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Home»Startup»How Do We Get More Than 1% Of Us Solving The Most Complex Challenges Of Our Time?
How Do We Get More Than 1% Of Us Solving The Most Complex Challenges Of Our Time?
November 16, 2022 No Comments5 Mins Read
By Umaimah Mendhro, founder of One League, aims to realize global impact by connecting global change-makers to world-class education & opportunities.
Growing up in the throes of a life of exile in Saudi Arabia and then in rural Pakistan, without access to proper schools, I was keenly aware of the power education held in both making and breaking my future. I never questioned what I understood as one of the fundamental truths of life: education is the greatest equalizer. To achieve advancement and prosperity, one needs not much other than grit, perseverance and a good education.
As I constructed my path from home- and self-schooling to an MBA at Harvard Business School, having fought for every little win along the way, and thanks to one role model and two champions in my life, I found myself prying open what felt like the locked doors of an elite society. A society within which things were just that much easier to achieve, success and accomplishment that much more within reach. A world where each entryway always led to another—a network of power, of movers and shakers, where companies were formed, funding flowed back and forth and opportunities cycled and recycled. A beautiful place bursting with potential. But much of that potential all looked the same.
Homogenous Voices
74% of the population that attends the most competitive colleges are raised in families in the top income quartile (not middle class, not upper middle class, but top income quartile), while less than 3% come from families in the bottom income quartile, notes Fareed Zakaria in In Defense of a Liberal Education. Anthony Carnevale et al. in The Merit Myth: How Our Colleges Favor the Rich and Divide America says, "The education system is an increasingly powerful mechanism for intergenerational reproduction of privilege." This system filters out right upfront who gets a seat at the table.
See also Now is the time to think about cyber insurance
Until recently, minorities had been largely left out of the boardroom. "Racial minorities held only about 12% of board seats with over 40% of all U.S. boards still including only white directors," according to the Harvard Law School Forum on Corporate Governance. While 2021 was a record-breaking year for venture capital, according to PitchBook, with U.S. venture investment reaching $300 billion, almost double the previous year, women founders secured only 2%, Black founders secured only 1.2% and Latinx-founded companies secured only 2.1% of venture funding.
Lack Of Representation Of Global Population And Challenges
While many arguments are made for improving diversity in corporate boards, venture financing and the workplace for the sake of equity and fairness, or improved business performance (perhaps because the first reason didn't get us all moving), I believe what we have to worry about is lack of representation of some of the most formidable challenges we face as a global society.
In today's increasingly complex and interconnected world, we are facing some of the worst existential crises in recent history.
As noted by the Federation of American Scientists, the threats of natural disasters, severe climate change and life-threatening diseases in one part of the world can create a ripple effect across the globe while our growing inequality is exacerbated by both the global pandemic and regional and global conflict. Yet, despite these threats, we are severely unprepared to manage our global risks.
We can't expect to solve our global challenges using the same kind of thinking we used to create them. We need to expand the sphere of solutions, the domain of ideas, the range of voices, and the seats at the table.
See also Microsoft survey studies the Gen Z approach to entrepreneurialism
We need much more than a mere 1% of our population to solve some of the most complex global challenges of our time.
So what can be done?
Unlocking The Sphere Of Solutions
It's people like you and me in positions and places of power—whether we were born into those places or benefited from a few role models or champions to get there—who can bring more seats to the table.
Could we, in our personal and professional lives, support and champion others not just by mentoring and advising them but by prying open the locked doors to opportunity?
Who can we bring to our board who doesn't look and think like us already, and who doesn't come from within our elite network of power? Who can we hire who represents the full breadth of our customer base, not just the customers we can already relate to?
If we benefited from a world-class institution's education and network, a coveted venture fund's financing and support, can we create a way in for others who would otherwise not have the same access?
Could we spend some of our social capital to bridge the path to privilege so that we can benefit from all the latent potential and talent in our global society? Because to solve humanity's most formidable challenges and build a better future for all of us, we need a lot more than a few of us.
challenges complex solving time
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Knights of Honor II: Sovereign gets trailer showing more medieval grand strategy
Whole Conflict: Warhammer III will get new trailer introducing Festus the Leechlord


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Cannomois is a group of plants in the Restionaceae described as a genus in 1828. The entire genus is endemic to Cape Province in South Africa.

 Species

References

Restionaceae
Poales genera
Endemic flora of South Africa
Flora of the Cape Provinces
Fynbos

redpajama_set_name: RedPajamaWikipedia


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We are delighted to be able to off this magnificent set which would add grandeur to any production of Phantom of the Opera.
Built by the Belgrade Theatre Coventry for the Taylorian Productions UK and Ireland amateur premiere of the show at the Gaiety Theatre on the Isle of Man.
This set could also be adapted to make productions of My Fair Lady, Oliver!, or The Addams Family Musical.


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Whereas coal mining in Shanxi Province expanded exponentially after 1949, coal production in South Wales experienced just a short period of growth after the Second World War due to the post- war construction effort.
Ten days after Carillion's collapse on 15 January, the contractor and consultant Mace published an analysis suggesting that if the productivity of construction had improved in line with that of manufacturing over the past two decades, ... and it should be seen as a canary-in-the-coal-mine event.
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Underground Coal Mining OPERATION. ... The intrinsically safe GSi units can be used in the potentially explosive underground environment. These units use a 2-pair cable for communication and power from a safe area. COMMUNICATIONS.


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ACCEPTED

#### According to
Index Fungorum

#### Published in
Jl S. Afr. Bot. 35: 291 (1969)

#### Original name
Nummularia helvetica Nitschke

### Remarks
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redpajama_set_name: RedPajamaGithub


When I was a young'un, some dame by the name of 'the tooth fairy' used to break in and take a tooth formerly in my mouth, now resting under my pillow, and exchange it for whatever coinage she had rolling around in her pockets, minus lint and bubble-gum wrappers. A few years on and lastnight I got an even better payoff, with It's A Dan's World receiving it's 200th subscribed follower.
Just like the tooth fairy (who for manliness sake I'm now claiming was The Watcher from the pages of The Fantastic Four), I didn't hear or feel the numbers tick over, but I am extremely grateful to all the readers out there, who have helped make this event happen. That includes you.
Feels good just soaking in that era of Justice League for a minute doesn't it? Forget the team's roster, click to enlarge the image and read the talent included on that lefthand banner - incredible!
Back off that tangent, with the IADW Facebook page developing a following of its own, Twitter, and the page visits to this the core site, I am blown away on a daily basis by the time you all put into visiting my rants and ravings. The fact you return, well that is just tops. 200 registered followers merely a reason for me to pause, exhale and say 'Thanks. Couldn't have done it without you'.
mazel tov mate! by the way that Thor trailer looks pretty damn good i'm glad Loki is in it. and thanks on the Plastique heads up i'll check that out.
Thanks Dave! Hope you're having a great weekend out there.
Congrats mate! Always a fun blog to visit... keep up the great work!
Big up congrats on reaching your milestone! Its often good fun going on this blog; a fine range of classic covers, images and features.
"a great weekend", hey mate it's only Wed. here.
Wow thanks for the great comments all - nice to have you all stop in!
TKM - you're right there, but who better to draw a milestone cover than the milestone guy right?
Dave, sorry about that - here it's a public holiday, and being on the eve of it felt like the weekend!
Dan, you are a gem. And I don't just love you cause you agree with me about Sandman and Legion.
I also love you because you introduced me to Saga.
For that alone I owe you a keg.
Thanks for the great words Random. Glad I could pass my addiction to SAGA on. Now if everyone I passed it on to would buy me a keg I'd have... Well probably still one keg, but what a keg it would be!


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Jewellery after the French Revolution.
Revival of Greek and Roman style
Georgian diamond, ruby and emerald ring, circa 1830.
The French revolution in 1789 affected jewellery wearing and production greatly. Escaping a fearful time when even the possession of jewellery or belt buckles could lead to the guillotine, there was a prevailing need for a new fashion. The new emerging styles reflected the dramatic change society had gone through, signalling the new philosophy of 'Liberté, égalité, fraternité' (liberty, equality, fraternity). There was a revival of Greek and Roman style due to the interest in the archeological movement and classical art. The neoclassical look popularised simple light and loose dresses emphasised with a tie or hem under the bodice, resembling ancient statues found in Italy and Greece. A style now commonly known as the 'Empire Silhouette'. Bare arms gave rise to the fashion of wearing bangles, always in pairs, around the upper arms. It became popular for ladies to wear rings on every finger of their hands, even the thumbs, further adorning themselves with large circular earrings.
Filigree jewellery and finely detailed gold work became fashionable. In part because of a shortage of money, goldsmiths had become inspired by French peasant jewellery, creating pieces finely detailed in gold that were lighter and cheaper to produce. The new-found ideals of the late 18th century did not permit much use of expensive materials such as diamonds and pearls. There was a larger emphasis on cheaper gemstones. Seed pearls became popular as did garnet, aquamarine, coral, amethyst, chalcedony, turquoise, amber and lapis lazuli.
The motifs used in the late 18th and early 19th century were predominantly classical with a high level of symmetry. Popular shapes were rosettes, festoons, half fluted pillars, vases, urns, hearts, ribbon bows, portraits and snakes. Later these were supplemented with crescent moon and star brooches,
From 1796 onwards, cameos became very fashionable. When in 1804 Napoleon was crowned as Emperor of France, such was his love for this style, he had his coronation crown decorated in cameos. Known as the 'Crown of Charlemagne' it was made up of eight half arches adorned with shell cameos and carved carnelians which met at a golden globe, on top of which was displayed a cross. During this period until his remarriage, his wife Josephine, Empress of France, was viewed as the ultimate style icon, maintaining the fashion that people would covet and copy.
Napoleon's passion for the antique world led to him establishing a school of gem engraving in 1805. With the new Empire, came some semblance of security and luxury trades were revived with enthusiasm. Jewellery became more abundantly worn and jewellery houses benefited from this rise in luxury, finding new clients in the prospering elite of French society. Paris remained the fashion capital and other countries still drew inspiration from the latest fashions which were in vogue there.
Amongst our extensive collection of ancient, antique and vintage collections at Berganza we have a rare selection of early jewels from the 17th and 18th centuries. Shop instore or online today to acquire your own piece of history.
Georgian emerald and diamond cluster ring, English, circa 1830.
Georgian precious topaz earrings, circa 1820.
Early hardstone cameo ring.
Amethyst and gold cannetille earrings, circa 1820.
Georgian antique ruby and diamond ring, circa 1820.
Georgian hair slide, English, circa 1800.
Georgian miniature and diamond brooch, French, circa 1800.

