text:
How cybersecurity solutions are helping curb America's opioid crisis
When the clock struck midnight on January 1, it didn't just mark the start of a new year. It also marked the first day of a historic national mandate to curb the opioid epidemic: starting in 2023, organizations and healthcare providers must electronically prescribe all Schedule II, III controlled substances. , IV and V covered by Medicare Part D.
The Electronic Prescribing Mandate for Controlled Substances (EPCS) requires additional steps to ensure secure prescribing practices. However, the additional digital demands of the mandate can cause clinicians to spend more time with technology, resulting in workflow inefficiencies. But it is not necessary. If implemented with the right strategy, the EPCS mandate can lead to safer prescribing without hindering the work that matters most: treating patients.
More than 100,000 Americans died from drug overdoses between July 2021 and July 2022, according to CDC data. Opioids — especially synthetic opioids like fentanyl — were responsible for most of those deaths. However, addiction often begins with prescription opioids. The federal EPCS mandate, along with similar state-level EPCS laws, are an important step in addressing the opioid epidemic because they address unauthorized prescribing at the source. Recognizing the widespread repercussions of addiction, healthcare organizations are beginning to adopt highly effective strategies to stop drug diversion in its tracks.
While compliance can be met through a variety of technologies, healthcare organizations should consider implementing advanced tools with capabilities that go beyond mandate requirements. Those that provide tightly coupled integrations often lead to more successful results. This is the key to achieving compliance, minimizing drug diversion and reducing over-prescribing without creating additional burdens for hospital staff.
Rationalization of workflows and EPCS compliance
Studies show that mandatory electronic prescribing reduces medication errors, improves patient outcomes, reduces the number of patient visits, and generates hundreds of billions of dollars in healthcare savings. Of course, hospitals seeking these benefits must first initiate detailed and highly collaborative cross-functional project plans to ensure full compliance with specific DEA requirements.
The federal EPCS mandate has two major components for healthcare organizations: multi-factor authentication (MFA) for providers who prescribe controlled substances, and comprehensive reporting that tracks prescribing events as they occur. occur. The AMF confirms a doctor's identity and authorization to prescribe a particular drug, while the reporting system tracks all prescribing activity and can feed algorithms to identify diversion.
It's easy to imagine how these demands could slow down the daily work of patient care. For example, many healthcare systems require a lengthy MFA process, such as re-entering a username and password. For reporting, many hospitals perform manual audits of prescribing patterns, cross-referencing electronic health record (EHR) reports with dispensing cabinet activity. While these methods do the job and ensure compliance, they can be time consuming. Additionally, manual processes are riddled with errors. Additionally, healthcare systems can face hefty fines if they are investigated and found to be non-compliant.
But with the right cybersecurity tools, healthcare organizations can confidently comply with EPCS compliance while having better control and removing burdens from healthcare personnel. By integrating digital identity solutions that work with their current EHRs, healthcare organizations can go beyond FDA and DEA compliance. Advanced tools can unlock better visibility into prescribing practices without limiting the efficiency of hospital staff.
Consider MFA, for example. With digital identity technologies integrated into the existing cyberinfrastructure, doctors can easily prescribe necessary medications while away from the hospital while complying with EPCS rules. And the authentication process doesn't need to be as tedious or tedious as typing in a password every step of the way: a wide range of convenient and innovative options such as hands-free authentication, notifications Token-push, fingerprinting, and facial biometrics allow providers to choose a DEA-compliant method that works best for them, while preventing unauthorized access and drug diversion.
Detecting drug diversion with artificial intelligence
While the AMF ensures accountability and safety in the prescribing process, the full reporting mandate creates an opportunity for healthcare organizations to combat drug diversion based on prescribing patterns.
Without the proper tools to analyze the immense amount of data from EHR distribution reports and cabinets, manually auditing these processes will be a major challenge. Manual monitoring is ineffective, if not impossible, given the constraints hospital IT staff already face. This is where AI and analytics-based cybersecurity platforms can make a crucial difference. AI can automate what would otherwise be a manual cross-reference between all systems and reports.
If a provider claims to prescribe a patient a medication for chronic pain, but only gives them half of the prescribed opioids while pocketing the rest, an AI-powered platform like Imprivata's FairWarning solution, will report their behavior as suspicious. This solution can be integrated into the healthcare system's existing digital identity strategy. For example, the same credentials a clinician uses to log into the EHR are the same credentials they use to prescribe medications and apply MFA. This provides a simplified way to analyze data and track prescription patterns.
The warrants set strict standards, but they alone will not stop diversion. The methods specifiers implement to achieve compliance will only be as effective as the organization's ability to integrate with their existing technology. Complying with federal EPCS mandates is just the start of an effective drug diversion strategy. By taking a proactive approach to uncovering drug diversion with AI and digital identity tools, hospitals can amplify the goal of the EPCS mandate to prevent future drug diversion and enable corrective action before the incident occurs. does not degenerate. This will be key to enabling a broader wave of digital transformation for prescribing controlled substances.
Daniel Fabbri, Ph.D.is the Chief Data Scientist at Imprivata.
Dr. Fabbri is also an Assistant Professor of Biomedical Informatics and Computer Science at Vanderbilt University. His research focuses on machine learning applied to electronic medical records, clinical data, and data privacy. Dr. Fabbri's research has been sponsored by the National Science Foundation, the National Institutes of Health and the US Department of Defense.
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text:
Colonial Daydreams
Submitted by ssmith on February 14, 2009 - 2:36pm
Over on Globe Books, Cormorant Books owner and publisher Marc Côté (the Globe erroneously leaves off the circumflex in his name) offers some very passionate, astute and sobering thoughts on why the average Canadian will buy an American or British novel long before they'll buy a Canadian one.
Writes Côté:
The obstacle Canadian publishers must overcome in their own markets isn't Indigo Books, or Costco, or amazon.ca, or any of the large retailers that account for probably 75 per cent of the book market. It isn't the staggering geography, the two official languages, the small population or the millions of immigrants. It's the simple fact that our English-language media is dominated by U.S. and British culture. ... This cultural colonialism is so pervasive as to seem normal.
He goes on to make a pretty solid argument about why Canadian publishers need financial assistance from the government.
...the Canadian book market is not dominated by Canadians but by [foreign-owned] publishers that benefited from the military and naval power of the British Empire and the protectionism of the United States in the 19th century. Canada did not conquer other parts of the world and install education systems identical to "back home." We were the colony.
The article is well worth reading for anyone who has ever wondered why they have such a strong sense of the narrative mythologies of the USA and UK, but have little or no sense of any Canadian narrative mythology.
This is something I've spend a lot of time pondering over the years, but it was not until earlier this year when I encountered Robert Bringhurst's volume The Surface of Meaning, which is a history of books and book design in Canada, that I began to really understand the reason why the Canadian identity is so elusive.
I've known for quite some time that one reason the Canadian identity remains obscure is that Canada does not have a founding narrative mythology. That is, we have no founding stories. We are not a nation that was forged in rebellion and revolution, as was the USA, or that traces its ancestry back aeons to legends steeped in magic and tradition, as is Britain. I'm embarrassed to say, but if I was travelling in another country, and someone were to ask me to tell them the story of how Canada was founded, I would be hard pressed to come up with a very interesting answer.
Then I encountered this line in Bringhurst's book:
"...the country went without a national library until 1953."
That kind of sealed the deal for me. If a national library is a repository of a nation's collective stories, how much narrative mythology could Canada have actually accumulated if it didn't feel the need to collect its stories together until 460 years after Italian-born explorer Giovanni Caboto (aka John Cabot), under the patronage of Henry VII of England, pushed the "play" button on what would eventually become Canada by landing on the Atlantic coast (Labrador, it is believed) in 1497?
Surely something happened in all that time. Did no one think to write it down? I'm being facetious, of course, but the question remains: Where are the great works of fiction that tell of the formation of this great land?
My own feeling is that mythologies are forged in the crucible of conflict. I think that because Canada did not have a separate identity to fight for, because it was essentially born out of squabbling and wars between the British, French and Americans over (Native) land, there were no moments in which heroic figures rose from the ashes to represent triumphant Canadian values. There were battles, treaties were signed, borders were establish, but all along we remained a colony. Our story became an extension of those British, French and American stories.
This is, I think, why great landscape painters like the Group of Seven and Tom Thompson, as well as Emily Carr, A.J. Casson, and David Milne, have become so integral to the Canadian identity. We may not have found our grand tales to tell about Canadian heroes because we were "just" a colony, but there has always been the land. The land, I think, has filled the void in our collective imagination left by our absent heroes. It was, and is, our story. Indeed, if I was travelling in a foreign nation and someone asked me about Canada, I would happily point to landscape paintings by such artists and say, "That's Canada. That's not just what it looks and feels like, but that's the mythology of Canada. The land itself. We have a visual and geographic mythology, not a narrative one. The land is our hero."
I wonder if it is too late to have a literary equivalent.
ssmith's blog
Join to Comment
ssmith on February 20, 2009 - 5:52pm said:
@calisaurus - It would be a great class. I'd love it if we could build up a mythical sense of our nation. Some of those figures you mention have been heroic, but I still don't think we have (m)any heroes of the larger-than-life variety here. The people you mention were all great figures, but I don't have the sense that they have risen above themselves to become part of the common consciousness and identity of Canada.
@sinckerzmom - Terry Fox is the sort of figure who may well be on his way to becoming mythological, IMHO. From all that I can tell, he was just an average guy who got it into his mind to do something amazing. He wasn't doing it for himself and his actions have come to represent something that is pretty humbling when you look at it. That's why I don't think the answer is that we're too nice. In fact, I think a significant part of the Canadian identity is sacrifice. Don't ask why, because I don't know. But I think that one of the reasons Fox resonates so strongly with Canadian's is that we look at him and say, "If only I could be so brave and selfless."
That's an interesting site, btw. Many great figures. (And one or two I'd argue strongly against.)
calisaurus on February 19, 2009 - 3:52pm said:
Actually, it just occurred to me that "Canadian mythology" would be a great class for Universities... and they certainly would need textbooks on the subject! Pearson... Oxford... are you listening?
There are many Canadian heroes - just think of June Callwood, Elijah Harper, Lester Pearson, Nellie McClung, Roger Obata... etc. I think the problem comes from the smaller Canadian market. We can assume that a book about Canada will sell the majority of its copies in Canada - and Canada has a far smaller population than the US or the UK, thus publishers are less willing to take risks on these sort of books - because it's a huge financial risk. If people just made books for the sake of reading, instead of for necessary profits, I think you would see a lot more books on Canadian mythology.
snickerzmom on February 18, 2009 - 8:12am said:
Wow! I have been sitting here for 15 minutes trying to think of a Canadian hero - from a historical perspective. All I could come up with was Terry Fox, and that is way too recent. So I thought, hey, "google"!
Not much help. Google the term "Canadian hero" and the best result you get is this site: http://www.canada-heros.com/in.... Which is interesting in that I didn't know a bunch of the people listed.
I think you have a real point, we are truly missing a Canadian narrative. Are we just too nice?
Mon November 1, 2010
November Book Giveaway: Glimpse: Selected Aphorisms by WIR George Murray
Submitted by clelia on November 1, 2010 - 8:13am
George Murray is Open Book's November 2010 Writer in Residence. Visit his WIR page to read his blog and his answers to the Proust Questionnaire and to check out his reading and website recommendations. You can send your questions and comments for George to writer@openbooktoronto.com or post them directly to his blog.
Shaun Smith
Shaun Smith is a novelist and journalist living in Toronto. His young adult novel Snakes & Ladders was published in January 2009 by the Dundurn Group.
Go to Shaun Smith 's Author Page

redpajama_set_name: RedPajamaCommonCrawl


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People looking for hip, premium, expensive, exclusive entertainment for their weddings, parties, corporate events, etc. hip, premium, expensive, exclusive are all good buzz words to consider when creating designs.


