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Mark Kushner: Yes.
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Prof. Murillo: Okay. Welcome everybody. Today's Mitzi talk. It's my great pleasure to introduce to you today, Dr. Frank Graziani from Lawrence Livermore National Laboratory
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Prof. Murillo: Frank got his bachelor's in physics from Santa Clara University, followed by a PhD from the University of California, Los Angeles.
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Prof. Murillo: Followed by a couple of postdocs at University of Colorado and University of Minnesota during that time. He was a very different person. He did cosmology and particle physics.
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Prof. Murillo: But that eventually led him to see the light, where he joined Lawrence Livermore National Laboratory in 1989 two switches area focus to radiation transport.
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Prof. Murillo: And plasma physics, for which he became very well known in those two communities for doing that and lead a lot of influential efforts during that time.
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Prof. Murillo: Today he's now the director of Lawrence Livermore is high energy density Sciences Center.
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Prof. Murillo: And focuses his technical work as you will see today on the micro physics of dense plasmas and and also high energy density physics education. He's very he's been
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Prof. Murillo: Central and trying to coordinate universities across the nation and actually across the world in building up a network of universities like those in Mitzi for high energy density physics education.
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Prof. Murillo: Throughout his, his career. He's had many, many leadership positions. He's been a group leader. He's been associated division leader.
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Prof. Murillo: He's led projects for verification and validation. He's been PCI on internal LD Artes of the lab. He was the national leader for the boost initiative.
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Prof. Murillo: He's edited books and led some very influential workshops, one that I got to be part of at the Institute for pure and applied mathematics at UCLA on computational higher density physics.
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Prof. Murillo: That was, that was amazing and and you won't be surprised that after all of that he's won many, many awards. He's won the deal we defense program awards of excellence.
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Prof. Murillo: Not once, not twice, not three times four times and also one Lawrence Livermore is most prestigious award for science and technology. The Livermore directors s&t award.
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Prof. Murillo: And finally, that was sort of capped off when Lawrence Livermore named him a distinguished member of the technical staff, which is the highest level of technical staff member that you can be at the laboratories
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Prof. Murillo: So with that, maybe we can well give welcome Frank here to Mitzi maybe you can all do like a thumbs up, just to the equivalent of a clap to welcome Frank. Welcome Frank. And I'll turn it over now to mark.
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Mark Kushner: Well thank you Michael and my part of this ceremony is to welcome Frank and thank him for spending his virtual day with us and to virtually present the Mitzi mug to to frame.
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Mark Kushner: As we all know, the mint tea mug is
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Mark Kushner: Perhaps the most cherished possession that a plastic is this can have until we're happy to transfer this this chalice to to Frank for his display and use the Great, thank you very much for being our seminar speaker today.
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Frank Graziani: Thank you very much, Mark, it's a it's a real pleasure. And thank you, especially for the mark. I really appreciate it and I'll be stepping out of it. During my seminar today and thank you very much to you and lipsey for the for this kind invitation.
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Frank Graziani: So today what I thought I would do is kind of give you a an overview, a journey of discovery and particularly personal discovery and quantum Hydra dynamics and particularly how it might relate to really tailored instability and in quantum many body systems.
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Frank Graziani: I will say that I, I owe part of that started this journey to Michael Murillo who
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Frank Graziani: At that time we were working get together in the on the Cimarron project, which was developing a molecular dynamics capability for for dense matter.
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Frank Graziani: And helium Stanton came to one of our meetings one day and start talking about this thing called quantum hydro dynamics and little did I know that at this relatively late stage of my life. I would be
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Frank Graziani: Doing research in this area and finding it absolutely fascinating. So thank you Michael for that.
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Frank Graziani: And thank you, university MICHIGAN. MICHIGAN STATE. I think it's wonderful that all of you have have invited me to this, this is if you if you don't know what I look like. This is a caricature by Mike Campbell so i i i have become infamous
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Frank Graziani: Did Mike but
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Frank Graziani: It's, it's not a bad, bad, bad likeness. So thank you very much again.
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Frank Graziani: So first of all, let's, let's talk a little bit about what high energy density science is and some of you may know this very well. Some of you may not
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Frank Graziani: But it's, it's, in some sense, it's different than what we would consider ordinary plasma physics that maybe I would have learned when I was a graduate student at UCLA.
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Frank Graziani: It certainly is. It's really studying mattered extreme conditions. And when we think of the variety of phenomenon can be everything for as mundane as micro meteorite and impacts on a spacecraft. And here we see a
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Frank Graziani: NASA Ames Research Center energy flash from a Hypervelocity Impact to doing high pressure experiments on water at laser for laboratory energetics are doing simulations of target being exposed to.
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Frank Graziani: Expel radiation.
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Frank Graziani: at El CLS up here right now, away from me in Palo Alto, or of course our holy grail of someday attaining thermal control thermal nuclear fusion. There is a time integrated x ray image of a hotspot.
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Frank Graziani: At the initial, initial confinement facility so hundreds Udacity science spans this wide range of wide range of phenomenon and we have a
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Frank Graziani: Variety of facilities for for understanding and investigating this material can be anything from inertial confinement facilities like NIF National Ignition Facility.
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Frank Graziani: Or the Omega facility, it can be diamond animal cells where we are looking at materials and probing equations stayed at very high pressure gas guns postpartum machines like the Z machine. And so there's a worldwide network of these types of facilities all probing
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Frank Graziani: Matter to extreme conditions at again high pressure high temperature and high pressure
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Frank Graziani: And I'm sorry I against me. So what does high pressure high density and high temperature mean. Well, the easiest way to, to, to think about it is in terms of the density versus temperature
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Frank Graziani: Plus plot. So right here, I've, I've plotted density on the on the vertical axis and particles per sec. So that's particle particle number
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Frank Graziani: On the horizontal axis I have temperature, both in EV and degrees Kelvin and I'll mostly in this talk. Be talking about electron volts, I rarely will be talking about degrees Kelvin.
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Frank Graziani: The conversion between an electron volts and Calvin electron volts about 10,000 degrees Kelvin.
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Frank Graziani: So we have kind of our exposure to ordinary materials like metals. So relatively cold room temperature and dense the surface of the sun, which is more
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Frank Graziani: But rather diffuse you have magnetic fusion, which is very hot here. You're talking about kilovolt or 10s of kilovolt of temperature but relatively
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Frank Graziani: Relatively rarefied and then we have the, the other phenomena such as planetary interiors. So these are the courses giant planets or exoplanets where you're relatively
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Frank Graziani: Cool temperatures, maybe on the orders of 10 EV to 100 BV but very high high density or inertial confinement fusion were burn is actually happening where you're at very high temperature and very high density
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Frank Graziani: So one of the things that characterizes the phenomena. We're going to talk about today. And that's going to be what I would call high energy density physics or high energy density matter.
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Frank Graziani: It's going to be phenomena at these extreme conditions were pressures are above a mega bar and so mega bar is about an electron volts per cubic Engstrom and and the pressures in this regime.