redpajama_set_name: RedPajamaCommonCrawl


text:
Why I Serve: Army Reserve Soldier goes the distance to serve
By Staff Sgt. David OversonApril 28, 2019
PAGO PAGO, American Samoa (April 27, 2019) - Many Americans know the struggle of a long commute to work, but how many travel thousands of miles to accomplish their tasks regularly?Lt. Col. Alejandro Buniag is the Commander of Theater Support Group American Samoa Detachment, American Samoa, but lives 3,600 miles away in Guam.Buniag's determination and commitment can be seen in his leadership philosophy."In order to convince and inspire others to follow and accomplish a mission, the leader must be a true believer in the mission," said Buniag. "Only when leaders at all levels understand and believe in a mission, can they pass that understanding and belief to their teams so they can persevere through challenges, execute, and win."Born and raised in Saipan, Buniag has spent the past 20 years calling Guam home, where he shares his life with his wife Amanda and their five children. However, those aren't the only children Buniag spends his time with, when not wearing an Army uniform, the self described math geek is a fifth grade school teacher on Anderson Air Force Base, Guam.Though the TSG may be a small unit in personnel, its responsibility is immeasurable. The TSG is responsible for the entire Pele U.S. Army Reserve Center, the new state-of-the-art Modular Small Arms Range, the Defense Enrollment Eligibility Reporting System for the island, and is responsible for the other American Samoa tenant units' wellbeing."I'm just so privileged to be part of history," Buniag added. "This is a fantastic unit, and now with the MSAR opening on American Samoa, the very first one in the entire Army Reserve, it's truly an honor."One might say the TSG team is still getting to know one another with Buniag being relatively new to his roll, and his executive officer, Capt. Ruth Fa'Atiga, just coming onboard within the past month as well."We're still getting to know one another," said Fa'Atiga. "However, so far it's great working with him. He gives clear guidance with clear expectations and I see nothing but greatness ahead for our unit."Even the most competent leaders can be overwhelmed if they try to tackle multiple, or a number of tasks simultaneously, so Buniag's motto is to "Keep things simple, prioritize, and execute.""Simplifying as much as possible is crucial to success," he added. "Keep plans, orders and communications simple, clear and concise, and we will see success."Buniag expressed that there is an unwritten rule about leadership. Leaders are responsible for everything in a business, operation, or mission. Implementing extreme ownership requires checking your ego at the door and operating with a high degree of humility."I'm very blessed," added Buniag. "This is a great unit, and though we may be small, we leave a very big footprint here in American Samoa."


Liraglutide is approved for use in patients with Type 2 diabetes but clinical studies have revealed that it is an effective weight loss agent and can be prescribed off label for patients who are overweight or obese. Marketed as Victoza, Liraglutide can be prescribed in the UK for immediate dispensing and next day delivery.
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Liraglutide is delivered by way of a daily subcutaneous injection and makes patients feel more full and satisfied with less food. It also reduces the speed by which the stomach empties, which also makes patients being treated with Liraglutide, crave food less.
In a Europe-wide trial of 554 people with a body mass index of between 30 and 40 (making them clinically obese), patients lost significant amounts of weight.
Can I Buy Victoza (Liraglutide)?
If you are a Type 2 diabetic then you will need to speak to your doctor about this medication to control your condition. You should not attempt to change your treatment regimen without speaking to your GP. If you are looking to be prescribed Victoza for weight loss then this is possible. Prescribing Victoza "off label" for weight loss is being done by some doctors as it is believed that the overall health benefits outweigh the risk of side effects.
What are the side effects of Liraglutide?
Most patients taking Liraglutide did not experience any side effects. Some patients reported nausea and vomiting but the events were not sufficiently grave to make them stop taking the medication and were generally short-lived.
There have however been concerns that Liraglutide may cause thyroid tumours after mice developed the cancer. However, it is thought that mice have a specific sensitivity, which does not apply to humans. There have also been reports of an increased likelihood of pancreatitis.
When will Liraglutide be available?
Liraglutide has already been approved by the European Medicines Agency for diabetes so approval as an obesity treatment should be easier than if it were a brand new drug. Clinical trials are on-going but preliminary results are extremely positive.
Please complete a weight loss consultation. There are two new weight loss medications under consideration by the European Medicines Agency at the moment and they both show some promise. We will keep you informed of any new developments in this field. We promise not to bombard you with e-mails and you can unsubscribe at any time.


text:
\section{\label{sct:intro}Introduction}
The concept of uncorrelated binary collisions\cite{note1} has played an important role in the development
of the kinetic theory of fluids.\cite{note2} It is a central idea in the Boltzmann
kinetic equation for dilute gases, the  Enskog equation for high density liquids, and generalizations of
Enskog theory that take into account the softness of the repulsive
potential.\cite{curtiss64-III,curtiss65-IV,curtiss65-V,Chapman,Hirschfelder}
The nature of the binary
collisions in the Boltzmann and Enskog theories are very different, involving in one case the entire
interparticle potential, and in the other case only the repulsive part of the potential, which is  idealized
as a hard sphere potential.

For kinetic theories of dense liquids that do take into account the effect of both the attractive and
repulsive parts of the interatomic potential on the dynamics, the different range, strength, and effects
of the two parts of the potential has motivated the construction of theories that treat these two parts
very differently.  Examples include the Rice-Allnatt theory\cite{allnatt61,rice61,sengers61} and the Karkheck-Stell
theory.\cite{karkheck80,karkheck81,karkheck82,stell83,karkheck85,karkheck88,stell89}

The equilibrium theory of the structure and thermodynamics of liquids shows similar developments. At
low density, both the radial distribution function and the thermodynamic properties are represented by
relatively simple expressions that contain the entire potential.  At higher density, various theories
have invoked the primacy of the repulsive forces for determining the
structure.\cite{widom64,reiss65,widom67,henderson67,barker67,andersen71,andersen83}
Theories that take into account both types of forces at high density often calculate the effect of
attractions using very different ideas from those used for repulsive forces.\cite{widom64,lebowitz65,henderson67,barker67,andersen72}

Theories of equilibrium structure and thermodynamics have been derived and formulated in a variety of ways,
including perturbation theory, integral equations, and cluster expansions.\cite{hansen-mcdonald,mayer76}
The Mayer cluster theory and its extensions\cite{mayer76,morita61,stell64,andersen77} provide a
unifying theoretical framework for deriving and analyzing many of these theories (and many others) and
relating them to one another. The Mayer theory represents the static correlation functions and free energy
of a fluid in terms of an infinite set of diagrams.  Approximations are typically developed by identifying
certain subsets of diagrams, summing them, and discarding the others. A standard tactic in
cluster theory is to decompose the objects in the theory that represent interactions into various pieces
and include the various pieces in different ways in developing
approximations.\cite{andersen71,andersen77,andersen83,stell64}

A focus on interparticle forces and their effects on dynamics is quite different from the perspective
provided by the fully renormalized kinetic theory of Mazenko and
co-workers.\cite{mazenko:72,mazenko:73I,mazenko:73II,mazenko:74III,mazenko77}
In the latter approach, the theory of the correlation functions for density fluctuations
in equilibrium liquids is formulated in such a way that the interparticle potential does not appear;
instead, only the equilibrium static correlation functions, such as the pair correlation function and its
generalization to more than two particles, appear in the final results.  In effect, the dynamics is
expressed in terms of the potentials of mean force rather than the `bare' potential.

One of the results of this theory was a generalized Boltzmann-Enskog memory
function (denoted GBE) for fluids and an approximation to this function that they called the
`moderate density approximation' to the generalized Boltzmann-Enskog memory function (denoted MGBE).
The latter can be interpreted physically as taking into account
uncorrelated binary collisions among the particles, with the potential that determines the dynamics being
the potential of mean force rather than the bare potential.

In recent work, Young and Andersen\cite{tomMDI,tomMDII} used molecular dynamics simulations to study
the behavior of several pairs of atomic liquids.  Each pair consisted of two systems at the same
density but different temperatures, with very similar pair correlation functions, and hence very similar
potentials of mean force, despite their very different interparticle potentials.  It was found that
some (but not all) of the features of the dynamics of such a pair of liquids were very similar,
suggesting that the potential of mean force is in some sense a more important determinant of
dynamics than is the bare potential itself.  This is consistent with the basic ideas of renormalized
kinetic theory.

A formally exact diagrammatic formulation of the kinetic theory of density correlation functions was
developed by one of us.\cite{andersen:dkt1,andersen:dkt2,andersen:dkt3}
It expresses the dynamic correlation functions of the
particle density in terms of an infinite series of diagrams in a way analogous to the infinite series in
the Mayer cluster theory for equilibrium static correlation functions. The vertices of the
graphs, which represent interactions among particles, are expressed in terms of static correlation functions,
and the theory is fully renormalized in the sense of Mazenko. This theory can provide a unifying theoretical
framework for deriving and analyzing kinetic theories of density fluctuations in liquids.

Using this graphical theory, Ranganathan and Andersen\cite{andersen:dkt4,andersen:dkt5}
considered the case of a fluid whose interatomic potential contained a continuous but very repulsive short
ranged part, as well as longer ranged attractions. Using the inverse of the strength of the short ranged
force as a small parameter, they determined the diagrams that are most important for the short time behavior
of the memory functions for the dynamic correlation function for density fluctuations. This analysis involved
a procedure much like the one discussed above for the Mayer theory, namely a decomposition of the fully
renormalized quantities that appear in the diagrams of the theory into various contributions with different
characteristics in the limit as the small parameter goes to zero. The most important diagrams
({\it i.e.} the most divergent diagrams) were identified. Summing this infinite series of diagrams exactly led
to an approximation for the memory function that is called the short time approximation (STA).
The structure of this approximation indicates that it takes into account uncorrelated binary
collisions involving the repulsive part of the potential, with a collision frequency that is determined
by the structure of the fluid (and hence is influenced by the attractive forces).

Using this short time approximation and neglecting the longer time contributions to the memory function gives
a kinetic theory that can be regarded as a generalization of the Enskog theory to dense fluids with continuous
repulsive forces as well as attractive forces.  The Enskog kinetic equation is local in time, with no memory
of the past history. As a result, the memory function in the Enskog theory is proportional to a Dirac
delta function in time. For fluids with continuous repulsive potentials, there is a part of the memory
function that is very large at very short times\cite{verlet} and that approaches a delta function in
time as the repulsive part of the potential approaches a hard sphere potential. The STA includes all
diagrammatic contributions to that large, short-lived part of the memory function. This is the sense in
which the STA is a generalization of the Enskog theory.