text:
Bella Hadid Just Admitted She Struggled 'Intensely' With Her Mental Health For The Past 5 Years
byKim Wong-Shing Nov 18, 2019 at 2:06 pm EST
Photo: Shutterstock.
Bella Hadid is 23 years old and at the top of the modeling world—she appears to be living the dream. But depression doesn't discriminate, and nobody is immune. Bella Hadid's mental health struggles were a focal point of her interview at the Vogue Fashion Festival on Friday. The model revealed that she's been dealing with her mental health for the past five years, and she's finally ready to talk about it.
Bella confessed that she often struggles to balance her mental health with her high-profile career. "For a while I just didn't want to talk about it, and I've gone through a lot in the past few years with my health," she told audience members during a live interview at the festival. "I feel guilty for being able to live this incredible life, have the opportunities that I do, but somehow still be depressed. It doesn't make sense."
To be fair, life in the spotlight is difficult. It's probably not a coincidence that Bella's mental health took a turn for the worse right when her career took off at age 18. Speaking of her early modeling days, she said: "I would cry every single morning, I would cry during my lunch breaks, I would cry before I slept. I was very emotionally unstable for a while when I was working 14-hour days for four months straight as an 18-year-old. I think I just wanted to breathe a little bit. And so it kind of put me in a spiral."
A post shared by Bella 🦋 (@bellahadid)
But that wasn't the end of her difficulties. She mentioned that while she's "good" now, she's been struggling with her mental health "very intensely" for the past five years, which is why it's so important to her to speak up about it on social media these days. "I just hope I can help in any way I can," she said.


text:
\section{Introduction}

\noindent
In a simple fluid the five conserved densities relax very slowly at small
wave vectors. The nonlinearities in the hydrodynamic equations provide a
further mechanism for slowing down the decay of fluctuations and are
responsible for the long-time algebraic tails of the correlation functions
that determine the transport coefficients [1]. The coefficients of the
algebraic tails arising from these long-wavelength or conventional
mode-coupling effects are, however, often much smaller than those observed
by molecular dynamics. In particular, the conventional mode-coupling
theory yields a coefficient of the long-time tail of the stress-tensor
autocorrelation function (STAF) of a dense hard-sphere fluid which is 500
times smaller than observed in simulations [2]. It has been shown that this
difference can be accounted for in terms of finite-wave-vector or
generalized mode-coupling effects [3]. In a dense fluid density fluctuations
relax slowly even on molecular length scales. As a consequence, density
nonlinearities lead to a renormalization of the transport coefficients via a
finite-wave-vector mode-coupling mechanism [4,3] that can qualitatively
account for the size of the coefficient of the long-time tail of the STAF
[2] for the shear-dependent viscosity [5,6] observed in computer simulations.
In addition they provide a good description of the behavior of the velocity
autocorrelation function at intermediate times [7]. A self-consistent
implementation of the generalized mode-coupling theory also predicts, at a
critical density, the transition to a glassy state where structural
relaxation is frozen [4,8-10]. On the other hand, a more detailed analysis
has shown that the transition is an artifact of the approximation used and
is cutoff when additional couplings are included [11]. The generalized
mode-coupling theory has, however, provided considerable insight into the
dynamical properties of dense simple liquids. In particular, it gives an
adequate description of the slowing down of the dynamics of dense fluids of
moderate viscosity, in a regime above the glass transition were the
relaxa- tion is highly cooperative and nonexponential [12,13]. \\

\noindent
In this paper we are interested in the mode-coupling
contribution to the shear viscosity and in the long-time behavior of the
stress-tensor autocorrelation function of a dense binary fluid mixture. In a
recent paper [14], Erpen- beck reported molecular-dynamic studies of an
isotopic hard-sphere mixture at moderate densities and concluded that in
this case the size of the coefficient of the long-time tail of the
time-correlation function for the shear viscosity observed in the
simulations agrees with the prediction of conventional mode-coupling theory.
In mixtures, as in one-component fluids, the conventional mode-coupling
contribution to the STAF only accounts for the kinetic part of the
correlation function. The kinetic contribution dominates at low and moderate
densities and in this regime the conventional mode-coupling theory agrees
with the simulations. Even in one-component fluids [15] it is only at high
densities $(n\sigma^{3}\geq 0.7)$ that the largest contribution to the
correlation function is the potential one and the observed coefficient
of the $t^{-3/2}$ tail
of the STAF is much larger than predicted by conventional mode-coupling
theory. The situation should not be different for mixtures. The agreement of
the simulations with conventional mode-coupling theory can only occur at
the relatively moderate densities considered by Erpenbeck. His
observations do not preclude the possibility of obtaining large
finite-wave-vector mode-coupling corrections to the stress-tensor
autocorrelation function at higher densities. To our knowledge such effects
have not yet been investigated in computer simulations of mixtures. \\

\noindent
Here we evaluate the generalized (to include finite-wave-vector effects)
mode-coupling contribution to the STAF and to the shear viscosity of a
binary mixture of hard spheres of different sizes and masses. The
generalized mode-coupling theory takes into account that in a dense
viscous liquid
structural rearrangements are very slow due to a close packing of the
molecules. The slow dynamics of density fluctuations on molecular length
scales affects the macroscopic transport properties through a nonlinear
mode-coupling mechanism [4,3]. At large wave vectors the most important of
these contributions involves the product of two density fluctuations
[16,3]: this contribution is the slowest decaying and has the largest
amplitude. In the case of a one-component
fluid, the dynamics slows down significantly at only one value of the wave
vector corresponding to the location of the first maximum of $S(k)$ [17]. In
contrast, in a dense mixture of spheres of different sizes fluctuations in
the mass densities of both species can be long-lived on molecular length
scales [18] and the dynamics can slow down over a range of wave vectors. In
paper I [18] we obtained a set of linear generalized hydrodynamic equations
that describe the dynamics of the density fluctuations of the two species
down to molecular scales. We found that when the packing fractions of the
two species are comparable and all three partial static structure factors
of the mixture are peaked, the density fluctuations of the two species decay
very slowly at two length scales corresponding to the molecular diameters
of the two species. We now consider the nonlinear generalized hydrodynamic
equations and discuss the renormalization of the transport coefficients by
density nonlinearities. The interplay of the density fluctuations of the two
components leads to finite-wave-vector mode-coupling corrections to the
correlation functions which determine the transport coefficients. \\

\noindent
We are interested in a binary mixture of hard spheres as the simplest model
where one can study in some detail the role of compositional disorder in
slowing down the fluid dynamics. Compositional disorder seems to be an
essential feature of the simplest glass formers (metallic glasses). Our work
is focused towards understanding how the structural relaxation of the two
species and interdiffusion affect the dense fluid dynamics. We choose to
study the mode-coupling correction to the stress- tensor autocorrelation
function which determines the shear viscosity as the simplest of the various
mode-coupling effects in a mixture.\\

\noindent
The self-consistent implementation of the generalized mode-coupling theory
of binary mixtures has been discussed recently by several authors as a model
of the glass
transition [19,20]. Our work differs from this work on mixtures because here
we simply study the mode-coupling corrections to lowest order in
perturbation theory, without attempting any self-consistent closure. We do
so for two reasons. First we believe that one needs to understand well
the simplest perturbation theory before making it self consistent.
Second, and most importantly, it has been
shown [3] that for one-component Ouids a naive mode-coupling tbeory of the
type we discuss here describes well the slowing down of the dynamical
properties above the glass transition, in a regime of moderate viscosity
where the dynamics is cooperative (and therefore nonexponential), but not
yet activated [12]. Our objective is to put the description of the dynamics
of dense mixtures in this regime on the same solid footing as that of
one-component fluids.\\

\noindent
The outline of the paper is as follows. In Sec. II the. basic theory used
here is described. The perturbation theory in the nonlincarities in the
hydrodynamic equations is then outlined in Sec. III, where an expression
for the renormalized shear viscosity is obtained. In Sec. IV we present our
results for the STAF and the shear viscosity for hard-sphere fluid
mixtures. We analyze the concentration dependence of the STAF for a
mixture and conclude the paper by discussing our results and by
making some remarks about connections to earlier theoretical studies
and computer simulations. \\~\\

\section{Nonlinear generalized hydrodynamics}

\noindent
The hydrodynamic properties of a binary fluid mixture are described in terms
of the six conserved densities: the mass densities $\rho_{1}({\bf r},t)$ and
$\rho_{2}({\bf r},t)$ p.(r,t) of the two species,
the total momentum density ${\bf g}({\bf r},t)$, and the energy density
$e({\bf r},t)$ [18]. We begin by
writing a set of hydrodynamic equations that have the same structure both at
large and short (molecular) length scales. The starting point for the
derivation of these generalized hydrodynamic equations is the free-energy
functional for an inhomogeneous equi- librium binary fluid mixture, given by

\renewcommand\theequation{2.1}
\begin{equation}
F = \int d{\bf r} \left[ \frac{g^{2}_{1}}{2\rho_{1}} + \frac{g^{2}_{2}}{2\rho_{2}} \right] + F_{V}.
\end{equation}

