & \approx 4 j + 2 + \frac{8}{\sqrt{3}\pi}\sqrt{j (j+1)}
\; \bar{\kappa}, \\
\label{eq_transPertc}
s_{0,2j-1} & \approx 4 j + 2 - \frac{8}{\sqrt{3}\pi}\sqrt{j (j+1)}
\; \bar{\kappa},
\end{align}
where $j=1$, 2, $\ldots$, and
\begin{align}
\label{eq_transPertd}
s_{l,\nu} & \approx 2 \nu + l + 1 + \frac{(-1)^{l+\nu} \;
\Gamma(l+\nu+1)}{\sqrt{\pi} \; \Gamma(l+3/2) \; \Gamma(\nu+1)}
\; \bar{\kappa}^l
\end{align}
for $l>0$.
For $l=0$, the lowest eigenvalue varies quadratically with
$\bar{\kappa}$ in the
small $\bar{\kappa}$ regime. The higher lying $s_{l,\nu}$ values for
$l=0$ appear in
pairs as $\bar{\kappa} \rightarrow 0$, with a splitting that is linear in
$\bar{\kappa}$.
For $l>0$, in contrast, all eigenvalues are non-degenerate as
$\bar{\kappa} \rightarrow 0$.  The lowest $l=1$ eigenvalue equals 2
for $\bar{\kappa}=0$,
crosses 1 for $\bar{\kappa} \approx 8.62$, and becomes purely imaginary for
$\bar{\kappa} \approx 13.61$.  The latter point indicates the onset of Efimov
physics~\cite{Petrov2003}.  For $8.62 \le \bar{\kappa} \le 13.61$,
non-universal three-body states can exist~\cite{Petrov2003,WernerCastinPRA2006,NishidaSonTan2008,DBlumePRL2010,DBlumePRA2010}.
Similar behavior is found for the lowest $s_{l,\nu}$ values for higher odd
angular momenta.  The $s_{l,\nu}$ values for even $l$ are greater than 1 for
all $\bar{\kappa}$, indicating the absence of non-universal and
Efimov physics in the even $l$ angular momentum channels.
For $l=1,3,5,\ldots$ [see Fig.~\ref{fig_sln}(b) for $l=1$],
the $s_{l,\nu}$ with $\nu>0$ begin one unit below the non-interacting value in
the $\bar{\kappa}=0$ limit and decrease to the next lower non-interacting value
in the limit $\bar{\kappa}\rightarrow\infty$.
For $l=2,4,6,\ldots$ [see Fig.~\ref{fig_sln}(c) for $l=2$],
the $s_{l,\nu}$ begin one unit below the non-interacting values in
the $\bar{\kappa}=0$ limit and approach the non-interacting values
in the limit $\bar{\kappa} \rightarrow \infty$.
In the limit $\bar{\kappa} \rightarrow \infty$, all states save those
that describe Efimov physics behave like non-interacting states.
The two masses are so heavy compared to the single light particle that
even infinitely strong interactions cannot mediate a lowering of the energy.

Now that we have the $s_{l,\nu}$, we can calculate
$Q_{1,2}$ and $Q_{2,1}$, and thus $b_{1,2}$ and $b_{2,1}$.
We restrict ourselves to the regime where Efimov physics is absent (i.e., we
limit ourselves to \
$\bar{\kappa} \le 13.61$) and we assume that the three-body system
behaves fully universal (i.e., we assume the absence of three-body resonances
in the regime $8.62 \le \bar{\kappa} \le 13.61$).  Under these assumptions,
the virial coefficient $b_{2,1}$, Eq.~(\ref{eq_v21}), can be written as
\begin{align}
\label{eq_b21a}
b_{2,1} = \sum_{l=0}^{\infty} b_{2,1}(l),
\end{align}
where
\begin{align}
\label{eq_b21}
b_{2,1}(l) = \sum_{\nu=0}^{\infty}\sum_{q=0}^{\infty}
(2l+1)\biggr[e^{-(2q+s_{l,\nu}+1)\tilde{\omega}}\nonumber \\
- e^{-(2q+s_{l,\nu}^{\rm{NI}}+1)\tilde{\omega}}
- e^{-(2q+1/2+2\nu+l+3/2)\tilde{\omega}} \nonumber \\
+ e^{-(2q+3/2+2\nu+l+3/2)\tilde{\omega}}\biggr].
\end{align}
The first and second terms in the square bracket on the right hand side
of Eq.~\eqref{eq_b21}
arise from the $Q_{2,1}/Q_{1,0}$ and $Q_{2,0}$ terms,
respectively, while the third
and fourth terms in the square bracket on the right hand side of
Eq.~\eqref{eq_b21} arise from the $-b_{1,1}Q_{1,0}$ term.
For $l>0$, the second and fourth terms cancel,
while the third term can be rewritten in terms of $s_{l,\nu}^{\rm{NI}}$.
Performing the sum over $q$ analytically, we find
\begin{align}
\label{eq_s12L}
b_{2,1}(l>0) = & \frac{e^{2\tilde{\omega}}}{e^{2\tilde{\omega}}-1} \times \\
& \biggl[(2l+1) \sum_{\nu=0}^{\infty}
\left(e^{-(s_{l,\nu}+1) \tilde{\omega}} - e^{-s^{\rm{NI}}_{l,\nu}
\tilde{\omega}}\right)\biggr] \nonumber.
\end{align}
For $l=0$, the second and fourth terms do not completely cancel and
contribute, after the sum over $q$ is done analytically, a term proportional
to $\exp(-3\tilde{\omega})$.
Combining the first and third terms in the square bracket on the
right hand side of Eq.~\eqref{eq_b21}, we find
\begin{align}
\label{eq_s12}
b_{2,1}(l=0) = & \frac{e^{2\tilde{\omega}}}{e^{2\tilde{\omega}}-1}
\biggl[e^{-3\tilde{\omega}} - e^{-2 \tilde{\omega}} +   \\
& \sum_{\nu=0}^{\infty}\left(e^{-(s_{0,\nu}+1)\tilde{\omega}}
- e^{-s^{\rm{NI}}_{0,\nu}\tilde{\omega}}\right)\biggr] \nonumber.
\end{align}
Since the mass ratio dependence only enters through the $s_{l,\nu}$,
Eqs.~\eqref{eq_b21a}-\eqref{eq_s12} remain valid for $b_{1,2}$.
In practice, we calculate the first 10,000 $s_{l,\nu}$ values for each $l$,
$l=0-50$, and determine the $b_{1,2}(l)$ and
$b_{2,1}(l)$ using Eqs.~(\ref{eq_s12L}) and
(\ref{eq_s12}).
We find that we obtain better convergence if we employ the same cutoff on the
terms that involve $s_{l,\nu}$ and $s_{l,\nu}^{\rm{NI}}$ than if we perform the
sum over $\nu$ that involves the non-interacting
$s_{l,\nu}^{\rm{NI}}$ values analytically.

Figure~\ref{fig_Lcontributions}
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig2.eps}
\vspace*{0.1cm}
\caption{(Color online)
Angular momentum contributions $b_{2,1}(l)$ for $l=0-4$ for
(a) $\bar{\kappa}=1$ and (b) $\bar{\kappa}=6.67$
as a function of $\tilde{\omega}$ at unitarity.
Solid, dotted, dashed, dash-dotted, and dash-dot-dotted lines show
$b_{2,1}(l)$ for $l$=0, 1, 2, 3, and 4, respectively.
}\label{fig_Lcontributions}
\end{figure}
shows the $b_{2,1}(l)$ for the first
five angular momenta for (a) $\bar{\kappa}=1$ and (b) $\bar{\kappa}=6.67$.
The $b_{2,1}(l)$ are negative for even $l$ and positive for odd $l$ for all
$\tilde{\omega}$.
The leading contribution for any mass ratio
comes from the solution with $\nu=0$. For odd $l$, we have
$s_{l,0}+1<s_{l,0}^{\rm{NI}}$ [see Fig.~\ref{fig_sln}(b) for $l=1$],
implying that the $b_{2,1}(l)$
are positive for all $\tilde{\omega}$ and $\bar{\kappa}$.
For even $l$, we have $s_{l,0}+1>s_{l,0}^{\rm{NI}}$
[see Fig.~\ref{fig_sln}(c) for $l=2$], implying that the $b_{2,1}(l)$
are negative.
For $l=0$, the $-\exp(-2\tilde{\omega})$ term gives the leading contribution,
implying that $b_{2,1}(l)$ is negative for all $\tilde{\omega}$
and $\bar{\kappa}$.
A similar analysis can be conducted for the $b_{1,2}(l)$.
The alternating sign of the $b_{1,2}(l)$ and the $b_{2,1}(l)$ implies that
the full virial coefficients $b_{1,2}$ and $b_{2,1}$ converge like an
alternating series as more $l$ terms are included.  We find that the
convergence rate decreases as $\bar{\kappa}$ increases.