The predictions of this theory have been tested by comparison with computer simulation studies of the
Lennard-Jones fluid.\cite{andersen:dkt5} The accuracy of the theory is different for different correlation
functions but is surprisingly good at high temperatures for the viscosity and diffusion constant.
The theory fails to describe the changes in correlation functions that take place as a dense high
temperature fluid is cooled to the triple point temperature.  In particular, the increasingly negative dip
in the velocity autocorrelation function and the self-longitudinal current are not well described,
suggesting that the dynamics in the vicinity of the triple point requires much more than uncorrelated
repulsive binary collisions for its description.

Ranganathan and Andersen also identified a more general set of diagrams whose sum leads to an approximation,
called the binary collision approximation (BCA), that describes the effect of uncorrelated binary collisions
that are determined by the potential of mean force. (The relationship of this to the moderate density
approximation of the generalized Boltzmann-Enskog memory function will be discussed in more detail below.)

In this paper, we discuss a variety of ways in which binary collisions might be defined for use in
approximations based on uncorrelated binary collisions.  We test several of them by comparing their
predictions with the results of molecular dynamics simulations of a dense Lennard-Jones liquid at various
temperatures, from the triple point temperature to four times that value. There were several motivations
for this work.

\noindent
1.\ \, There are many different possibilities for specifying the nature of the binary collisions in dense
fluids. Some of them are mentioned above. Others are suggested by the possibility of breaking the
potential of mean force, which appears in fully renormalized kinetic theory, into various contributions
and describing the effect of the various contributions in different ways, in analogy to the way in which
the bare potential is decomposed in both kinetic theory and the equilibrium theory of fluids.
Moreover, the diagrammatic kinetic theory suggests some straightforward extensions of binary collision
approximations (as is discussed below).

\noindent
2.\quad It would be worthwhile to test the various approximations by direct comparison of their predictions
with computer simulation studies in order to determine which of them provides, in some sense,
the best starting point for further development of kinetic theory.

\noindent
3.\quad The mode coupling theory of G\"otze and coworkers, and the kinetic theory that provides the basis
for it,\cite{sjogren:80I,sjogren:80II,sjogren:80III}
focusses on the long time behavior of the memory function for density
correlations in a fluid, but it assumes that there is a short time part that represents brief and
presumably binary collisions between the atoms.  However, the theory does not provide a microscopic
expression for this short time part, and in practice its contribution is described by a simple empirical
function of time obtained by fitting to simulation or experimental data. It would be worthwhile to
understand the nature of the true short time behavior of the memory function as a starting point for
understanding the approximations made in mode coupling theory and attempting to go beyond those
approximations.

Our discussion will rely heavily on the diagrammatic kinetic
theory\cite{andersen:dkt1,andersen:dkt2,andersen:dkt3,andersen:dkt4} as a
unifying framework for stating and comparing the various approximations as well as for deriving some
of them.  We restrict attention to dense atomic liquids (rather than gasses) with short range forces
(rather than Coulombic forces).  We also restrict attention to theories that calculate and use a
memory function to calculate the
equilibrium time correlation functions of the density in phase space.  Finally,
the correlation functions we focus on in this paper are self (or incoherent) correlation functions.

\section{\label{sct:bca}Binary collision approximations and the diagrammatic theory}

\subsection{Self correlation function}
We are concerned with the kinetic theory of the self correlation function of density fluctuations in
a dense atomic fluid.  The incoherent scattering function and self-diffusion are experimental
observables associated with this function. (For additional studies that deal with other correlation
functions, see Ref.~\onlinecite{jen:thesis}.)

The system of interest is a single component classical atomic fluid at equilibrium. Its Hamiltonian is
$$H({\bf r}^N,{\bf p}^N)=\sum_{i=1}^N|{\bf p}_i|^2/2m+\sum_{i<j=1}^Nu(|{\bf r}_i-{\bf r}_j|)$$
The self part of the two-point time correlation function for density fluctuations in single particle phase
space is
$$  C^{(s)}(11',t)\equiv
\frac{1}{\rho}\left\langle\sum_{i=1}^N \delta f_i(1,t)\delta f_i(1',0)\right\rangle $$
where $\rho=N/V$ is the number density,
$$ f_i(1,t) = \delta(\mathbf R_1-\mathbf r_i(t))\delta(\mathbf P_1-\mathbf P_i(t)) $$
is the density in single particle phase space for particle $i$, and
$$ \delta f_i(1,t) \equiv f_i(1;t)-\langle f_i(1;t) \rangle$$
is a density fluctuation.
We use numbers like 1, 2, {\it etc.} to represent points in single particle phase space,
{\it e.g.,} $1=\mathbf R_1\mathbf P_1,\ 2' = \mathbf R_{2'}\mathbf P_{2'}$. Angular brackets denote
equilibrium ensemble averages.

The kinetic equation for this correlation function is
\begin{eqnarray}
  \lefteqn{\frac{\partial}{\partial t}C^{(s)}(11',t)} \nonumber\\
\label{eq:kineticEquation}
      &=& \int_0^t dt'\,\int d3\, M^{(s)}(13;t-t')C^{(s)}(31',t')
\end{eqnarray}
where the self part of the memory function $M^{(s)}$ is the sum of two terms
$$  M^{(s)}(11',t) = M_f(11',t) + M_c^{(s)}(11',t) $$
The flow term is
$$ M_f(11',t) = -\frac{\mathbf P_1}{m}\cdot\nabla_{\mathbf R}\delta(11')\delta(t) $$
We use the notation $\delta(11') = \delta(\mathbf R_1-\mathbf R_{1'})\delta(\mathbf P_1-\mathbf P_{1'})$
and $\nabla_{\mathbf R}\delta(11')$ is the gradient of $\delta (11')$ with respect to the first position
argument. We make approximations for $M^{(s)}_c$, the collisional part of the self memory function,
which we shall simply refer to as the memory function.

The initial condition for eq.~(\ref{eq:kineticEquation}) is
$$  C^{(s)}(11',0) = \rho M_m(1)\delta(11') $$
where  $M_m$ is the Maxwell-Boltzmann distribution of momenta
$$ M_m(1)\equiv (2\pi mk_BT)^{-3/2}\exp\left(-|\mathbf P_1|^2/2mk_BT\right)$$
and $g(11')\equiv g(|\mathbf R_1~-~\mathbf R_{1'}|)$
is the pair correlation function of the fluid.

\subsection{Diagrammatic series}
The diagrammatic kinetic theory provides several closely related expressions for the correlation
function $C^{(s)}$ and the memory function $M^{(s)}$ as the sum of the values of an infinite series of
diagrams of a certain topological structure.\cite{andersen:dkt1,andersen:dkt2,andersen:dkt4}
The diagrams contain propagators that
describe the time evolution of density fluctuations and vertices that represent renormalized interactions
among those fluctuations. We shall focus on the two formulations of these series that are most useful for
the present discussion.

The first form of diagrammatic series\cite{andersen:dkt2} contains $\chi^{(0)}$ propagators,
which describe density fluctuations at sets of one or more points in the fluid, and $Q$ vertices.

The second form of the diagrammatic series\cite{note5}
contains $\chi^{(fp)}$
propagators, each of which describes the propagation of a single density fluctuation associated with a
single particle that moves according to free particle motion, and $Q^{(c,p)}$ vertices, which describe
the localized interactions among such density fluctuations.

The two series are each formally exact. The second is derived from first. The first is useful for proving
some formal properties of the series, such as symmetry properties. In practice, the second is more useful
for deriving practical approximations. In the second series it is possible to identify density fluctuations
in different parts of a diagram and interpret them as being caused by the presence (or absence) of the
same particle. The vertices in the theory are calculated from static correlation functions of the fluid,
and so it is possible to identify how many distinct correlated particles are associated with a vertex.
The identification of the number of particles associated with a vertex is in some cases ambiguous,
for reasons that we will discuss, but it is still possible to identify terms that correspond to,
for example, the interaction of two particles, and hence it is possible to discuss whether diagrams
describe binary collisions.

Additional diagrammatic series can be derived from the second series by expressing the vertices as sums
of various contributions.  For example, in vertices that contain the potential of mean force, the latter
can be decomposed into a short ranged repulsive part and a longer ranged oscillatory part, leading to
the original vertex being expressed as a sum of two new vertices.  The resulting diagrammatic series
can then be used to develop approximations in which the effects of the two parts of the potential are
taken into account in different ways. (When an approximation is made to a diagrammatic series for a
correlation function, by neglecting some diagrams and/or approximating certain vertices, the resulting
approximation may or may not be consistent with known symmetry properties of the correlation function,
such as stationarity and time reversal symmetry. See Appendix~\ref{appendix:sym} for a discussion.)

\subsection{Diagrammatic characterization of various approximations}
\paragraph{Generalized Boltzmann-Enskog memory function.}
This memory function, denoted GBE, is generated in Mazenko's theory
by neglecting a function $\Gamma$, which represents renormalized dynamical interactions, in the equation of
motion for a function in terms of which the memory function of interest is
expressed.\cite{note6}
In terms of the first diagrammatic series, it represents the sum of all diagrams in the series for the memory
function in which there are at most two density fluctuations propagating at any time.

This approximation satisfies important symmetry properties exactly (see Appendix~\ref{appendix:sym}).
Moreover, it gives exactly the correct value of the
memory function at $t=0$.  However, it is not clear how it might be calculated in any practical fashion,
and to our knowledge numerical calculations of the GBE memory function have never been performed.
The physical meaning of the approximation is hard to specify but the mathematical meaning in the context
of the diagrammatic theory is clear.  It contains contributions to the memory function that describe
two density fluctuations that interact with each other and propagate. The identities of the two particles
associated with the two fluctuations can be different at different times.
All interactions and propagations of the two fluctuations are
treated exactly. The approximation that is made  is to neglect the possibility of three or more simultaneous
fluctuations.

\paragraph{Binary collision approximation.}
This approximation, denoted BCA, was defined
in Ref.~\onlinecite{andersen:dkt4} in terms of the second diagrammatic series.
It is the sum of all diagrams in the series for
the GBE memory function in which the same pair of particles is associated with the fluctuations at all times.

