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Todd Randall Allen: Alright, so welcome everyone to the Friday afternoon nuclear engineering colloquium, so this is kind of a special event for a number of reasons, so we're holding our.
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Todd Randall Allen: Annual graduate student recruiting weekend, we had to do it virtually this this year, unfortunately, but we've invited our visiting students to listen to the colloquium so that's special number one, so thank you, thank you all for joining us.
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Todd Randall Allen: To this is the first chance for our newest Professor Professor Scott ballroom who recently joined us from the University of iowa.
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Todd Randall Allen: To give a talk so will be the hearing about his work, for the first time since he joined us as in as a faculty Member so we're really excited about that, and third, we just we're co sponsoring this with the Michigan Institute for plasma and engineering.
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Todd Randall Allen: And so, hopefully we'll have a bunch of visitors listening from gypsy also and I understand that there are certain traditions associated with a mitzi meeting, and then I should turn it over to mark kushner take care of that those mitzi like ceremonies before we get started.
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Todd Randall Allen: it's all yours.
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Mark Kushner: Thank Thank you very much it's a hearty welcome to start as a new member of the Michigan and a member of mitzi a member of nurse we're all extremely happy that you have decided to join us and to give this joint mitzi nurse.
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Mark Kushner: colloquium now had we been in person, there would have been the mitzi mug ceremony.
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Mark Kushner: So we will do it virtually Scott ED.
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Mark Kushner: and recognition of your giving the joint symposium, we are pleased to present you with the Nazi mug.
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Mark Kushner: just arrived.
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Scott David Baalrud: Thank you.
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Scott David Baalrud: Does it arrived by Amazon.
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Mark Kushner: Thank you very much.
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Scott David Baalrud: Okay, thank you so ready to get started.
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Todd Randall Allen: yeah it's all yours.
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Scott David Baalrud: Well, thank you all for the Nice warm welcome and welcome to all the prospective students.
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Scott David Baalrud: There we go is the the slides coming up okay.
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Todd Randall Allen: yep let me see i'm fine thanks.
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Scott David Baalrud: yeah so I had the privilege to talk to several students, today I hope.
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Scott David Baalrud: Your interactions with everybody went really well.
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Scott David Baalrud: it's unfortunate that we can't be in person, but we're I think we're doing the best we can.
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Scott David Baalrud: So the title of my talk today is is this even a plasma the physics of strongly couple plasmids So when I talk about this topic of strongly couple plasmas i'm frequently asked.
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Scott David Baalrud: Particularly by colleagues who've written textbooks about plasmas well is this even a plasma so we're just going to hit this one head on and ask the question and talk it through.
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Scott David Baalrud: let's see so the way we're usually introduced to plasma is a state of matter goes something like this if you start with something that sufficiently cold in depth.
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Scott David Baalrud: Like us, it's solid like ice.
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Scott David Baalrud: like an iceberg, and if we add heat to the ice, eventually, it will melt into water, which is a liquid.
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Scott David Baalrud: If we keep adding heat eventually the liquid will evaporate into a vapor or a gas, and if we keep adding heat or some form of energy source.
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Scott David Baalrud: Eventually, some of the electrons will be stripped out of the atoms in the gas, and you will have a highly electrically conductive gas, which we call the plasma.
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Scott David Baalrud: And so, in that sort of way to describe different States of matter, we can quantify the different States of matter by a basic properties.
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Scott David Baalrud: So one of the basic properties, we usually use to classify states of matter is whether it takes the shape of its container Okay, so a solid won't take take the shape of its container but liquids gases do.
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Scott David Baalrud: Another property, we might ask is if it fills the volume of the container that it's in so solid or liquid would not fill the volume of the container whereas a gas would so these can be distinct these properties can distinguish solid liquid and gaseous states of matter.
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Scott David Baalrud: Another question, we might ask is if it's substantially changes its volume when compressed if it's a highly compressible so we'd expect solids and liquids are usually not highly compressible whereas gases are.
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Scott David Baalrud: And in the conventional description of a plasma like on the previous diagram.
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Scott David Baalrud: We expect plasmas have a lot of the properties of gases, so they take the shape of their container fill their volume and their compressible but they have the additional property of being highly electrically conducting so that's what distinguishes them from a gas.
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Scott David Baalrud: solids liquids may or may not be.
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Scott David Baalrud: So we're getting a little bit of talk if somebody can mute.
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Scott David Baalrud: Okay, so to contrast with it, these.
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Scott David Baalrud: Conventional descriptions of the four states of matter i'm going to talk about strongly couple plasmas today.
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Scott David Baalrud: And like gaseous state plasmas they're highly electrically conducting but maybe or maybe not like gaseous state plasmas they may or may not.
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Scott David Baalrud: Have they may have gas liquid or solid like properties in terms of their material properties of either taking the shape or volume of their containers or being compressible.
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Scott David Baalrud: And the way we're going to quantify whether a plasma is weekly coupled meaning gas like or strongly coupled is through a coupling parameter.
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Scott David Baalrud: So the coupling parameter applies to other states of matter, besides plasmas and, in general, it's the average potential energy of interacting particles.
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Scott David Baalrud: At their inner particles facing So this is the force, this is a potential associate with a force that particles feel from one another at their entire particle spacing a so the coupling parameters, the ratio of this potential energy to the mean kinetic energy or the temperature.
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Scott David Baalrud: So weakly coupled system would have a small coupling parameter and a strongly couple system, a large coupling parameter.
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Scott David Baalrud: So apply to a plasma the potential energy of interaction between particles is the coombe energy.
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Scott David Baalrud: which has shown here at evaluated at the average in our particles facing so we see that this kula uncoupling parameter is proportional to the charge state squared.
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Scott David Baalrud: To the density to the one third power and inversely proportional to the temperature.
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Scott David Baalrud: So that tells us that there are three potential ways to make a strongly coupled plasma one, it could be low very low temperature.
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Scott David Baalrud: To it could be very high charge date or three have very high density and, in fact, you can make a strongly couple plasma through each of these three mechanisms, and that is done.
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Scott David Baalrud: So the low temperature avenue are there's two common experiments one is not neutral plasma experiments So these are either trapped up here on or electron plasmas.
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Scott David Baalrud: Which are typically magnetized and confined in traps, they can have temperatures over quite a broad range of conditions like one to 100 million kelvin and their gamma parameters can be highly scalable so they can go from very weakly couple of all the way.