Figure \ref{fig_signChange234}
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig3.eps}
\vspace*{0.1cm}
\caption{(Color online)
Solid, dashed, dash-dotted, and dash-dot-dotted
lines show the virial coefficient
$b_{2,1}$ as a function of $\tilde{\omega}$ for
$\bar{\kappa}=2,3,4$ and 6.67. The thin dotted line at 0 is
a guide to emphasize the sign change of $b_{2,1}$ for $\bar{\kappa}=4$ around
$\tilde{\omega}\approx 1.67$.
}\label{fig_signChange234}
\end{figure}
shows the virial coefficients
$b_{2,1}$ for $\bar{\kappa}=2,3,4$ and $6.67$ (from bottom to top) as a
function of $\tilde{\omega}$.
The virial coefficient $b_{2,1}$ for $\bar{\kappa}=2$ and 3 is negative
for all $\tilde{\omega}$.
The virial coefficient for $\bar{\kappa}=4$ is negative for large
$\tilde{\omega}$, changes sign at $\tilde{\omega} \approx 1.67$,
and is positive for small $\tilde{\omega}$.
The virial coefficient for $\bar{\kappa}=6.67$
changes sign at a much lower temperature, i.e., at
$\tilde{\omega} \approx 5.59$.
The sign change of $b_{2,1}$ can be attributed to the fact that the negative
$l=0$ contribution increases slower in magnitude than the positive $l=1$
contribution with increasing $\bar{\kappa}$
(see Fig.~\ref{fig_Lcontributions}), so that
the positive $l=1$ contribution dominates for sufficiently large
$\bar{\kappa}$.
In the high temperature limit, $b_{2,1}$ first becomes positive for
$\bar{\kappa}\approx 3.11$.
In the low temperature regime, $b_{2,1}$ remains negative for
$\bar{\kappa}<8.62$ and changes sign when $s_{1,0}=1$, i.e.,
the $\bar{\kappa}$ value beyond which three-body resonances can
appear~\cite{Petrov2003,WernerCastinPRA2006,NishidaSonTan2008,DBlumePRL2010,DBlumePRA2010}.
We find that the virial coefficient $b_{1,2}$ is negative for all
$\bar{\kappa}$ and $\tilde{\omega}$.
We note that a general framework for calculating $b_{1,2}$ and $b_{2,1}$ for
unequal-mass systems is given in Ref.~\cite{Leyronas2011};
however, this reference did not provide numerical values for
$b_{1,2}$ and $b_{2,1}$.
The fact that the third order virial coefficient $b_{2,1}$ changes sign
as $\bar{\kappa}$ increases suggests that
the higher order virial coefficients might also change sign.
Throughout this paper, we restrict our analysis to systems with
$n_1+n_2 \le 3$.

In the high temperature limit, we Taylor expand the $b_{n_1,n_2}$
($n_1+n_2=3$)~\cite{auxMaterial},
\begin{align}
\label{eq_bhighT}
b_{n_1,n_2} = b_{n_1,n_2}^{(0)} + b_{n_1,n_2}^{(2)}\tilde{\omega}^2
+ b_{n_1,n_2}^{(4)}\tilde{\omega}^4
+ \cdots.
\end{align}
To our calculated accuracy, we find that the odd powers vanish, i.e., the
virial coefficients $b_{1,2}$ and $b_{2,1}$
appear to be even functions in $\tilde{\omega}$, similar to
their single-component counterparts.
For the equal-mass system, the expression~\eqref{eq_bhighT} was considered
in Ref.~\cite{LiuPRL2009}.
The $b_{n_1,n_2}^{(0)}$ term is independent
of $\tilde{\omega}$ and thus universal.
The higher order corrections $b^{(n>0)}_{1,2}$ and $b^{(n>0)}_{2,1}$ are
non-universal, since they depend on the harmonic trapping potential through
$\tilde{\omega}$.
The supplementary material~\cite{auxMaterial}
tabulates the $b_{1,2}^{(n)}$ and $b_{2,1}^{(n)}$,
$n=0,2,4$ and 6, as a function of $\bar{\kappa}$.
The $b_{1,2}^{(n)}$ and $b_{2,1}^{(n)}$ with $n=0,2,$ and 4 are shown in
Fig.~\ref{fig_orders024} as a
function of $\bar{\kappa}$, where $\bar{\kappa}$ is shown on a log scale.
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig4.eps}
\vspace*{0.1cm}
\caption{(Color online)
Circles show the universal parts $b_{1,2}^{(0)}$ ($\bar{\kappa}<1$)
and $b_{2,1}^{(0)}$ ($\bar{\kappa}>1$) of the virial coefficients
as a function of $\bar{\kappa}$.
The inset (the axes use the same labels as the main panel) shows
the first non-universal corrections $b_{1,2}^{(2)}$ and $b_{2,1}^{(2)}$
(squares) and
the second non-universal corrections $b_{1,2}^{(4)}$ and $b_{2,1}^{(4)}$
(triangles)
as a function of $\bar{\kappa}$.
The lines are guides to the eye.
}\label{fig_orders024}
\end{figure}
The universal virial coefficient $b_{2,1}^{(0)}$ has an infinite slope at
$\bar{\kappa}=13.61$ where Efimov physics sets in.
For $\bar{\kappa} > 13.61$ (not shown), $b_{2,1}^{(0)}$ is expected to diverge
due to the contributions of the infinite sequence of Efimov trimers.
In the limit $\bar{\kappa}\rightarrow0$,
the $b_{1,2}(l)$ with $l>0$ vanish and we have
\begin{align}
\label{eq_b12w0limit}
b_{1,2}(\bar{\kappa}=0) = & \frac{-e^{2\tilde{\omega}}}
{(e^{\tilde{\omega}}+1)^2(e^{2 \tilde{\omega}} + 1)} \nonumber \\
\approx & -\frac{1}{8} + \frac{3}{32}\tilde{\omega}^2
- \frac{3}{64}\tilde{\omega}^4 + \mathcal{O}(\tilde{\omega}^6).
\end{align}

\section{Thermodynamics of the trapped mass-imbalanced two-component fermi gas}
\label{sec_thermo2}
This section considers the thermodynamics of the harmonically trapped
two-component Fermi gas at unitarity as a function of $\kappa$. For the
equal-mass case, we refer the reader to Refs.~\cite{LiuPRL2009,LiuPRA2010,HuNewJPhys2010,SalomonNature2010,TokyoScience2010}.
We measure our temperature $T$ in units of the semi-classical Fermi temperature
$T_F$ of a single-component Fermi gas with $N/2$ particles of mass $m_1$,
$T_F=[6(N/2)]^{1/3}\hbar \omega/k_B$~\cite{notation3}.
Correspondingly, our energy is measured in units of $E_F$, where $E_F=k_BT_F$.
Our starting point is the thermodynamic potential $\Omega^{(2)}$\!,
Eq.~\eqref{eq_thermPotHighTa}, with $n_1 + n_2 \le 3$ (i.e., including
$b_{1,1}$, $b_{1,2}$, and $b_{2,1}$).

The two coupled number equations, derived from
$N_i=-\partial \Omega^{(2)} / \partial \mu_i$ ($i=1$ and 2), are
\begin{align}
\label{eq_N1}
\frac{N_1}{Q_{1,0}} = f_1(z_1)
+  b_{1,1}z_1z_2 + 2 b_{2,1}z_1^2z_2 + b_{1,2}z_1z_2^2
\end{align}
and
\begin{align}
\label{eq_N2}
\frac{N_2}{Q_{1,0}} = f_2(z_2)
+  b_{1,1}z_1z_2 + b_{2,1}z_1^2z_2 + 2 b_{1,2}z_1z_2^2.
\end{align}
Throughout, we use the full temperature dependent expressions for
the virial coefficients, and
\begin{align}
\label{eq_fz1}
f_1(z_1) = \sum_{n_1=0}^{\infty} n_1 b_{n_1,0} z_1^{n_1}
\end{align}
and
\begin{align}
\label{eq_fz2}
f_2(z_2) = \sum_{n_2=0}^{\infty} n_2 b_{0,n_2} z_2^{n_2}.
\end{align}
Equations~\eqref{eq_fz1} and~\eqref{eq_fz2} can be employed if
$z_i < e^{3\tilde{\omega}/2}$, $i=1,2$.
For larger $z_i$,
useful parameterizations of Eqs.~\eqref{eq_fz1} and~\eqref{eq_fz2} are
\begin{align}
\label{eq_fz}
f_i(z_i) & \approx -\mbox{Li}_3(-z_i)  \\
& + \tfrac{1}{8} \tilde{\omega}^2
\left[-\mbox{Ln}(1+z_i)-\mbox{Li}_3(-z_i) \right] \nonumber \\
& + \tfrac{1}{1920} \tilde{\omega}^4 \left[ \frac{17 z_i}{(1+z_i)^2} -
30 \mbox{Ln}(1+z_i) - 13 \mbox{Li}_3(-z_i) \right] \nonumber \\
& + \mathcal{O}(\tilde{\omega}^6), \nonumber
\end{align}
where Ln is the natural logarithm function and Li is the polylogarithm
function.
For the temperatures considered in this paper, terms of order
$\tilde{\omega}^6$ and higher are negligible.
Equations~\eqref{eq_fz1} and~\eqref{eq_fz2} treat the non-interacting
``reference pieces'' $f_1(z_1)$ and $f_2(z_2)$ as an infinite series
in the fugacities and the interacting pieces at third order in the fugacities.
Alternatively, one may consider truncating the non-interacting and interacting
pieces at the same order in the fugacities.
We have opted for the former approach for two reasons:
(i) In the limit that $P=1$ or $-1$, we recover the exact results of the
single-component Fermi gas.
(ii) In the regime where the $z_i$ are small, i.e., in the regime where the
virial equation of state up to third order is expected to provide reliable
results, we find fairly small differences
[e.g., $<0.06\%$ in Fig.~\ref{fig_fugacities}(a)]
between the approaches that
treat the non-interacting piece at infinite order and third order in the
fugacities.
For a given temperature $T$, polarization $P$, and total number of particles
$N$,
we solve Eqs.~\eqref{eq_N1} and \eqref{eq_N2}
self-consistently for $z_1$ and $z_2$.