This approximation does not satisfy important symmetry properties exactly (see Appendix~\ref{appendix:sym}).
However, it does give exactly the correct value for $M^{(s)}$ at $t=0$.  A practical algorithm
exists to evaluate it, as is discussed below.

The physical meaning of the approximation is that it contains contributions to the memory function that
describe two density fluctuations, associated with two specific particles, that interact with each other
and propagate, without having three fluctuations present at any time. All interactions and propagations
of the two fluctuations are treated exactly. The approximation here is not only to neglect the possibility
of three or more simultaneous fluctuations (as in the GBE) but also to neglect diagrams in which the two
fluctuations at one time represent a different pair of particles than the two fluctuations at other times.
This characterization is the reason why it can be regarded as a binary collision approximation,
{\it i.e.} one that involves only two particles.

\paragraph{The moderate density approximation to the generalized Boltzmann-Enskog memory function.}
This memory function, denoted MGBE, was defined by Mazenko as an
approximation to the GBE.  Various quantities in the GBE are defined in terms of static correlation functions
of the fluid. To construct the MGBE, the contributions to these quantities that contain three and four
particle correlation functions are deleted.\cite{note7}

The deletion of the three and four particle correlation functions to define the MGBE is closely related to,
but not equivalent to, what is done in defining the BCA to include diagrams in which fluctuations
representing at most two distinct particles appear.  (See Appendix~\ref{appendix:leftVertex}.)
The MGBE is almost
equivalent to the BCA, except for one factor in every diagram.  From a diagrammatic point of view,
the MGBE can be regarded as an approximation to the BCA in which the vertex $Q_{12}$ that appears
in each diagram is replaced by an approximation that contains the potential of mean force.
(A detailed expression for the MGBE is given below, together with a comparison to the BCA.)

This approximation satisfies important symmetry properties exactly (see Appendix~\ref{appendix:sym}).
It does not give exactly the correct value of the memory
function at $t=0$. A practical algorithm exists to evaluate it, as is discussed below.

The physical meaning of the MGBE is equivalent to that of the BCA.  The only difference is that the
additional approximation made to the $Q_{12}$ leads to a binary collision approximation that satisfies
the symmetry condition.

\paragraph{Short time approximation.}
This will be referred to as the STA. This was derived\cite{andersen:dkt4} from the second diagrammatic series
using the procedure discussed above, namely expansion of the diagrams in powers of  a small parameter
representing the inverse of the strength of the repulsive part of the potential and retaining those terms
that at short times are most divergent. The diagrams that appear are those of the BCA, as defined above,
but with vertices that contain the repulsive part of the interatomic potential, rather than the potential
or the potential of mean force. (See below for a more precise statement of the value of the STA.)

The STA satisfies the symmetry property approximately, but not exactly. (In the limit of infinitely
repulsive forces, it satisfies the symmetry property exactly.  See Appendix \ref{appendix:sym}.)
It does not give
exactly the correct value of the memory function at $t=0$. A practical algorithm exists to evaluate it.

\paragraph{Other binary collision approximations.}  Below we shall discuss several approximations whose
diagrammatic meaning is straightforward. Like the STA, they can be defined graphically by starting with
the BCA, separating each of the vertices into two terms, and retaining only the first term.
The physical meaning of such an approximation depends on the nature of the separation. The reason for
investigating such approximations is the desire to develop better binary collision approximations as a
starting point for a more general kinetic theory.

\paragraph{Beyond the binary collision approximation.}
A major restrictive assumption of all the approximations mentioned above is that the  possibility of
having three or more simultaneous density fluctuations is ignored. The pair of particles that are associated with
the two density fluctuations interact with each other, but the presence of other particles is taken into
account only in some average sense, if at all, due to the appearance of static pair correlation functions
of the fluid in the various vertices.

\begin{figure}
\epsfig{figure=sbca1_may14,width=8cm}
\caption{The first two diagrams are members of the diagrammatic series for the binary collision
memory functions. (The remaining diagrams in the binary collision memory functions have
additional $Q_{22}^{(c,2)}$ vertices on the two horizontal particle paths.) The third and
fourth diagrams are examples of diagrams that are not in a binary collision memory function.
One or both of the two particles is interacting separately with a third particle in an STA
approximation. The parts of the diagrams that have these STA interactions are indicated with
dashed lines and dashed circles. (See the text and Fig.~\ref{fig:diagrams-sbca2} for additional
discussion.)}
\label{fig:diagrams-sbca1}
\end{figure}

\begin{figure}
\epsfig{figure=sbca2_may14,width=8cm}
\caption{Examples of diagrams in a binary collision approximation augmented with $M_{STA}^{(s)}$ memory
functions on the particle paths to represent interactions with third particles of the type shown in
Fig.~\ref{fig:diagrams-sbca1}. The first diagram here includes the third diagram of
Fig.~\ref{fig:diagrams-sbca1}. The second diagram here includes the fourth diagram of
Fig.~\ref{fig:diagrams-sbca1}. The generic member of this series is obtained by taking the generic
member of the series for a binary collision approximation (see the first two diagrams in
Fig.~\ref{fig:diagrams-sbca1} for examples) and inserting an arbitrary number of $M_{STA}^{(s)}$
memory functions on each of the lines. (See the text and Fig.~\ref{fig:diagrams-sbca1} for
additional discussion.)}
\label{fig:diagrams-sbca2}
\end{figure}

It is relatively simple, in the context of a graphical theory, to construct approximations that relax these
assumptions in a physically meaningful way. The first two diagrams in a binary collision approximation are shown
as the first two diagrams in Fig.~\ref{fig:diagrams-sbca1}. The generic BCA diagram consists of two particle
paths\cite{andersen:dkt4} that propagate from the
$Q_{21}^{(c,1)}$ vertex on the right to the $Q_{12}^{(c,1)}$ vertex on the left. In between these vertices
are zero or more $Q_{22}^{(c,2)}$ vertices. These diagrams describe the propagation of a pair of particles
that interact with each other using the potential that is contained within the $Q^{(c,2)}_{22}$ function.
To develop the $K$ approximations, consider including diagrams in the exact graphical series that are similar
to these and that are constructed by inserting STA memory functions into the propagators
on the particle paths of the original diagrams of the BCA. The diagrams that result are those
of the original BCA diagrams with any number of STA memory functions inserted.
See Fig.~\ref{fig:diagrams-sbca2}.

When the diagrams with STA memory functions are evaluated, the two points have different times associated with
them and the STA memory function is  a function of the time difference. Since the collisions are brief, we
approximate the effect of these vertices using a BGK model.\cite{krook} The BGK model memory function is
$$M_K(11^\prime;t) =\nu m_K(11^\prime)\delta(t)$$
where
$m_K(11^\prime) =-\delta({\bf P}_1-{\bf P}_1^\prime)+M_m({\bf P}_1)$
 and $\nu$ is a collision frequency.  The BGK model is an approximate memory function that is instantaneous
in time. It approximates a collision as having the
effect of replacing the momentum of the particle by a new random momentum drawn from a Maxwell-Boltzmann
distribution.  Because of its simplicity, its effect can easily be incorporated into the algorithm for
evaluating binary collision approximations, as is discussed below.

This procedure allows any binary collision approximation to be converted into one that also
includes brief uncorrelated binary collisions of each of the two particles with third particles or
fluctuations in the surrounding fluid.  Graphically this is done by inserting $M_K$ vertices, as
approximations for STA memory functions, into the $\chi^{(fp)}$ propagators that by themselves
describe free particle propagation.  When this procedure is applied to any approximation, the
resulting approximation will be given a name with ``K'' as a prefix.  Thus, for example, the MGBE
approximation, when modified in this way, will be called the K-MGBE approximation.  In constructing
such an approximation, the value of the parameter $\nu$ must be chosen.  This will  be discussed below.

We shall refer to this class of approximations as $K$ approximations. $K$ approximations are mathematically
and physically similar to a type of approximation suggested by Mazenko and Yip.\cite{note8}

\subsection{Analytic formula for the BCA and related approximations}
The various approximations mentioned above, with the exception of the GBE approximation, can be expressed
in terms of very similar analytic formulas for $M^{(s)}$.  As a result, all of them can be evaluated with
a similar algorithm. Here we will state the formulas. The algorithm will be discussed later.

For the BCA, the memory function is
\begin{eqnarray}
\label{eq:MBCA-generalForm}
  \lefteqn{M^{(s)}_{BCA}(1t,1't') =\int d2\,d3\,d4\,d5\,}\\
&\times& Q_{12}^{(c,1)}(1;23)\chi_{BCA}(23t;45t') Q_{21}^{(c,1)}(45;1')\nonumber
\end{eqnarray}
where the $Q$ vertices represent interaction terms and the $\chi_{BCA}$ term is a response function.
The detailed equations for these functions are
\begin{eqnarray}
\label{eq:orig_Q12}
 \lefteqn{ Q_{12}^{(c,1)}(1;1'2') = \nabla_{\mathbf R}V_{12}(12')\cdot\nabla_{\mathbf P}\delta(11') }\\
\label{eq:orig_Q21}
  \lefteqn{Q_{21}^{(c,1)}(12;1')}\\ &=& \frac{M_m(1)}{M_m(1')} \rho(2)M_m(2)g(12)
         \nabla_{\mathbf R} V_{21}(12)\cdot \nabla_{\mathbf P}\delta(11')\nonumber\\
\label{eq:orig_chiBCA}
\lefteqn{\chi_{BCA}(12t;1'2't')}\\
 &=& e^{-i\mathcal L(12)(t-t')} \delta(11')\delta(22')\Theta(t-t')\nonumber
\end{eqnarray}
The Liouville operator $\mathcal L$ is
\begin{eqnarray}
\label{eq:fullL}
  \mathcal L(12) &=& L_0(1) + L_0(2) + \tilde L_1(12)\\
\label{eq:L0}
  L_0(1) &=& -i\frac{\mathbf P_1}{m(1)}\cdot\nabla_{\mathbf R_1}\\
\label{eq:L1}
  \tilde L_1(12) &=& i\nabla_{\mathbf R}V_{22}(12)\cdot(\nabla_{\mathbf P_1} -\nabla_{\mathbf P_2})
\end{eqnarray}
These formulas contain three functions $V_{12}$, $V_{21}$ and $V_{22}$ that are two particle potentials.
For the BCA, they are
\begin{equation}
\label{eq:allV}
 V_{12}(12)=u(12);\quad V_{21}(12)=V_{22}(12)=v_{MF}(12)
\end{equation}
where $u(12)$ is the interparticle potential and $v_{MF}(12)=-k_BT\ln\,g(12)$ is the potential of mean force,
and $g(12)$ is the pair correlation function.