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Frank Graziani: Mean that things like our ordinary concepts of chemistry material science plasma physics or altered.
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Frank Graziani: So this these phenomena can have any number of characteristics and this is I've listed some of them radiation, especially at the hot tub. The high temperatures above a kilovolt
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Frank Graziani: About five kilowatts radiation pressure becomes a dominant an important factor and pushing material around and you can see that I have the pressure nega bars.
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Frank Graziani: At 45.7 to the fourth routines and kilovolt so once you start getting into the five kilowatt regime or even 10 kilowatt James You're now talking about gigabytes of pressure, they're strong interactions between particles. This can happen, particularly in the, what we call it in the cooler.
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Frank Graziani: But dense regime of planetary or interiors, it's multiple species us sometimes we have to deal with electrons, protons deuterium and tritium highs aeons such as xenon and gold. It can be highly non equilibrium
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Frank Graziani: Where you have temperatures out of equilibrium have shocks running through your problem.
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Frank Graziani: You can have Fermi degeneracy, which is the case for example in planetary interiors, you can have quantum nonlocality which is different, a different aspect of quantum mechanics that people don't think about that often.
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Frank Graziani: But it's it's it's the idea that there's wave packet spreading and this is important, particularly in in
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Frank Graziani: In more hotter plasmas that you would see in inertial confinement fusion. And of course, and all the systems, you could have bound states and Tomic kinetics. So it's a very dynamic very
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Frank Graziani: non equilibrium and it can be a very non equilibrium environment.
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Frank Graziani: So in terms of looking at these these phenomena. What are some of the static and dynamic quantities of interest that you'd be you'd be interested in and looking at. And so I've just kind of
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Frank Graziani: Taken a sample. And there's many, many more that we can talk about. But of course, burn physics you know the the holy grail of trying to control from nuclear Burns is critically important.
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Frank Graziani: There's the idea of transport and coupling offices would be things like viscosity electron I and coupling species diffuse 70 of those those types of models will be
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Frank Graziani: are critically important. The equation of state, which is more of an equilibrium phenomena. But how, how easy is to compress something or how stiff, is it the equation of state is very important in these extreme
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Frank Graziani: extreme environments on stopping power that is for example in deuterium and tritium fusion. Fusion I produces 14 MTV neutron, but a three and a half me V alpha
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Frank Graziani: That plows through the plasma like a charged locomotive and heats up the rest of the plasma. So understanding stopping power tells you something about the energy and momentum transfer
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Frank Graziani: From the charged particles to the rest of the plasma and, of course, one of the things that's very interesting is mixing and turbulence. This is something we see
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Frank Graziani: In in in supernova. This is something we see in inertial confinement fusion. So it happens at all scales and
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Frank Graziani: In the past, I've always, I was talking to Carolyn, the smart in the past, I've always shown this slide.
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Frank Graziani: And I've always had kind of dead physicists as representing these different models and I decided today. Take a little different course in this show that the. There's a lot of young people.
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Frank Graziani: In a diverse group of people doing some wonderful, wonderful physics in this area. So there's a lot of different questions we'd like to know more about in these extreme environments.
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Frank Graziani: So let's go back to our kind of our faith based plot and we can kind of characterize
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Frank Graziani: The, what I would call the, you know, I've got the Jovan interiors and the inertial confinement fusion.
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Frank Graziani: Logos there. So let's talk first about the ideal plasma conditions. So these are plasmas that we refer to as weekly couple. So the plasma coupling parameter
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Frank Graziani: Is the ratio of the potential energy to the kinetic energy. So these types of plasmas individual scattering between particles relatively weak.
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Frank Graziani: The potential energy plays is a preservative in nature, it's dominated by thermodynamics. Here also the Fermi degeneracy factor.
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Frank Graziani: Is not a factor at all the electrons behave. Classically, it's a Maxwell Boltzmann distribution and particles or weekly couple
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Frank Graziani: This is the regime that typically when one takes a course at least when I did in plasma physics in school.
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Frank Graziani: This is regime, you typically studied this would be the regime of things like the aurora borealis tokamak plasmas etc etc.
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Frank Graziani: And this is regime where a lot of our theories or well developed Fokker plonk equations of last off equations last last off max Maxwell's equations, that's where these are our well well well developed
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Frank Graziani: I will also refer to this as hot, dense matter in this regime here.
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Frank Graziani: So there's another aspect of high energy density physics, which is what I would call the warm dense matter plasma conditions.
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Frank Graziani: And here the plasma coupling tends to be on the order of one or greater. So thermodynamics doesn't play as much of the effects of individual particle particle correlations, which are important.
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Frank Graziani: In addition, electrons tend to be very degenerate. So the Fermi the jack for me direct distribution plays an important role. And I've tried to capture that here.
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Frank Graziani: With these two ISO contour lines where you can see the plasma coupling parameter. So contour line of point one.
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Frank Graziani: everything to the right of that is what we would call the ideal plasma conditions everything up and to the left would be you're getting to the more non ideal.
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Frank Graziani: Or warm dense matter plasma conditions. The ISO contour line of n lambda cubicles one tells you, Atlanta. He cube is the thermometer broadly wave like that gives you a sense of when
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Frank Graziani: Particles are be behaving more Maxwell Boltzmann like versus more for me direct like and that's what I'm referring particularly to electrons so
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Frank Graziani: below that line, you have more classical like electrons. And since they have a maximum bolts Boltzmann distribution above that line, you typically have degeneracy effects from a direct distribution plays an important role. So this is a way of at least characterizing the systems.
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Frank Graziani: Now just to convince you that, in fact,
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Frank Graziani: That it's not one or the other. In fact, if you look at something like the trajectory of the gas region of an imploding inertial confinement fusion capsule.
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Frank Graziani: It actually starts its life in the warm dense matter region and evolves into the ideal plasma condition region as it's burning and fusing
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Frank Graziani: And so it's under it's very important to understand materials in both of these conditions, even for burn experiments like the the initial confinement fusion facility.
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Frank Graziani: So,
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Frank Graziani: This talk the rest of talk is going to be really focusing about this interesting part of the problem, which is the left part of that.
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Frank Graziani: That faith based plot which is again warm dense matter. And what's interesting about warm dense matters. It's really at the meeting point of several distinct physical regimes in particular, it's not condensed matter physics.
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Frank Graziani: It's, it's really, if you want to call it a high temperature condensed matter it's it's operating it
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Frank Graziani: Attentive ease or hundreds of Ev temperature scale. So the, the standard condensed matter, matter models typically don't work very well there.
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Frank Graziani: It's not really plasma physics either it's it's a cool plasma so like we were saying there's degeneracy there's strong particle strong coupling between particles.
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Frank Graziani: So it's it's really not a plasma physics as as maybe one learned it in graduate school. So this is really the challenge that well developed models. When extended to mourn dense matter really face severe problems.