Figure~\ref{fig_fugacities}
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig5.eps}
\vspace*{0.1cm}
\caption{(Color online) Trapped system at unitarity.
Solid and dashed lines (dash-dotted and dotted lines)
show the fugacities $z_1$ and $z_2$, respectively, for $\kappa=1$
($\kappa=13$) as a function of the polarization $P$ for
$N=5\times 10^7$ and
(a) $T/T_F=1$ and
(b) $T/T_F=1/2$.
}\label{fig_fugacities}
\end{figure}
shows the fugacities $z_1$ and $z_2$ at unitarity
as a function of the polarization for
(a) $T/T_F=1$ and (b) $T/T_F=1/2$ for $\kappa=1$ (solid and dashed lines)
and $\kappa=13$ (dash-dotted and dotted lines).
The fugacities for $T/T_F=1$ are smaller than
0.35 for both mass ratios considered,
suggesting that the virial equation of state can
likely be trusted for $T/T_F \ge 1$.
For $T/T_F=1/2$, in contrast, the fugacities are as large as 3.5,
suggesting that the description is at best qualitatively correct.
On the scale shown, the fugacities for $T/T_F=1$ and $\kappa=1$ are nearly
indistinguishable from those at the same temperature but higher mass ratio.
This makes sense intuitively, since the fugacities are small
in the high temperature regime,
thereby suppressing higher order terms in the virial expansion.
At lower temperatures [see Fig.~\ref{fig_fugacities}(b)], the higher order
virial coefficients become more important, as evidenced by the fact that the
fugacities show a notable dependence on the mass ratio.

In the following, we use the virial equation of state to
determine the free energy $F$,
$F = \Omega^{(2)} + \mu_1 N_1 + \mu_2 N_2$, the entropy $S$,
$S=-\partial \Omega^{(2)}/\partial T$
(calculated for fixed $\omega$, $\mu_1$ and $\mu_2$),
and the energy $U$, $U=F+TS$, as a function of the
mass ratio $\kappa$, the polarization $P$, and the temperature $T$
at unitarity.
Solid and dotted lines in Fig.~\ref{fig_freeEnergy1}
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig6.eps}
\vspace*{0.1cm}
\caption{(Color online) Free energy per particle $F/N$ ($N=5\times 10^7$)
as a function of the polarization $P$ for
(a) $T/T_F=1$ and
(b) $T/T_F=1/2$
for the trapped system at unitarity.
Alternating solid and dotted curves, from top to bottom,
show the free energy for $\kappa=1-13$ in units of 1.
The dashed lines connect the $(F_{eq},P_{eq})$ points for different $\kappa$.
}\label{fig_freeEnergy1}
\end{figure}
show the free energy per particle $F/N$ as a function of the polarization for
(a) $T/T_F=1$ and (b) $T/T_F=1/2$.
In both panels, the alternating solid and dotted lines, from top to bottom,
correspond to mass ratios $\kappa=1,2,\ldots,13$.
For the equal mass system, the minimum of the free energy occurs for $P=0$,
i.e., for equal numbers of spin-up and spin-down fermions.
As $\kappa$ increases, the minimum of the free energy moves to positive
polarizations, i.e., to a majority of heavy particles.  We refer to the
$(F,P)$ values at which the free energy is minimized as $(F_{eq},P_{eq})$.
Dashed lines in Figs.~\ref{fig_freeEnergy1}(a) and \ref{fig_freeEnergy1}(b)
connect the $(F_{eq},P_{eq})$ values for different $\kappa$.
For both temperatures, $P_{eq}$ increases with increasing $\kappa$,
which can be understood by realizing that the three-body system
consisting of two heavy atoms and one light atom has a lower energy than
the three-body system consisting of one heavy atom and two light atoms.
Moreover, for a given $\kappa$, $P_{eq}$ increases with decreasing temperature.
This trend makes sense, since we expect interaction
effects to become more important as the temperature decreases.

To obtain an analytical expression for $P_{eq}$, we take the derivative of
$F$ with respect to $P$ at fixed temperature and fixed number of particles.
We find that the extremum of $P$ occurs when the two fugacities are equal
(i.e., $z_1=z_2=z_{eq}$).
This restriction allows for a straightfoward calculation of $z_{eq}$, namely
\begin{align}
\label{eq_zeq}
\frac{N}{Q_{1,0}} = f_1(z_{eq})+f_2(z_{eq})+2 b_{1,1}z_{eq}^2
+ 3(b_{2,1}+b_{1,2})z_{eq}^3
\end{align}
can be solved self-consistently for fixed $N$ and $T$.
Up to third order in $z_{eq}$, $P_{eq}$ can be written as
\begin{align}
\label{eq_Feq}
P_{eq} = \frac{N}{Q_{1,0}} (b_{2,1}-b_{1,2}) z_{eq}^3.
\end{align}
$P_{eq}$ is equal to
zero for all $\kappa$ if the virial equation of state terminates at the
second order in the fugacity.  At the third order, we find $P_{eq}=0$ for
$\kappa=1$, since $b_{1,2}$ equals $b_{2,1}$ in this case.
For $\kappa>1$, however, $P_{eq}$ is positive since $b_{2,1}>b_{1,2}$ for all
$\kappa>1$.
Interestingly, the sign change of $b_{2,1}$ in the high temperature
regime for $\kappa \approx 3.11$ does not lead to a discontinuity or a sign
change of $P_{eq}$ since $b_{1,2}$ is finite and negative for all $\kappa$.
For small deviations from $\kappa=1$, we find that $P_{eq}$ and $F_{eq}$ change
linearly and quadratically, respectively, as a function of $\kappa$ for
fixed temperature and fixed number of particles. We find that
$\partial^2 F/\partial P^2$, calculated for fixed $T$ and $N$, is greater
than zero for all polarizations and mass ratios considered, which implies
that the system is stable at this level of approximation.

To discuss the dependence of the entropy $S$ and the energy $U$ on the
temperature and the polarization, we focus on the mass ratio $\kappa=6.67$,
i.e., on the experimentally realizable K-Li system at unitarity~\cite{EWillePRL2008,MTaglieberPRL2008,FMSpiegelhalderPRA2010,LCostaPRL2010,ARidingerEPJD2011,DNaikEPJD2011,ATrenkwalderPRL2011}.
Thin dotted and solid lines in Fig.~\ref{fig_thermoCombined}(a)
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig7.eps}
\vspace*{0.1cm}
\caption{(Color online) Thermodynamic observables $F/N$, $S/N$,
and $U/N$ as a function
of the polarization $P$ for $\kappa=6.67$ and $N=5\times 10^7$
for the trapped system at unitarity.
Alternating dotted and solid lines show
(a) the free energy per particle $F/N$
for $T/T_F=1,1.2,\ldots,1.8$ (from top to bottom),
(b) the entropy per particle $S/N$
for $T/T_F=0.95,0.96,\ldots,1.04$ (from bottom at $P\approx 0$
to top at $P \approx 0.5$), and
(c) the energy per particle $U/N$
for $T/T_F=0.99,1,\ldots ,1.06$ (from bottom at $P \approx 0.5$
to top at $P\approx0.1$).
In panel (a), the dashed line connects
$(F_{eq},P_{eq})$ values for different $T$.
In panels (b) and (c), the dashed lines show $S/N$ for $U/N=2.91E_F$ and
$U/N$ for $S/N=5.73k_B$, respectively.
In all three panels,
the vertical lines indicate the value of $P_{eq}$ for $T/T_F=1$.
}\label{fig_thermoCombined}
\end{figure}
show the free energy per particle as a function of the polarization for
$T/T_F=1,1.2,\ldots,1.8$ (from top to bottom).
The dashed line connects the $(F_{eq},P_{eq})$ values for different $T$
(but fixed $\kappa$ and $N$).
The vertical solid line marks the value $P_{eq}$ for $T/T_F=1$,
where $F/N=-2.82E_F$, $S/N=5.73k_B$, and $U/N=2.91E_F$.
Thin solid and dotted lines in
Fig.~\ref{fig_thermoCombined}(b) show the entropy per particle $S/N$ as
a function of the polarization for $T/T_F=0.95,0.96,\ldots,1.04$ (from bottom
at $P\approx 0$ to top at $P \approx 0.5$).
Similarly, thin solid and dotted lines in Fig.~\ref{fig_thermoCombined}(c)
show the energy per particle $U/N$ as a function of the polarization for
$T/T_F=0.99,1,\ldots ,1.06$ (from bottom at $P \approx 0.5$
to top at $P\approx0.1$).
The dashed lines in Figs.~\ref{fig_thermoCombined}(b)
and~\ref{fig_thermoCombined}(c) connect points of constant $U/N=2.91E_F$
and $S/N=5.73k_B$, respectively (i.e., the values of $U/N$ and $S/N$
at $P_{eq}$ for $T/T_F=1$).
As expected, the entropy for a fixed energy and the energy for a fixed
entropy are concave and convex functions of the polarization, respectively.
It can be seen that the entropy per particle $S/N$ is maximized at $P_{eq}$
while the energy per particle $U/N$ is minimized at $P_{eq}$.
The fact that $-\partial^2 S/\partial P^2$ ($\partial^2 U / \partial P^2$) is
greater than zero for fixed $U$ and $N$ (for fixed $S$ and $N$) is
consistent with the fact that the system is stable as concluded earlier from
the fact that $\partial^2 F / \partial P^2$ is greater than zero for fixed
$T$ and $N$.  Lastly, we note that the maximum of the isotherm $S/N$
moves to smaller $P$ with decreasing $T$ and that the minimum of
the isotherm $U/N$ moves to larger $P$ with decreasing $T$.