The BCA is exact in the low density limit and corresponds to the linearized Boltzmann equation in that
limit on appropriate length and time scales. The $\chi_{BCA}$ describes two particle scattering with
an effective interparticle potential that is the potential of mean force, which also appears in $Q_{21}$.
At low density, the potential of mean force approaches the bare potential. The BCA describes binary
collisions (on microscopic length and time scales) in much the same way as the Boltzmann equation, with
some many body effects accounted for in the effective potential.

The MGBE is of the same form as eqs.~(\ref{eq:MBCA-generalForm})--(\ref{eq:L1}), with (\ref{eq:allV})
replaced by
\begin{equation}
\label{eq:MGBE-allV}
     V_{12}(12)= V_{21}(12) = V_{22}(12) = v_{MF}(12)
\end{equation}
The STA is of the same form as eqs.~(\ref{eq:MBCA-generalForm})--(\ref{eq:L1}), with (\ref{eq:allV}) replaced by
$$V_{12}(12)=V_{21}(12)=V_{22}(12)=u_r(12)$$
where $u_r(12)$ is the repulsive part of the potential $u(12)$.

Any other binary collision approximation, as discussed above, leads to results of the form of
eqs.~(\ref{eq:MBCA-generalForm})--(\ref{eq:L1}),
with the  potentials $V_{12}$, $V_{21}$, and $V_{22}$ being the functions used in the approximation.
See Table~\ref{table:listApprox} for a list of various approximations we have considered.

\begin{table}
\begin{center}\begin{tabular}{l|cccc}
Approximation & $V_{12}$ & $V_{21}$ & $V_{22}$ & $\hat M_{\hat k\hat
k}(\mathbf k,0)$ \\ \hline
STA  & $\ u_r(r)\ $ & $\ u_r(r)\ $ & $\ u_r(r)\ $ & -332 \\ \hline
BCA  & $u(r)$   & $v_{MF}(r)$  &  $v_{MF}(r)$ &  -281 \\
K-BCA  & $u(r)$   & $v_{MF}(r)$  &  $v_{MF}(r)$ & -281 \\ \hline
MGBE & $v_{MF}(r)$ & $v_{MF}(r)$  &  $v_{MF}(r)$  &  -289 \\
K-MGBE & $v_{MF}(r)$ & $v_{MF}(r)$  &  $v_{MF}(r)$ & -289 \\ \hline
RPMF& $\ v_{rMF}(r)\ $ & $\ v_{rMF}(r)\ $  & $\ v_{rMF}(r)\ $  & -215 \\
K-RPMF & $v_{rMF}(r)$ & $v_{rMF}(r)$  &  $v_{rMF}(r)$  & -215 \\  \hline
hybrid &              &               &                &      \\
MGBE/STA & $v_{MF}(r)$ & $v_{MF}(r)$ & $u_r(r)$ & -289
\end{tabular}\end{center}
\caption{A table of all approximations studied, stating the necessary potential functions that define
a binary collision approximation for the memory function and the value
of the $\hat k\hat k$ matrix element of the memory function at $t=0$.
The BCA value for $\hat M_{\vec\mu\vec\nu}(\mathbf k,0)$ is exact. Note that $u(r)$
is the bare potential, $u_r(r)$ is the repulsive part of the bare potential, $v_{MF}(r)$ is the potential
of mean force, and $v_{rMF}(r)$ is the repulsive part of the potential of mean force.
Details about approximations not discussed in this text can be found in Ref.~\onlinecite{jen:thesis}.}
\label{table:listApprox}
\end{table}

Any $K$ approximation, as discussed above, leads to results of the form of
eqs.~(\ref{eq:MBCA-generalForm})--(\ref{eq:L1}) with eq.~(\ref{eq:fullL}) replaced by
\begin{eqnarray}
\label{eq:Kapprox}
 \lefteqn{\mathcal L_K(12)}\\
   &=& L_0(1) + L_0(2) + \tilde L_1(12) + \tilde L_{BGK}(1) + \tilde L_{BGK}(2) \nonumber
\end{eqnarray}
The symmetry properties of the various approximations are discussed in Appendix~\ref{appendix:sym}.

\section{Calculations}
\subsection{\label{subsct:basisFuncExpan} Basis function expansion of the kinetic equation}
Using the expressions for the $Q$ vertices and $\chi_{BCA}$ response function, the expression in
eq.~(\ref{eq:MBCA-generalForm}) leads to
\begin{eqnarray*}
\lefteqn{M^{(s)}_{BCA}(1t;1't') = \int d2\,d3\,d4\,d5\, \frac{M_m(4)}{M_m(1')}\rho(5)M_m(5)g(45)}\nonumber\\
&&\times\left(\nabla_{\mathbf R}V_{12}(13)\cdot\nabla_{\mathbf P}\delta(12)\right)
\cdot\left(\nabla_{\mathbf R}V_{21}(45)\cdot\nabla_{\mathbf P}\delta(41')\right)\nonumber\\
&&\times\, e^{i\mathcal L(23)(t-t')}\delta(24)\delta(35)\Theta(t-t')
\end{eqnarray*}

It is convenient to take the Fourier transform of $C^{(s)}$ and $M^{(s)}$ with regard to the difference in
their position arguments and to represent their momentum dependence in terms of a set of Hermite polynomial
basis functions. The details of the Hermite polynomial expansion are discussed in
Ref.~\onlinecite{jen:thesis}. From eq.~(\ref{eq:MBCA-generalForm}), we can obtain the following expression
for the Fourier transformed matrix elements of $M^{(s)}$
\begin{eqnarray}
\label{eq:Mmunu}
 \lefteqn{\hat M^{(s)}_{\vec\mu\vec\nu,BCA}(\mathbf k,t) = -\int d\mathbf P_1\,d\mathbf R_2\,d\mathbf P_2 \,
    g(|\mathbf R_1-\mathbf R_2|)} \nonumber\\*
&\times& M_m(\mathbf P_1)\rho M_m(\mathbf P_2)
(\nabla V_{Q_{21}}(|\mathbf R_1-\mathbf R_2|)\cdot \nabla h_{\vec\nu}(\mathbf P_1))\nonumber\\*
&\times& e^{-i\mathbf k\cdot\mathbf R_a} (\nabla V_{Q_{12}}(|\mathbf R_a-\mathbf R_b|)\cdot
    \nabla h_{\vec\mu}(\mathbf P_a))
\end{eqnarray}
The $\vec\mu$ and $\vec\nu$ are labels for the Hermite polynomial basis functions. Each is an ordered
triplet of nonnegative integers. (In the following, we shall use `$\vec 0$' as an abbreviation for the
triplet `000' and `$\hat k$' as an abbreviation for `001'. Note that $ M^{(s)}_{\vec\mu\vec\nu}=0$ if either
$\vec\mu=0$ or $\vec\nu=0$. $M^{(s)}_{\hat k\hat k}$ is the single most important matrix element of the
memory function. In the limit of small wave vector and/or short time, it is the only matrix element that
contributes to the correlation function, and in general it is the matrix element that most strongly affects
the time dependence of the correlation function.) In eq.~(\ref{eq:Mmunu}), $\mathbf R_1$ is defined to be the
origin and the phase points $\mathbf R_1\mathbf P_1\mathbf R_2\mathbf P_2$ are the initial conditions for
two particles that evolve forward in time according to Hamilton's equations of motion with $V_{22}$ as the
interatomic potential to the final phase points $\mathbf R_a\mathbf P_a\mathbf R_b\mathbf P_b$ at time $t$.
The matrix elements of the memory function are calculated using the two particle trajectory calculation method
of Ranganathan and Andersen.\cite{andersen:dkt5} See Appendix~\ref{appendix:numCalc}
for details and for a discussion of the error analysis.

The correlation functions of interest in this work are the self correlation functions: the incoherent
intermediate scattering function (IISF)
$$ \hat F_s(\mathbf k,t) = \hat C^{(s)}_{\vec0\vec0}(\mathbf k,t)
    = \frac{1}{N} \left\langle\sum_{i=1}^N e^{-i\mathbf k\cdot(\mathbf r_i(t)-\mathbf r_i(0))} \right\rangle$$
\iffalse the velocity autocorrelation function (VACF)
$$ \Psi(t) = \frac{1}{N} \left\langle \sum_{i,j=1}^N p_{i,z}(t)p_{i,z}(0)\right\rangle$$ \fi
and the self longitudinal current correlation function (SLCC)
\begin{eqnarray*}
 \hat J_{ls}(k,t) &=& \hat C^{(s)}_{\hat k\hat k}(k_z,t) \\
  &=& \frac{1}{N} \left\langle\sum_{i=1}^N \frac{p_{iz}(t)}{m}\frac{p_{iz}(0)}{m}
                e^{-ik_z(r_{iz}(t)-r_{iz}(0))}\right\rangle
\end{eqnarray*}
In these equations it is to be understood that the wave vector $\mathbf k$ is in the $z$-direction.
The IISF and SLCC functions are related in the following way
$$ \hat J_{ls}(\mathbf k,t) =
         -\frac{1}{|\mathbf k|^2}\frac{\partial ^2}{\partial t^2}\hat F_s(\mathbf k,t) $$
In the graphs below, the SLCC is normalized so that its value at unity is $t=0$.
We focus on the SLCC because we have found its qualitative features to be sensitive to errors in binary
collision approximations. For a discussion of additional correlation functions, see
Ref.~\onlinecite{jen:thesis}.
\iffalse We note that $J_{ls}(\mathbf k,t)$ is equal to $\ddot F_s(\mathbf k,t)$, except for a constant
numerical factor, and that in the limit $k\rightarrow 0$, $J_{ls}(k,t)$ is equal to the velocity
autocorrelation function, except for a constant numerical factor. When results are presented below,
$\Psi(t)$ and $J_{ls}(k,t)$ will be normalized so that their values at $t=0$ are equal to unity.\fi

For some of the discussion we will need two related functions. The first is the total correlation function
$$ C(11',t) = \langle \delta f(1,t)\delta f(1',0)\rangle$$
where
$$ f(1,t)=\sum_{i=1}^N \delta(\mathbf R_1-\mathbf r_i(t))\delta(\mathbf P_1-\mathbf p_i(t))$$
is the total particle density and
$$ \delta f(1,t) \equiv f(1,t)-\langle f(1,t)\rangle$$
is a density fluctuation. The second function is the Fourier transform of $C(11',t)$, the coherent intermediate
scattering function (ISF)
$$ \hat F(\mathbf k,t) = \hat C_{\vec0\vec0}(\mathbf k,t) =
  \frac{1}{N}\left\langle \sum_{i,j=1}^N e^{-i\mathbf k\cdot(\mathbf r_i(t)-\mathbf r_j(0))} \right\rangle $$

\subsection{Solution of the kinetic equations}
Use of the Hermite polynomial basis set for the momentum arguments converts the kinetic equation for
$\hat C^{(s)}({\bf k},t)$ into an infinite dimensional matrix equation for the matrix elements of
$\hat C^{(s)}$. We solve this infinite dimensional equation numerically using the method of Ranganathan and
Andersen.\cite{andersen:dkt5} In this method, the equations for successively larger finite sets of
coupled equations are solved. Each of these finite sets is chosen such that its solution agrees with the
exact solution of the infinite set up to order $t^n$ for small $t$, and successive finite sets have
increasing values of $n$. When the successive solutions no longer change with increasing $n$, the last
result is regarded as the solution to the infinite set. A detailed discussion of the method can be found in
Ref.~\onlinecite{mr:thesis}.