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Frank Graziani: And some of the references I refer to you there, the one called the theory of quantum liquids is the book by David pines, which is
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Frank Graziani: Which is wonderful, which is, I believe, has been reissued so this was such an important area that the fusion energy sciences advisory committee report in 2009
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Frank Graziani: Called out this area of science has been something that's it's challenging intellectually very interesting and challenge, both to experimental us and theorists. And so we're going to basically talk about that more today.
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Frank Graziani: So let's let's focus on the theorists and non experimental. So the rest of this talk is really going to be about
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Frank Graziani: Quantum hydro dynamics and in the theory aspects of what we're trying to do. So let's again focus on the on the upper left hand side of this this plot, let's focus on warm dense matter.
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Frank Graziani: So let's think about what the challenges are, when one has to develop a theoretical description or theoretical model for this.
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Frank Graziani: For the phenomenon this regime and what the challenges are well there's quantum effects there can be diffraction or we call it quantum nonlocality that's not sorry for the misspelling it's not on locality.
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Frank Graziani: For me statistics bounced states. There's also classical effects are strong correlations and there's finite temperature all going on, same time
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Frank Graziani: So there's been a variety of different theoretical approaches to addressing looking at some of the
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Frank Graziani: Some of the questions that we talked about just a few moments ago. In other words, what's the equation state. What are the transport coefficients. What are some of the properties of this mass of matter in this regime.
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Frank Graziani: density functional theory of is, is the workhorse for a lot of people, both ordinary static equilibrium density functional theory, but time dependent density functional theory.
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Frank Graziani: Semi classical electrodynamics on Michael Morello and I and variety of others of us in Cimarron project we're developing molecular dynamics codes.
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Frank Graziani: To to explore this regime quantum Monte Carlo appropriately modified kinetic equations with what we call a local field corrections or another approach. But one of the most exciting from my standpoint and is is quantum hydro dynamics and
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Frank Graziani: And and why
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Frank Graziani: What's interesting about it is that in some sense quantum hydrodynamic emerges from hierarchical description of matter and you can think of it going from the micro which is a talk at a mystic
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Frank Graziani: To the mezzo which is statistical to macro which is hydrodynamic so when we think about trying to solve the electronic structure of the quarter the quantum many body problem. We certainly have the ability to solve the end body Schrodinger equation.
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Frank Graziani: That's a really hard problem to do. It's, it's, you can't deal with very large systems and it's very time intensive
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Frank Graziani: Another approaches statistical. This is the regime of kinetic theory. So the so called visionary equation, which we'll talk about in a moment.
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Frank Graziani: Which is we've gone from this n complex valued wave functions to six dimensional face base description was simplifies the problem considerably, but it's still a challenge.
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Frank Graziani: And then of course hydrodynamic and the beauty of the hydrodynamic description is that you've basically reduced your end body system to a set of five field variables so dense, the velocity or more momentum and energy so
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Frank Graziani: That the, one of the things I want to mention also, I should mention really quickly is that transitioning from kinetics to Hydra dynamics.
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Frank Graziani: Can be done via moments and we'll, we'll be doing that. So the Zero Moment of a distribution function is just the particle density
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Frank Graziani: First order moment is the velocity second order moment is related to the pressure sensor. So there's a way to go through this go from one one level to the other. I should mention, for those of you who are experts. This is
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Frank Graziani: A pictorial representation of the bbg Ky hierarchy. So there's a rigorous
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Frank Graziani: Approach to going from all three of these levels from Adam mystic to hydro dynamics. So what's the beauty to with the hydro dynamic approach is that really the hope is that it would offer
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Frank Graziani: Speed of computation, along with accuracy, while also including all the all the features that we just talked about, which is strong correlations
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Frank Graziani: Quantum diffraction or quantum nonlocality and firming the Genesee so might so the hope is that it offers a an easier way to solve the quantum anybody problem in the same way that the density functional theory has done so.
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Frank Graziani: What's old is always new again. And that's, that's true here. So with with the amazing thing about quantum Hydra dynamics, it really emerged.
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Frank Graziani: Right after the first publications by Schrodinger and and Heisenberg, it's just a remarkably fertile period for for science. So Matt along
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Frank Graziani: Came up with a description of single particle quantum mechanics in terms of just density and velocity fields and these are basic set of equations.
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Frank Graziani: Basically a conservation equation for density a conservation equation for momentum say look just like the Euler equations one knows very well.
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Frank Graziani: The other approach was a statistical approach. This was a little bit later around 1932 and this was Eugene Wagner, which developed a completely independent face based distribution approach to quantum mechanics. And we'll talk about that, about this in a moment.
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Frank Graziani: So like I said,
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Frank Graziani: What was old, old is always new again. And that's true in this case here. So these are the various people that have enlightened me over the recent couple of years and
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Frank Graziani: And certainly are have written some wonderful papers in this in this field, I refer you to a lot of their literature. So to tell you, in some sense, the excitement in this field and the interest
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Frank Graziani: First, let's talk about a little bit about the what is Hydra dynamics. And just to remind you, and I think a lot of you are in the engineering department and have seen these equations, many, many times.
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Frank Graziani: I of course going through a physics education was not exposed to these equations until I came to Lawrence Livermore National Laboratory where of course Hydra dynamics and rad height radiation hydrogen hammocks plays a big role, but the equations. Basically, in some sense,
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Frank Graziani: Are for plasma is our continuity equation of all the conservation mass conservation of momentum with involving divergence of stress energy transfer
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Frank Graziani: Grading if a pressure and the conservation of energy on the right hand side of basically who have type of collision terms. So this is a general form of hydrodynamic equations.
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Frank Graziani: And so
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Frank Graziani: In some sense, quantum hydrodynamic can be broken up into what I would call three different types of what I've called families.
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Frank Graziani: And one is the so called way function on thoughts and we're going to go through all three of these briefly.
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Frank Graziani: This is the original approach by Matt along and years later by bone which was a particular way function on thoughts as you see there plugging it back in the Schrodinger equation and getting a set of hydrodynamic equations.
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Frank Graziani: The original development was for single particle Hydra dynamics. The approached by Felix block which was I think 19
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Frank Graziani: Was shortly after his shoulders and Heisenberg's papers came out was more of a top down approach, he was he was concerned about many body systems and he his was phenomenal logical in the census.
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Frank Graziani: He said, I'm going to assume there's a set of hydrodynamic equations momentum and density and that there's some sort of
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Frank Graziani: divergence of generalized force term on the right hand side. And what that is is is TBD.
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Frank Graziani: What he did was he in his original paper took the gradient of the Thomas Fermi pressure. And so he had a one of the simplest models for a fluid model for quantum mechanics, particularly applicable to dance plasmas. And then of course there's the kinetic theory approach of Wagner and
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Frank Graziani: And we'll talk about this a little bit. A little bit more, but his kinetic equation moments of that kinetic equation.
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Frank Graziani: Can be, can you can derive a set of hydrodynamic equations. Again, involving density and velocity fields.