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Scott David Baalrud: solidification and here i'm showing an example of a plasma crystal on the right and and plasma, so these experiments are used an Anti matter experiments, such as at CERN but also in quantum information science.
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Scott David Baalrud: Another example is ultra neutral plasma So these are start as ultra cold atomic gases which are cool than magneto optical traps.
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Scott David Baalrud: And so, these are the types of experiments people use to make bose-einstein condensates and reach the lowest temperatures on earth, but if you start with that ultra cold atomic gas and eyes it with a laser you can create an ultra cold state of plasma.
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Scott David Baalrud: Due to some heating effects that naturally occur when you is the medium.
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Scott David Baalrud: It heats to something like 50 milla kelvin for the ions.
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Scott David Baalrud: And a one to 100 kelvin for the electrons so you can get to a State where ions are moderately coupled and to the point, one to 10 range and electrons are typically weekly coupled.
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Scott David Baalrud: Okay, the second avenue for a strong coupling that we mentioned was a high charged date so examples.
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Scott David Baalrud: Of plasmas through this avenue are dusty plasmas So these are plasmas which are otherwise weekly coupled and gashes like but they contain many small particles which are usually have a micron size.
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Scott David Baalrud: And those dust particles collect electron charges more rapidly than on charges, and so they charge to something like 10,000 electronic charges and they're about room temperature.
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Scott David Baalrud: They can be highly scalable in terms of their coupling strength from from week to start very strong and crystals.
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Scott David Baalrud: Some advantages of these types of experiments are that they're extremely well diagnosed, because the dust moves around slow enough and it's large enough to track with optical cameras.
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Scott David Baalrud: And they can be controlled with lasers some disadvantages are that gravity also works against the dust so they tend to be two dimensional systems and they're also not purely strongly correlated systems as they're embedded in a weekly couple of plasma.
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Scott David Baalrud: Okay, and then the third avenue which we're going to concentrate on mostly in this talk is high density, and this is the field of high energy density plasmas.
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Scott David Baalrud: Which are extreme states of matter that exist at hundreds of giga joules per meter cubed that's that's a really high energy density and that corresponds to millions of atmospheres of pressure.
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Scott David Baalrud: exotic effects happen at this state of matter where the extreme pressure can cause electrons to be stripped from Adam so it's not just impact ionization like it is another gaseous like states.
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Scott David Baalrud: These are plasmas that are solid density to several times compressed and can range over a huge range of temperatures.
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Scott David Baalrud: From room temperature to kill electron volts, and so they have highly varying coupling strengths and so these type of systems arise naturally in astrophysics such as.
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Scott David Baalrud: The core of giant gas planets or the atmosphere of white dwarf stars, but they can also be produced in the laboratory such as inertial confinement fusion experiments or plasma is made using high intensity lasers.
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Scott David Baalrud: And so, these type of high energy density facilities our flagship facilities of US science, so I give a couple of examples here one is a busy pulse power facility, which is hundreds of millions of dollars of experiment.
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Scott David Baalrud: And another is the national ignition facility, which was a multibillion dollar experiment on the Left it's a.
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Scott David Baalrud: pulse power using current as a driver and on the right it's a pulse power using lasers as a driver.
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Scott David Baalrud: And I give a pitch here for Michigan that the high intensity laser technology such as makes the national ignition facility possible was awarded behalf of the Nobel Prize a couple years ago for work done right here at the University of Michigan.
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Scott David Baalrud: And so, as a motivation for this talk predicting the outcome of these important experiments is often limited by our ability to model equation of state and transport properties of dense plasmids.
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Scott David Baalrud: And that's going to be the motivation for our work.
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Scott David Baalrud: But i'd like to a quick plug sense it's a graduate recruiting day.
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Scott David Baalrud: That and I stole this slide from Professor mcbride from a couple days ago, so you might recognize it.
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Scott David Baalrud: But strongly couple of classes are made in the laboratories in nurse routinely so One example is the pulse power and microwave laboratory with professors gilligan Bach mcbride and low Another example is the.
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Scott David Baalrud: High intensity laser facilities like Hercules or Zeus Professor crucial neck or Professor currents and Thomas.
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Scott David Baalrud: And another as we'll see in the talk is even a some low temperature plasma experiments atmospheric pressure and and liquids can reach strongly a couple of plasma conditions so there's the low temperature plasma science and technology laboratories of Professor foster and Krishna.
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Scott David Baalrud: Okay, one other parameter to introduce last dimension, this parameter for this talk, I think.
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Scott David Baalrud: Is the degeneracy parameter so plasma is that a really dense can be influenced not only by strong correlation effects but also degeneracy of the electrons and when that happens.
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Scott David Baalrud: Or that happens when the inner particle spacing so the distance between atoms is less than or about the probably wavelength.
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Scott David Baalrud: So, in that sense, you you don't talk about interacting particles anymore, but you talking about interacting wave functions.
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Scott David Baalrud: And that is quantified often by the Fermi degeneracy parameter, which is the ratio of the temperature of the system to the Fermi energy which corresponds to the ratio of that inner particle spacing at to the durably wave like squared.
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Scott David Baalrud: And just to point out that's proportional to the temperature and inversely proportional to the density of the two third power so we'll see that.
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Scott David Baalrud: Small values of this data indicate high levels of degeneracy so that's not quite intuitive I would have maybe defined it the opposite way, but so just remember small feta means highly degenerate okay.
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Scott David Baalrud: And when electrons are degenerate, we need to modify our definition of the luck, of the cool low coupling strength for electrons.
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Scott David Baalrud: Because interactions are now characterized by the Fermi energy rather than the temperature.
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Scott David Baalrud: And so, our our basic definition of the coupling parameter being the ratio of the potential energy and kinetic energy sticks.
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Scott David Baalrud: But now the mean kinetic energy is determined by the Fermi energy rather than the temperature and so we, we see that that leads to a fundamentally different scaling.
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Scott David Baalrud: Of the coupling parameter any degenerate regime and, namely i'd like to point out that higher density means weaker coupling if you're in a quantum degenerate regime.
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Scott David Baalrud: Okay, so here's sort of the 30,000 foot view of plasma and other States of matter so showing on the vertical axis 15 orders of magnitude in temperature and on the horizontal axis 35 and density.
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Scott David Baalrud: And so you'll find all of your favorite plasmas on this diagram so the gaseous like states of plasma that we typically talk about live in the upper left corner of the diagram above the gamma equals one line and so those are things like the solar corona the aurora lightning.