The thermodynamic quantities shown in
Figs.~\ref{fig_fugacities}-\ref{fig_thermoCombined}
have been obtained for $N=5 \times 10^7$ particles.
For fixed $T/T_F$, the actual temperature $T$ decreases with decreasing $N$,
since
$T_F$ decreases with decreasing $N$.  Correspondingly, $\tilde{\omega}$
increases with decreasing $N$ (again, assuming fixed $T/T_F$).
Since the leading
order non-universal corrections scale with $\tilde{\omega}^2$
[see Eqs.~\eqref{eq_bhighT} and~\eqref{eq_fz}],
the non-universal corrections are expected to be negligible for large $N$,
but not for very small $N$.
Figure~\ref{fig_percentDiff}
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig8.eps}
\vspace*{0.1cm}
\caption{(Color online) Percent difference between $P_{eq}$ for
$N=5 \times 10^7$ and $P_{eq}$ for $N=500$ as a function of
the mass ratio $\kappa$ for the trapped system at unitarity.
Solid, dashed and dotted lines show the percent difference for $T/T_F=1/2,1$
and 2, respectively.
}\label{fig_percentDiff}
\end{figure}
shows the percentage difference
between $P_{eq}$ for $N=5\times 10^7$ and $P_{eq}$ for $N=500$ as a function
of $\kappa$ for $T/T_F=2,1$ and 1/2.  The non-universal corrections are
largest,
i.e., on the order of 2\%, for $\kappa=1$ and $T/T_F=1/2$
[solid line in Fig.~\ref{fig_percentDiff}].
For larger $N$, the non-universal corrections are even smaller, implying
that Figs.~\ref{fig_fugacities}-\ref{fig_thermoCombined} show essentially
universal, $N$-independent thermodynamic quantities of the trapped
two-component Fermi system with equal trapping frequencies.

We conclude this section by noting that $P_{eq}$ is quite small for the
K-Li system
down to $T/T_F=1$, but then changes more rapidly as the temperature is lowered
further.
The dependence of $P_{eq}$ on the mass ratio $\kappa$ may be enhanced by
switching to a trap geometry for which the single-particle
energies are dependent
on the mass $m_1$ or $m_2$.  One such system is, as detailed in the
next section,
the homogeneous system, which favors much larger polarizations than the trapped
system for comparable $T/T_F$.

\section{Thermodynamics of the universal homogeneous mass-imbalanced two-component fermi gas}
\label{sec_thermo3}

\subsection{Local Density Approximation}
This section shows how the homogeneous system can be related to
the harmonically trapped system via the local density approximation~\cite{LiuPRL2009,CMenottiPRL2002}.
In the high temperature limit, the thermodynamic potential
$\Omega^{(2)}_{\rm{hom}}$ of the homogeneous system can be written as
\begin{align}
\label{eq_homHighT}
\Omega^{(2)}_{\rm{hom}} = \Omega^{(1)}_{1,\rm{hom}}
+ \Omega^{(1)}_{2,\rm{hom}}
+ \Delta\Omega_{\rm{hom}},
\end{align}
where
\begin{align}
\label{eq_homSingCompFGa}
\Omega_{1,\rm{hom}}^{(1)} & = -k_B T \frac{V}{\lambda_{m_1}^3}
\sum_{n_1=1}^{\infty} b_{n_1,0,\rm{hom}}  \; z_1^{n_1}, \\
\label{eq_homSingCompFGb}
\Omega_{2,\rm{hom}}^{(1)} & = -k_B T \frac{V}{\lambda_{m_1}^3}
\sum_{n_2=1}^{\infty} b_{0,n_2,\rm{hom}}  \; z_2^{n_2},
\end{align}
and
\begin{align}
\label{eq_homThermPotHighTb}
\Delta \Omega_{\rm{hom}} = & - k_B T  \frac{V}{\lambda_{m_1}^3}
\sum_{n_1=1}^{\infty} \sum_{n_2=1}^{\infty} b_{n_1,n_2,\rm{hom}}
z_1^{n_1} z_2^{n_2}.
\end{align}
Here, the $b_{n_1,n_2,\rm{hom}} $ denote
the virial coefficients of the homogeneous system,
and $z_1$ and $z_2$ are the fugacities of the homogeneous system.
The virial coefficients $b_{n_1,n_2,\rm{hom}}$ are defined by
Eqs.~\eqref{eq_v11}-\eqref{eq_v21}
with the $Q_{n_1,n_2}$ interpreted as the partition functions of the
homogeneous systems.
Equations~\eqref{eq_homHighT}-\eqref{eq_homThermPotHighTb} are analogous to
the high temperature limits of
Eqs.~\eqref{eq_thermPotHighTa}-\eqref{eq_thermPotHighTb}.
In Eqs.~\eqref{eq_homHighT}-\eqref{eq_homThermPotHighTb}, we used that the
high temperature limit of $Q_{1,0,\rm{hom}}$
is given by $V/\lambda_{m_1}^3$~\cite{HuangStatMech},
where $V$ is the volume of the system and where
$\lambda_{m_1}$ is the thermal de Broglie wavelength of a particle of the
heavy species,
\begin{align}
\label{eq_deBroglieWL}
\lambda_{m_1} = \sqrt{\frac{2 \pi \hbar^2}{m_1 k_B T}}.
\end{align}

In the local density approximation~\cite{LiuPRL2009,CMenottiPRL2002},
the thermodynamic potential $\Omega^{(2)}$
of the trapped system is given by a weighted average of
$\Omega^{(2)}_{\rm{hom}}(\vec{r})$ over all space,
\begin{align}\label{eq_N1trap}
\Omega^{(2)} = \frac{\int \Omega^{(2)}_{\rm{hom}}(\vec{r})
d^3 \vec{r}}{\int d^3 \vec{r}},
\end{align}
i.e., we assume that the thermodynamic potential at each location $\vec{r}$
in the trap can be approximated by the corresponding bulk value.
This implies that the $z_i$ in
Eqs.~\eqref{eq_homSingCompFGa}-\eqref{eq_homThermPotHighTb}
are replaced by $z_i(\vec{r})$,
$z_i(\vec{r})=\exp[\mu_{i,loc}(\vec{r})/(k_B T)]$,
where the local chemical potentials $\mu_{i,loc}(\vec{r})$ are defined as
\begin{align}\label{eq_chemPotLoc}
\mu_{i,loc}(\vec{r}) = \mu_i - \tfrac{1}{2}m_i\omega^2\vec{r}^2.
\end{align}