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Frank Graziani: And all three of these approaches involve some sort of pressure Thomas Fermi pressure quantum nonlocality and possibly correlations and exchange and exchange meaning just they have a cemetery of the way function.
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Frank Graziani: So,
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Frank Graziani: These are. This is kind of the excitement that I that I was referring to earlier. And again, the last, I would say, certainly the last decade, but even more so there's been a variety of papers coming out.
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Frank Graziani: And I would refer you to any of these papers to get kind of a general background on the on the field. Everything from the Madeline approach.
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Frank Graziani: To the block approach, including the book by Fernando Haas on quantum plasmas. And there's been enough international interest that there's a workshop on quantum hydrating hammocks and Strasburg that was originally planned for September, but it's been rescheduled for for next year.
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Frank Graziani: So let's spend a few minutes on on each of these methods so Madeline's approach was fairly direct which is he was
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Frank Graziani: He took the single particle and again emphasize single particle shorter equation.
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Frank Graziani: He used on sites where he had a an amplitude that was dependent on position and time and then a face factor or for those of you versed in class mechanics, it's it's similar to an action.
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Frank Graziani: So when he did that and you stick that you make the hotspots and put that into the shorter equation, you get a setup to couple of equations.
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Frank Graziani: And if you identify the momentum with the gradient of the action or the gradient of the face factor and the density with the amplitude squared of the
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Frank Graziani: Of the way function. You get a set of hydrodynamic equations. And that's what scene to the right. And so you get your conservation of
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Frank Graziani: Density and you get your constant conservation of momentum. What's interesting about this equation is that you won't see anything like a necessarily pressure term.
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Frank Graziani: You get something like a gradient of a potential, which is the external potential is the force term.
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Frank Graziani: But what's very interesting is you can have the quantum mechanics in some senses completely localized in the so called bone potential or quantum diffraction.
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Frank Graziani: And this is this is reflective of the quantum nonlocality. So remember, we haven't talked anything about quantum degeneracy. This is for single wave functions. So, this is this this pressure term is coming from the fact that the wave packet spreads.
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Frank Graziani: It has a width to. It's not a delta function and so
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Frank Graziani: word of warning that should not be extended to many particle systems like plasma. So I'm going to talk about that in a moment. So what about Wagner, so they are defined what what basically as a free transform of a
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Frank Graziani: Split density matrix or density opera here size star sigh, and if you apply the time derivative to both sides of that equation and you use shorter. Here's the equation.
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Frank Graziani: Y la you get something that looks very similar to a kinetic equation using plasma physics, although something very different about this. It's non local
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Frank Graziani: So you have this integral that tells you something about that that reflects the fact that quantum mechanics is a non local theory that in fact wave packets spread
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Frank Graziani: Now, interesting enough, if you do an expansion of this term on the right hand side in h bar.
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Frank Graziani: You in fact get something that looks just like a blast off like equation with the forcing term and infection term.
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Frank Graziani: And one of the things I want to point out about this particular function is it's not strictly a probability distribution function, you can interpret it that way, it in fact can be negative.
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Frank Graziani: And that negative is a reflection of any quantum diffraction or quantum not nonlocality features in the in the system. So again, this is single particle. Now, interesting enough, you can actually apply
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Frank Graziani: Moments to this to this equation, just like you do classically if you've taken a course in plasma physics or statistical physics.
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Frank Graziani: You will know that if I take moments of the distribution function, I can get things like a density a momentum and a pressure, pressure sensor. So you do you do that. You get a set of equations here and it looks similar. But the question you might ask is, where's the bone potential
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Frank Graziani: What we saw it. We just saw it. So where does it disappear.
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Frank Graziani: Now one thing I do want to say that the advantage of this type of an approach is that it will yield an energy equation. Now, you might have asked
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Frank Graziani: You know, in the modeling approach. Whereas if I want to do an energy equation. And where is it the beauty of the bigger approaches you can take moments higher order moments and you can get those moments.
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Frank Graziani: Now there is a closure problem, which is one. The lower order moments are coupled to the higher order moments, but you can't get it.
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Frank Graziani: So this was a quandary for me. And then you can do a very simple exercise which is if you just take the on salts that Madeline gives you show that in the upper left you take you substitute that into the bigger the function
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Frank Graziani: And then you evaluate the stress tensor you in fact find that the divergence of that stress tensor is nothing more than the gradient of the bump potential
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Frank Graziani: So the bump potential is there. It's just hidden and more insidious way. But what's nice about the finger approach is that it generalized is the concept of a quantum potential
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Frank Graziani: nonlocality that reflects the physics of nonlocality it generalized. Is it because I didn't have to make this onslaughts. This is an onsite. This is a very specific choice for the way function. So that's something that I think is very interesting.
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Frank Graziani: So we've explored hydrodynamic descriptions of single particle quantum mechanics. But what about many body systems. So if you look at the equations I had just arrived. There's no equation of state.
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Frank Graziani: There's no particle particle action interactions and there's no quantum statistics. And where the heck did that go. So what do you have to do.
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Frank Graziani: So there's been a number of papers out in the last decade, talking about this some more recent than others about the formulation of quantum hydro dynamics for for many body systems.
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Frank Graziani: And here I'm talking about something more what I would call bottoms up fundamental derivation. The, the block approach, of course, is what I would call a top down approach where you you have you basically
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Frank Graziani: Positive posit a set of hydrodynamic equations and say, what is the functional on the right hand side will go into that a little bit more detail.
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Frank Graziani: So a student of mine, David Victor did something very similar to the Madeline and what he did was, he said, I'm going to find a manatee body wave function, but I'm going to properly anti Semite tries it
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Frank Graziani: Because he's, he was interested in electrons and for me on systems. When you do that, you get a generalization of the hydrodynamic equations that Madeline derived
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Frank Graziani: In fact, you see those on the, on the right hand side. And in fact, because of this anti Semite translation and the many body nature of the way function, you get a Thomas for me pressure which is characteristic of
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A
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Frank Graziani: degenerate matter, you get the bomb potential appearing and this is characteristic again of quantum nonlocality you get
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Frank Graziani: Dirac you get the exchange term, which was the anti Semitism. The way function, but you also get Columba interactions particle particle interactions of the is just some sort of external potential that can be zero.
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Frank Graziani: So you can get it out of the Madeline approach if you're very careful about what you do.
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Frank Graziani: As I've been saying that block in 1933 and then Yang in 1974 took advantage of particularly Yang and 1974 took advantage of all the developments destiny functional theory, the big advantage there was
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Frank Graziani: What Yang and blocked it much earlier was they said, I'm going to pause it again, that there's a set of hydrodynamic equations.
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Frank Graziani: And there's going to be a divergence of generalized force and this generalized force is going to be chosen to, uh, to involve the gradient of the variation of the free energy functional
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Frank Graziani: The advantage of this type of approach is that in equilibrium, you basically retain all of the properties of density functional theory, meaning
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Frank Graziani: All the infrastructure that goes into determining the in the electronic ground state properties are Barry in Omega are buried in that functional
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Frank Graziani: And this has been in some sense the workhorse for a lot of people recently.