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Scott David Baalrud: But there are other plasmas on this diagram that live in different parts of the face face such as the ultra cold plasmas we introduced.
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Scott David Baalrud: The inertial confinement fusion plasmas or the highly dense.
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Scott David Baalrud: astrophysical plasmas.
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Scott David Baalrud: And so we usually think of plasmas is living up here.
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Scott David Baalrud: condensed matter is living here near solid density and you know room temperature.
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Scott David Baalrud: And what i'll talk about today is is this intermediate state of matter which we sometimes call warm dense matter, which is a strongly coupled state of matter.
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Scott David Baalrud: Which is too hot for condensed matter theories to apply, but too cold for plasma theories to apply, and so what we're going to be talking about is how to go from plasma, you know realms and work our way into this warm dense maturation.
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Scott David Baalrud: And so the questions were after asking answering are what are the structural and transport properties of strongly couple plasmas and how can we describe them.
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Scott David Baalrud: And will attack the problem using two different approaches the first is a simulation approach and.
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Scott David Baalrud: Particularly will use a molecular dynamics simulations and those do two things for us those provide fundamental data, since their first principles approaches to understand basic processes that happen and Mormons matter.
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Scott David Baalrud: A second where we'll use it as to guide theory and to test our theories.
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Scott David Baalrud: Okay, and then the second approach, besides the simulations is going back to the fundamentals and trying to come up with new approaches to plasma kinetic theory so starting at the beginning and.
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Scott David Baalrud: Driving new approaches based on new expansion.
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petrov: parameters.
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Scott David Baalrud: And so we'll talk a little bit about that, and how we have proposed approaches to extend plasma theory into strongly correlated plasma regions.
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Scott David Baalrud: Okay, and i'll talk.
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Scott David Baalrud: A lot of the first part of the talk here about the one component plasma model and the reason we study this model is because it's entirely characterized by the gamma parameter.
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Scott David Baalrud: So this model consists of classical point particles of the same sign of charge, but you assume that there's some not interacting background that makes it overall charge neutral, but only one sign of charges mobile or interacting.
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Scott David Baalrud: And this is them if you normalize time in terms of the plasma frequency and distance in terms of the average in a particle spacing is entirely characterized just by that cool uncoupling parameter.
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Scott David Baalrud: So we can vary the gamma parameter from small to large values and understand how the system changes or study how it changes.
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Scott David Baalrud: And it turns out, it also represents some physical systems so it'll turn out to represent ions and an ultra cold plasma, for instance, or dustin, it is the plasma.
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Scott David Baalrud: Okay i'll show you some results of molecular dynamics simulations first and these molecular dynamics simulations solve newton's equations of motion for classical particles interacting on a periodic box.
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Scott David Baalrud: Are these are typically simulations, we need to do a high performance computers, we usually simulate something like 10s to millions of particles, depending on the conditions were interested in looking at.
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Scott David Baalrud: Oh, we use advanced algorithms to make it as efficient as possible, such as this particle particle particle mesh algorithm.
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Scott David Baalrud: And we do our computations on a.
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Scott David Baalrud: Large computers, like the comic cluster from nsf exceed or local high performance computing machines at the universities.
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Scott David Baalrud: And what what's limiting about the molecular dynamics simulations is that you can typically only accurately simulate simple systems such as this one component plasma.
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Scott David Baalrud: And also noteworthy is that they become more difficult, as you get to weaker coupling so it's a good way to simulate strongly correlated systems, but it breaks down and we for coupling.
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Scott David Baalrud: Okay, so on to a bit of the physics.
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Scott David Baalrud: One of the most important parameters to understand for the rest of this talk on the theory side is the pair distribution function.
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Scott David Baalrud: So the pair distribution function as a measure of spatial correlations in the plasma and so it's defined this way, so if you take.
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Scott David Baalrud: The background density of the plasma so that's the uniform density times this function G, which is the pair distribution function as a function of the radius around a particle say at the origin.
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Scott David Baalrud: Times, the volume is of a spiritual Shell at a distance are away from that particle at the origin.
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Scott David Baalrud: and any set that equal to the total number of particles in that spherical Shell that's how this radio distribution function is defined so you can SIP think about it simply as a density profile around a test charge at the origin.
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Scott David Baalrud: Okay, and then this is how we can define it and computed in the simulations so we can track the positions of all of the particles in the system and average over all of them at statistical equilibrium.
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Scott David Baalrud: So a system that completely lacks correlation will be quantified by a radial distribution or a pair distribution function of one that means the particles are having a particle the origin does not influence anything about the background.
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Scott David Baalrud: I like to point out, though, that even weekly couple plasmas you know any system is going to have some level of correlations Okay, and in a weekly couple of plasma.
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Scott David Baalrud: we're familiar with correlations in the sense of Dubai shielding Okay, so if you take one charge in the plasma.
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Scott David Baalrud: there's a polarization and cloud from the electrostatic interaction with the other charges around and that polarization cloud is the density profile of the background around the discharge.
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Scott David Baalrud: And so, in fact, at we coupling the radio, the pair distribution function or call it radio distribution function to and the talk it's just the exponential of the divine potential that we're familiar with in in plasma one.
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Scott David Baalrud: So it doesn't completely lack correlations even in a weekly couple plasma but it's mostly you know, for most distances, the radio distribution function, we close to one and it's only when you get within it by length of a particle that you see the correlations.
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Okay.
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Scott David Baalrud: Okay So what does the radio distribution function look like, as you range from week to strong coupling so on the left here i'm showing.
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Scott David Baalrud: A molecular dynamics simulation at the top of particle motion of the one component plasma a coupling strength of point one so that's we coupling and that's a dilute gas like medium where particles move around largely freely having small week interactions with one another sort of continuously.
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Scott David Baalrud: If we raise the coupling strength up to about 10 we see that now particles don't move so freely.
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Scott David Baalrud: They you know they're mostly they can make a transitions of many inner particle space things, but then they have much more large angle strong interactions with other particles in the system.
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Scott David Baalrud: And if you look at the radio distribution function in this more dense gas like state you'll see that there's.
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Scott David Baalrud: A lack of particles for a core which we sometimes called the cooler core.
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Scott David Baalrud: Before you get to peak numbers above one means it's more likely than average to find a particle of that position at a position that's near the average in her particle spacing So this is the onset of what we call strong correlation physics.