\noindent
The potential part $F_{V}$ of the free-energy functional can be written
exactly in the form of an expansion in the fluctuations $\delta n_{s}({\bf r},t)-n_{0s}$ of the number density $n_{s}({\bf r},t)$ of species $s$, with
$s=1,2$, from its equilibrium value $n_{0s}$, [21]
\renewcommand\theequation{2.2}
\begin{eqnarray}
&&\frac{\delta F_{V}}{\delta\rho_{s}} = \frac{k_{B}T}{m_{s}}
\left\{\ln [\lambda^{3}_{Ts}n_{s}({\bf r})] - \sum^{2}_{s^{\prime}=1}
\int d{\bf r}^{\prime}C_{ss^{\prime}} (|{\bf r} - {\bf r}^{\prime}|)\delta n_{s^{\prime}}({\bf r}^{\prime}, t)\right.\nonumber\\
&&-\left.\frac{1}{2} \sum^{2}_{s^{\prime}=1} \sum^{2}_{s^{\prime\prime}=1}
\int d{\bf r}^{\prime} \int d{\bf r}^{\prime\prime}C_{ss^{\prime}s^{\prime\prime}} ({\bf r}, {\bf r}^{\prime}, {\bf r}^{\prime\prime})\delta n_{s^{\prime}}({\bf r}^{\prime}, t)\delta n_{s^{\prime\prime}}({\bf r}^{\prime\prime}, t) + \cdots \right\}.
\end{eqnarray}

\noindent
Here $m_{s}$ is the mass of a particle of type $s$, $T$ is the temperature,
$\lambda_{Ts} = (\hbar^{2}/2\pi m_{s}k_{ B}T)^{1/2}$ is the thermal
wave-lngth of particles of type $s$ and
$C_{ss^{\prime}}(|{\bf r}-{\bf r}^{\prime}|)$ and
$C_{ss^{\prime}s^{\prime\prime}}({\bf r},{\bf r}^{\prime}{\bf r}^{\prime\prime})$ are the equilibrium two-particle and three-particle direct
correlation functions. From this free energy, nonlinear fluctuating
hydrodynamic equations can be derived by the standard Poisson bracket method
outlined in Appendix A [22]. Here we neglect temperature fluctuations
since they do not play an
important role in determining the renormalization to be discussed in Sec.
III. The equations for the partial mass densities of the two species and the
total momentum density are

\renewcommand\theequation{2.3}
\begin{equation}
\partial_{t}\rho_{1}({\bf r},t)+{\bf nabla}\cdot \left[\frac{\rho_{1}}{\rho}{\bf g}({\bf r},t)\right] = -G_{1}({\bf r},t) + f_{1}({\bf r},t),
\end{equation}

\renewcommand\theequation{2.4}
\begin{equation}
\partial_{t}\rho_{2}({\bf r},t)+{\bf nabla}\cdot \left[\frac{\rho_{2}}{\rho}{\bf g}({\bf r},t)\right] = -G_{2}({\bf r},t) + f_{2}({\bf r},t),
\end{equation}

\renewcommand\theequation{2.5}
\begin{eqnarray}
\partial_{t}g_{\alpha}({\bf r},t)+\delta_{\beta} \left[\frac{g_{\alpha}g_{\beta}}{\rho}\right] &&+ \sum^{2}_{i=1} \rho_{s}\partial_{\alpha} \left[ \frac{\delta F_{V}}{\delta\rho_{s}} \right]\nonumber\\
&&= -G_{g_{\alpha}}({\bf r},t) + f_{g_{\alpha}}({\bf r},t).
\end{eqnarray}

\noindent
Here $G_{1}({\bf r},t)$ and $G_{2}({\bf r},t)$ account for diffusive
dissipation of species type 1 and 2, respectively, and $G_{g_{\alpha}}({\bf r},t)$ corresponds to viscous dissipation. We write these terms in the form

\renewcommand\theequation{2.6}
\begin{equation}
G_{s}({\bf r},t) = \partial_{\alpha} \sum^{2}_{s^{\prime} = 1} \int d{\bf r}^{\prime}\gamma^{\alpha\beta}_{ss^{\prime}} ({\bf r},{\bf r}^{\prime}; n_{s})\partial^{\prime}_{\beta} \left[\frac{\delta F}{\delta\rho_{s^{\prime}}({\bf r}^{\prime})}\right],
\end{equation}
for $s=1,2$, and
\renewcommand\theequation{2.7}
\begin{equation}
G_{g_{\alpha}}({\bf r},t) = \int d{\bf r}^{\prime}\Gamma_{\alpha\beta} ({\bf r},{\bf r}^{\prime}; n_{s}) \frac{\delta F}{\delta g_{\beta}({\bf r}^{\prime})}.
\end{equation}

\noindent
From the equation of continuity for the total density fluctuations
$\rho({\bf r},t)=\rho_{1}({\bf r},t)+\rho_{2}({\bf r},t)$, it follows that
$G_{1}({\bf r},t)+G_{2}({\bf r},t)=0$. As we have seen in $I$, the only
fluctuations that are
slowly varying on molecular length scales are the fluctuations of the
partial densities of the two species. Large-wave-vector momentum
fluctuations relax on short-time scales and momentum behaves like a
nonhydrodynamic (nonconserved) variable. We have
therefore written $G_{1}({\bf r},t)$ and $G_{2}({\bf r},t)$ in a
form explicitly consistent with the general
structure of dissipative terms pertaining to conserved variables. The form
of $G_{g_{\alpha}}({\bf r},t)$ corresponds to that of dissipation associated
with a non-conserved
variable [23]. On the other hand, it should be kept in mind that our
dissipative kernels $\gamma^{\alpha\beta}_{ss^{\prime}}$ $\Gamma_{\alpha\beta}$
are nonlocal and reproduce the well-known transport
coefficients in the long-wavelength limit. \\

\noindent
In the above equations $f_{1}({\bf r},t),\;\;f_{2}({\bf r},t)$, and
$f_{g_{\alpha}}({\bf r},t)$ are Gaussian random fluctuating forces which
are related to the corresponding linearized dissipative kernels
through the fluctuation-dissipation theorem. The random forces have zero
average and correlations given by

\renewcommand\theequation{2.8}
\begin{equation}
\langle f_{s}({\bf r},t)f_{s^{\prime}}({\bf r}^{\prime},t^{\prime})\rangle =
2k_{B}T\gamma^{\alpha\beta}_{ss^{\prime}} ({\bf r},{\bf r}^{\prime})\partial_{\alpha}\partial^{\prime}_{\beta}\delta ({\bf r}-{\bf r}^{\prime})\delta (t-t^{\prime}),
\end{equation}

\renewcommand\theequation{2.9}
\begin{equation}
\langle f_{g_{\alpha}}({\bf r},t)f_{g_{\beta}}({\bf r}^{\prime},t^{\prime})\rangle =
2k_{B}T\Gamma_{\alpha\beta} ({\bf r},{\bf r}^{\prime})\delta ({\bf r}-{\bf r}^{\prime})\delta (t-t^{\prime}).
\end{equation}

\noindent
Equations (2.3)-(2.5) follow from general considerations
if one neglects temperature fluctuations and non-Markovian effects. \\

\noindent
To complete the equations, one needs to specify the dissipative kernels. We
will neglect dissipative nonlinearities in our analysis. For a dense
one-component fluid a theory that only retains thermodynamic nonlinearities
predicts a $t^{-3/2}$ tail for the STAF with a coefficient that is a
factor of 2 smaller than that observed in simulations [7].
The inclusion of dissipative nonlinearities accounts for this factor-of-2
difference [17]. For simplicity here we neglect dissipative
nonlinearities and assume $\Gamma_{\alpha\beta}({\bf r},{\bf r}^{\prime};
n_{s})\simeq \Gamma_{\alpha\beta}({\bf r},{\bf r}^{\prime};n_{s0})$ and
$\gamma^{\alpha\beta}_{ss^{\prime}}({\bf r},{\bf r}^{\prime};n_{s}) \simeq
\gamma^{\alpha\beta}_{ss^{\prime}}({\bf r},{\bf r}^{\prime};n_{s0})$. The
simple fluid results indicate that even with this approximation
the theory will provide a semiquantitative description of the behavior of
the STAF at intermediate times. We also neglect convective non-linearities
which are responsible for the asymptotic $t^{-3/2}$ tails in the STAF. At high
density the coefficient of the asymptotic tail is much smaller that the
contribution discussed here. \\

\noindent
It is convenient for the discussion below to consider the Fourier transform
of the generalized hydrodynamic equations (2.3)-(2.5). The Fourier
components of momentum fluctuations are separated into longitudinal and
transverse parts,

\renewcommand\theequation{2.10}
\begin{equation}
{\bf g}({\bf k},t) = \hat{\bf k}g_{l}({\bf k},t) + \sum_{i=1,2} \hat{\bf k}^{i}_{\perp}g_{Ti}({\bf k},t),
\end{equation}
where $\hat{\bf k}={\bf k}/k$ is a unit vector along ${\bf k}$ and
$\hat{\bf k}^{i}_{\perp}$ are unit vectors normal to ${\bf k}$.
After neglecting convective and dissipative nonlinearities and retaining
only quadratic nonlinearities in the density fluctuations, the
Fourier-transformed equations are given by

\renewcommand\theequation{2.11}
\begin{equation}
\partial_{t}\rho_{s}({\bf k},t) + ik\frac{\rho_{s}0}{\rho 0} g_{l}({\bf k},t)=
\frac{k^{2}}{\beta} \sum_{s^{\prime},s^{\prime\prime}=1}^{2} \gamma_{ss^{\prime}}(K)[\delta_{s^{\prime}{s^{\prime\prime}}} - n_{s^{\prime}}C_{s^{\prime}s^{\prime\prime}}(k)] \frac{\delta n_{s^{\prime\prime}}({\bf k},t)}{\rho_{s^{\prime}0}} + f_{s}({\bf k},t),
\end{equation}
for $s=1,2$, and
\renewcommand\theequation{2.12}
\begin{eqnarray}
[\partial_{t}+\gamma_{L}(k)]g_{l}({\bf k},t)+ik && \sum^{2}_{s,s^{\prime}=1}
[\delta_{ss^{\prime}} - n_{s}C_{ss^{\prime}}(k)]\delta_{n_{s^\prime}}({\bf k}, t)\nonumber\\
&&=I_{L}({\bf k},t) + F_{L}({\bf k},t)
\end{eqnarray}

\renewcommand\theequation{2.13}
\begin{equation}
[\partial_{t}+\gamma_{T}(k)]g_{T_{i}}({\bf k},t)=I_{T_{i}}({\bf k},t)+f_{T_{i}}({\bf k},t).
\end{equation}