The integrations on the right hand side of Eq.~\eqref{eq_N1trap} are
straightforward.
We first perform the integration involving $\Delta\Omega_{\rm{hom}}(\vec{r})$.
The integration in the denominator introduces a factor of $V$,
which cancels with the factor of $V$ in the numerator.
This leaves, after performing the angular integrals,
\begin{align}
\label{eq_locDensityInt}
\Delta\Omega = & -4 \pi \frac{k_B T}{\lambda_{m_1}^3}
\sum_{n_1=1}^{\infty} \sum_{n_2=1}^{\infty}
b_{n_1,n_2,\rm{hom}} z_1^{n_1} z_2^{n_2} \times \\
&\int_0^{\infty}  \!\!\!
\exp{[-\tfrac{1}{2}(m_1 n_1 + m_2 n_2)\omega^2 r^2/(k_B T)]r^2 dr}. \nonumber
\end{align}
Performing the remaining integration and
expressing the result in terms of the mass ratio $\kappa$, we find
\begin{align}
\label{eq_O1trapA}
\Delta\Omega = & -k_B T \left(\frac{k_B T}{\hbar \omega}\right)^3 \times
\nonumber \\
& \sum_{n_1=1}^{\infty} \sum_{n_2=1}^{\infty}
\frac{b_{n_1,n_2,\rm{hom}} }{(n_1 + n_2 /\kappa)^{3/2}} z_1^{n_1} z_2^{n_2}.
\end{align}
The prefactor $[k_B T/(\hbar\omega)]^3$ coincides with the high $T$ limit of
$Q_{1,0}$ for the trapped system.
Comparison of Eqs.~\eqref{eq_thermPotHighTb} and~\eqref{eq_O1trapA}
allows us to relate the universal virial
coefficients $b_{n_1,n_2}^{(0)}$ of the trapped system
to the virial coefficients $b_{n_1,n_2,\rm{hom}}$ of the
homogeneous system,
\begin{align}
\label{eq_vcHomToTrap}
b_{n_1,n_2,\rm{hom}}  = (n_1 + n_2 /\kappa)^{3/2} b_{n_1,n_2}^{(0)}.
\end{align}
Considering the integrations over $\Omega^{(1)}_{1,\rm{hom}}(\vec{r})$ and
$\Omega^{(1)}_{2,\rm{hom}}(\vec{r})$, one finds analogous relationships between
$b_{n_1,0,\rm{hom}} $ and $b_{n_1,0}^{(0)}$, and between
$b_{0,n_2,\rm{hom}} $ and $b_{0,n_2}^{(0)}$.

\subsection{Universal thermodynamics of the homogeneous system}
\label{sec_thermoHom}
This section considers the universal, $N$-independent thermodynamics of the
homogeneous two-component
Fermi gas at unitarity as a function of $\kappa$.
We measure our temperature $T$ in units of the semi-classical Fermi temperature
$T_F$ of a single-component Fermi gas with $N/2$ particles of mass $m_1$,
$k_B T_F=[6 \pi^2 (N/2)/V]^{2/3} \hbar^2/(2 m_1)$~\cite{HuangStatMech}.
Notice that the Fermi temperature scales inversely with the mass $m_1$.
This implies that the Fermi temperature of $N/2$ light particles with mass
$m_2$ is $\kappa$ times larger than the Fermi temperature of $N/2$ heavy
particles.  The different Fermi temperatures of the single-component heavy and
light particles are, of course, a direct consequence of the mismatching Fermi
surfaces of unequal-mass Fermi gases or, equivalently, a direct consequence of
the fact that the single-particle energies of a mass $m_i$ particle in a box
with periodic boundary conditions scale with $1/m_i$.
The high temperature virial equation of state of the homogeneous system can be
applied down to $T/T_F$ about $1$ for $\kappa=1$.  For unequal masses, the
applicability regime, using the heavy mass to define $T_F$ (see above),
is limited to $T/T_F \gtrsim \kappa$.

The coupled number equations of the
homogeneous system, up to third order, read
\begin{align}
\label{eq_homN1}
\frac{N_1 \lambda_{m_1}^3}{V} = & -\mbox{Li}_{3/2}(-z_1) + b_{1,1,\rm{hom}}
z_1 z_2 \nonumber \\
& + 2 b_{2,1,\rm{hom}}  z_1^2 z_2 + b_{1,2,\rm{hom}}  z_1 z_2^2
\end{align}
and
\begin{align}
\label{eq_homN2}
\kappa^{3/2} & \frac{N_2 \lambda_{m_1}^3}{V} = -\mbox{Li}_{3/2}(-z_2)
+ \kappa^{3/2} b_{1,1,\rm{hom}}  z_1 z_2 \nonumber \\
& + \kappa^{3/2} b_{2,1,\rm{hom}}  z_1^2 z_2 + 2
\kappa^{3/2} b_{1,2,\rm{hom}}  z_1 z_2^2.
\end{align}
Dotted and dash-dotted lines in Fig.~\ref{fig_homFugacities}
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=80mm]{fig9.eps}
\vspace*{0.1cm}
\caption{(Color online) Fugacities $z_1$ and $z_2$ of the homogeneous
system as a function of the polarization $P$ for
(a) $\kappa=1$ and $T/T_F=6.67$,
(b) $\kappa=6.67$ and $T/T_F=6.67$,
(c) $\kappa=1$ and $T/T_F=2$, and
(d) $\kappa=2$ and $T/T_F=2$.
Dotted and dash-dotted lines show $z_1$ and $z_2$, respectively,
for the unitary system obtained by solving Eqs.~\eqref{eq_homN1}
and~\eqref{eq_homN2} self-consistently.
For comparison, solid and dashed lines show
$z_1$ and $z_2$, respectively, for the non-interacting system.
}\label{fig_homFugacities}
\end{figure}
show $z_1$ and $z_2$, respectively, as a function of the polarization $P$
for the unitary system obtained by solving Eqs.~\eqref{eq_homN1}
and~\eqref{eq_homN2} self-consistently for fixed $\kappa$ and $T$.
The panels show the fugacities for
(a) $\kappa=1$ and $T/T_F=6.67$,
(b) $\kappa=6.67$ and $T/T_F=6.67$,
(c) $\kappa=1$ and $T/T_F=2$, and
(d) $\kappa=2$ and $T/T_F=2$.
For the equal mass case, Eqs.~\eqref{eq_homN1} and~\eqref{eq_homN2}
are symmetric and thus $z_1$ and $z_2$ are symmetric about $P=0$ for
any temperature [see~Figs.~\ref{fig_homFugacities}(a)
and~\ref{fig_homFugacities}(c)].
For unequal masses, in contrast, Eqs.~\eqref{eq_homN1} and~\eqref{eq_homN2}
are not symmetric and the crossing of the fugacities $z_1$ and $z_2$ is shifted
to positive polarizations [see Figs.~\ref{fig_homFugacities}(b)
and~\ref{fig_homFugacities}(d)].
For comparison, solid and dashed lines in Fig.~\ref{fig_homFugacities} show
the fugacities $z_1$ and $z_2$, respectively, for the corresponding
non-interacting systems.
Even for the non-interacting system, the fugacities $z_1$ and $z_2$ are not
symmetric with respect to $P=0$; this is a direct consequence of the
$\kappa^{3/2}$ factor in Eq.~\eqref{eq_homN2}.
For $-1 < P < 1$, the fugacities for the non-interacting systems lie above
the respective fugacities for the unitary systems.
For $\kappa > 1$, this implies that the unitary system has a smaller
equilibrium polarization $P_{eq}$
than the non-interacting system for the same $T$ and $\kappa$.
Intuitively, this makes sense since we expect that the attractive interactions,
at least for mass ratios not too much larger than $1$, push the system
toward an equal mixture of heavy and light particles.

Figure~\ref{fig_homF_vs_P}
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig10.eps}
\vspace*{0.1cm}
\caption{(Color online) Free energy per particle $F/N$ of the homogeneous
system as a function
of the polarization $P$ for $T/T_F=10$ and mass ratios $\kappa=1,2$, and 6.67.
The curves are grouped in sets of two, each labeled by their corresponding
mass ratio.
The solid lines are calculated using the virial equation of state at unitarity.
For comparison, the dotted lines show $F/N$ for the non-interacting system.
}\label{fig_homF_vs_P}
\end{figure}
shows the free energy per particle $F/N$ as a function of the polarization $P$
for $T/T_F=10$ and three different mass ratios $\kappa$.
The solid lines are calculated using the virial equation of state
of the unitary system.
For comparison, dotted lines show $F/N$ for the non-interacting system.
Since the free energy is a concave function with respect to $P$
for fixed $N$ and $T$, the equilibrium polarizations
are stable minima.