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Scott David Baalrud: And if you continue to say gamma of 100, so this is a very strongly coupled system now you see that particles are almost nearly trapped in their nearest neighbor wells.
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Scott David Baalrud: But they you know can migrate out of them, sometimes still and that, if you look at a comparison of this to say a liquid.
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Scott David Baalrud: you'll find that this is very similar to the atomic motions in a liquid, so we might think of this as a liquid like plasma state.
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Scott David Baalrud: And here that cool own hole extends a bit out of the way out to almost an average and particle spacing and the peaks and troughs associated with longer range correlations are stronger.
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Scott David Baalrud: And then, finally, if we go all the way to gamma value of 175 in the cool uncoupling parameter for the one thing about a plasma you see it actually crystallizes into a body summer centered qubit crystal.
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Scott David Baalrud: Okay, so we want to concentrate mostly on those fluids like states of plasma we won't talk much about the solid state so strongly couple plasma were more coming from.
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Scott David Baalrud: A traditional plasma physics and getting into the more dense strongly correlated regions and what we're ultimately interested in.
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Scott David Baalrud: Describing are the thermodynamic and transport properties as those are the unknowns and hydrodynamic descriptions of the plasma as a fluid, so, in particular, we still expect the basic structure of fluid equations to apply to you know liquid like states, as well as gas like states.
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Scott David Baalrud: But where the unknowns now are now are things like the thermodynamic properties so say the pressure internal energy and temperature of the system.
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Scott David Baalrud: As well as the transport properties so things like the sheer and bulk viscosity is thermal conduction and electrical conduction So these are the properties of the material properties of the plasma that we want to describe.
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Scott David Baalrud: It turns out, at least in the classical limit that thermodynamic properties are entirely determined by this parameter we introduced the pair distribution function.
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Scott David Baalrud: So you can use the framework of classical statistical mechanics to describe things like the equation of state entirely in terms of.
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Scott David Baalrud: ideal parts so say the ideal component of the equation, the State, as well as the radial distribution function and potential through which particles interact, so, in our case, this is the cool own potential.
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Scott David Baalrud: And so, if you know, at least in the classical limit the pair distribution function, you can, in principle, compute all of the thermodynamic functions Okay, so you can have a similar relationship as the pressure for other thermodynamic quantity is like the internal energy and the compressor.
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Scott David Baalrud: Okay, and I, like to point out that each of the thermodynamic quantity is will have some ideal component and some excess component.
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Scott David Baalrud: And in the weekly couple the regime, it will be the ideal component that's the dominant one and we'll see that as we transition to strong coupling it's the excess component that becomes the dominant one.
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Scott David Baalrud: So it's the interactions between particles described by that inner part interaction potential that's the dominant force, leading to the macroscopic material properties.
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Scott David Baalrud: So, fortunately, excellent approximations exist for computing the radial distribution function at it essentially any coupling strength.
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Scott David Baalrud: i'm showing you one example which we don't need to get into the details of called the hyper another chain approximation.
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Scott David Baalrud: But just the takeaway here is that it's just a closed set of equations that you can solve a very quickly on your laptop and you can get results that are very close to what you compute from molecular dynamics and so here i'm showing a comparison of the of the computed.
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Scott David Baalrud: pair distribution function from this approximation in comparison to molecular dynamics and you see that it agrees very well that it breaks down a little bit as it gets a very strong coupling but it turns out there's even fixes for this model that make it almost exact.
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Scott David Baalrud: So we basically consider that a solved problem, so we can model, the radio distribution function, at least in the classical limit that determines equation of state property, so we kind of consider the thermodynamics.
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Scott David Baalrud: solved and so we're going to concentrate, most of our attention on in this talk are the transport so coefficients so we'll be looking for instance.
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Scott David Baalrud: At the diffusion coefficient so the diffusion coefficient relates a thermodynamic force, which in this case is a concentration gradient.
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Scott David Baalrud: To the thermodynamic flux, that is produced in response in this case of particle flux and the constant of proportionality in that relationship is what we call the diffusion coefficient.
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Scott David Baalrud: Is we'll look at a few of these linear transport laws, the other one will look at as viscosity so newton's law viscosity relates to shear stress to assure velocity gradient.
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Scott David Baalrud: And the proportionality, as the sheer viscosity and the final one is he conductivity so for his law if he can take conductivity relates a temperature gradient to the heat flux.
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Scott David Baalrud: And so it turns out, if we can characterize each of these transport coefficients Oh, we can we can determine either the hydro dynamics description or the image description in the case of a magnetized plasma.
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Scott David Baalrud: Okay.
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Scott David Baalrud: And so, to start with, we can use our molecular dynamics tools to compute these things Okay, and how we do that is through a set of relationships called green qubo relations.
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Scott David Baalrud: And these are exact relationships that relate the fluctuation properties of particles at equilibrium, so if you have a simulation done at equilibrium.
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Scott David Baalrud: You can track the positions and velocities of the particles in time and then relate time correlation functions of certain quantities to those transport coefficients so that's how we do the computation in molecular dynamics so, for example.
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Scott David Baalrud: A correlation of the velocities of particles will lead to the diffusion coefficient.
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Scott David Baalrud: For viscosity it will be a correlation of of terms that have to do with the velocities of particles, as well as the potential at which they interact, so their positions and the forces.
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Scott David Baalrud: thermal conductivity will have contributions from kinetic parts potential parts and terms that are mixed, which we call the real terms.
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Scott David Baalrud: And so, like the thermodynamic properties, the kinetic terms of the transport coefficients dominate at we coupling.
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Scott David Baalrud: But the potential and the various components dominate at strong coupling okay so show you some results here we compute these trajectories of particles from molecular dynamics and we can compute each of these transport coefficients and so here are the results.
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Scott David Baalrud: So this is the on the left of the self diffusion coefficient in the middle, the sheer viscosity coefficients and on the right, the thermal conductivity coefficient, of the one component plasma is a function of the coolest coupling strength.
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Scott David Baalrud: Okay, and so here's the data, so this is how we use the molecular dynamics simulation, so now we have like a characterization of the basic transport properties as a function of of the coupling strength.
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Scott David Baalrud: We can go a little bit further in the molecular dynamics simulations and pull apart different contributions.
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Scott David Baalrud: To the transport coefficients from those kinetic or potential parts and so here for say sheer viscosity.