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Frank Graziani: Because it avoids the idea of way function. That was the idea of taking moments of things that you basically just say, Okay, I'm going to assume I have a set of equations that are hydrodynamic in nature.
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Frank Graziani: I'm going to utilize all the infrastructure from from density functional theory and and that's going to be my set of hydrodynamic equations quantum hydrodynamic equations for the quantum anybody problem. And this has been used with with great success recently.
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Frank Graziani: Michael and his student at do have taken this type of approach, adding in this has forced terms.
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Frank Graziani: To calculate things like the dynamic structure factor of aluminum at warm dense matter conditions. It's also been done by
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Frank Graziani: By Dirk Eric key and
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Frank Graziani: And Gianluca Grigori and larger.
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Frank Graziani: Different approach. But again, calculating the ionic dynamic structure factor. So these types of techniques are finding a lot of us.
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Frank Graziani: David in his thesis. Second part of his thesis, I should say did use the block to HD and what he did was he used block Q HD to calculate the electronic structure and used ions to
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Frank Graziani: Model the ions with molecular dynamics and he looked at stopping power in warm dense matter. And this is an example of stopping power and warm dense matter.
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Frank Graziani: Here is the proton stopping with a very simple equation state these type of electron fluid fluctuations. The bright spots you see there in the middle plot are
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Frank Graziani: Ions that fast one moving from the lower left to the upper right is the the fast moving charged particle and that clouds of stuff represent density of the electron.
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Frank Graziani: Fluid. If you want to think of it that way. You also looked at what happens when you take a Thomas Fermi equation to state and looked at stopping power and you get these very interesting
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Frank Graziani: Wakes in front of the charged particle
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Frank Graziani: So,
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Frank Graziani: How do we extend the previous approaches to higher order moments. So we've just remember, think about this. Again, we've only talked about density and we've only talked about
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Frank Graziani: Momentum, we have not talked about energy or temperature. So when we talk about worrying about the thermal energy in the plasma, where's that going to come from. So the question is, can we extend Madeline on, block, block.
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Frank Graziani: In my opinion, the many, you know, I may be in the minority but many body vigor offers hope and so am I. And I won't go through this because it's a it's a whole other talk
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Frank Graziani: But I think the marriage of the Wagner formalism was second quantization where you extend the Wagner kinetic equation too many body theory too many body physics.
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Frank Graziani: And we've done that Michael Joe Bauer. And I did that in the paper called kinetic molecular dynamics for hot, dense matter a lot of that physics was laid out in the wonderful book called statistical physics by commodity, which
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Frank Graziani: It's also talked about a little bit in quantum kinetic theory by Michael bonnets, but this idea of marrying the visionary equation with second quantization produces a second
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Frank Graziani: Set of hydrodynamic equations that includes quantum nonlocality particle correlations and degeneracy. So that is something for another day. Maybe next year, but this is ongoing research. So that's something that we should think about
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Frank Graziani: So I would say that a another type of approach we're exploring to study the many body system is kind of taking a step back from Q HD and saying,
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Frank Graziani: Let's treat the electrons with kinetic theory. So the middle part of that hierarchy and the islands with molecular dynamics.
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Frank Graziani: And so this is something that Matthew link because doing. He's a graduate student working with myself. Andrew Chris leap and Michael Cirillo, and I do have to say I grabbed that picture of Andrew Chris leap, because that's the only time I've seen him and shirt and a tie.
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Frank Graziani: But that's, I do. I did have to point that out. So, Matthew is doing some wonderful work looking at stopping power in
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Frank Graziani: In one day in class in dense plasmas using this type of approach again what we call kinetic theory, MD.
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Frank Graziani: Kinetic theory, ie Wagner for electrons ions been treated with molecular dynamics. And this is one of the new new results. He has looking at the to stream instability.
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Frank Graziani: Of last off versus Wagner and you can see one of the things that comes out of viggers the softening of the of the very sharp features in the to strip instability problem. And this is related to the fact that
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Frank Graziani: It's it's a reflection of the uncertainty principle or the wave packet spreading softening the these sharp gradients.
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Frank Graziani: So let's get to the last 15 minutes of this talk, and let's talk about hydrodynamic instabilities and for any of you involved in the HDD community. And any of you involved in the pulse power or
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Frank Graziani: I CF communities hydrodynamic instabilities are something, one has to live with. It's one of the principal reasons why National Ignition Facility is not acquired ignition yet is the the instability is growing and
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Frank Graziani: Destroying the performance of the capsule. So in general, there's kind of three different approach or three different categories. One is the really tailor instability. A second has mesh cough. The third is Calvin Helmholtz
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Frank Graziani: Really tailor is probably the most well known amongst all of you. This is for example a heavy fluid on light fluid with an acceleration constant acceleration going downward rectifier mesh cough is
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Frank Graziani: A Shockwave passing through an interface between light fluid and heavy flood Kelvin Helmholtz something you see
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Frank Graziani: fairly frequently and Glock cloud layers, for example, and this is whereas velocity gradient between the interfaces of to materials and creates those type of roll ups the bubble structures that you see below so
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Frank Graziani: The interesting, the one of the questions I had is, what would quantum hydrodynamic say about something like this. So one of the things we're trying to do is we're trying to model really tailor instabilities
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Frank Graziani: At hydrodynamic instabilities in general at at at Livermore, and the other national laboratories
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Frank Graziani: And I thought, well, let's, let's see what this means for something like quantum hydrodynamic system. And we're going to talk about that now.
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Frank Graziani: So we're going to consider a stratified heterogeneous electronic fluid and so that that initial condition will be will be heterogeneous in the z direction will be homogeneous in the x y direction will have a downward constant acceleration g
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Frank Graziani: That it can be anything you'd like, but the gist of the initial distribution of of pressure and and density can be. I'm sorry of density and momentum can be anything that you'd like
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Frank Graziani: So we're going to assume that our quantum hydrodynamic equations are the following. We're going to assume a neutralizing background a proton. So the protons are going to be some in some sets of a fixed framework.
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Frank Graziani: And the electron is going to be a fluid moving in this fixed framework we're going to assume as simple Thomas for a refresher, which is appropriate for degenerate electrons at its temperature
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Frank Graziani: WE'RE GOING TO HAVE A HEART. RETURN meaning and the Posada equation. So particles are going to be interacting through a simple us on equation we're going to assume for a simplicity of analysis that it's incompressible. I'm not justifying that physically at all here. I'm doing it mostly
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Frank Graziani: As a comparison with existing literature, particularly Chandra say cars book and the books by Paul Drake.
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Frank Graziani: And so that's something that needs to be investigated further, but that's something. So this is the set of equations. We're going to investigate.
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Frank Graziani: One of the things I want to point out to you is that at least for uniform density. The bomb potential vanishes. So I want to point that out to him.