To better understand the interplay of the second and third order virial
coefficients for the homogeneous system, we calculate
$z_{eq}$ analogously to Eq.~\eqref{eq_zeq} and find
$P_{eq}$ for the homogeneous system to be
\begin{align}
\label{eq_Peq}
P_{eq} = & \frac{N\lambda_{m_1}^3}{V}\Bigg(-\mbox{Li}_{3/2}(-z_{eq})
\left[1-\frac{1}{\kappa^{3/2}}\right] \nonumber \\
& + [b_{2,1,\rm{hom}}-b_{1,2,\rm{hom}}]z_{eq}^3\Bigg).
\end{align}
Eliminating the polylogarithm function, $P_{eq}$ can be rewritten as
\begin{align}
\label{eq_PeqOpt}
P_{eq} & = \frac{\kappa^{3/2}-1}{\kappa^{3/2}+1} - b_{1,1,\rm{hom}}
\frac{\kappa^{3/2}-1}{\kappa^{3/2}+1} \frac{2V}{N\lambda^3_{m_1}} z_{eq}^2 + \\
& + \Bigg[ b_{1,2,\rm{hom}}^{(0)} \frac{1-2\kappa^{3/2}}{\kappa^{3/2}+1}
+ b_{2,1,\rm{hom}}^{(0)}\frac{2-\kappa^{3/2}}{\kappa^{3/2}+1} \Bigg]
\frac{2V}{N\lambda^3_{m_1}} z_{eq}^3. \nonumber
\end{align}
As expected, the $z_{eq}^0$ term depends only on the mass ratio $\kappa$
and is greater than zero for $\kappa>1$; in the limit
$\kappa\rightarrow \infty$, the non-interacting
system prefers to have only heavy particles, i.e., $P_{eq} \rightarrow 1$.
The $z_{eq}^2$ term of the unitary system, in contrast,
is negative for all $\kappa>1$.
This implies that the second order virial coefficient of the unitary system
acts to decrease $P_{eq}$ compared to the $P_{eq}$ of the non-interacting
system.
Interestingly, the $z_{eq}^3$ term of the unitary system is positive for
$\kappa < 5.40$ and negative for $\kappa>5.40$, independent of temperature.
This implies that the third order virial coefficients act to increase
(decrease)
$P_{eq}$ for $1 < \kappa < 5.40$  ($\kappa > 5.40$) compared to the $P_{eq}$
calculated using the virial equation of state up to second order.

Figure~\ref{fig_homFeqA}
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig11.eps}
\vspace*{0.1cm}
\caption{(Color online) Equilibrium polarization $P_{eq}$ of the
homogeneous system at unitarity as a function of
the mass ratio $\kappa$ for $T/T_F=10$.
The dashed and solid lines show $P_{eq}$ calculated using the virial
equation of state up to second and third order, respectively.
The dashed and solid lines are nearly indistinguishable on the scale shown.
For comparison, the dotted line shows $P_{eq}$ for the
non-interacting system.
The solid line in the inset shows the difference between the $P_{eq}$'s
calculated using the virial equation of state up to the
third and second orders (the dashed line shows zero as a reference).
}\label{fig_homFeqA}
\end{figure}
shows the equilibrium polarization $P_{eq}$ as a function of $\kappa$
for $T/T_F=10$.
The dashed and solid lines show $P_{eq}$ calculated using the virial equation
of state of the unitary system up to second and third order, respectively.
The second and third orders are nearly indistinguishable on the scale shown.
For comparison, the dotted line shows $P_{eq}$ for the non-interacting system.
The inset of Fig.~\ref{fig_homFeqA} shows the difference of $P_{eq}$
calculated up to the second order subtracted from $P_{eq}$ calculated up to the
third order. Note that the zero crossing occurs at $\kappa=5.56$,
and not at $\kappa=5.40$, since the equilibrium fugacity determined from the
virial equation of state up to third order is slightly smaller than that
determined from the virial equation of state up to second order.

Figure~\ref{fig_homFeqB}
\begin{figure}
\vspace*{+1.5cm}
\includegraphics[angle=0,width=70mm]{fig12.eps}
\vspace*{0.1cm}
\caption{(Color online) Equilibrium polarization $P_{eq}$ of the
homogeneous system at unitarity as a function of
the temperature $T$ for $\kappa=6.67$.
The dashed and solid lines show $P_{eq}$ calculated using the virial
equation of state up to second and third order, respectively.
The dashed and solid lines are nearly indistinguishable on the scale shown.
For comparison, the dotted line shows $P_{eq}$ for the
non-interacting system.
The solid line in the inset shows the difference between the $P_{eq}$'s
calculated using the virial equation of state up to the
third and second orders.
}\label{fig_homFeqB}
\end{figure}
shows the equilibrium polarization
$P_{eq}$ as a function of $T$ for $\kappa=6.67$. The dashed and solid lines
show $P_{eq}$ at unitarity calculated using the virial equation of state
up to second and third order, respectively.
The second and third orders are nearly indistinguishable on the scale shown.
For comparison, the dotted line shows $P_{eq}$ for the
non-interacting system;
it is independent of $T$ and equals $0.89$.
The inset of Fig.~\ref{fig_homFeqB} shows the difference of $P_{eq}$
at unitarity
calculated up to the second order subtracted from $P_{eq}$
at unitarity calculated up to the third order.
Since the mass ratio used is larger than $5.40$, the third order correction
acts to further decrease $P_{eq}$ for $T/T_F > 5$.
Note that the virial equation of state is likely unreliable
for $T/T_F \lesssim 6.67$.
As the temperature decreases, interactions become more important
and the deviations between $P_{eq}$ for the unitary and the non-interacting
system increase.

\section{Conclusions}
\label{sec_conclusion}
This paper considers the high temperature virial equation of state of
unequal-mass two-component Fermi gases at unitarity.
For the trapped system with equal frequencies and $\kappa>1$, we found that
the equilibrium polarization $P_{eq}$ at unitarity is relatively small for
the temperatures considered.
The small deviations from $P_{eq}=0$ are introduced by the third order virial
coefficients.
For the homogeneous system, in contrast, we found comparitively large $P_{eq}$
for $\kappa>1$.
The value of $P_{eq}$ is determined by an intricate interplay between the
second and third order virial coefficients.
For the cases investigated, $P_{eq}$ increases with decreasing temperature for
the trapped system while $P_{eq}$ decreases with temperature for the
homogeneous system.
The fourth-order virial coefficients $b_{3,1}$ and $b_{2,2}$ for unequal-mass
systems are presently unknown, preventing us to estimate the importance
of higher-order terms in the virial equation of state.

At the temperatures considered, the curvature of the free energy
is positive for all polarizations, thus
implying the absence of any first-order phase transition.
If the unequal-mass system is prepared with a polarization $P$
different from $P_{eq}$, then the free energy lies above the value
it would have for $P_{eq}$.  This suggests that the system
prepared with $P \ne P_{eq}$ may phase separate when the temperature
falls below a certain critical value.

In future work, it will be interesting to investigate the high temperature
thermodynamics of unequal mass systems when the light and heavy species are
confined by harmonic potentials with different frequencies via the local
density approximation.
Moreover, the treatment of cylindrically symmetric traps will be relevant for
ongoing experiments.
Other future extensions of our work include the treatment of unequal mass
systems away from unitarity and at lower temperatures.

\section{Acknowledgements}
Support by the NSF through grant PHY-0855332
is gratefully acknowledged.
We also acknowledge discussions with Stefano Giorgini on the use of the local
density approximation.

redpajama_set_name: RedPajamaArXiv


Thanks... guess I'll keep an eye out for thoughts from players in the group when it comes out. I suppose in either case I've probably got a fair amount new to enjoy given how long it's been since I've played an installment.
What do you think about gameplay in recent installments, if you've been playing? How long does it hold your interest? How repetitive vs. original, varied, and exciting does it seem?
Overall it is... fine. Diablo 3 showed me how much stronger it could be though. Going to kingdom mode really helped stop the "I'm bored" as no longer did you do 50 Wu Hefei's with the exact same events and a bad story for 90% of them. You do most battles once, some twice if they have a hypo condition/been affected by it, and only doing 10 stories (Wei, Wu, Shu, Jin, Lu Bu with a historical and hypo) helps with storytelling (though quality varies widely).
Yeah, while I really liked the fact that they put in more events and missions into the new Dynasty Warriors games it does kinda sacrifice re-playablity a bit. Unless you are the kind of person who must beat a level in every difficulty you probably won't play each level more then 3-4 times in my opinion.
You will probably play through once to get through it, another time to make sure you succeed in every objective, one time messing around trying to win while failing every objective and perhaps one more time in case you really liked the level, perhaps on another difficulty. At least that is how I did it, some of you might enjoy replaying them again and again.
I mean I replayed the Shu story line a couple of more times to make sure I got all of the missions done, but I only played Yi Ling twice total. I got all of the objectives but do I really want to lose and barely survive again and again? Same with Wu He Fei, unless you are going to hypothetical route where you win do I really want to flee from Wei over and over again after I finished it?
I will try to keep an open mind about the open world (get it ) and wait for more information to be released. I will be disappointed if we get Cheng Pu, for Wu and like Cao Hong for Wei cause then I will know that they aren't bothering to expand the latter years battles which is what I really want. I want more Jin v Wu, I want Jiang Wei's disasters northern campaigns!
I think DW were right to go down the "only need to play 2/3 times depending if you want 5th/6th weapons", it was no fun having to do stages 20 times. 2/3 keeps the fun and that some battles change widely depending on the conditions, that provides surprise. I do think adding in "fight battle that your going to lose" is a good idea, having the odd one where the objective is simply to get out of there alive does allow for variety.
LiuBeiwasGreat wrote: Same with Wu He Fei, unless you are going to hypothetical route where you win do I really want to flee from Wei over and over again after I finished it?
Most in the west would have liked Liao Hua including myself from what I can tell. I was hoping for latter Wu expanding myself and yes, would be nice for matter latter Shu.
LiuBeiwasGreat wrote: I will try to keep an open mind about the open world (get it ) and wait for more information to be released.
Perhaps you are right, grinding levels just to get through the campaign did get tiring in the earlier games, however I feel like I stop playing the newer ones faster then I did the old ones.
As for the fights where you lose, I guess Shu shouldn't have the only missions where you do that Now that I think about it each faction as at least one battle that involves running away. Chang Ban, Wan, Hei Fei, and.......you know my mind if blanking on the name of the Jin mission but the one where you invade Han Zhong and are forced to retreat.
I did find that one silly while playing it though, as I was cutting through Shu-Han forces at every step and didn't feel like I really needed to run away.
LiuBeiwasGreat wrote: Perhaps you are right, grinding levels just to get through the campaign did get tiring in the earlier games, however I feel like I stop playing the newer ones faster then I did the old ones.
LiuBeiwasGreat wrote: As for the fights where you lose, I guess Shu shouldn't have the only missions where you do that Now that I think about it each faction as at least one battle that involves running away. Chang Ban, Wan, Hei Fei, and.......you know my mind if blanking on the name of the Jin mission but the one where you invade Han Zhong and are forced to retreat.
Destiny Mode was an awesome idea and they could have really expanded upon it in future games had they continued to develop it, but it's just like Koei to leave good ideas on the curb and ignore them in future installments. The open world does lend itself more to this sort of air though, admittedly.
Destiny Mode was my favourite mode in any 3K series ever. I would definitely love to see it explored in more depth in an open world setting.
Destiny Mode was fun but needing building up, I also liked the way the officers would treat you in their own, sometimes flirty, style.