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Scott David Baalrud: We can we plot the kinetic part which is the purple triangles and compared to the potential part which is the green triangles and we see as expected.
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Scott David Baalrud: When the plasma sufficiently weekly couple it's the kinetic parts that dominate the transport and when the plasma sufficiently strongly couple it's the potential parts that dominate the.
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Scott David Baalrud: Transport and actually in the case of sheer viscosity there's a transition.
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Scott David Baalrud: And that transition is marked by a minimum of the total viscosity coefficient.
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Scott David Baalrud: and similarly for thermal conductivity there will be a transition from the kinetic parts to the potential parts.
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Scott David Baalrud: And in this case there's also a minimum of the thermal conductivity.
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Scott David Baalrud: Okay, so we can also use the simulations to tease out other you know basic properties of the plasma so, for instance, we can notice that.
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Scott David Baalrud: at certain conditions of coupling strength, particularly when it's bigger than about 50 certain properties that we typically attribute to liquids are satisfied.
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Scott David Baalrud: So you're a couple of examples, so the stokes Einstein relation pertains to liquids, and that is a statement that the product of the diffusion coefficient and the sheer viscosity coefficient are constant.
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Scott David Baalrud: So that is definitely not seem to be observed that week coupling, but when we get to a coupling strengths above 50 it's well satisfied.
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Scott David Baalrud: We can also see that the radius law viscosity which relates, a concept of access entropy to the sheer viscosity coefficient.
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Scott David Baalrud: As well satisfied and when by the one component plasma when the coupling strength is about 50 but not below, so this is another way that the simulations can teach us stuff about what's, the most important properties of the system Okay, so it gives us some fundamental data to work with.
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Scott David Baalrud: Okay, but ultimately what we're after is developing a theoretical description of these properties.
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Scott David Baalrud: Okay, so.
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Scott David Baalrud: Just for this theoretical development, let me first in one slide review how plasma kinetic theories are usually developed from the boltzmann equation in the weekly couple the region.
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Scott David Baalrud: So the textbook plasma theories have these basic properties.
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Scott David Baalrud: It starts from a basic expansion parameter in terms of weak correlation so a small ask them topically small gamma parameter and how that comes into the theory is through a couple of approximations.
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Scott David Baalrud: So first off, let me say here and kinetic theory we're trying to describe what we call the face space distribution function of particles.
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Scott David Baalrud: So this is describing you know, say, at a given position in space what the distribution of velocities of particles are at that position.
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Scott David Baalrud: And so what we ultimately want is a description of that which is the lowest order of the face face distribution functions.
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Scott David Baalrud: The next order will describe within a certain volume the interaction of two bodies and the so that would be F two.
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Scott David Baalrud: And, and so on to higher orders, with three body interactions and for body interactions and so on, so what we need to do in in kinetic theory is reduce the complexity and order to come up with an approximation for that lowest order distribution function.
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Scott David Baalrud: And how that usually goes about is to.
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Scott David Baalrud: use this week coupling as an expansion parameter Okay, and so in the bolts derivation of the boltzmann equation, we say that there's no triplet correlations.
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Scott David Baalrud: Okay, so what that means and mathematically is that the three body distribution function is zero so we just say only two bodies and interact at any given time.
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Scott David Baalrud: Bodies here being the particles, we also make use of what's called the molecular chaos approximation, which is to say that when particles interact two particles they're not correlated with one another at their initial time.
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Scott David Baalrud: Both of these approximations require that the coupling strength vs and topically week.
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Scott David Baalrud: A third approximation, we need to derive the boltzmann equation.
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Scott David Baalrud: Is that the particles are only interact through a finite range.
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Scott David Baalrud: And that's not a limitation of the boltzmann equation applied to gases atomic gases, but it is the cooling systems, because the cool enforce.
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Scott David Baalrud: extends over infinite range formally so at least the divergence is the theory and so you actually have to do is kind of fix up the theory the boltzmann to impose the by screening, so you actually have to impose some correlation.
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Scott David Baalrud: Okay, and anyway that that's the basic summary of what leads to plasma kinetic theory.
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Scott David Baalrud: And so, if we do that, so we do those steps we derive the boltzmann equation, and we apply it to plasmas here are the results, so we often call this land of spitzer theory.
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Scott David Baalrud: After two of the pioneers of the theory, and so we see if we're at sufficiently week coupling it agrees very well with the molecular dynamics data.
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Scott David Baalrud: And so, like I said, we can only go so so weak a coupling with a molecular dynamics, because it becomes too computationally expensive, but we can go low enough in coupling that we can reach the awesome tonic regime of usual plasma theory.
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Scott David Baalrud: But we noticed that, once the coupling strength is bigger than about point one or so the theory sort of rapidly diverges it breaks down abruptly.
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Scott David Baalrud: Okay, so what we're after is trying to go further and coupling strength.
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Scott David Baalrud: And we're going to particularly focus on the regime or the coupling strength is not in that liquid state, so we want to, we want to go from.
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Scott David Baalrud: The gas like state a gamma of like point one up through strong strongly couple the regime but we're not necessarily targeting that liquid like regime that we saw earlier in the simulations.
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Scott David Baalrud: And the reason we're motivated in that direction is that's most relevant to the experiments so most of the high energy density plasma experiments or the ultra cold plasma experiments are in this type of region.
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Scott David Baalrud: We want to theory, because the molecular dynamics simulations are too expensive computationally to really deal with real systems, we can only do those.
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Scott David Baalrud: We can relate to those simple systems, and we really want to look at more complicated systems and what we need to do that as a fundamentally new approach to kinetic theory.
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Scott David Baalrud: And so, in order to try to do that we've kind of asked ourselves, is there a deeper principle we can pay attention to in our development of the boltzmann equation.
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Scott David Baalrud: And what we suggest is that if you consider the equilibrium state is a known quantity, that you can construct the kinetic theory that extends into the strongly coupled regime.
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Scott David Baalrud: And so we're going to take advantage of the fact that we, we know how to characterize the radio distribution function at any coupling strength.
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Scott David Baalrud: Because it's an equilibrium quantity.
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Scott David Baalrud: And instead of expanding the hierarchy of equations that we get by saying the triplet correlation is zero we're going to expand, to say the triplet correlation is equal to its value at equilibrium.
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Scott David Baalrud: Okay, so we have a new expansion parameter that rather than measuring the strength of correlations measures, the deviation of the strength of correlations from their value add equilibrium so, even if the correlations are strong, we know that the equilibrium value.