\noindent
In writing the above equations we have used that for an isotropic
fluid
\renewcommand\theequation{2.14}
\begin{equation}
\Gamma_{\alpha\beta}({\bf k}) = \rho_{0}[\hat{k}_{\alpha}\hat{k}_{\beta}\gamma_{L}(k)+(\delta_{\alpha\beta}-\hat{k}_{\alpha}\hat{k}_{\beta})\gamma_{T}(k)],
\end{equation}

\renewcommand\theequation{2.15}
\begin{equation}
\gamma^{\alpha\beta}_{ss^{\prime}}(k) = \delta_{\alpha\beta}\gamma_{ss^{\prime}}(k).
\end{equation}

\noindent
In the long-wavelength-limit the viscous kernel $\Gamma_{\alpha\beta}({\bf k})$
reduces to the usual combination of gradients and bare shear and bulk
viscosities $\eta_{s}$ and $\zeta$,
\renewcommand\theequation{2.16}
\begin{equation}
\lim_{k\rightarrow 0}\Gamma_{\alpha\beta}({\bf k})=k_{\alpha}k_{\beta}(\zeta + \frac{4}{3}\eta_{s})+(k^{2}\delta_{\alpha\beta} - k_{\alpha}k_{\beta})\eta_{s}.
\end{equation}

\noindent
The diffusion kernels $\gamma_{ss^{\prime}}(k)$ are related to the
diffusion constants $D_{ss^{\prime}}$ of a binary mixture, according to

\renewcommand\theequation{2.17}
\begin{equation}
D_{ss^{\prime}} = \lim_{k\rightarrow 0} \frac{k_{B}T}{\sqrt{m_{s}\rho_{s}m_{s^{\prime}}\rho_{s^{\prime}}}}\gamma_{ss^{\prime}}(k).
\end{equation}

\noindent
Finally, the nonlinearities in Eqs. (2.12) and (2.13) are given by

\renewcommand\theequation{2.18}
\begin{eqnarray}
I_{L} ({\bf k},t) = \frac{1}{2} \sum^{2}_{s,s^{\prime}=1}&& \int \frac{d{\bf q}}{(2\pi)^{3}}[V^{L}_{ss^{\prime}}({\bf k},{\bf q}) + V^{L}_{s^{\prime}s}({\bf q},{\bf k})]\nonumber\\
&&\times \delta_{n_{s}} ({\bf q},t)\delta_{n_{s^{\prime}}}({\bf k}-{\bf q},t),
\end{eqnarray}

\renewcommand\theequation{2.19}
\begin{eqnarray}
I_{T_{i}} ({\bf k},t) = \frac{1}{2} \sum^{2}_{s,s^{\prime}=1}&& \int \frac{d{\bf q}}{(2\pi)^{3}}[V^{T_{i}}_{ss^{\prime}}({\bf k},{\bf q}) + V^{T_{i}}_{s^{\prime}s}({\bf q},{\bf k})]\nonumber\\
&&\times \delta_{n_{s}} ({\bf q},t)\delta_{n_{s^{\prime}}}({\bf k}-{\bf q},t),
\end{eqnarray}
with

\renewcommand\theequation{2.20}
\begin{eqnarray}
V_{ss^{\prime}}^{T_{i}} ({\bf k},{\bf q}) +  V_{s^{\prime}s}^{T_{i}} ({\bf q},{\bf k}) = \frac{\hat{\bf k}^{i}_{\perp}\cdot{\bf q}}{\beta} [C_{ss^{\prime}}(|{\bf k}-{\bf q}|) - C_{ss^{\prime}}(q)],
\end{eqnarray}

\renewcommand\theequation{2.21}
\begin{eqnarray}
V_{ss^{\prime}}^{L} ({\bf k},{\bf q}) +  V_{s^{\prime}s}^{L} ({\bf q},{\bf k}) &=& \frac{\hat{\bf k}^{i}_{\perp}\cdot{\bf q}}{\beta} [C_{ss^{\prime}}(|{\bf k}-{\bf q}|) - C_{ss^{\prime}}(q)],\nonumber\\
&+&\frac{k}{\beta} \sum^{2}_{s^{\prime\prime}=1} n_{0_{s^{\prime\prime}}}C_{ss^{\prime}s^{\prime\prime}}({\bf q}-{\bf k}; - {\bf q}).
\end{eqnarray}

\noindent
The linearized form of Eqs. (2.3)-(2.5) is identical to the linear
equations obtained in paper I for a binary fluid of hard spheres from the
Enskog theory. For the case of hard-sphere fluids one can derive explicit
analytical expressions for the dissipative kernels. These expressions were
obtained in I. The viscous kernels are given by
\renewcommand\theequation{2.22}
\begin{equation}
\gamma_{L}(k) = \frac{2}{3\rho} \sum_{s,s^{\prime}} \frac{2\mu_{ss^{\prime}}\sqrt{n_{s}n_{s^{\prime}}}}{t_{E_{ss^{\prime}}}} d_{1}(k\sigma_{ss^{\prime}}),
\end{equation}

\renewcommand\theequation{2.23}
\begin{equation}
\gamma_{T}(k) = \frac{2}{3\rho} \sum_{s,s^{\prime}} \frac{2\mu_{ss^{\prime}}\sqrt{n_{s}n_{s^{\prime}}}}{t_{E_{ss^{\prime}}}} d_{2}(k\sigma_{ss^{\prime}}),
\end{equation}
where $t_{E_{ss^{\prime}}}$, is the Enskog mean free time between collisions,
\renewcommand\theequation{2.24}
\begin{equation}
\frac{1}{t_{E_{ss^{\prime}}}} = \frac{4\sqrt{\pi n_{s}n_{s^{\prime}}}\sigma^{2}_{ss^{\prime}}}{(2\beta\mu_{ss^{\prime}})^{1/2}} \chi_{ss^{\prime}},
\end{equation}
with $\mu_{ss^{\prime}}=m_{s}m^{\prime}_{s}/(m_{s}+m_{s^{\prime}})$ and
$chi_{ss^{\prime}}$ the pair correlation
function of species $s$ and $s^{\prime}$ evaluated at contact,
$chi_{ss^{\prime}}=g_{ss^{\prime}}(r=\sigma_{ss^{\prime}})$. The functions
$d_{1}(x)$ and $d_{2}(x)$ are defined
in terms of the spherical Bessel functions $j_{l}(x)$ as
$d_{1}(x)=1-j_{0}(x)+2j_{2}(x)$ and $d_{2}(x)=1-j_{0}(x)-j_{2}(x)$.
The four ditTusion kernels $\gamma_{ss^{\prime}}(k)$ are related via the
Onsager reciprocity relations, which require $\gamma_{11}(k)=\gamma_{22}(k)=-\gamma_{12}(k)=-\gamma_{21}(k)$. Therefore there is only one independent
diffusion constant in a binary mixture. For hard-sphere fluids it is
given by
\renewcommand\theequation{2.25}
\begin{equation}
\gamma_{11}({\bf k}) = \frac{3}{2} \left[\sum_{s,s^{\prime}} \frac{2\mu_{ss^{\prime}}}{\sqrt{m_{s}m_{s^{\prime}}\rho_{s}\rho_{s^{\prime}}}t_{E_{ss^{\prime}}}} d_{ss^{\prime}}(k\sigma_{ss^{\prime}})\right]^{-1}\;
\end{equation}
where
\renewcommand\theequation{2.26}
\begin{equation}
d_{ss^{\prime}}(x) = \frac{\rho_{s}}{\rho_{s^{\prime}}} - (-1)^{s+s^{\prime}}[j_{0}(x) - 2j_{2}(x)]\;
\end{equation}

\noindent
This concludes the derivation of the nonlinear generalized hydrodynamic
equations. In the next section we describe the perturbation theory to
evaluate the mode-coupling contribution to the STAF. \\~\\

\section{Mode-coupling theory}

\noindent
We are interested in evaluating the generalized mode-coupling contribution
to the coefficient of shear viscosity $\eta_{s}$ and to the corresponding
STAF, $\rho_{\eta}(t)$. They are related by the Green-Kubo formula,
\renewcommand\theequation{3.1}
\begin{equation}
\eta_{s} = \beta \int^{\infty}_{0} dt \rho_{\eta}(t).
\end{equation}

\noindent
The shear viscosity governs the decay of the transverse momentum
fluctuations,
$C_{T}({\bf k},t)=\langle g_{T}({\bf k},t)g_{T}(-{\bf k},0)\rangle$.
When nonlinearities are neglected, the generalized hydrodynamic equations
discussed in Sec. II give
\renewcommand\theequation{3.2}
\begin{equation}
[\partial_{t} + \gamma_{T}(k)]C^{(0)}_{T}({\bf k},t) = 0,
\end{equation}
where
\begin{eqnarray*}
\lim_{k\rightarrow 0}\gamma_{T}(k) = k^{2}\eta_{E}/\rho_{0},
\end{eqnarray*}
with $\eta_{E}$ the Enskog shear viscosity of a mixture [24,25].
The solution to (3.2) is obtained immediately,

\renewcommand\theequation{3.3}
\begin{eqnarray}
C^{(0)}_{T}({\bf k},t) &&=e^{-\gamma_{T}(k)t}C_{T}(k,0)\nonumber\\
                       &&\simeq e^{\eta_{E}k^{2}t/\rho_{0}}C_{T}(k,0),
\end{eqnarray}
where the second equality applies in the limit $k\rightarrow 0$ and $C_{T}(k,0)=\rho /\beta$. When quadratic nonlinearities are retained in Eq. (2.13),
we obtain

\renewcommand\theequation{3.4}
\begin{equation}
[\partial_{t}+\gamma_{T}(k)]C_{T}({\bf k},t)=\frac{1}{2} \sum^{2}_{s,s^{\prime}=1} \int^{\prime} \frac{d{\bf q}}{(2\pi)^{3}}[V^{T}_{ss^{\prime}}({\bf k},{\bf q}) + V^{T}_{s^{\prime}s}({\bf q},{\bf k})]\langle\delta n_{s}({\bf q},t)\delta n_{s^{\prime}}({\bf k}-{\bf q},t)g_{T}(-{\bf k},0)\rangle ,
\end{equation}
where the prime on the integral sign denotes a short distance cutoff.
The effect of the nonlinearities on the right-hand
side of Eq. (3.4) can be approximately incorporated by a perturbation theory
that reexpresses the correlation function on the right-hand side of Eq.
(3.4) in terms of $C_{T}({\bf k},t)$. The theory is described in Appendix B.
Equation (3.4) is then recast in a form where the contribution from
nonlinearities enters as a non-Markovian renormalization of the transverse
damping $\gamma_{T}$,