The IISF, which corresponds to $\hat C^{(s)}_{\vec 0\vec 0}(\mathbf k,t)$, is converged with the
$\mathcal O(t^9)$ approximation. The SLCC, which corresponds to
$\hat C^{(s)}_{\hat k\hat k}(\mathbf k,t)$, is converged with the $\mathcal O(t^7)$ approximation. We present
only these solutions in the results that follow. Moreover, we concentrate on the results for the SLCC since
none of the approximations are able to consistently reproduce all of the features of this function, especially
the temperature dependence of the negative region for the function.
Since the SLCC is proportional the second derivative of the IISF, the accuracy of an
approximation for the SLCC function is consistent with its accuracy for the IISF.

\subsection{Molecular dynamics calculations}
The correlation functions calculated from the binary collision approximations for the memory function will
be compared to results generated from molecular dynamics simulations of a 500-particle, one component
Lennard-Jones liquid
at equilibrium.\cite{tomMDI,tomMDII,stein} The states of interest are reduced density $\rho=0.85$
(approximately the triple point density) at reduced temperatures $T = 0.723,\, 1.554,\, 3.000$
(the triple point temperature is approximately $0.723$). Results for temperatures between 0.723 and 1.554
at $\rho=0.85$, and high temperature results at $\rho=0.75$ can be found in Ref.~\onlinecite{jen:thesis}.
The usual reduced units for the Lennard-Jones fluid are used.\cite{AllenTildesley}
The correlation function results from molecular dynamics simulation and the binary collision approximations
have small statistical error whose values are generally smaller than the differences between the simulation
data and the theoretical curves. The details of the molecular dynamics simulations can be found in
Refs.~\onlinecite{tomMDI} and \onlinecite{tomMDII}.

\section{Results}
\subsection{BCA}
Although the BCA approximation gives exactly the correct $t=0$ value of the memory function, this approximation
leads to correlation functions that oscillate rapidly and with sharply increasing magnitude as $t\rightarrow\infty$.
Although the shape of the BCA memory function
looks reasonable, there is insufficient area under the curve for the solution to the kinetic equation to be
well-behaved. We suspect that this is related to its failure to satisfy the symmetry property discussed in
Appendix~\ref{appendix:sym}. (Recall the form of this approximation,
eqs.~(\ref{eq:MBCA-generalForm})--(\ref{eq:L1}).)

\subsection{STA, MGBE, and K-MGBE}

\subsubsection{STA}
Overall, for the high densities studied, the STA approximation does reasonably well at predicting correlation
functions at all temperatures and wave vectors, with increasing quantitative accuracy with high temperature
and large wave vector. The short time predictions of the STA are reasonably accurate, as is the overall
longer time behavior. The STA approximation is able to reproduce qualitative trends of the various
correlation functions with changes in wave vector. The trends with changes in temperature are also reproduced,
except for the SLCC curve where the
shifting and increasing magnitude of the negative dip is not well reproduced. In fact, the STA approximation
consistently fails to predict the temperature dependence of the negative dips in the SLCC curve.
A complete discussion of the results of the STA approximation can be found in
Refs.~\onlinecite{andersen:dkt5} and \onlinecite{jen:thesis}.

\subsubsection{MGBE}
The MGBE approximation is more accurate at short times than the STA, as shown in Fig.~\ref{fig:st-SLCC-lowTk},
and is in essentially perfect agreement with the simulation data for times less than about
0.1. This statement holds for the self correlation functions at all
temperatures and wave vectors studied.  The short time part of the SLCC function is largely determined by
the value of the $\hat k\hat k$ matrix element of the memory function at $t=0$. The MGBE does not have
exactly the correct zero time values of the memory function matrix elements but it is more accurate than
the STA, and this accounts for its accuracy at short times.   The MGBE result begins to become inaccurate at
$t\approx0.1$, rising significantly above the simulation data.

\begin{figure}
\epsfig{figure=st-SLCC-0723T-075k,width=8cm}
\caption{Normalized SLCC function. Although the MGBE and K-BCA approximations do poorly at representing
the simulation result for longer times, for times $t<0.1$ we see that these are the most quantitatively
accurate approximations. The statistical error bars for the MGBE correlation function are shown and are
barely discernable on this scale.
All of the theoretical correlation functions have similarly small statistical errors.}
\label{fig:st-SLCC-lowTk}
\end{figure}

\begin{figure*}
\epsfig{figure=lt-SLCC-0723T-075k,width=8cm}
\epsfig{figure=lt-SLCC-0723T-675k,width=8cm}
\caption{Normalized SLCC function for the STA, MGBE and K-MGBE approximations for $T=0.723$ at small and
large wave vector. The STA and MGBE curves lie on top of each other at large wave vector.
High temperature results are shown in Fig.~\ref{fig:SLCC-lt-compare-highT}.}
\label{fig:SLCC-lt-compare-lowT}
\end{figure*}

\begin{figure*}
\epsfig{figure=lt-SLCC-1554T-075k,width=8cm}
\epsfig{figure=lt-SLCC-1554T-675k,width=8cm}
\caption{Normalized SLCC results for the STA, MGBE, and K-MGBE approximations for $T=1.554$ at small and
large wave vector. See Fig.~\ref{fig:SLCC-lt-compare-lowT} for low temperature results. }
\label{fig:SLCC-lt-compare-highT}
\end{figure*}

For longer times, $t\ge0.1$, the MGBE gives very inaccurate results.
See Figs.~\ref{fig:SLCC-lt-compare-lowT} and \ref{fig:SLCC-lt-compare-highT}.  For large
wave vectors at both high and low temperatures, the MGBE behaves reasonably, with a single minimum in the
SLCC at approximately the right time, but an overall quantitative disagreement with the data.  For small wave
vectors at all temperatures, it predicts oscillation in the SLCC for times of about 0.2-0.3, a positive
maximum in the function at $t\approx0.3$, and a very slow decay of the correlation function to zero from
above, in qualitative disagreement with the simulation results.  The latter have a single minimum, followed
by an approach to zero from below on a shorter time scale than the MGBE result approaches zero.

The origin of this unusual behavior, especially the oscillations, is easily understood from the nature of
the MGBE.  The memory function in the MGBE is a time correlation function for derivatives of the potential
of mean force in a two particle system for which the two particles interact according to the potential of mean
force of the fluid and move according to Hamiltonian dynamics. See eqs.~(\ref{eq:MBCA-generalForm}),
(\ref{eq:MGBE-allV}), (\ref{eq:Mmunu}) and Appendix~\ref{appendix:numCalc}.
This idea, which follows from the explicit formula eq.~(\ref{eq:Mmunu}),
is the basis for the algorithm used to evaluate the matrix
elements of the memory function.  The potential of mean force at high density is oscillatory and extends
to distances of several atomic diameters.  Thus, collisions that are relevant for the MGBE are of long duration
and the correlation function that gives the memory function is nonzero for a very long time.

\begin{figure}
\epsfig{figure=MF001001-all-0723T-075k,width=8cm}
\caption{Self part of the $\hat k\hat k$ matrix element of the memory function.
The STA approximation generates a non-oscillatory memory function while both MGBE and K-MGBE approximations
yield oscillatory memory functions. These oscillations are the cause of the oscillations in the
MGBE and K-MGBE results for the SLCC.}
\label{fig:MF-shortTime-comp}
\end{figure}

Figure~\ref{fig:MF-shortTime-comp}
shows the time dependence of the principal matrix element of the memory function for the MGBE
approximation for a low temperature state. There is a pronounced oscillation that persists for times
longer than the time scale for the true correlation function to decay.  For comparison, the STA memory
function decays to zero very quickly, within times of the order of 0.1.  As we shall discuss below in the
next section, the large positive feature in the memory function and the subsequent oscillation are not
present in the actual memory function.

These results show that the MGBE approximation, although quantitatively correct at short times,
is qualitatively incorrect at intermediate and long times.

\subsubsection{K-MGBE}
The collisions described above that are the basis of the MGBE approximation, namely Hamiltonian dynamics
of a pair of particles whose interaction potential is equal to the potential of mean force of the liquid,
seem inappropriate for determining the dynamics of a dense liquid.  The particles surrounding a pair of
particles should do more to determine the dynamics of the particle pair than merely to renormalize
the bare potential into the potential of mean force. In Mazenko's theory, this would be taken into account
by including the quantity called $\Gamma$, which represents renormalized dynamical interactions of the pair.
In the cluster theory, this would be taken into account by including the coupling of the pair to fluctuations
of the density of third particles.

The K-MGBE approximation discussed above provides a tractable way of including these interactions, provided
they are regarded as independent uncorrelated repulsive collisions of the two particles with third particles,
approximated using the BGK model memory function.  This approximation sums, in an approximate way, a class of
diagrams in the exact diagrammatic theory.  We assume that these collisions involve the short range repulsive
part of the interatomic potential, so we pick the parameter $\nu$ so that the BGK memory function has a
strength that is related to that of the STA memory function. Since the most important matrix element of the
memory function is the $\hat k\hat k$ matrix element, we choose $\nu$ so that this matrix element for the BGK
model corresponds to the actual matrix element for the STA.  More precisely, we choose $\nu$ such that
\begin{equation}
\label{eq:nu}
\nu = \int_0^\infty dt'\,\hat M^{(s)}_{\hat k\hat k,STA}(\mathbf0,t')
\end{equation}
While this is a crude approximation overall, it was devised to give some idea of the effects of third particles
on the dynamics of the pair, in the hope of yielding an improved version of the MGBE.