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Scott David Baalrud: So it's a perturbation away from equilibrium, rather than a perturbation and strength of interactions.
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Scott David Baalrud: We also modify that molecular chaos approximation to use this radio distribution function to describe the initial correlation of particles when they interact.
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Scott David Baalrud: And it turns out that also resolves one of the shortcomings of weekly couple of plasma theory and that itself consistently determines the the range over which particles interact so particles only happened within a certain collision volume that's finite.
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Scott David Baalrud: And in doing this we enforce that the theory has the exact equilibrium properties.
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Scott David Baalrud: Okay, so.
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Scott David Baalrud: The result of this is an equation that looks a lot, like the boltzmann equation, but with some important differences.
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Scott David Baalrud: Now particles, instead of interacting through the the coolest force interact through the potential of mean force and the potential of mean for us is the statistical potential you get if you take two particles at a fixed position, but you average over all of the other particles at equilibrium.
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Scott David Baalrud: And that is directly related to this, the natural log of the radio distribution function which we know.
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Scott David Baalrud: In the weekly couple limit we return the familiar divided by huckle potential that we would expect.
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Scott David Baalrud: A second difference is that there's another term in the kinetic equation that fixes the equation of state properties.
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Scott David Baalrud: So it enforces not just the boltzmann equation, we have only an ideal gas equation of state, but by enforcing that we have the exact equilibrium will get another term and the equation in the kinetic equation, which, when you take a fluid limit.
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Scott David Baalrud: provides the equation of State, so it gives you the access term exactly the equation of state.
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Scott David Baalrud: Okay, and so here are the results for the transport coefficients.
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Scott David Baalrud: So i'm showing the same molecular dynamics data, as I did earlier, but now i've put the theoretical predictions on.
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Scott David Baalrud: And so we see that, indeed, this theoretical approach can extend plasma kinetic theory orders of magnitude and coupling so keep in mind, this is a logarithmic scale.
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Scott David Baalrud: And instead of breaking down a gamma of about point one and extends up to gamma values of about 10 or 20 depending on the particular transport coefficient.
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Scott David Baalrud: So we targeted this regime of you know gamma of point one to 20.
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effectively.
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Scott David Baalrud: We can dig in a little bit more to why the theory breaks down, eventually, and we see that if we if we compare the theory just with the kinetic part of the transport coefficients it agrees really well, even at higher coupling and what we're really.
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Scott David Baalrud: Missing there eventually when you get into the liquid like regime.
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Scott David Baalrud: Is that the presumption that particle interactions are statistically in.
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Scott David Baalrud: Independent when they when they start their collision breaks down Okay, so we still have some notion of we correlations but it extends to much higher values of correlation strength than the boltzmann equation does.
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OK.
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Scott David Baalrud: OK, and then with the remaining time I want to just show you a few tests we've done of the theory from both experimental and real warmness matter systems.
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Scott David Baalrud: So first ultra called neutral plasmas so here's an example of an experiment from tompkins group at rice university where they make an ultra whole neutral plasma in there, maybe the optical trap and then use lasers to.
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Scott David Baalrud: pump certain quantum states of atomic states, and then they can diagnose those atomic States individually in Washington relax and time Okay, so they pump away from equilibrium and then watch them relax and doing so they can measure.
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Scott David Baalrud: Momentum relaxation rates, which is showing at the top here, as well as diffusion coefficient so here i'm showing the same theories we've been looking at previously so here's the plasma theory and here's the.
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Scott David Baalrud: What we call this mean force kinetic theory and we see that the experiments are able to distinguish.
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Scott David Baalrud: The strong coupling effects.
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Scott David Baalrud: though they could only get to coupling strengths of a few.
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Scott David Baalrud: This study of ultra cool plasmas also led to a sort of novel prediction of a physical effect that we were able to.
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Scott David Baalrud: test with molecular dynamics simulations which is, if you apply the standard kinetic theory from weekly couple plasmas.
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Scott David Baalrud: You find that there's a sort of a fundamental symmetry which is particles the collision rate between different species in the plasma is independent of the side of the charges okay.
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Scott David Baalrud: So it means the electron Ion collision rate would be predicted to be the same as the positron collision right, for instance okay so there's a symmetry of charge sign symmetry.
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Scott David Baalrud: And so what we were able to predict is that that's charged science symmetry breaks down when you get into a strongly correlated region, and you actually get a higher conversion rate from electron collisions, then you would from a positron collision.
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Scott David Baalrud: And this give us additional can you know convincing argument that it's the force in a collision that is modified by strong correlation effects, and not just the range, because if you only modify the range of the interaction you don't get this sign cemetery breaking.
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Scott David Baalrud: Okay, so that's sort of a neat validation of the ideas.
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Scott David Baalrud: Okay, so onto a real kind of high energy density systems rather than more model systems.
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Scott David Baalrud: Now, I just want to set the stage a little bit now we're going to talk about dense plasma, so we need to fold in what we introduced early in the talk of electron degeneracy so you're showing some quantum molecular dynamics simulations.
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Scott David Baalrud: But now they're solving for electron probability distribution functions, rather than.
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Scott David Baalrud: political positions.
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Scott David Baalrud: I never Nevertheless, we can apply this theory and we've done some quantum generalizations were recently, but here i'll talk about Ion transport.
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Scott David Baalrud: Where the degenerate electronic effects are all wrapped into the radio distribution function and having to do with an electron density profile surrounding islands.
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Scott David Baalrud: Okay, so if we start first by taking the radio distribution function from a simulation from a quantum electrodynamics simulation using that as the input to the theory.
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Scott David Baalrud: And then, comparing the result, with the result of the simulation for a transport coefficient that's what we first did to try to validate the concepts.
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Scott David Baalrud: And so here i'm showing you some results from deuterium at solid density, the four grams per cubic centimeter over a range of temperatures of one to 100 electron levels.
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Scott David Baalrud: And so the theory the theory and the simulations are compared with the data points in the black line.
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Scott David Baalrud: So we see as long as we get this right input to the theory of this radio distribution function that the theory is accurately able to produce the outcome of the experiment for the transport coefficient.
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Scott David Baalrud: And just to point out this, this is a range of coupling strengths from week to strong and degeneracy parameters from classical to degenerate.