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Frank Graziani: Okay, so that's the set of equations that we're solving deaths, the momentum Hassan and the definite and there's the definition of the so called bone potential
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Frank Graziani: Okay. And I want to refer you to these three excellent books of the one by Paul, I don't know if Paul is online, but these three books to my Bibles through doing this type of this work.
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Frank Graziani: I will say that a Chandra say cars book was the first one I was exposed to as a young student and I don't think I appreciate it, then like I do now, but it's a it's a treasure treasure trove.
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Frank Graziani: So the first thing we're going to do is we're going to perturb and initially static state. So we're going to have the stratified heterogeneous electronic fluid.
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Frank Graziani: And we're just going to perturb it so. Okay, so we'll assume that the initial velocity zero and the z direction. But there's some sort of heterogeneity and all the other quantities and delta you delta and etc etc represents small perturbations to each of those physical quantities.
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Frank Graziani: So you've got a set of linear equations. So we'll assume that these these perturbations or week so we can linear eyes.
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Frank Graziani: The equations. And this is if you if you know anything about linear response or driving dispersion relations. This is in the same same subject area. So you basically get a porter a perturbation for a time evolution equation for the number density and the the momentum.
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Frank Graziani: Perturbation equations. Alright, so we're going to take those alright so what do we do with those
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Frank Graziani: So like any good scientists are applied mathematician, we're going to apply a flat or normal mode analysis to those equations.
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Frank Graziani: And we're going to look at the growth rates particularly omega omega being imaginary, for example, will tell you that in fact that linear response is bounded
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Frank Graziani: And it's stable. If omega is positive.
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Frank Graziani: And and real than we know in fact that that the response is unstable. Alright, so we're gonna stick those those qualities in, and after a bit of algebra.
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Frank Graziani: We got a set of we are able to reduce those equations to a couple set of velocity and potential perturbation equations and so alpha z represents the perturbation of loss in z direction.
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Frank Graziani: Gamma represents the perturbation in the posts on the the potential. So one of the things I want to point out to you is the terms that are zero.
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Frank Graziani: If Pusan is not present, or zero. If the bomb term is zero. So if you if you take those terms that I've bracketed and read and let them be zero. That's a, that's the problem that is solved in Chandrasekhar his book, so he calculates the growth rate for this type of stratified configuration.
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Frank Graziani: Well, what's next.
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Frank Graziani: So he took those two equations. He applied them to two different problems. One was the heavy on light which is to uniform fluids.
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Frank Graziani: But once heavier than the other and calculates the growth rate and this is he calculates the well known fact that the growth rate goes as a square root of the wave number
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Frank Graziani: With the so called that would number and this is also beautifully explained in Paul's book also the other
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Frank Graziani: Problem that he talks about is an exponentially varying density. And this is where the initial density
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Frank Graziani: Goes as some initial value eat to the BC, so be can be either negative or positive, so it can be
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Frank Graziani: More rarefied as you're going in the positive direction and become more more dense and you don't in the positive direction. So we're going to study this example. And I'll show you the results of what the growth rate looks like for this exponentially stratified medium.
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Frank Graziani: So when you take that the latter approach, then you get a set of equations. Now what's interesting about this is, let's look at the velocity equation.
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Frank Graziani: That is exactly the equation that Chandrasekhar derived if
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Frank Graziani: gamma zero N FP is one. So let's take gamma equal to zero know plus on know particle particle interactions cool vomit particle particle interactions.
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Frank Graziani: You in fact see that there's this new term called which is p FK squared omega squared that modifies the growth rate and this is purely
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Frank Graziani: Due to the quantum effects due to the bone term and this is we've been able to there by isolate the effect of the quantum terms on the growth of the perturbations.
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Frank Graziani: Now if you throw put something in there, then the it becomes more complicated and I'm in this now of solving this system numerically. I have not been able to solve it analytically, other than the
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Frank Graziani: Other than when the term is not present, but these are the set of equations, one would would solve. So what do you get
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Frank Graziani: So you can you can drive a growth rate equation that is
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Frank Graziani: Is shown there, which is the growth rate omega squared is equal to the acceleration times b and member B is inverse length.
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Frank Graziani: And it is a, it tells you something about the variation of the heterogeneity of the exponential configuration.
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Frank Graziani: You get a stabilizing quantum term. Notice the time the sign change it's minus k squared. It's not plus k squared. So the growth rate is stabilized. So if we look at be less than zero, which is
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Frank Graziani: Exponential configuration where you're starting dance at the at the ground level and becoming more and more rarefied your growth rate is imaginary for all modes. So it really doesn't matter.
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Frank Graziani: Okay, I want you to also realize that if h bar square to zero. This is the problem that's in Chandra safeguards book. This is the exactly the answer he gets
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Frank Graziani: It. He sparred squared is not equal to zero but and b is greater than zero, so be now is is stratified where you have a heavy layer on top of lighter layers going down and down further down to the ground. You get of course the plotting omega squared vs. K. Oh, shoot.
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Mark Kushner: Well, I think we've lost Frank just momentarily. So let's give them a few minutes to reconnect.
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Easy on them. This semester.
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Frank Graziani: My apologies, everybody. This is the. Can everybody hear me.
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Mark Kushner: Yes, we can hear you.
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Frank Graziani: All right, thank you for your patience.
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Frank Graziani: We're I'm almost at the end I lost internet connection, so that's that's what happened. That's why I didn't, I didn't drop dead or anything so
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Frank Graziani: What I wanted to point out here was that if you look at the the growth rate as a function of the wave number
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Frank Graziani: You have the classically always unstable configuration. And this is what you normally would expect, of course, I mean, you have the heavier fluid on top and it with this exponential stratification to lighter fluid.
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Frank Graziani: But the bone pressure term adds a stabilizing term. And you can see that here in this in this growth rate as a function of wave number
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Frank Graziani: And in fact, it forces at some high enough wave number two to turn over and become negative. So if you look at omega squared being positive. So that's basically the the
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Frank Graziani: Unstable modes and negative. That's the stable most where you'd expect the growth rate to be imaginary. So one of the things you can ask yourself. Now ask yourself is, well, how realistic is this and I in this and take this with a grain of salt.
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Frank Graziani: Because we've left a lot of physics out, but at least I think this gives you some sense of what the at least physically qualitatively what the bone pressure term would be so if we put plot the log of the ratio of of
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Frank Graziani: The growth rate due to just the quantum term to just the classical growth rate as derived by chance to say car. And I've written those out there. I just put some numbers together. And so if we have a
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Frank Graziani: Configuration that's maybe one centimeter and height, the exponential variation goes the factor there goes 10 seven centimeter inverse
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Frank Graziani: And then you can pick various accelerations. So these accelerations 10 minus one microns per second squared to 10 minus seven to 10. These are the 10 minus one is
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Frank Graziani: Typically at the low end of the accelerations they observe on really tailor experiments that they that they do for the Omega facility, but this at least gives you a sense of where the crossover point would be and what length scales, you might you might see this
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Frank Graziani: And for example, at 10 minus one microns per nanosecond squared there, you're going to have a wavelengths, on the order of 10 minus six centimeters or 100th of a micron.