\renewcommand\theequation{3.5}
\begin{equation}
[\partial_{t}+\gamma_{T}(k)]C_{T}({\bf k},t)= \int^{t}_{0} d\tau \sum_{\eta}
({\bf k},t - \tau) C_{T}({\bf k},\tau) = 0,
\end{equation}
with
\renewcommand\theequation{3.6}
\begin{equation}
\sum_{\eta}({\bf k},t) = \frac{1}{2} \sum_{\mu,\nu=\pm} \int^{t} \frac{d{\bf q}}{(2\pi)^{3}}|A_{\mu v}({\bf k},{\bf q})|^{2} e^{[z_{\mu}({\bf q})+z_{\nu}({\bf k}-{\bf q})]t}
\end{equation}
and
\renewcommand\theequation{3.7}
\begin{equation}
|A_{\mu,\nu} ({\bf k},{\bf q})|^{2} \sum^{2}_{s,s^{\prime}=1}
\sum^{2}_{s^{\prime\prime},s^{\prime\prime\prime}=1}
[V^{T}_{ss^\prime} ({\bf k},{\bf q}) + V^{T}_{s^{\prime}s} ({\bf q},{\bf k})]
\alpha^{\mu}_{ss^{\prime}}({\bf q})
[V^{T}_{s^{\prime\prime}s^{\prime\prime\prime}} ({\bf k},{\bf q}) + V^{T}_{s^{\prime\prime}s^{\prime\prime\prime}}({\bf q},{\bf k})]
\alpha^{\nu}_{s^{\prime\prime}s^{\prime\prime\prime}}({\bf k}-{\bf q}).
\end{equation}

\noindent
Explicit expressions for the amplitudes $\alpha^{\mu}_{ss^{\prime}}$ are
given in Appendix B.\\

\noindent
It is convenient to consider the Laplace transform of (3.5),

\renewcommand\theequation{3.8}
\begin{equation}
[z + \gamma^{R}_{T}(k,z)]\tilde{C}_{T}({\bf k},z)=C_{T}({\bf k},t=0),
\end{equation}
where
\renewcommand\theequation{3.9}
\begin{equation}
\tilde{C}_{T}({\bf k},z) = \int^{\infty}_{0}dt\;e^{-zt} C_{T}({\bf k},t)
\end{equation}
for ${\rm Re}(z)>0$. The renormalized transverse dissipative
kernel $\gamma^{R}_{T}$ is the sum of its bare part $\gamma_{T}(k)$ and a
mode-coupling contribution $\delta\gamma^{\rm mc}_{T}$
\renewcommand\theequation{3.10}
\begin{equation}
\gamma^{R}_{T}(K,z) = \gamma_{T}(k) + \delta\gamma^{\rm mc}_{T} (k,z),
\end{equation}
with
\renewcommand\theequation{3.11}
\begin{equation}
\delta\gamma^{\rm mc}_{T}(K,z) = \int^{\infty}_{0}dt\;e^{-zt}\sum_{\eta}({\bf k},t).
\end{equation}

\noindent
The long-wavelength limit of the transverse dissipative kernal $\gamma^{R}_{T}({\bf k},z)$ defines the frequency-dependent shear viscosity $\eta_{s}^{R}(z)$
according to
\renewcommand\theequation{3.12}
\begin{equation}
\lim_{k\rightarrow 0}\gamma^{R}_{T}({\bf k},z) = k^{2}\eta^{R}_{s}(z).
\end{equation}

\noindent
The mode-coupling contribution to the renormalized viscosity,
$\eta_{s}^{R}(z)=\eta_{E}+\delta\eta^{\rm mc}_{s}(z)$, is then given by

\renewcommand\theequation{3.13}
\begin{equation}
\delta\eta^{\rm mc}_{s}(z) = \lim_{k\rightarrow 0} \frac{1}{k^{2}}\int^{\infty}_{0} dt\;e^{-zt} \sum_{\eta}({\bf k},t).
\end{equation}

\noindent
By comparing Eqs. (3.1) and (3.13), we see that the long-wavelength limit
of $\sum_{\eta}({\bf k},t)$ also determines the mode-coupling contribution
to the STAF $\rho_{\eta}(t)$ according to

\renewcommand\theequation{3.14}
\begin{equation}
\delta\rho^{\rm gen}_{\eta}(t) = \lim_{k\rightarrow 0} \frac{1}{k^{2}}\sum_{\eta}({\bf k},t).
\end{equation}

\noindent
Finally, $\delta\rho^{\rm gen}_{\eta}(t)$ and $\delta\eta^{\rm mc}_{s}$ are
explicitly given by

\renewcommand\theequation{3.15}
\begin{eqnarray}
\delta\rho^{\rm gen}_{\eta}(t) = \frac{1}{2}\sum_{\mu ,\nu = \pm}&& \int^{t}
\frac{d{\bf q}}{(2\pi)^{3}} |A_{\mu\nu}({\bf 0},{\bf q})|^{2}\nonumber\\
&&\times e^{[z_{\mu}({\bf q})+z_{\nu}(-{\bf q})]t}\;,
\end{eqnarray}
and
\renewcommand\theequation{3.16}
\begin{equation}
\delta\eta^{\rm mc}_{s} = \frac{1}{2}\sum_{\mu ,\nu =\pm}\int^{t} \frac{d{\bf q}}{(2\pi)^{3}} \;\;\frac{|A_{\mu\nu}({\bf 0},{\bf q})|^{2}}{[z_{\mu}({\bf q})+z_{\nu}(-{\bf q})]}\;.
\end{equation}

\section{Results}

\noindent
In this paper we have used a generalized mode-coupling theory that takes
into account molecular scale effects to compute the STAF of a dense
binary-fluid mix- ture. As in the case of a one-component fluid of packing
fraction $\eta=(\pi /6)n\sigma^{3}=0.46$, in a dense mixture the
coefficient of the long-time tail of the STAF evaluated from generalized
mode-coupling theory is about two orders of magnitude larger than the
corresponding coefficient obtained from conventional mode-coupling theory. \\

\noindent
Here, we consider a mixture of hard spheres of sizes $\sigma_{1}$ and
$\sigma_{2}$, with $\sigma_{2} > \sigma_{1}$. As discussed
in paper I, the slowing down of the dynamics of the dense mixture on short
length scales is directly related to the short-range correlations that
characterize the fluid structure and that show up as peaks in the partial
static structure factors. These are in turn determined by the relative
packing fractions of the two species, $\eta_{1}=\pi n_{1}\sigma^{3}_{1}/6=
\eta x_{1}\alpha^{3}/(x_{2}+x_{1}\alpha^{3})$ and
$\eta_{2}=\pi n_{2}\sigma^{3}_{2}/6=\eta x_{2}/(x_{2}+x_{1}\alpha^{3})$,
where $\eta =\eta_{1}+\eta_{2}$ is the total
packing fraction and $x_{l}=n_{1}/n$ and $x_{2}=n_{2}/n$, with
$n=n_{1}+n_{2}$, are the concentrations. Notice that in our
calculation we keep $\eta$ constant $\eta = 0.46$ in all the data discussed
below), while changing the concentration $x_{2}$ of the large spheres.
Finally, in this
paper we only present results for mixtures of spheres of equal masses. This
is done mainly to dissociate the role of the structural parameters from the
role of the mass ratio in slowing down the dynamics. All times are
normalized to an Enskog time $t_{E}$ given by

\renewcommand\theequation{4.1}
\begin{equation}
\frac{1}{t_{E}} = \frac{4\sqrt{\pi}(n_{1}+n_{2})\sigma^{2}_{12}}{(2\beta\mu_{12})^{1/2}}\;\chi_{12}\;.
\end{equation}

\noindent
This time scale is neither the Enskog mean free time of a hard-sphere fluid
of type-l nor type-2 spheres. It is chosen because it depends only weakly on
the concentration of each species. \\

\noindent
We first discuss the conventional mode-coupling contribution to the STAF
of a mixture, denoted by $\delta\rho^{\rm conv}_{\eta}(t)$. As in
one-component fluids, one finds $\delta\rho^{\rm conv}_{\eta}(t)
\simeq At^{-3/2}$ [26]. This algebraic tail arises from the slow decay
of fluctuations in the conserved densities at long wave-lengths and the
feedback from these long-lived fluctuations through the convective
nonlinearities in the hydrodynamic equations. It describes the true
asymptotic long-time behavior of the correlation function. The
coefficient $A$ was evaluated many years ago by Pomeau
[26] and, as in one-component fluids, it was found to be very small.
In Fig. 1 we show $\delta\tilde{\rho}^{\rm conv}_{\eta}(t)=(\beta t_{E}/\eta_{E}/\eta_{E})\delta\rho_{\eta}^{\rm conv}(t)$ as a function of
$(t/t_{E})^{-3/2}$ for three values of the concentration $x_{2}=n_{2}/n$
of larger particles. Here $\eta_{E}$ is the Enskog shear viscosity of
the mixture [24]. The coefficient of the long-time tail, given by the
slope of the straight lines of Fig. 1, has a weak dependence on the
concentration $x_{2}$. The slope is also quite independent on the total
packing fraction $\eta$.\\

\noindent
The conventional mode-coupling contribution to the STAF should be compared
to the generalized mode- coupling contribution in Eq. (3.15) and denoted
here by $\delta\rho^{\rm gen}_{\eta}(t)$. Equation (3.15) can be written as
a one-dimensional integral over the magnitude $q$ of wave vector.
This integral has been evaluated numerically using an upper cutoff
$q_{\rm max}\sigma_{12}=20$. We have checked that the contribution to
the integral from reduced wave vectors larger than 20 is negligibly small.
In Fig. 2 we show the generalized mode-coupling contribution
$\delta\tilde{\rho}^{\rm gen}_{\eta}(t)$ as a function of
$(t/t_{E}^{-3/2}$ for a packing fraction $\eta = 0.46$. The curves are
well approximated by
straight lines over a range of time scales, indicating that the generalized
mode-coupling contribution to the STAF appears to decay as $\sim t^{-3/2}$
in this region. The generalized mode-coupling contribution evaluated here
describes the decay of the correlation function over the intermediate time
scales that are generally accessible to computer simulations. It is very
hard to probe by simulations the truly; asymptotic long-time behavior of
the correlation functions described by the conventional long-time tails.\\

\noindent
The apparent failure of our curves in Figs. 1 and 2 to
reduce to the result obtained for a one-component fluid in the limit of
either small $(x_{2}=0.01)$ or large $(x_{2}=0.9)$ concentration of large
spheres is only an artifact of our choice of the units of time [here we
have scaled time with $t_{E}$ given by Eq. (4.1)]. If in each case
$(x_{2} << 1$ and $x_{2}\simeq 1)$ we scale the time
with the Enskog mean free time of the majority component,
$t_{E11}$ and $t_{E22}$, respectively
[see Eq. (2.24)], then our results for $x_{2}=0.01$ and $x_{2}=0.9$
are essentially identical to those for a one-component fluid. \\