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Scott David Baalrud: Okay, so that's great so we we know if we have the right input to the theory this equilibrium properties that para distribution function that we.
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Scott David Baalrud: are confident we can predict transfer coefficients but it's not very convenient if you need to run the quantum molecular dynamics simulation.
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Scott David Baalrud: In order to get the input to the theory fortunately there are great groups that have been studying equation of State for a long time.
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Scott David Baalrud: And they have come up with really accurate and fast to evaluate models for this radio distribution function that account for the quantum nature of the electrons.
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Scott David Baalrud: And so we've teamed up with this group at Los Alamos Charlie start and DDA Simone.
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Scott David Baalrud: who have developed an average Adam to component plasma model they call it, but you can just think of it as like a quantum generalization of that hyper netted chain approximation introduced earlier.
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Scott David Baalrud: So the point is that it accurately predicts the radio distribution function in these warm beds matter conditions.
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Scott David Baalrud: And so it turns out, if we use their model to rapidly provide the input to our kinetic theory, we can get very similar agreement, as we saw with similar system with simple systems like the one component plasma.
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Scott David Baalrud: And so we've looked at a number of materials now here we're looking at aluminum because it's a common material use that high energy density plasma experiments.
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Scott David Baalrud: showing this self diffusion coefficient over a range of warmness matter temperatures at solid density on the top and 10 times a solid density on the bottom.
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Scott David Baalrud: The theoretical predictions are the solid black lines and then all the other data are various types of quantum molecular dynamics simulations.
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Scott David Baalrud: And so we see a similar range of agreements in terms of the cooler uncoupling strength, as we saw for the simple systems.
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Scott David Baalrud: And just to point out, you know this aluminum data over this range of temperatures can get kind of exotic so the charge date has a wide range of values, three to 12 the coupling strength is wide over this range and here you get effects like pressure ionization becoming important.
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Scott David Baalrud: Okay, we looked at iron because it's important in modeling giant planet interiors and exoplanets.
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Scott David Baalrud: iron at solid density over a wide range of temperatures same punch line similar.
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Scott David Baalrud: range of applicability of the theory so here i'm showing self diffusion and share viscosity.
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Okay.
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Scott David Baalrud: The theories have been applied a bit and astrophysics so there's some stellar models that have adapted the the theory Mesa is one of them.
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Scott David Baalrud: I here's one of our own studies, looking at white dwarfs so we were looking at the sedimentation of heavy elements through the atmosphere of a white dwarf.
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Scott David Baalrud: So when you look at a white dwarf, like any astrophysical object basically all you're seeing is the light coming out of the surface.
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Scott David Baalrud: And so there's a mystery in this field, about what the population of different heavy elements is in the surface, because you think.
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Scott David Baalrud: The strong gravity would pull them through the atmosphere really quickly, and so they really some of the models for the evolution of white dwarfs depend kind of heavily on.
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Scott David Baalrud: Knowing the transport of different elements through the outer atmospheres, which are dense plasmas and So here we looked at silicon through helium and calcium through helium which are impurities that the astrophysicists measure.
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Scott David Baalrud: And, and that was a informing some models and white dwarf physics.
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Scott David Baalrud: Okay, then one just last thing this theoretical construct generalizing the boltzmann equation is not just applicable to plasmas but we've also applied it to dense gases.
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Scott David Baalrud: And so here's one example of, so this is a neutral gas instead of a charge gas now where we applied the theory to the Leonard Joan system, so the letter Jones is a type of interaction potential it's a model.
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Scott David Baalrud: But here i'm showing the phase diagram with the letter Joan system, so the vertical axis is a normalized temperature and the horizontal is normalized density.
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Scott David Baalrud: And here's the phase diagram so here will be the gas like state or not like the gaseous state here's a liquid state down here a solid and up here as a supercritical fluid.
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Scott David Baalrud: And so here i'm showing some data, where we compare the new theory predictions, which are the.
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Scott David Baalrud: Blue triangles with the molecular dynamics data, the Red circles and then the traditional boltzmann equation, and so, you see, as expected, as you get into the supercritical fluid state the boltzmann equation breaks down because this the medium is too dense.
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Scott David Baalrud: But we do see a similar to the plasma states the strongly couple plasma states the mean force kinetic theory is able to accurately predict this the diffusion coefficient in this instance at the higher densities relevant to the supercritical fluids day.
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Scott David Baalrud: and
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Scott David Baalrud: It is it similarly fails, when you get to the liquid state.
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Scott David Baalrud: So it's very analogous to the cooling system in terms of the coupling strength, where you may think of this regime between damn of point one and 20 is is more of like a supercritical plasma state.
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Scott David Baalrud: Okay, so that's the talk so conclusions, I thought I hope I can see a little bit that strongly couple puzzles are a neat thing, whether or not you call them the plasma or not they're cool they're fundamentally different than weekly couple plasmas.
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Scott David Baalrud: In our work, we use molecular dynamics simulations to provide basic data and to guide or theoretical development we propose new approaches to a plasma kinetic theory that we can then tests using the molecular dynamics simulations.
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Scott David Baalrud: And i'd like to just thank some of the students and collaborators who contributed to this work over the years, so up until last month, it was it was mentioned, I was at the University of iowa in a couple of graduates.
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Scott David Baalrud: The rest are still in the group so thanks to the students for all the work here, and just a plug censuses graduate recruiting day that we have some open graduate research positions in these topics, so please i'll let me know if you're interested.
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Todd Randall Allen: All right, thank you alright thanks thanks a lot Scott lots of virtual flapping you can't hear it, but that's the problem with the Internet apologize, so there was a few questions that dropped in as you were talking, so let me just go to a few and.
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Todd Randall Allen: Let me start with sack wolf so zach he has a lot of questions so he's gonna pick which one he wants to ask you, and then we'll see if we have time to come back to him later so you're upset great.
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Zachary Wolff: Thank you for the talk Professor ballard it very enlightening I had something regarding the when you're talking about the advanced computer systems, you were working with you were talking about how you.
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Zachary Wolff: simulate it molecular dynamics and you're talking about how they were good for a thousands of particles for specifically one component systems, what could you give us a practical example of what you would consider a like a plasma system that'd be one component in this case.
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Scott David Baalrud: Okay sure, maybe there's two parts of the question i'll try, both on So what is the computational question of what sets the number of particles, we need to simulate.
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Scott David Baalrud: And so that's not where it when we do molecular dynamics simulations we're not we're definitely not simulating a whole experiment right so millions of particles is a microscopic volume what we're trying to do is simulate.