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Frank Graziani: But then, of course, the, the, the chances of observing it get better as you go to lower and lower accelerations as principally because
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Frank Graziani: You're not giving the classical acceleration time enough to develop. So it's dominated by the quantum term.
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Frank Graziani: So again, I said, take this with a grain of salt because there's other physics that could dominate and not play a big role here so has viscosity electric fields could could play a big role.
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Frank Graziani: We haven't included ions. This is just for the electron part but from my standpoint is still useful as a verification test problem.
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Frank Graziani: So anybody who's got a quantum hydro code that wants to put together a configuration with the long term unstable fluid configuration can can use this as a verification test problem at the very least.
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Frank Graziani: So as I said, we, as I told you, just a few minutes ago. I've been unsuccessful in finding analytics solution to that growth rate with the song term added because that electric fields could play a big role and dominate over the long term. So I'm solving that
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Frank Graziani: Numerically, so it's basically it's time to wrap this up. And so what I'd like to do is talk to you a little bit on this last slide.
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Frank Graziani: And again, thank you for sticking around. I know it's late there. And thank you for sticking around. And thank you for your patience. I'm in my idea in my mind. There's many areas and hit cutie that need to be researched a
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Frank Graziani: Bigger seems to from my from my standpoint, the only approach that allows us to extend it beyond both dancing velocity. So for interested in finite temperatures
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Frank Graziani: In warm dense matter if that plays a role. We need to extend these equations and Michael I talked about this many years ago with student, David.
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Frank Graziani: Can kind of bigger approach for Q he produced a closed set of equations for moments is the many body Madeline, the only option.
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Frank Graziani: And so there's an interesting paper by appeared on where they close. They use the beginner approach and close it by doing something like a minimization of the entropy
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Frank Graziani: The second colonization method.
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Frank Graziani: is under development, we've been able to sit down and write down what the operator form of the hydro equations are and that's what you do get you guys set up to HD equations.
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Frank Graziani: That are set of operator like equations for dense the momentum and energy. So that's something all TBD. And it could be that one sticks with the block approach and or the mental approach and is able to
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Frank Graziani: Do more there. So there's a variety of things that one should could do with ghd one is dispersion relations.
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Frank Graziani: Looking at the Q HH t plus, plus, plus on system, looking at the hydrodynamic properties of the of the system with dispersion relations. One of the things that we're doing right now I'm doing with Chris scholars is adding QH key capabilities to Miranda so Miranda is actually a multi species.
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Frank Graziani: rad hydro code. It's all varying with adaptive mesh refinement. But if you look at the especially the block equate the block approach to Q HD
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Frank Graziani: It's something where you could take an existing hydrodynamic code and put in things like the bone terms and some n plus on terms and and you basically have something like a Q HD equation.
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Frank Graziani: Now what's nice about Miranda, you can treat the electrons. So, like that. And then the ions can be treated as a typical classical fluid. So that's something that I think would be something that we're doing that could be very interesting.
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Frank Graziani: One of the things I left open is how relevant is the bomb potential chairman instabilities. I think that's there's, there's a lot of things that have to be done before one can take that realistically
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Frank Graziani: Again, I'm looking at the dynamic looking at the analytic growth rates. I'm sorry, looking numerically at the growth rates and the RT instability and some of the things that you can explore with Q HD when you have this capability can do things like
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Frank Graziani: Dirt Carrick he did and Michael and others with the dynamic structure factor stopping power.
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Frank Graziani: Is another thing you can look at shocks and warm dense matter and like I said hydrodynamic instability. So there's a lot of very interesting things, one can do
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Frank Graziani: One could do here. So we are done. Thank you for your patience and waiting around as I scrambled to get the internet working again and I think my many friends and colleagues at Michigan. Michigan State environment today. So thank you very much.
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Mark Kushner: Thank you very much, Frank. Are there any questions and feel free to unmute yourself to ask your questions or you can type them into the chat box and we can go from there.
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Alexander Thomas: Okay. Well, I guess I'll start off
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Alexander Thomas: Alec toss
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Frank Graziani: So shy, Alex.
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Alexander Thomas: Thank you. That's a
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Alexander Thomas: Very nice and inspiring inspiring talk. Actually, I am intrigued by this I'm it's trying to wrap my head around one of the concepts you talking about, though, you
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Alexander Thomas: So it's this idea of including not quantum nonlocality
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Alexander Thomas: The thing I'm trying to get my head wrapped around is that hydrodynamic theories. It's a isn't it local. By definition, so how how is it. How's the quality included
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Frank Graziani: Yeah, I think one has to think of it in terms of length scales. So this is nonlocality in the sense of the width of the way function. So, one has to think of things like thermal the wavelength has been the relevant scale there. So even though
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Frank Graziani: It's non local in that sense in a hydrodynamic sense, I think it's local. So I think one has to think of it that way.
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Frank Graziani: And so
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Frank Graziani: So, but I think that's a, that's a really good point.
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Alexander Thomas: So in a sense, it's sort of
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Alexander Thomas: Non
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Alexander Thomas: Because I you know I'm sort of familiar with normal character quality in the in the in the classical kinetic sense right
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Alexander Thomas: Which can which you can include by some sort of Colonel methods but but here. Every, everything here is defined locally in terms of just
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Alexander Thomas: A local position but and local gradients. That's, that's what I'm trying to
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Frank Graziani: Oh, I see what you're saying.
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Frank Graziani: Yeah, yep, yep, yep, yep, yep, yep.
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Frank Graziani: Yeah, so the, so yeah, so that's that's a very good question, which is basically if we're talking about quantum not nonlocality how's it appearing in the in the queue HD equations which are set of
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Frank Graziani: Equations basically for so you know velocity dancing and energy and some gradients. So where's the nonlocality come in there.
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Frank Graziani: It comes in. Typically, it comes in, it's represented by the bone potential term so that Boehm term that that pressure term that's proportional h r squared is is key in some sense.
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Frank Graziani: Localized is all the effects of the quantum nonlocality in that term, it introduces a
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Frank Graziani: If you want to think about it, it's almost an artificial. I hate to use where our official but it introduces additional pressure term into the equations.
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Frank Graziani: which reflects the fact that the wave packet. You can't, it's, it's not going to be a delta function. So I can be a classical particle, you can't squeeze it down to a delta function, so it's
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Frank Graziani: Effectively in the hydra equations treat is that there's a pressure pushing that way function pushing that way functionality.
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Alexander Thomas: Thank you.
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Frank Graziani: And there's a wonderful book. But why last name is why it's called. I think it's called
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Frank Graziani: Quantum trajectory method, I think, Springer published it is called an introduction of quantum hydro dynamics that goes into detail, particularly first single particle quantum mechanics, but I can send that to you.