Re: PLEASE HELP! PORTABLE HARD DRIVE DOESN'T WORK ANYMORE! PLEASE! HELP!
is it on amazon? how does it look?
does it connect the same way the sata does?
where's the best one to get?
Links to some of the best ones??


So I've not made time for the blog the last couple of weeks and I'm sorry. But surely you can understand, I've not had time for anything at all lately, not even sleep, let alone the blog. I've been working quite a number of weekends. It's my first time doing several weekends in a row and it is quite a shock to the body. My feet (actually all the way up to my calves) are constantly feeling a bit achy and I've not seen bigger, darker circles under my eyes. I worry they might never go away. Help! So my first Sunday off I exercised and got a haircut. Not the best of cuts, miscommunication on my part, but that's a story for another day.
You know you're busy when the highlight of your weekend was lying in bed, catching up on last month's Vogue magazine.
Anyway, I've been feeling really out of touch, not being able to get my daily dose of Pinterest and R29. Getting dressed in the morning still feels exciting (yes, I'm a girlie girl), but when you're sleepy and don't feel like torturing your feet anymore in heels, the task can be quite a challenge. Especially so if you're vertically challenged like myself.
Images from here, here, here, here, here, here, here, here, here, here, here, here, here and here.
Images from here, here, here, here, here, here, here, here, here, here, and here.


Mise en place is French for "putting in place" (thank you Wikipedia). It is a phrase used in the culinary world, typically referring to the preparation and organization of ingredients. When I am cooking, I always feel so much better when I take the time to prep – to mise en place – my ingredients.
Just this past week we celebrated both my baby girl Clare's first birthday, which also happens to be her great-grandmother's birthday. We were enormously blessed to have Grammie, as we are fondly calling Ben's grandma, visit us and to be here for their birthdays. Typically for birthdays we let the birthday person pick the dinner. Since Clare is only 1, we decided Grammie might make a better dinner choice. She asked for something Italian, which is great for me and my cooking comfort zone. I'm not sure what I would have done if she had said, "You know, I'd love some Thai food." I'd like to think I would have figured something out, but it probably would have kept me up all night wondering if I could pull it off.
Anyway, long story short, I decided to make stuffed shells. Basically, think lasagna but instead of layers like a cake, the ricotta cheese and sausage/ground beef mix is stuffed into a jumbo noodle shell and then baked in a marinara sauce and covered in more cheese – delicious! But, very time consuming if you don't plan and mise en place well. Thank goodness for Grammie who is so helpful in the kitchen.
Interestingly, I head a talk a number of months ago from a chef about how the practice of mise en place is more than just ingredient preparation, it can actually be a model for life. A big point was not about the preparation, but the clean up. This chef talked about how you can't have a good, orderly kitchen if the dishes are stacked up. If he had a chef that chopped vegetables perfectly every time but consistently forgot to clean up the cutting board and knives used, that chef would not go far.
I'm the kind of person that is great at the prep, not so great at the immediate clean up. It always gets cleaned up, but not usually in a timely manner. Grammie, on the other hand, is an immediate cleaner. We can create a lot of dishes, quickly, especially when I'm cooking something with a number of components. On a typical day, all those dirty dishes would pile up in a somewhat haphazard stack, all waiting to be taken care of at the end of the day. With Grammie here, my stack would magically diminish each time I went to add to it. In it's place were neatly organized clean dishes on the drying rack, waiting to be put away. It was delightful!
Grammie showed me this week the virtue of really completing a task, start to actual finish. How easy is it to be distracted and to not finish what you were doing. If you have kids (or even if you don't, just insert your own examples), some days it feels like it is impossible to actually complete anything. Then, at the end of the day, you look around and there are rolls of toilet paper sitting on the printer, laundry baskets with folded clothes on the couch, dishes in the sink, and legos mimicking a mine field on the carpet.
The choices I make throughout the day impact my mood, my relationships and my sleep, basically everything. When I am on top of completing the things I start, I feel better. I look around my house and I see that the dishes are done, the floor is swept, the laundry is put away. I have a greater abundance of patience, tolerance and compassion for my children because it isn't being used up by needing patience, tolerance and compassion for myself and my unfinished tasks.
So thank you Grammie, for giving me such a beautiful example of how to complete tasks. The time spent finishing the task is well worth the extra minutes. Actually, I think that the tasks are done faster when you go start to finish, rather than start, stop, start, stop, start, stop, finish at the end of the day or the next day.
Do you find the same for you?


text:
Family Divorce Research Paper
According to Wikipedia – the free encyclopedia[4], "From 2002 to the latest survey data as of 2012,[2] 29% of first marriages among women aged 15–44 were disrupted (ended in separation, divorce or annulment) within 10 years. [3]" This is alarmingly a high rate of marriage failure. After reading the novel by Jordan Sonnenblick, "Notes From the Midnight Driver" [1], I am submitting a research paper on the topic, "Family Divorce" using the situation of the novel as an example.
Alex's dad, in the novel, ran away from his marriage with Alex's 3rd Grade teacher. He was perhaps disappointed with Alex's mom. Alex's mom, as a result, was mad at his dad. Her frustration was justifiable but after completing the novel it became clear that the couple's frustrations and disappointments were immediate and did not last for ever.
From this example, we can say that often couples are frustrated with their immediate disappointments and want to take divorce. However, in their immediacy, they can't see or remember a long term effect of their decision. In this research paper, I wanted to raise three questions about family divorce and answer them in order they are raised.
Why people opt for divorce?
Some divorce are genuine separations where one of the spouse is violent or addicted to drug but NOT ALL. Some divorces are the result of quick upsets or immediate disappointments or infidelity such as is the case with Alex's dad.
How divorce are realized?
Getting into relationship is very easy. Getting out of it is not that easy. Our Governments grant marriages readily but they do not un-grant them into divorce readily. There is a legal process that couples have to go through the court system. They not only have to pay the court fee but the divorce lawyer's fee as well. This is a big expense. It takes away money from the everyday expenses and lowers the living standard of the family involved in the divorce case. Also, as per Wikipedia[4], 'The legal process for divorce may also involve issues of spousal support, child custody, child support, distribution of property and division of debt, though these matters are usually only ancillary or consequential to the dissolution of the marriage.' What this means is the couple will have to pay big bills to many services such as accounting services and other lawyers' services to take care of all those complicated issues. This is a drain on time and money.
What are the after effects of the divorce?
When Alex was arrested he blamed all other things that was happening in...
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I started making these for PumpkinSpice and WAS working on an animated version, but php and stuff got in the way. So, before they end up in the "abandoned and forgotten" section of my C Drive, here they are.
These windows are modular - by using the one tile window and the sidelight in combination with each other you can get a lot of variation. See the attached pictures.
These windows are slaved to the Breeze-easy Base Game windows, so any recolours you may have in your downloads folder will work on these as well.
Credits: Simpe, PumpkinSpice for asking, iCad for the recolours.
Ah, the Window-maker has returned, triggering my download-addiction! Thanks Lee, they look wonderful!
I feel an urge to recolour some windows coming on.
The return of the Window Fish! And a beautiful return it is, too!


text:
Q: Edit configuration file through python I need to edit a configuration file through python and i tried searching on stackoverflow and google and they don't cover my situation, since i need to replace lines in the file and perform matches in my search.
Also, what i found covers how to do it for one line, i will be performing at least 8 line replacements in the file and I would like to know if there is a cleaner and more elegant way of doing this than putting 10 replace(foo, bar) lines altogether.
I need to "match" lines like "ENABLEPRINTER", "PRINTERLIST", "PRNT1.PORT".
I want to match thesse text and ignore whatever follows (ex: "=PRNT1, PRNT2").
So i would do something like
replace('ENABLEPRINTER', 'y')
replace('PRINTERLIST', 'PRNT3)

The file looks like this:
ENABLEPRINTER=n
PRINTERLIST=PRNT1, PRNT2

PRNT1.PORT=9600
PRNT1.BITS=8

Also note these files are about 100 lines and i need to edit about 10 of them.
Thank you very much for your help.
UPDATE:
Using the code posted by @J.F. Sebastian, i'm now getting the following error:
configobj.ParseError: Parse error in value at line 611.