\noindent
By comparing Figs. 1 and 2 we see immediately that at the large values of
the density considered here the generalized mode-coupling contribution to
the STAF is about two orders of magnitude larger than the conventional
mode-coupling contribution (the vertical scales in Figs. 1 and 2
differ by a factor of $10^{3}$). We also find that
$\delta\rho^{\rm gen}_{\eta}(t)$ depends strongly on
the fluid density. For $\eta =0.23$ (the value used in Erpenbeck's
simulations) $\delta\rho^{\rm gen}_{\eta}(t)$
is smaller than $\delta\rho^{\rm conv}_{\eta}(t)$. This is consistent
with the fact
that the generalized mode-coupling contribution accounts for the potential
part of the correlation function that dominates at high density. One of the
factors that controls its size is the height of the first peaks of the
static structure factors of the fluid, which in turn are large only at
high density. \\

\noindent
The generalized mode-coupling contribution given in Eq. (3.15) is the sum of
four terms, corresponding to the coupling of two density modes (- -), the
coupling of two diffusion modes (+ +), and two equal terms corresponding
to the coupling of one diffusion mode and one density mode (+ - and - +). In
Figs. 3-5, we show separately these contributions to the STAF as well as
their sum for three values of the concentration $x_{2}$ For a small $(x_{2}=0.01$, Fig. 3) and a large $(x_{2}=0.9$, fig. 5) concentration of large spheres the
generalized mode-coupling contribution to the STAF is dominated by only
one of these four terms. For $x_{2}<< 1$ the density mode\\

\noindent
$(z-)$ describes the relaxation of total density fluctuations which are
mainly determined by the majority component (type 1, in this case). The
coupling of two of these density modes gives then the dominant
generalized mode-coupling contribution to the STAF. For $x_{2} \simeq 1$
 it is the mode we label diffusion mode $(z+)$ that describes the
relaxation of density fluctuations and the coupling of two of these
modes dominates the mode-coup1ing contribution to the STAF. At an
intermediate value of the concentration of large spheres
$(x_{2}=0.2$, Fig. 4) we find that the contributions from
all four terms in Eq. (3.15) are of comparable magnitude. For large or
small values of the concentration of one of the two species (here the
large spheres), the mixture resembles a one-component fluid and the density
fluctuations of the majority component govern the slow structural relaxation
of the fluid. The effect of diffusion of the low-concentration component on
slowing down the decay of the STAF appears to be small. For the
intermediate value of the concentration the packing fractions of the two
species are comparable and the mixture structurally resembles a metallic
glass. The mixture is structurally very disordered and all three partial
static structure factors are peaked at well-separated values of the wave
vector, determined by the sizes of the two species. As discussed in paper I,
fluctuations in the density of both species decay slowly at molecular length
scales. The two extended modes of diffusion and total density fluctuations
cannot be decoupled: both soften appreciably at large wave vectors and
contribute to the mode-coupling integral. There is therefore a range of
large wave vectors where density fluctuations decay slowly. This should be
contrasted to what happens in the case of a simple fluid, where the
mode-coupling integrals are dominated by the single value of the wave vector
corresponding to the first maximum of the static structure factor. \\

\newpage

\noindent
We now discuss in some detail the effect of compositional disorder on the
decay of the STAF in a dense mixture. For the present discussion we
normalize all times to $t^{*}$ given by

\renewcommand\theequation{4.2}
\begin{equation}
\frac{1}{t^{\star}} = \frac{4\sqrt{\pi}(n_{1}\sigma^{2}_{1}+n_{2}\sigma^{2}_{2})}{(2\beta\mu_{12})^{1/2}}\;(x^{2}_{1}\chi_{11} + x^{2}_{2}\chi_{22} + 2x_{1}x_{2}\chi_{12})\;.
\end{equation}

\noindent
This time scale, unlike $t_{E}$, has the feature of going to the
one-component fluid limit for $x_{2}<< 1$ and $x_{2}\simeq 1$.\\

\noindent
We first consider the case of conventional mode-coupling. In Fig. 6(a) we
show $\delta\rho^{rm conv*}_{\eta}(t) = (\beta t^{*}/\eta_{E})\delta\rho^{rm conv}_{\eta}(t)$ as a function of $(t/t^{*})$ for three small values of the
concentration $x_{2} = n_{2}/n$ of larger particles. Increasing the
concentration of large particles in a
fluid of small spheres slightly shifts the characteristic time scale
governing the decay of the STAF to larger times. For a given value of
$(t/t^{*})$, the STAF of the mixture is then larger than that of a
one-component fluid. This is, however, a very small effect and it is
entirely due to the concentration dependence of the Enskog shear
viscosity of the fluid. A similar behavior is observed when small
particles are added to a fluid of large particles [see Fig. 6(b)]. \\

\noindent
The concentration dependence of a generalized mode-coupling contribution to
the STAF is quite different. In Figs. 7(a) and 7(b) we display $\delta\rho^{\rm gen*}_{\eta}(t)=(\beta t^{*}/\eta_{E})\delta\rho^{\rm gen}_{\eta}(t)$ as a function of $(t/t^{*})$ for various values of the concentration
$x_{2}=n_{2}/n$ of larger particles. For
values of $x_{2}$ close to 1 [Fig. 7(b)], when the large spheres are much more
abundant than the small spheres, increasing the fraction of small spheres
results in a shift of the characteristic time regime where the mode-coupling
contribution to the STAF is large to shorter times. The shift is in
the opposite direction as compared to that observed for the conventional
contribution and it is more pronounced. A similar behavior is also observed
for small values of $x_{2}$ [Fig. 7(a)].\\

\noindent
A comparison of Figs. 6 and 7 shows that the concen- tration dependence of
$\delta\rho^{\rm gen}_{\eta}(t)$ is stronger than that of
$\delta\rho^{\rm conv}_{\eta}(t)$. This indicates that the degree of
compositional disorder in the fluid plays a role in the dense fluid
dynamics. \\

\noindent
In order to assess more precisely when compositional disorder enhances
large-wave-vector mode-coupling effects, we have also evaluated the
mode-coupling contribution to the shear viscosity,
$\delta\rho^{\rm gen}_{\rm mc}$. In Fig. 8
we show $\delta\rho^{\rm gen}_{\rm mc}/\eta_{E}$
as a function of $x_{2}$ for a few values of the size ratio $\alpha$.
Here $\eta_{E}$ is the Enskog shear viscosity of the mixture. In order to
properly interpret the results displayed in Fig. 8 it is important to
first clarify the concentration dependence of the Enskog shear viscosity.
At the high densities considered here the Enskog shear viscosity of a
hard-sphere fluid is dominated by the potential contribution. For a
one-component fluid this is given by
$\eta_{E}^{V} = \frac{4}{15}\sqrt{\pi m/\beta}[(n\sigma^{3})^{2}\chi /\sigma^{2}].$ If the reduced density $n\sigma^{3}$ is kept fixed, then
$\eta^{V}_{E}\sim 1/\sigma^{2}$ and the viscosity decreases as the
sphere diameter increases (the pair correlation at contact $\chi$ only
depends on $n\sigma^{3}$). The Enskog shear viscosity $\eta_{E}$ of a binary
mixture depends on the concentration $x_{2}$ of larger particles and
reduces to the Enskog viscosity of a fluid of hard spheres of size
$\sigma_{1}$, $\eta_{E1}$, for $x_{2}=0$, and to the Enskog viscosity
of a fluid of hard spheres of size $\sigma_{2}>\sigma_{1}$, $\eta_{E2}$,
for $x_{2}=1$, with $\eta_{E2}<\eta_{e1}$. Compositional disorder always
increases the bare viscosity in the sense that for $x_{2}<<1,\;\;\eta_{E}>\eta_{E1}$ and for $x_{2}\simeq 1,\;\eta_{E}>\eta_{E2}$.\\

\noindent
In contrast we see from Fig. 8 that the concentration dependence of the
generalized mode-coupling contribution $\delta\eta^{\rm mc}_{s}$ (relative
to $\eta_{E}$) is quite different. The dashed line in the
figure indicates the value of $\delta\eta^{\rm mc}_{s}/\eta_{E}$ for a
one-component hard-sphere fluid with $n\sigma^{3}=0.88$ as obtained
before by other researchers [27]. Our results show that in a mixture with
a small fraction of large spheres (a colloidal suspension) the generalizai
mode-coupling corrections are larger than in a one-component fluid.
This corresponds to the fact that the motion of large spheres in a dense
background fluid is very slow. Shear relaxation also gets slower as the size
difference increases (smaller $\alpha$). On the other hand, in a mixture
containing a small fraction of small spheres in a fluid of large spheres
$(x_{2}\sim 1)$ the generalized mode-coupling correction to the shear
viscosity, while still larger than the one-component
value, is considerably smaller than obtained in a
colloidal-fluid-like mixture $(x_{2}<< 1)$. This is because small
spheres can move rather easily through a matrix of large spheres. These
considerations are strictly meaningful only for $\alpha =0.5$ and
$\alpha=0.7$, that is, for mixtures of spheres of rather disparate sizes.
The $\alpha =0.9$ value shows how the curves converge to the one-component
fluid limit. Our findings are in agreement with the results of simulations
of mixtures by Mountain and Thirumalai [28]. These authors reported that a
mixture with a small fraction of small particles can crystallize more easily
than a mixture with a few large particles, due to the fact that small
particles can move more easily in a matrix of large particles compared to
large particles in a matrix of small particles. \\

\section{Discussion}

\noindent
In this paper we have formulated a generalized mode-coupling theory for
a dense binary fluid mixture that takes into account the slowing down of
the fluid dynamics at molecular scales due to the closely packed fluid
structure. This has been used to evaluate the renormalization of the
STAF due to density nonlinearities in the
hydrodynamic free energy. As in the case of a one-component fluid [3], in a
dense fluid mixture the long-time tails in the STAF are about two orders
of magnitude larger than the conventional long-wavelength contributions.
The concentration dependence of $\delta\tilde{\rho}^{\rm gen}_{\eta}(t)$
is stronger than that of $\delta\tilde{\rho}^{\rm conv}_{\eta}(t)$.
This clearly indicates that at high densities the degree of
compositional disorder in the fluid affects the static fluid structure and
plays an important role in slowing down the fluid dynamics. In this
section we make a few comments on the results obtained in this paper.\\

\begin{enumerate}
\item [(1)] In our analysis we have considered a dense binary mixture of
different sizes and equal masses at different values of the concentration.
The packing fraction of each species is the main parameter controlling
the static structural properties of dense fluids at molecular length
scales. For this reason concentration and size ratio have qualitatively
the same effect on the short-wavelength dynamics. We find that at
intermediate values of the concentration when the fluid is structurally
disordered, there is a finite
range of wave vectors over which the dynamics slows down and consequently
the mode-coupling integrands are large over a range of length scales. This
is reflected in the fact that all possible couplings of the density
fluctuations of the two species make comparable contributions to the total
$\delta\tilde{\rho}^{\rm gen}_{\eta}(t)$. The presence of a range of length
scales and hence a
distribution of time scales for a disordered dense binary mixture that we
predict is a characteristic feature of glass-forming liquids [29]. In
contrast, in a simple fluid the mode-coupling integrands are very sharply
peaked at the position of the first maximum of the static structure
factor. The presence of compositional disorder shifts the time regime
where the generalized mode-coupling contribution to the STAF is large to
shorter times. In order to test our results one needs to perform
molecular-dynamics simulations for mixtures of dissimilar sizes at high
densities.