Figure~\ref{fig:MF-shortTime-comp}
shows that the K-MGBE memory function has less pronounced oscillations and a more rapid decay to
zero than the MGBE memory function.  Figures~\ref{fig:SLCC-lt-compare-lowT} and \ref{fig:SLCC-lt-compare-highT}
show the SLCC correlation functions given by the K-MGBE
approximation. Introducing the BGK collisions extends the initial time interval over which the approximation
is valid, for low and high temperatures and for small and large wave vector. The intermediate time behavior
of the K-MBGE is unsatisfactory, however, with a peak in the correlation function at times of about 0.3
appearing under all conditions studied.  The long time behavior of the K-MGBE is at best a slight improvement
over that of the MGBE.

\subsection{Additional approximations}
In an attempt to find improved binary collision approximations, we investigated a number of other
possibilities.

\paragraph{RPMF approximation.}

The STA memory function, which uses the repulsive part of the potential in each of the vertices, has the
virtue of decaying to zero rapidly without oscillations at long time that produce unphysical behavior of the
correlation function at long times.  The rapid decay is a result of the fact that the two particle dynamics
in the calculation of the memory function involves purely repulsive interactions (and hence brief collisions)
and the range of the repulsive potential is so short.

An approximation, which we denote RPMF, based on using the repulsive part of the potential of mean force in
each vertex, was constructed and studied.  While the memory function  generated by this approximation has
the feature of rapid decay, as discussed for the STA, its magnitude and its integrated area are smaller than
that of the STA and it generates an SLCC that falls from its initial value significantly more slowly that the
STA result.  This basic flaw in the RPMF approximation is evident in the zero time value of the memory
function (see Table~\ref{table:listApprox}).

\paragraph{K-RPMF approximation.}
The K-RPMF approximation adds BGK collisions with third atoms to the RPMF approximation.  This memory
function decays more slowly than that of the RPMF, but it has the same initial value as the RPMF, and
overall  is not a significant improvement over the RPMF.

\paragraph{A hybrid of the MGBE and STA approximations.}
We attempted to construct approximations that have accurate values of the memory function at zero time but
that were based on relatively brief collisions.  In one such approximation, we used the potential of mean
force in the $Q_{12}^{(c,1)}$ vertex and the $Q_{21}^{(c,1)}$ vertex (as in the MGBE approximation) but the
repulsive part of the potential in the $Q_{22}^{(c,1)}$ vertex (as in the STA).  We also considered the
approximation that supplemented this approximation with BGK collisions with third particles. This hybrid
is not a significant improvement over the MGBE or STA approximations
and had some artifacts for small wave vector that are inconsistent with the simulation results.

\paragraph{Other approximations.}
A variety of other unsuccessful approximation were also studied.  See Ref.~\onlinecite{jen:thesis} for details.

\section{Extraction of the memory function}
Thus far, we have tested approximate kinetic theories by calculating their memory functions, solving the
memory function equation for the related correlation functions, and comparing the predicted correlation
functions with computer simulation results. These comparisons give only an indirect test of the accuracy of
the approximate memory function approximations.

In this section we consider a different set of questions.  Given a correlation function calculated from
simulation data,
what is the memory function that corresponds to it? What are the properties of the `exact' memory function?
How do the approximate memory functions compare with the `exact' memory function?

The numerical technique we use for extracting the memory function from the correlation function data
employs a Fourier series representation of the memory function as a function of time.  The Fourier
coefficients in the series were adjusted to minimize the differences between the SLCC calculated from the
memory function and the SLCC obtained from computer simulation.  For small wave vector, the $\hat k\hat k$
matrix element of the memory function plays the dominant role in determining the SLCC
function.\cite{jen:thesis}
The reason for this is that the coupling of the correlation function to other elements of the memory function
involves positive powers of the wave vector.  Thus we need to construct a Fourier representation of only this
matrix element of the memory function and solve for the Fourier coefficients of just one function rather
than many.  The coefficients are found by a minimization of the sum of the squares of the deviations of the
calculated correlation function and the simulated function.  See Ref.~\onlinecite{jen:thesis} for details.

\begin{figure}
\epsfig{figure=eMF-compare-2temps,width=8cm}
\caption{Extraction results (dashed lines) for the $\hat k\hat k$ matrix element of the self memory
function compa STA curves (solid lines) at $\rho=0.85$ and $|\mathbf k|=0.75$.
The top set of lines at $t=0$ corresponds to $T=0$. The lower set at $t=0$ corresponds to $T=3.000$.}
\label{fig:extractMF-results}
\end{figure}

The extracted $\hat M^{(s)}_{\hat k\hat k}({\bf k},t)$ for small wave vector is shown in
Fig.~\ref{fig:extractMF-results} for the
lowest and highest temperatures studied, where they are compared with the STA predictions for the same
function.  The extracted memory function at low temperature has none of the oscillations and high maxima
that are predicted by the MGBE and K-MGBE approximations.
Compare Fig.~\ref{fig:MF-shortTime-comp}. The STA results are qualitatively
similar to  the extracted results, but there are significant differences.  1.~The magnitude at zero time
is larger for the STA than for the extracted memory function.
2.~The STA curves fall rapidly to zero, while the extracted curves
decay to zero more slowly, from above at the higher temperature, and from below at lower temperatures.

The differences between the extracted memory function and the STA approximation is plotted for high, medium,
and low temperatures in Fig.~\ref{fig:extractMF-diff}. At the highest
temperature, the memory function decays to zero from above.  This is presumably a manifestation of the
hydrodynamic vortex effect that is seen for hard spheres.\cite{alder} The long time negative part
of the memory function at lower temperatures is presumably a manifestation of the cage effect, and is related
to the long slow decay of the simulation correlation function at low temperature and small wave vector
(see Fig.~\ref{fig:SLCC-lt-compare-lowT}).
The very long time part of the memory function is shown in Fig.~\ref{fig:mctTest}.

\begin{figure}
\epsfig{figure=differenceOnlyPlot,width=8cm}
\caption{The difference between the extracted and STA memory functions,
$\hat M_{kk,STA}(0.75,t)-\hat M_{kk,ext}(0.75,t)$, for three different temperatures.}
\label{fig:extractMF-diff}
\end{figure}

\begin{figure}
\epsfig{figure=mctTestPlot-0723T,width=8cm}
\caption{The longer time part of the memory function predicted by mode coupling theory is compared to the
memory function numerically extracted from the simulation data for the SLCC at $T=0.723$
and $|\mathbf k|=0.75$.}
\label{fig:mctTest}
\end{figure}

The actual memory function at low temperatures clearly has a rapidly decaying part, falling to 2\%
of its initial value at $t\approx0.5$, and a  more slowly decaying part that falls to zero (within the
statistical error) at about $t=1.5$.  None of the binary collision approximations we have studied is
qualitatively consistent with the true long time behavior of the memory function.
Moreover, none of them provides a quantitative description of the initial rapid decay.

\iffalse
The ({\bf TODO}: Perhaps put this paragraph in the conclusions section.)
({\bf attn:} Hans, see the changes in last two sentences. Please see my email for an explanaion.)
STA provides the best overall description of this rapidly decaying part, in that it leads to memory functions
that decay rapidly and on the right time scale.
The areas under the STA memory functions are too large in magnitude at short times, however, which leads to
the failure of the STA to correctly predict the approach, location and magnitude of the negative dips and their
temperature dependence in the SLCC curves.
Moreover, the failure to describe the long time behavior of the memory function is the reason why the STA
does not correctly describe the exit from the negative dip, that is, the shape of the negative dip for times
$t>0.2$. \fi

\section{Mode coupling description of the long time part of the memory function}
The mode coupling theory of G\"{o}tze\cite{gotze} can be used to calculate, among other things, a
correlation function denoted $\phi^{(s)}_q(t)$ that is proportional to
$\hat C^{(s)}_{\vec0\vec0}(\mathbf k,t)$, the Fourier transform of a specific matrix element of
$C^{(s)}(11',t)$. The theory is based on the Mori-Zwanzig projection operator method,\cite{mori:I}
which relates this
function to its second order memory function, denoted $M^{(s)}_q(t)$. The assumptions of the theory lead
to a mode coupling relationship that expresses the long time part of $M^{(s)}_q(t)$ as a bilinear
functional of $\phi^{(s)}_q(t)$ and $\phi_q(t)$, where the latter is proportional to
$\hat C_{\vec0\vec0}(\mathbf q,t)$. See eq.~(3.34) of Ref.~\onlinecite{gotze}.

The kinetic theory presented here relates $\hat C^{(s)}_{\vec0\vec0}(\mathbf k,t)$ to an infinite
dimensional matrix of memory functions $\hat M^{(s)}_{\vec\mu\vec\nu}(\mathbf k,t)$. For small
wave vector $\mathbf k$, $\hat C^{(s)}_{\vec0\vec0}(\mathbf k,t)$ is coupled primarily to
$\hat M^{(s)}_{\hat k\hat k}(\mathbf k,t)$, and in the limit as this wave vector goes to zero, this is the
only element that affects the value of $\hat C^{(s)}_{\vec0\vec0}(\mathbf k,t)$. Our numerical work
confirms that $|\mathbf k|=0.75$ is small enough for this to be the case. It is straightforward to show
that when this is the case, the relationship between $\hat M^{(s)}_{\hat k\hat k}(\mathbf k,t)$
and $\hat C^{(s)}_{\vec0\vec0}(\mathbf k,t)$ is precisely of the form generated by the Mori-Zwanzig
theory. In other words, for small wave vector, the single matrix element
$\hat M^{(s)}_{\hat k\hat k}(\mathbf k,t)$  is the Mori-Zwanzig second order memory function of
$\hat C^{(s)}_{\vec0\vec0}(\mathbf k,t)$.