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Scott David Baalrud: A large enough microscopic volume that it represents a fluid element basically of the system, and if we can do that, we given enough particles to simulate a fluid elements, then we can.
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Scott David Baalrud: associate the properties of that fluid element, with the larger macroscopic fluid.
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Scott David Baalrud: So that's that's what kind of sets our scale of how many particles, we need so that turns out to be set by.
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Scott David Baalrud: The largely by the range over which particles interact, something we call the collision volume.
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Scott David Baalrud: And so, at strong coupling that collision volume is set by you know sort of several inner particle space things and so that leads to simulations that are are done with like thousands of articles like 5000 is a common number for us, but as you get to weaker coupling.
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Scott David Baalrud: it's more the device scale sets the scale of the interaction, rather than the inner particle spacing and, by definition, a weekly couple plasma will contain a large number of particles ended up I like.
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Scott David Baalrud: In a device sphere, so that makes it a requirement that you have to have a lot more particles that weaker coupling and that's actually why we can't get the very weak coupling with our simulations.
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Scott David Baalrud: Because computationally for the length of time, we need to solve the equations emotion and we can maybe do like a million or 10 million tops but that's our limit.
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Scott David Baalrud: Can there's a second part of the question.
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Scott David Baalrud: Maybe you can remind me of.
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Zachary Wolff: I was wondering what the practical example I think you actually already answered it, but a practical example, what you were what you qualified as a one component system in these in these test.
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Scott David Baalrud: Okay, thank you okay so.
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Scott David Baalrud: We use this one component system, mostly is like a model test of the theory, but it does also directly pertain to the one can or sorry the the electrical blossom experiments.
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Scott David Baalrud: So it turns out, since electrons are sort of not equilibrium and hot compared to the islands and those experiments they're almost like just a neutralizing background, and so I dynamics and those experiments are very close closely approximated another one component possible.
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Scott David Baalrud: But other systems are more complicated.
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Zachary Wolff: Perfect Thank you so much, I really appreciate it.
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Scott David Baalrud: thanks for the question.
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Todd Randall Allen: yeah Okay, so I think Tom Tom know Horn, you had one do you want to ask yours.
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Tom Mehlhorn: yeah let me unmute So let me ask you this kind of a combined question when you were talking about simulating things as a one component plasma I was wondering whether it was implicitly then for hydrogen.
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Tom Mehlhorn: ions or when you're working with so, then you know, later on, you went on when you're actually working with hires the ions.
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Tom Mehlhorn: I guess when do you when does your work.
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Tom Mehlhorn: overlap with finite temperature density functional theory with pseudo potentials and whatnot as opposed to the you know structure factors that you're working with.
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Scott David Baalrud: yeah so it would be the latter half of the latter segment of my talk, where I had comparisons with quantum molecular dynamics So those are those are done using density functional theory.
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Scott David Baalrud: that's exactly what what those are.
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Scott David Baalrud: And so, so the first question is, how does the charge date factor in when you're doing the one can want and plasma model.
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Scott David Baalrud: And that, in that case it's only through the coupling parameter so remember that coupling parameter how to charge state squared it and, in that one component plasma model that's the only parameter that.
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Scott David Baalrud: That you need to determine the system to define the system and so it's wrapped into that parameter, so you can.
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Scott David Baalrud: use it to look at systems with high charge dates that's fine but it's just a model system right it's not like the physical Mormons matter plasma with different charge dates.
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Scott David Baalrud: And so, in fact, that's why we had to go from the model system to this more complicated system, you know, in the latter part of the talk, where we have to do more complicated things like you say.
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Scott David Baalrud: and solve the quantum mechanics problem in order to compare the theory.
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Scott David Baalrud: And there you're kind of yourself consistently.
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Scott David Baalrud: Solving for the charge that.
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Scott David Baalrud: sounds like getting your question.
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Todd Randall Allen: Okay, so I think we get we have time for i'll give you one comment and then one question, so the comment, and this is clearly just to make your head swell it says that was awesome to hear, thank you for the great presentation.
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Todd Randall Allen: But the question then is, would you say this is from john foster would you say that the head of a streamer is close to being strongly coupled.
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Scott David Baalrud: Thanks john so despite giving the plug for low temperature possums I didn't quite circle back so it's kind of an amazing thing to think we usually think of air in in this room, or whatever room you're sitting in.
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Scott David Baalrud: As a week very loosely coupled system and it is from the top, from the.
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Scott David Baalrud: neutral Adam interaction perspective but it turns out, when you if you iron eyes it so it just take air and say just a theoretical actress I say you fully eyes the air and the ions are still at room temperature.
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Scott David Baalrud: It turns out, because you're dramatically increasing the range over which, in the particles interact with one another, because now you have this cool and force which is long range that they're actually strongly coupled.
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Scott David Baalrud: And so, so air in the room, would have a gamma Ion about 10 it turns out okay and that's quite strongly couple if you do a streamer it's not going to be fully Ionized okay it's going to be maybe.
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Scott David Baalrud: One part and 10,000 eyes or something, but it turns out, if you if you look at the scaling of that coupling parameter with density is only to the density of the one third power.
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Scott David Baalrud: So even if you're at an organization fraction of only one part of the million you're still in the regime that's a gamma bigger than like point one where the theory start to break down so Indeed I I do argue that the head of a streamer.
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Scott David Baalrud: Can not just be close to strongly couple, but it can be strongly.
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Scott David Baalrud: That that answer your question.
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You asked him.
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Todd Randall Allen: He praises, why did you talk about his work, his his day is.
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John Edison Foster: done now, I was struggling to find the meat, but then me but yeah that thinks that is the question.
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John Edison Foster: yeah.
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Todd Randall Allen: john it was like but it sounded like a real scientific question, it was.
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brilliant.
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Todd Randall Allen: All right, so anyway, so I want to thank a bunch of people, so thank all the students who visited us today thanks hope you had a good time I hope you're impressed will become joining us.
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Todd Randall Allen: Thanks mitzi for co sponsoring this I think it's great we do this more often we've got common topics and then special thanks to Scott we're glad you're here.
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Todd Randall Allen: i'm glad you're part of the department and really appreciate you giving this talk so with that we'll call it Friday hope you all enjoy your weekend looks nice out there, so get out and enjoy the song thanks everybody.
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Thanks.