Line 611 of the file is:
log4j.appender.dailyRollingFile.DatePattern='.'yyyy-MM-d

So the problem is with the ' character.
If I comment out that line, the script is working fine with the code posted by @J.F. Sebastian.

A: import re
pat = re.compile('ENABLEPRINTER|PRINTERLIST|PRNT1.PORT')

def jojo(mat,dic = {'ENABLEPRINTER':'y',
                    'PRINTERLIST':'PRNT3',
                    'PRNT1.PORT':'734'} ):
    return dic[mat.group()]

with open('configfile','rb+') as f:
    content = f.read()
    f.seek(0,0)
    f.write(pat.sub(jojo,content))
    f.truncate()

Before:
ENABLEPRINTER=n
PRINTERLIST=PRNT1, PRNT2

PRNT1.PORT=9600
PRNT1.BITS=8

After:
y=n
PRNT3==PRNT1, PRNT2

734=9600
PRNT1.BITS=8

Too simple to be definitive. Say what are the errors or weaknesses.
The advantage of regexes is they can be modulated easily to particular cases.
.
EDIT:
I've just seen that:
"what i want to do is assign a new value to the variable "
you could inform of that earlier !
Could you give an exemple of file before / after , please.
.
EDIT 2
Here's the code to change the values of certain variables in a file:
import re
from os import fsync

def updating(filename,dico):

    RE = '(('+'|'.join(dico.keys())+')\s*=)[^\r\n]*?(\r?\n|\r)'
    pat = re.compile(RE)

    def jojo(mat,dic = dico ):
        return dic[mat.group(2)].join(mat.group(1,3))

    with open(filename,'rb') as f:
        content = f.read()

    with open(filename,'wb') as f:
        f.write(pat.sub(jojo,content))

#-----------------------------------------------------------

vars = ['ENABLEPRINTER','PRINTERLIST','PRNT1.PORT']
new_values = ['y','PRNT3','8310']
what_to_change = dict(zip(vars,new_values))

updating('configfile_1.txt',what_to_change)

Before:
ENABLEPRINTER=n
PRINTERLIST=PRNT1, PRNT2

PRNT1.PORT=9600
PRNT1.BITS=8

After:
ENABLEPRINTER=y
PRINTERLIST=PRNT3

PRNT1.PORT=8310
PRNT1.BITS=8

A: If the file is in java.util.Properties format then you could use pyjavaproperties:
from pyjavaproperties import Properties

p = Properties()
p.load(open('input.properties'))

for name, value in [('ENABLEPRINTER', 'y'), ('PRINTERLIST', 'PRNT3')]:
    p[name] = value
p.store(open('output.properties', 'w'))

It is not very robust, but various fixes for it could benefit people who come next.

To replace multiple times in a short string:
for old, new in [('ENABLEPRINTER', 'y'), ('PRINTERLIST', 'PRNT3')]:
    some_string = some_string.replace(old, new)

To replace variables names in a configuration file (using configobj module):
import configobj

conf = configobj.ConfigObj('test.conf')

for old, new in [('ENABLEPRINTER', 'y'), ('PRINTERLIST', 'PRNT3')]:
    conf[new] = conf[old]
    del conf[old]
conf.write()

If by replace('ENABLEPRINTER', 'y') you mean assign y to the ENABLEPRINTER variable then:
import configobj

ENCODING='utf-8'
conf = configobj.ConfigObj('test.conf', raise_errors=True,
    file_error=True,           # don't create file if it doesn't exist
    encoding=ENCODING,         # used to read/write file
    default_encoding=ENCODING) # str -> unicode internally (useful on Python2.x)

conf.update(dict(ENABLEPRINTER='y', PRINTERLIST='PRNT3'))
conf.write()

It seems configobj is not compatible with:
name = '.'something

You could quote it:
name = "'.'something"

Or:
name = '.something'

Or
name = .something

conf.update() does something similar to:
for name, value in [('ENABLEPRINTER', 'y'), ('PRINTERLIST', 'PRNT3')]:
    conf[name] = value

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text:
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Maimonides
Power Respect Space Greatness Now
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Life Live Time Space Me Story
For years, I longed to hear Armstrong describe what it was like to contemplate Earth from 238,900 miles away. Former Space Center director George Abbey once told me that many NASA astronauts felt that looking at Earth was akin to a religious experience.
Douglas Brinkley
Experience Space Me Looking Earth
I work for ABC. If the thing that ABC is paying me for is storytelling - not to make sure that a costume is exactly right or all those other things - then it is up to me to find the most creative space possible so that that function of my job can happen.
Shonda Rhimes
Work Creative Job Space Me Find
I will tell you what I can't abide - and I think the Internet has really created a space for it - women criticizing other women and mothers criticizing other mothers.
Women Space You Think Internet Will
Consciousness-one level is understanding where we are in space. Consciousness two is where we understand our position in society: who's top dog, who's underdog and who's in the middle. And type-three consciousness is simulating the future. And type-three consciousness, only humans have this ability to see far into the future.
Dog Future Space Society Underdog
I cannot join the space program and restart my life as an astronaut, but this opportunity to connect my abilities as an educator with my interests in history and space is a unique opportunity to fulfill my early fantasies.
Christa Mcauliffe
Life History Opportunity Space Early
Journalism: an ability to meet the challenge of filling the space.
Rebecca West
Challenge Space Journalism Meet
The extent of your consciousness is limited only by your ability to love and to embrace with your love the space around you, and all it contains.
Love Space You Embrace Consciousness
When we developed written language, we significantly increased our functional memory and our ability to share insights and knowledge across time and space. The same thing happened with the invention of the printing press, the telegraph, and the radio.
Time Knowledge Space Memory Language
Now and then, someone is able to look at an empty space, conclude it would be a great place to start a revolution, and bravely go forward.
Great Start Space Revolution Place
I quickly learned that asking if an interview space was wheelchair accessible was a bad idea; it gave a potential employer an immediate bad impression. It was either a black mark against my name, or a straight up discussion of why I wouldn't be able to work there because they had no wheelchair access.
Stella Young
Work Space Potential Black Name Bad
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Focus Service Space Kitchen Worry
My weekends are oases of time and space, where I am able to draw a breath and dive into the stuff I couldn't get to that week - the great article I bookmarked, the friend whose emails I kept dropping, the blog post I'd meant to write on a subject that wasn't timely but was still important.
Rachel Sklar
Friend Time Great Space I Am Week
I made about fifty-four dollars a week and spent it on two flying lessons every week at the age of sixteen and was able to get a license then pretty early and knew that that's what I wanted to do, some kind of a career in aviation. I did know about space flight, but at that point, it was still pretty far out there.
Kevin A. Ford
Age Space Flying Know Week Career
If your ego starts out, 'I am important, I am big, I am special,' you're in for some disappointments when you look around at what we've discovered about the universe. No, you're not big. No, you're not. You're small in time and in space. And you have this frail vessel called the human body that's limited on Earth.
Attitude Time Space I Am Small Ego
The thing about hip-hop today is it's smart, it's insightful. The way they can communicate a complex message in a very short space is remarkable.
Today Space Smart Short Hip-Hop Way
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Architecture Space Fashion Place You
I claim that space is part of our culture. You've heard complaints that nobody knows the names of the astronauts, that nobody gets excited about launches, that nobody cares anymore except people in the industry. I don't believe that for a minute.
Culture Space People Believe You
Fifty thousand dollars' worth of cabinets isn't going to make you a better cook; cooking is going to make you a better cook. At the end of the day, you can slice a mushroom in about three inches of space, and you can carve a chicken in a foot and a half. So it doesn't matter how big the kitchen is.
End Cooking Day Space Kitchen You
The reason we have the stars twinkle at night is because the light is being kind of blurred by the atmosphere around the Earth. That is why the Hubble Space Telescope is so good, because it is above the atmosphere. So it is kind of like looking at the sun from the bottom of a swimming pool, versus looking at the sun above the swimming pool.
Michael J. Massimino
Good Stars Light Sun Space Night


A major part of Schoolboy Q's Blank Face rollout has been a narrative video trilogy, featuring short films for the songs "By Any Means," "Tookie Knows II," and now, "Black THougHts," as Stereogum points out. The suite has centered on tense interactions with the cops before and after a pawn shop robbery. The "Black THougHts" video, directed by Jack Begert and Dave Free of tHe little Homies, finds several suspects slumped in a courtroom, before Q imagines two alternative futures: one in which he's sent to prison, and another where he remains free to spend time with his daughter. Check out all three videos below.