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Carolyn Christine Kuranz: Thanks, Frank. This is Carolyn.
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Carolyn Christine Kuranz: That was a really nice talk. Hi. I just had a question. So I'm going to do that thing where I asked about how your work actually applies to my work.
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Carolyn Christine Kuranz: In my in kind of my thought about this, I would think so, you know, I do a lot of
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Carolyn Christine Kuranz: Instability work in a GED, and I would think, in order to get into the
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Carolyn Christine Kuranz: The quantum regime naively, I would think that you have to increase. There's a pressure increase or or subsequent density increase. Do you know how kind of what sort of
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Carolyn Christine Kuranz: Or is there some sort of dimension, this number, perhaps analogous to the plasma beta for magnetic pressure where you would say, Okay, you actually need to include these quantum effects and those
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Frank Graziani: Yeah, there's
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Frank Graziani: There's a slide I yeah I'm sorry I didn't even
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Carolyn Christine Kuranz: Know, and I also I had connectivity issues as well. So I might have missed it. If you did.
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Frank Graziani: Yes, I answered your question.
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Frank Graziani: Completely
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Carolyn Christine Kuranz: And then detail here. Yes.
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Frank Graziani: There is, and I took the slide out but that you can if you take the, the very simple form of the queue HD equations. The one that we
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Frank Graziani: showed you in the last part of this slide in the RT instability part you can actually applies, you can scale.
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Frank Graziani: All the dimension full parameters. So you can scale things like the dense. The the
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Frank Graziani: Position. So the position can be scaled by the Fermi length.
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Frank Graziani: You can scale time by the plasma frequency, etc, etc. So you can you can write down a set of dimension euless hydrodynamic equations with the dimensional is
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Frank Graziani: Posada equation and then there's one term that you can play with and that term involves the bone potential
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Frank Graziani: Which is the thing we're interested in, and there's this parameter outside of it which is called the brookner parameter which is the ratio of the inner particle distance to the border radius. And that thing is dependent on the density
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Frank Graziani: And that Bruckner parameter will tell you it's proportional to the plasma coupling parameter, where the plasma current
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Frank Graziani: Plasma coupling parameter is evaluated the Fermi energy. So, this the Brooklyn parameters in some sense the analog of the usual plasma coupling parameter that we know and love. But for called degenerate matter.
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Frank Graziani: And so one dense matter. And so I think that parameter, they're looking at dimension plus form of the equations tells you, I think would give insight into kind of what experimental conditions, one would need to see something like this and
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Frank Graziani: So I I I I hesitate to go too far down that
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Frank Graziani: Avenue until I've determine what's the importance of the electric fields in this problem and
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Frank Graziani: And
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Frank Graziani: And and anything else like viscosity. So I'd have to compare things like What are this constructor surface tension effects, but I think my ultimate goal would be if we could design an experiment that we could shoot an LLC or Neff, or i j laugh or Z or any to see something like this.
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Carolyn Christine Kuranz: Yeah. I think that'd be really interesting. If you could do essentially do you know our regime where it was important that wasn't IMPORTANT TO SEE THE DIFFERENCE. BECAUSE IT'S JUST if it's a stabilizing effect. Sorry. Now I'm outside
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Carolyn Christine Kuranz: But yeah, thank you, Frank. That's really sure.
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Mark Kushner: Any other questions.
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Mark Kushner: I have is
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Mark Kushner: This is mark of a strictly to the procedural question. All right, from an implementation standpoint parallelization. Are there any unique considerations here beyond what you would do on a standard plasma hydro code.
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Mark Kushner: As you may have some kernels, you need to go off and computer turn at top
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Frank Graziani: Yeah, I mean at the at the
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Frank Graziani: At the simplest level mark. It's, it's, there's very little one has to do in terms of existing hydro code. I mean, you would kind of zero level what you have to do is put in the proper equation state. So let's say is Thomas for me with a direct exchange term. So you put that in.
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Frank Graziani: You put in a bump potential term which is a function of density. It's a, it's a, it's a, it's a, you know, a funny term. It's a gradient of a divergence of this greater density over squared density. So it could cause a problem where we're sharks are in terms of numerical
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Frank Graziani: Issues.
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Frank Graziani: And that's, that's, and then you'd have to add in the person term. I don't know if I mentioned that, too. So I think those three factors that you put in this
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Frank Graziani: Equation state appropriate for degenerate terms you put in the long term, and then you put in a persona term. And so, Miranda is equipped to handle that it's multi species.
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Frank Graziani: So we were going to do electrons enhance separately. So the ions would be treated as a classical fluid, the electron fluid would have these additional terms.
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Frank Graziani: The electrons and eyes would interact through the posts on term and that I think was one of the things we were going to use to explore some of the characteristics of this kind of a system.
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Frank Graziani: And that's in fact why I was looking at, you know, this type of a problem to give us some insight about whether or not we're doing the physics. Right.
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Frank Graziani: If it's implemented correctly. That's why I was saying, it could be an interesting verification test problem, but from a parallel station standpoint it's it's it fits in very nicely to the existing Miranda code which is already massively parallel
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Mark Kushner: Is it with the post on term which you know sometimes doesn't appear and plasma hydro
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Right.
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Mark Kushner: Do you end up having to do something implicit or 70 implicit
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Frank Graziani: Yeah, that's it. That's a good question. In fact, I didn't realize this would be as big of a challenge to some of the to them as to the Miranda group, but they had just put in a
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Frank Graziani: Pause on solver and the biggest issue there was not the software itself, but the boundary conditions.
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Frank Graziani: Which I believe they put in. I don't know if it was implicit or explicit but i i believe it was a
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Frank Graziani: Well, it doesn't depend on time. But I think the biggest issue that they had to fix was the boundary conditions being consistent with the type of problem. I want to run
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Frank Graziani: I think that was the biggest session, but they, it was not part of the package that was routinely run. I just as you were saying it's not something that's typically typically included
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Frank Graziani: In that, and that's actually a good question. I mean, maybe I'd like to know from other people.
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Frank Graziani: Why, if it's not why it's typically not maybe others are more knowledgeable than I would could say something about why it's not. But certainly, if you look at something like the present ski equations which
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Frank Graziani: After all, it just moments of the kinetic equation valid for hydro, everything is their
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Frank Graziani: Electric fields. If you have magnetic fields. It's there to sources and the collision terms.
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Mark Kushner: Anything. Thank you.
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Sherman
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Mark Kushner: Anything. Any last questions.
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Mark Kushner: Well, if not pray. Thank you very much for this excellent seminar and for finding your way back through
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Mark Kushner: The internet
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Frank Graziani: Thank you.
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Frank Graziani: Thank you, Mark.
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Mark Kushner: Thank you. And I declare the seminar done for today and thank you audience for holding on to the end of the seminar.
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Frank Graziani: Yes. Thank you. God bless you.
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Frank Graziani: Thank you. Bye bye.
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Liam Stanton: Thanks a lot.
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Mark Kushner: Thank you.