When the mode coupling relationship mentioned above is expressed in terms of the present kinetic theory,
the result is of the following form
$$ \hat M^{(s)}_{\hat k\hat k}(\mathbf k,t)
   = \int d\mathbf q\,A(\mathbf k,\mathbf q)
     \hat C_{\vec0\vec0}(\mathbf q,t)\hat C^{(s)}_{\vec0\vec0}(\mathbf k-\mathbf q,t) $$
where $A(\mathbf k,\mathbf q)$ is a nonnegative function. The mode coupling theory asserts that this
holds for times longer than the duration of brief binary collisions, which in the present theory
corresponds roughly to the time at which the STA memory function has decayed to zero.

Applying this result to $|\mathbf k|=0$, we find
\begin{equation}
 \hat M^{(s)}_{\hat k\hat k}(\mathbf 0,t) =
  \int d\mathbf q\,A(\mathbf 0,\mathbf q)
           \hat C_{\vec0\vec0}(\mathbf q,t)\hat C^{(s)}_{\vec0\vec0}(\mathbf q,t)
\end{equation}
Each of the two time dependent functions on the right depends on the magnitude, but not the direction,
of its wave vector argument. The function $A(\mathbf 0,\mathbf q)$ is zero if $|\mathbf q|=0$, small
near $|\mathbf q|=0$, and highly peaked near $|\mathbf q|=q_{max}\approx 2\pi/\sigma$. Assuming that the
integral above is dominated by the range of wave vectors whose magnitude is near this value, we find
\begin{equation}
\label{eq:mctPrediction}
  \hat M^{(s)}_{\hat k\hat k}(0,t) \approx
     (constant) \times \hat C_{\vec0\vec0}(q_{max},t)\hat C^{(s)}_{\vec 0\vec0}(q_{max},t)
\end{equation}
where the constant is positive. Since $\hat M^{(s)}_{\hat k\hat k}(\mathbf k,t)$ is an even differentiable
function  of the wave vector argument, it is very slowly varying near $|\mathbf k|=0$, and we will apply
this result for $|\mathbf k|=0.75$, the smallest nonzero wave vector studied in this work.

We can test this relationship by comparing the time dependence of the extracted
$\hat M^{(s)}_{\hat k\hat k}(0.75,t)$ obtained from the
simulation data with the time dependence of the right side of eq.~(\ref{eq:mctPrediction}), which contains
functions that are routinely obtained from simulation data. Figure~\ref{fig:mctTest} shows the extracted
memory function $\hat M^{(s)}_{\hat k\hat k}(0.75,t)$ for longer times ($t>0.1$) for the lowest temperature
studied. At that time, the STA memory function has decayed almost completely to zero. The extracted memory
function has oscillations that are an artifact of the extraction procedure (note the vertical scale compared
with the graph of Fig.~\ref{fig:extractMF-results}). Figure~\ref{fig:mctTest} also shows the right side of
eq.~(\ref{eq:mctPrediction}), with the constant chosen so that the results matches the value of the extracted
memory function for $t\approx 1$. With this choice of the constant, the time dependence of the right side
accounts remarkably well for the time dependence of the memory function matrix element.

Note that this test was only for the smallest wave vector at the lowest temperature studied. For high
temperatures, eq.~(\ref{eq:mctPrediction}) is qualitatively incorrect ({\it i.e.,} the predicted sign
of the memory function is incorrect). This is presumably because the behavior is dominated by a hydrodynamic
vortex effect,\cite{alder} rather than the cage effect described by mode coupling theory.
For higher wave vectors
at low temperatures, the test cannot be performed, since the simple relationship between
$\hat M^{(s)}_{\hat k\hat k}(\mathbf k,t)$ and the second order memory function of the Mori-Zwanzig theory
does not hold. Note, also, that mode coupling theory allows, in principle, the calculation of the constant
of proportionality, but we have not done this. Nevertheless, the agreement in Fig.~\ref{fig:mctTest} is
striking, suggesting that mode coupling theory accounts quantitatively for the time dependence of the memory
function for long times ({\it i.e.,} for times longer than the time of a brief binary collision) for the
lowest temperature studied, which corresponds to the liquid state near the triple point.

\section{Summary and Conclusions}
\setcounter{paragraph}{0}
\paragraph{The accuracy of various approximate theories.}We have investigated various binary collision
approximations for the memory function of the correlation function of density fluctuations in a dense Lennard-Jones
liquid at equilibrium and compared their predictions with computer simulation results for the self
correlation functions, especially the self longitudinal current correlation function (SLCC).

The BCA of Ranganathan and Andersen is exact at $t=0$, but it leads to divergent results at long times,
presumably because of its failure to satisfy important symmetry properties.

The moderate density approximation of the generalized Boltzmann-Enskog memory function (MGBE) of Mazenko and
Yip\cite{mazenko77} is accurate at very short times. However, its intermediate time behavior is very different
from the correct results, and its predicted correlation functions display unphysical oscillations at intermediate
times.

The STA of Ranganathan and Andersen\cite{andersen:dkt4,andersen:dkt5} is not exact at $t=0$. It describes the
initial drop of the SLCC to zero reasonably accurately, but it does not accurately describe the shape and depth
of the minimum in that function and its temperature dependence. At high temperatures, it gives accurate values
of the transport coefficients.\cite{andersen:dkt5}

The K-MGBE approximation, which introduces uncorrelated brief collisions of the colliding pair with third
particles, is more accurate than the MGBE at short times but its intermediate time behavior is still
very inaccurate.

No other binary collision approximation discussed in this paper is a systematic improvement over the STA.

\paragraph{The behavior of the extracted memory function.}
For small wave vector, the most important matrix element of the memory function can be extracted from the
simulation results for the SLCC.

At short time, the extracted memory function drops rapidly to values close to zero, in a way similar to the STA
result, and then has a very small, long-lived, nonoscillatory tail that is positive at high temperatures and
negative at low temperatures.  At low temperatures, the negative tail is important for determining the shape of
the minimum in the SLCC.  This tail is probably the result of physical effects that are not included in any
binary collision approximation.  The time dependence of the long time tail of the memory function is in striking
agreement with what is predicted by mode coupling theory, but this is only a limited confirmation of that theory
since the extracted memory function can be obtained only at small wave vector.

\paragraph{Conclusions and speculations.}
The negative feature of the SLCC for the dense Lennard-Jones fluid indicates that the velocity of a particle is
anticorrelated with its velocity at times that are more than about 0.1 time units earlier.  An important part of
the basis for this feature can be described by binary collision theories that take into account the finite
duration of very repulsive collisions.  For hard spheres, the short time feature of the memory function is a Dirac
delta function, which by itself would generate monotonic decay of the SLCC from above.  In the STA, the short
time feature is spread over a range of times of the order of the duration of a repulsive collision, and the
resulting correlation function has a negative dip. However, the increase of the depth of the negative feature as
the temperature is lowered is caused not by this short time feature but rather by a small, negative, temperature
dependent long time tail in the memory function.  This is presumably a manifestation of what is commonly regarded
as caging of atoms at low temperatures and cannot be described by binary collision approximations.
Instead, mode coupling theory or something else that goes beyond binary collision approximations is needed to
describe this aspect of the negative feature.

This conclusion is not at all surprising.  Theories that  use mode coupling commonly include a binary collision
term in the memory function but do not discuss in detail the nature of that term.  It is clear from the present
work that the short time part of the memory function has behavior associated with brief binary {\em repulsive}
collisions, such as those described by the STA.  Collisions that include attractive as well as repulsive
interactions, such as those of the MGBE, have a much longer duration, and theories that include them have
memory functions that decay to zero much too slowly to provide a good first approximation for the correlation
functions at intermediate times.

We have not been able to derive any binary collision theory that describes brief binary repulsive collisions and
that has exactly the correct value of the memory function at $t=0$.  There is a straightforward expression for
the zero time value of the memory function\cite{note9}
in terms of static correlation functions, and this expression involves the entire potential,
the entire potential of mean force, and the pair correlation function.  Any graphical approximation for the
memory function that includes all this at $t=0$ would have more physics in it than binary repulsive collisions.

This leads us to speculate that the memory function for density fluctuations can be  usefully regarded as a
sum of at least three parts: a contribution from repulsive binary collisions (the STA or something similar
to it), another short time part that is related to all the other interactions (but whose nature is not
understood), and a longer time slowly decaying part that describes caging (of the type predicted by mode
coupling theory).

\acknowledgments
The authors would like to thank Dr.\ Thomas Young and Richard Stein for providing the molecular dynamics data
used in this work. This work was supported by the National Science Foundation grant CHE-0408786.
One of the authors (J.N.-V.) would like to acknowledge support from a DOE CSGF fellowship.

redpajama_set_name: RedPajamaArXiv


For decades, millions of women were instructed to take hormone therapy (usually estrogen plus progestin) after menopause, primarily to protect them from heart disease.
The reasoning behind that recommendation? After women passed through menopause, their risk of heart disease accelerated and became equal to that of men.
Obviously, said the promoters of hormone therapy, this postmenopausal jump in the incidence of heart disease is tied to women's falling estrogen levels — even though there was no direct evidence of this.
Then, in 2002, came the disconcerting news from the Women's Health Initiative study that taking postmenopausal hormones actually increased women's risk of heart disease, including blood clots, heart attacks and stroke. Subsequent studies have confirmed that finding (and added other risks, like that of breast cancer and dementia).
And now, as reported Tuesday in BMJ (formerly known as the British Medical Journal), researchers have found that the half-century-old paradigm about menopause increasing the risk of heart disease is, well, not true.
For the new study, led by Dhanajay Vaidya, an assistant professor of medicine at Johns Hopkins University School of Medicine in Baltimore, Md., researchers analyzed long-term data collected on American, English and Welsh men and women born at three different periods between 1916 and 1945.
They found no sudden increase in the death rates of women as each of the three groups passed through the menopausal transition years. Instead, they found that the number of women who died from heart disease increased at a steady pace of about 8 percent per year.
In other words, aging, not menopause (or dropping estrogen levels), increases a woman's risk of dying from heart disease — probably because "the cells of the heart and arteries are aging like every other tissue in the body," Vaidya said in a statement released with the study.
"Efforts to improve cardiac health in women should focus on lifetime risk rather than risk only after menopause," he and his colleagues wrote in the study's conclusion.
These findings also suggest that the medical community should have been focusing on men, not women to explain gender disparities in heart disease.
The data revealed that men are at an accelerated risk of dying from heart disease until the age of 45, when the rate slows to about 5 percent, similar to that of postmenopausal women.
"Instead of looking at menopause, what we should be looking at is what is happening biologically to men over time," Vaidya said.
And, I might add, we should finally stop medicalizing menopause.