\item [(2)] In this paper we only considered mixtures of spheres of equal masses.
This choice was motivated by our desire to focus on the role of frustration
caused by size difference in slowing down the fluid dynamics. In addition,
both our previous work [18] and computer simulations [20] have indicated
that the slowing down of the dynamics in a dense, closely packed binary
mixture depends more weakly on the mass ratio than on the size ratio of the
two species.

\item [(3)] The nonlinear fluctuating hydrodynamic equations presented here can also
be used to discuss the mode-coupling theory of the glass transition in
fluid mixtures. In this case the mode-coupling contribution to the time-
dependent density-density correlation function is evaluated
self-consistently. The self-consistent mode-coupling theory of mixtures has
been discussed by other authors [19]. It involves a considerable amount of
computation that tends to obscure the physics. It would be interesting to
consider a simplified model obtained, for instance, by replacing the static
partial factors of the mixture by constant values with appropriate
wave-vector cutoffs or by Lorentzians. A calculation of this type has been
carried out by Mazenko and Valls [30] for a one-component fluid.
\end{enumerate}

\newpage
\begin{center}
{\bf Acknowledgments}\\
\end{center}

\noindent
This work was supported by the National Science Foundation under Contract
No. DMR-91-12330. \\

\begin{center}
{\bf Appendix~~~A}\\
\end{center}

\noindent
In deriving the nonlinear tluctuating generalized hydrodynamic equations
of a binary tluid mixture we follow the approach described by Ma and Mazenko
[22]. The Langevin equation governing the time evolution of the hydrodynamic
variables denoted by $\psi_{i}$ may be written formally as

\renewcommand\theequation{A1}
\begin{equation}
\delta_{t}\psi_{1}(t)=V_{i}[\psi] - L_{ij} \frac{\delta F}{\delta\psi_{j}(t)} +
f_{i}(t)\;.
\end{equation}
The reversible part of the dynamics is given by
\renewcommand\theequation{A2}
\begin{equation}
V_{i}[\psi]=Q_{ij}[\psi] \frac{\delta F}{\delta\psi_{j}(t)} - \frac{1}{\beta}
\frac{\delta{\bf Q}_{ij}[\psi]}{\delta\psi_{j}(t)}\;,
\end{equation}
where ${\bf Q}_{ij}[\psi]=\{\psi_{i},\psi_{j}\}$ is the Poisson bracket
between the variables $\psi_{i}$ and $F[\psi]$ is the free-energy functional
given in Sec. II. The damping coefficients $L_{ij}$ and the thermal noise
$f_{i}$ pertain to the dissipative part of the dynamics. The noise $f_{i}$
obeys the fluctuation dissipation relation

\renewcommand\theequation{A3}
\begin{equation}
\langle f_{i}(t)f_{j}(t^{\prime})\rangle = 2k_{B}TL_{ij}\delta (t-t^{\prime})\;.
\end{equation}

\noindent
For our binary mixture the relevant hydrodynamic variables are
$\{\psi_{i}\} = (\rho_{s},g)$. The Poisson brackets needed in the
hydrodynamic equations are

\renewcommand\theequation{A4}
\begin{equation}
\{\rho_{s}({\bf r}),g_{\alpha}({\bf r}^{\prime})\}
= -\partial_{\alpha}\delta({\bf r}-{\bf r}^{\prime})\rho_{s}({\bf r}^{\prime})\;,\end{equation}

\renewcommand\theequation{A5}
\begin{equation}
\{g_{\alpha}({\bf r},g_{\alpha}({\bf r}^{\prime})\} = \partial^{\prime}_{\alpha}\delta({\bf r}-{\bf r}^{\prime})g_{\beta}({\bf r})-\partial_{\beta}({\bf r}-{\bf r})g_{\alpha}({\bf r}^{\prime})\;.
\end{equation}
\vspace*{2.5cm}

\begin{center}
{\bf Appendix~~~B}\\
\end{center}

\noindent
In this appendix we brietly describe the perturbation theory that leads to
Eq. (3.5). In order to close Eq. (3.4) we need to construct an equation for
the correlation function of three phase functions appearing on the
right-hand side of Eq. (3.4). This is done most simply by using the
invariance of equilibrium correlation functions under time translations,

\renewcommand\theequation{B1}
\begin{eqnarray}
\langle\delta n_{s}({\bf q},t)\delta n_{s^\prime}({\bf k}-{\bf q},t)g_{T}(-{\bf k},0)\rangle \nonumber\\
=\langle\delta n_{s}({\bf q},0)\delta n_{s^{\prime}}({\bf k}-{\bf q},0)g_{T}(-{\bf k},-t) \rangle\;.
\end{eqnarray}

\noindent
By letting $t \rightarrow -t$ and ${\bf k}\rightarrow -{\bf k}$ in Eq. (2.13),
multiplying the resulting equation with two density fluctuations, and taking
the ensemble average, we then obtain

\renewcommand\theequation{B2}
\begin{eqnarray}
[-\partial_{t} + \gamma_{T}(k)] \langle\delta n_{s}({\bf q},0)\delta n_{s^\prime}({\bf k}-{\bf q},0)g_{T}(-{\bf k},-t)\rangle \nonumber\\
=\frac{1}{2} \sum^{2}_{s^{\prime\prime},s^{\prime\prime\prime}=1}
\int \frac{d{\bf q}^{\prime}}{(2\pi)^{3}} [V^{T}_{s^{\prime\prime}s^{\prime\prime\prime}}(-{\bf k},{\bf q}^{\prime})+V^{T}_{s^{\prime\prime\prime}s^{\prime\prime}}({\bf q}^{\prime},-{\bf k})]\nonumber\\
\langle\delta n_{s^{\prime\prime}}({\bf q}^{\prime},-t)\delta n_{s^{\prime\prime\prime}}(-{\bf k}-{\bf q},-t)\delta n_{s}({\bf q},0)\delta n_{s^{\prime}}({\bf k}-{\bf q},0) \rangle\;.
\end{eqnarray}

\noindent
We then perform a cluster expansion of the four-point correlation function on the right-hand side of Eq.(B2) and neglet cumulants. With this approximation Eq.(B2) becomes
\renewcommand\theequation{B3}
\begin{eqnarray}
&&[\partial_{t} + \gamma_{T}(k)] \langle\delta n_{s}({\bf q},0)\delta n_{s^\prime}({\bf k}-{\bf q},0)g_{T}(-{\bf k},-t)\rangle \nonumber\\
&&\simeq\frac{1}{2} \sum^{2}_{s^{\prime\prime},s^{\prime\prime\prime}=1}
\int \frac{d{\bf q}^{\prime}}{(2\pi)^{3}} [V^{T}_{s^{\prime\prime}s^{\prime\prime\prime}}(-{\bf k},{\bf q}^{\prime})+V^{T}_{s^{\prime\prime\prime}s^{\prime\prime}}({\bf q}^{\prime},-{\bf k})]\nonumber\\
&&\langle\delta n_{s^{\prime\prime}}({\bf q}^{\prime},t)\delta n_{s}({\bf q},0)\rangle \langle\delta n_{s^{\prime\prime\prime}}(-{\bf k}-{\bf q},t)\delta n_{s^{\prime}}({\bf k}-{\bf q},0) \rangle\;.
\end{eqnarray}

\noindent
The time dependence of the density-density correlation on the right-hand side
of Eq.(B3) is now evaluated from Eqs.(2.11) and (2.12) by neglected
nonlinearities. The most important contribution to the mode-coupling
integrals comes from wave vectors near the peaks of the static structure
factors. Momentum fluctuations of such large wave vectors decay
rapidly and one can neglect the time derivative in Eq.(2.12) as compared
to the viscous damping $\gamma_{L}$ in treating all except the largest time
scales. The longitudinal momentum can then be eliminated between Eqs.(2.11)
and (2.12). The resulting two coupled linear equations for density
fluctuations are given by

\renewcommand\theequation{B4}
\begin{equation}
\partial_{t}\delta n_{s} ({\bf q},t)+q^{2}b_{ss^{\prime}}(q)\delta n_{s^{\prime}}({\bf q},t)=0\;,
\end{equation}
where summation over $s^{\prime}$ is intended, and
\renewcommand\theequation{B5}
\begin{equation}
b_{ss^{\prime}}(q) = \frac{n_{s}g_{s^{\prime}}(q)}{\beta\rho\gamma_{L}(q)} + \frac{\gamma_{ss^{\prime}}}{\beta m_{s}} \frac{f_{s^{\prime}}(q)}{\rho_{s^{\prime}}}\;,
\end{equation}
with
\renewcommand\theequation{B6}
\begin{eqnarray}
g_{1}(q) = \frac{1}{\Delta (q)} [S_{22}(q) -\sqrt{n_{2}/n_{1}}S_{12}(q)]\;,\nonumber\\
g_{2}(q) = \frac{1}{\Delta (q)} [S_{11}(q) -\sqrt{n_{1}/n_{2}}S_{12}(q)]\;,\\
f_{1}(q) = \frac{1}{\Delta (q)} \left[S_{22}(q) + \frac{m_{1}}{m_{2}} \sqrt{n_{1}/n_{2}}S_{12}(q)\right]\;,\nonumber\\
f_{2}(q) = \frac{1}{\Delta (q)} \left[S_{11}(q) + \frac{m_{2}}{m_{1}} \sqrt{n_{2}/n_{1}}S_{12}(q)\right]\;,\nonumber
\end{eqnarray}
with $\Delta(q)=S_{11}(q)S_{22}(q)-S^{2}_{12}(q)$. The approximate modes
obtained from Eqs.(B4) were discussed in paper I, where it was shown that
this simple approximation is indeed accurate at large wave vectors. The time
decay of the density-density correlation function is then governed by
two relaxation rates that we label density mode $[z-(q)]$ and diffusion
mode $[z+(q)]$,
\renewcommand\theequation{B7}
\begin{equation}
z\pm (q) = \frac{q^{2}}{2} (b_{11}+b_{22})\pm \frac{q^{2}}{2} [(b_{11}-b_{22})^{2}+4b_{12}b_{21}]^{1/2}\;.
\end{equation}
The density-density correlation functions are given by
\renewcommand\theequation{B8}
\begin{equation}
\langle \delta n_{s}({\bf q},t)\delta n_{s^{\prime}}(-{\bf q},0)\rangle =
\sum_{\mu =+,-} \alpha^{\mu}_{ss^{\prime}}({\bf q})e^{z_{\mu}(q)t}\;,
\end{equation}
where
\renewcommand\theequation{B9}
\begin{eqnarray}
\alpha^{\mu}_{1s^{\prime}}({\bf q})=\mu \frac{\sqrt{n_{1}n_{s^{\prime}}}S_{1s^{\prime}}(q)(b_{22}+z_{\mu})-\sqrt{n_{2}n_{s^{\prime}}}S_{2s^{\prime}}(q)b_{12}}{[(b_{11}-b_{22})^{2}+4b_{12}b_{21}]^{1/2}}\;,\nonumber\\
\alpha^{\mu}_{2s^{\prime}}({\bf q})=\mu \frac{\sqrt{n_{2}n_{s^{\prime}}}S_{2s^{\prime}}(q)(b_{11}+z_{\mu})-\sqrt{n_{1}n_{s^{\prime}}}S_{1s^{\prime}}(q)b_{21}}{[(b_{11}-b_{22})^{2}+4b_{12}b_{21}]^{1/2}}\;.
\end{eqnarray}

\noindent
The partial static structure factors are defined as $\langle\delta n_{s}({\bf q})\delta n_{s^{\prime}}(-{\bf q})\rangle$\newline $=\sqrt{n_{s}n_{s^{\prime}}}S_{ss^{\prime}}(q)$. Finally the eigenvalues given in Eq.(B7) can be
rewritten in the more transparent form presented in paper I and rewritten
here for completeness,

\renewcommand\theequation{B10}
\begin{eqnarray}
&& z\pm (q) = -\frac{q^{2}}{2}\left[\frac{1}{\rho\chi_{T}(q)\gamma_{L}(q)}
+ \frac{\gamma_{11}(q)}{\beta} \left\{\frac{f_{1}(q)}{m_{1}\rho_{1}}+ \frac{f_{2}(q)}{m_{2}\rho_{2}}\right\}\right]\nonumber\\
&&\pm \frac{q^{2}}{2}\left\{\left[\frac{1}{\rho\chi_{T}(q)\gamma_{L}(q)}
- \frac{\gamma_{11}(q)}{\beta} \left[\frac{f_{1}(q)}{m_{1}\rho_{1}}+ \frac{f_{2}(q)}{m_{2}\rho_{2}}\right]\right]^{2}
- \frac{4\gamma_{11}(q)}{\rho\beta^{2}\gamma_{L}(q)}\left[\frac{f_{1}(q)}{m_{1}}- \frac{f_{2}(q)}{m_{2}}\right]^{2}\right\}^{1/2}\;.\nonumber\\
&&
\end{eqnarray}
Here $\chi_{T}(q)$ is the generalized isothermal compressibility, defined as
\renewcommand\theequation{B11}
\begin{equation}
\chi_{T}(q) = \chi^{0}_{T} \frac{s_{11}(q)S_{22}(q)-S^{2}_{12}(q)}{x_{2}S_{11}(q)+x_{1}S_{22}(q)-2\sqrt{x_{1}x_{2}}S_{12}(q)}\;,
\end{equation}
with $chi^{0}_{T}=\beta /n$ the ideal-gas compressibility.\\

\noindent
Finally, by inserting the solution of Eq.(B3) on the right-hand side of
Eq.(3.4) and replacing $C^{(0)}_{T}({\bf k},\tau)=(\rho /\beta )e^{-\gamma_{T}(k)\tau}$ with $C_{T}({\bf k},\tau)$ in the mode-coupling integral, we
immediately obtain Eq.(3.5). The derivation described here is entirely
phenomenological. It can, however, be shown that the same result is
obtained by using diagrammatic methods [1,22].

redpajama_set_name: RedPajamaArXiv


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The defending SEC and Conference USA soccer champions will square off this evening in the first athletic contest of the 2015-2016 academic year.
First kick between Rice and No. 5 Texas A&M is set for 7 p.m. at Halloway Field in Houston. Tickets are $8 for adults and $6 for youths.
The Aggies won the SEC regular season and tournament championships and advanced all the way to the NCAA Tournament Final 4 last season. They are picked to finish second in the conference this year behind Florida.
Rice has never beaten A&M on the soccer field. The Aggies lead the series 3-0 with wins in 2002, 2007 and 2008. A&M has outscored Rice 12-2 in those matches.
Adams has a pretty darn good idea of Aggie expectations as a former standout soccer player at the institution of higher education. She scored 57 goals from 1997 to 2001, the second most in Texas A&M history. She is one of six former Texas A&M student-athletes that will be inducted into the Lettermen's Association Hall of Fame next month.
Entering her sixth year as the head coach, Adams has raised the level of expectations at Rice. The Owls are coming of a 14-4-3 season, which include a C-USA Tournament title and the first NCAA Tournament appearance since 2005. Rice did not drop a single match at home last season, going 6-0-2.
Adams has 11 starters back from last year's squad, including the two all-team leading goal scorers in school history, Lauren Hughes and Holly Hargreaves.
Hughes, a senior from Ottawa, was voted by C-USA coaches as the Preseason Offensive Player of the Year. Rice junior Jenny Fichera was selected as the C-USA Preseason Defensive Player of the Year. Hargreaves, who already owns three school records, joined Hughes and Fichera on the preseason all-conference team.
While the Owls are an experienced squad, tonight's game will mark the first collegiate experience for six Rice freshmen, including two from Texas: The Woodlands native Gretchen Cahill and Fairview native Dominique Mazero.
"I can't think of a better opponent to start off your college career," Adams said.
The Aggies return three starters, including 2014 All-SEC Second Team midfielder Janae Cousineau and 2014 SEC All-Freshman defender McKayla Paulson.


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Urgent Saudi Arabia announces a new vacation that will last for several days for the first time and start on this date and for this reason
Trends Aamal Trend Wednesday 25th January 2023 04:13 AM REPORT
With the resumption of students in the Kingdom for the second term, which began on the fourth day of December of the year 2022, when the first term holiday ended on the 30th of Rabi` al-Thani, multiple searches began by both parents and students to find out the extended vacation schedule for schools in Saudi Arabia and the date of the next extended vacation. This is what we show you in a number of detailed lines, so follow us.
Extended leave schedule
Urgent, a new two-day vacation in Saudi Arabia that will start soon on this important occasion
Vacancies in Qatar for citizens and residents of any nationality with high salaries provided by Qatar Foundation .. Terms and application link
Through what was launched by the Ministry of Education in the Kingdom about the date of the extended vacation in Saudi Arabia 2022-2023, the academic calendar was legalized after taking a number of amending steps for the various semesters of study. It is mentioned that the study calendar includes the school leave, including three of the most prominent holidays:
Foundation day holiday for the Hijri year 1444.
First extended leave.
Second extended vacation.
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According to the various questions that swept the communication sites about the date of the upcoming extended vacation, the extended vacation schedule that was launched by the educational body included several holidays, where the vacation is one or two days in addition to that Friday and Saturday as a regular vacation. We explain to you in bullet points the details:
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The first extended vacation for the weekend will be on the eighteenth of December 2022, according to the twenty-fourth Hijri calendar of Jumada al-Awwal, corresponding to Sunday, so that the next vacation that students are waiting for will be.
The second extended weekend for the weekend in the fifteenth and sixteenth centuries of January 2023 corresponding to the twenty-second Hijri and the twenty-third of Jumada al-Akhir 1444 corresponding to Sunday.
The "Foundation Day" holiday is on the twenty-second and twenty-third of February of the year 2023 corresponding to the second Hijri and the third of Sha'ban for the year 1444.
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When is the first extended holiday?
First extended weekend of December 18th, 2022.
When is the second extended holiday?
The second extended weekend of January 16th, 2023.
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Foundation Day holiday on the 22nd and 23rd of February 2023.
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These puppies can't live in high temperature whether and also hard to stay in cold environment as well. They are facing in to squash up nasal passages, also respiratory problems.
Pugs has lot of issues in there breathing. For the reason that of their squashed faces and shortened airways will cause to some breathing matters, wheezing and sniffing. This will getting worst in hot and cold weather.
Moreover, we need to be in alert on these puppies rather than other types due to their size.Because if the dogs were to get caught up under human feet in the home and stepped on.
What are weeducated about this teacup pugs ?
Yes, but always needs tob reed with tiny dogs like Chihuahuas.
Yes, nevertheless they are not real Pugs.
Whether it is toy Pug dog, dwarf Pug dog, pocket Pug, micro Pugs there is a problems.
In conclusion, we'll talk about miniature Pug facts, some more facts about the miniature Pug vs Pug.
As soon as we heard tea cup pugs it's going to be a problematic subject. You are both purchasing an unhealthy Pug or fit dog which isn't a Pug. If wants to adopt a real Pugs which stay tiny miniature Pug, runt Pug is the one and only option, and again it is a bad idea as well as no guarantee.
So the best choice is to choose between a real Pug and a teacup Pug which will crossedamong the Chihuahua and Pug. Whichever go larger and met all features you want, otherwise stay tiny and adore the odd tiny little Chug for what it really is.


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text:
title: Hierarchy
---
**Hierarchy**
Differences between items shown visually.
*   [A Closer Look at Hierarchy in Web Design](http://www.onextrapixel.com/2010/06/24/a-closer-look-at-hierarchy-in-web-design/)
*   [Creating Better Typographic Hierarchy](https://medium.com/designed-thought/creating-better-typographic-hierarchy-1148a46b2fc)
*   [Visual Design and Usability Yellow Brick Road](http://uxmag.com/articles/visual-design-and-usability-yellow-brick-road)

redpajama_set_name: RedPajamaGithub