WEBVTT 1 00:00:03.040 --> 00:00:07.420 Mark Kushner: Is professor today. Speaker. 2 00:00:07.560 --> 00:00:26.020 Mark Kushner: Oh, in the Department of Physics and Astronomy at West Virginia University, where he's also the Associate director for the Center for experimental, theoretical, and integrated computational platform physics which spells kinetic as the center acronym. 3 00:00:26.570 --> 00:00:35.510 Mark Kushner: Paul got his Ph. D. At University of Maryland College Park, and was a postdoctoral researcher at University of Delaware before during West Virginia. 4 00:00:35.820 --> 00:00:44.020 Mark Kushner: This research focuses on magnetic reconnection and you might recognize the Asseq reconnection rates. 5 00:00:44.270 --> 00:00:53.530 Mark Kushner: that's one of his many contributions to the field plus material turbulence shocks keep plasma physics processes in various plasma settings. 6 00:00:53.600 --> 00:01:03.050 Mark Kushner: These methodologies include analytical techniques and numerical simulations on the theory and modeling side as well as using institute plastic measurements. 7 00:01:04.129 --> 00:01:24.070 Mark Kushner: Paul has received numerous recognitions for his work. He's a Fellow of the American Physical Society, Aps, the American Geophysical Union Agu, and was honored with Agu Fred Scarf award for his thesis work, and later the procedures, mackerel, Wayne metal which recognizes mid career scientists. 8 00:01:24.340 --> 00:01:36.110 Mark Kushner: He's done an impressive amount of community service, including high profile committees, such as the NASA Heliophysics Advisory committee which he now shares, following our own monthly month 9 00:01:36.150 --> 00:01:41.559 Mark Kushner: and numerous agu committees ranging from awards to science policy. 10 00:01:41.920 --> 00:01:50.480 Mark Kushner: His research teaching service and outreach activities have all been recognized by internal awards from West Virginia University. 11 00:01:51.500 --> 00:01:59.130 Mark Kushner: His talk is titled 25 dynamics of plasma out of local thermodynamic equipment. 12 00:01:59.240 --> 00:02:25.540 Mark Kushner: and before. I give the floor to you. I have something else to give you the infamous mixing mug, and you should really respect this. I was asked to buy someone who really wanted this mug, and I used all my powers to squeeze one without her giving a talk, and was not successful. So you should really respect this and have this 13 00:02:25.540 --> 00:02:37.069 Mark Kushner: in a very honorable place to be rob. 14 00:02:38.320 --> 00:02:42.549 Mark Kushner: Thank you. 15 00:02:42.750 --> 00:02:50.390 Mark Kushner: Thank you very much to you for that welcome. How's that? 16 00:02:50.990 --> 00:03:20.649 Mark Kushner: Okay? So it's good to be here. The last time I was at Michigan and gave a talk I joked about how Michigan sports teams kept hiring coaches from Wvu. Fortunately, that's not a problem anymore. Anyone happy to be here, and thanks to Mike for canceling class so that we can get a nice big group in here. So go Mike's class. Yeah, alright. So today, I want to talk to you about 17 00:03:20.930 --> 00:03:48.030 Mark Kushner: some work we've done recently on quantifying dynamics of plasmas when they're not in local thermodynamic equilibrium. And I'll explain all that stuff in a moment. But I want to highlight. In particular. My former grad student and current post docs on verb. We have a lot of the work I'll be sharing with you today. And some other collaborators here. And of course, acknowledgements for the funding folks. 18 00:03:48.680 --> 00:04:11.569 Mark Kushner: okay, so just a quick overview mostly won't make sense until I get into the details and explain them to you. But we'll talk about how plasmas are often not in local thermodynamic equilibrium. And so what I'm gonna tell you about is some work we've done to try to quantify. 19 00:04:12.180 --> 00:04:17.230 Mark Kushner: It's on my PC, no, should I unmute 20 00:04:17.240 --> 00:04:21.049 Mark Kushner: mute on your PC, okay, yeah. 21 00:04:21.529 --> 00:04:22.720 Mark Kushner: okay, good. 22 00:04:26.260 --> 00:04:26.930 Mark Kushner: Okay? 23 00:04:27.420 --> 00:04:29.779 Mark Kushner: So yeah, I'll 24 00:04:29.800 --> 00:04:52.900 Mark Kushner: tell you about some work we've done to try to quantify systems or especially plasmids that are not in local thermodynamic equilibrium and I'll tell you about a measure with dimensions of power density. This won't make much sense right now. And so far we've applied it to fundamental plasm processes like reconnection and turbulence 25 00:04:53.270 --> 00:04:56.129 Mark Kushner: and land outdamping 26 00:04:56.680 --> 00:05:13.349 Mark Kushner: there. And but I'll tell you also that I think there's more broad applications. In many other areas of plasma physics including low temperatures, maybe even high energy density plasma physics. But even beyond plasma physics, for sure. 27 00:05:13.360 --> 00:05:21.269 Mark Kushner: I don't know every this is plasma. Physics talk right. But maybe some of you're interested in things outside of plasma physics, too. So I'll tell you about some of that. 28 00:05:23.120 --> 00:05:38.469 Mark Kushner: So the main theme that we often think about in a lot of settings for plasmas is energy conversion. So source, that's obvious for things like fusion. But even an astrophysical and heliophysical plasmas 29 00:05:38.470 --> 00:06:01.509 Mark Kushner: think of your favorite process. And really, you know, understanding what's going on really often comes down to saying what's going on with the energy. How's it being converted? And there's a number of processes that can cause the conversion of of energy. So I mentioned magnetic reconnection. Where magnetic field lines effectively break and cross, connect and release their energy to the plasma 30 00:06:01.900 --> 00:06:27.600 Mark Kushner: collisionless shocks turbulence. These are things where bulk flow. Energy usually gets converted into potentially magnetic energy also into thermal energy. Internal energy. You have wave particle interactions, which, of course, is process that gives energy conversion between the fields in the, in the plasma get transport, and of course, waves and instabilities. So these are all things you would study in a plasma class. 31 00:06:27.610 --> 00:06:35.449 Mark Kushner: And fundamentally, again, you can talk about all of these collisions as well. You can think about it in terms of you know, where's the energy going? 32 00:06:36.640 --> 00:06:56.699 Mark Kushner: So historically, the simplest thing to do is to think of a plasma as a fluid and so when we think of a plasma as a fluid, we're tacitly thinking of it as being in local thermodynamic equilibrium and what that means is that effectively, there's a well defined temperature. That's one way to think of it. 33 00:06:56.700 --> 00:07:12.440 Mark Kushner: And so when you have a plasma that's in local thermodynamic equilibrium the simplest model is that the energy conversion is just adiabatic. So, in other words, pressure over density to the gamma power ratio specific. Heats is a constant. 34 00:07:12.920 --> 00:07:16.740 Mark Kushner: That's great. It works pretty well a lot. 35 00:07:16.790 --> 00:07:37.709 Mark Kushner: of course, if you're not, this is a autological statement here. If you're not in local thermodynamic equilibrium. If you're near local thermodynamic equilibrium, it means you're you're not very far away from it. It turns out other effects. Come in. If you have collisions. So like viscosity and resistivity. Conduction, thermal conduction. Things like that 36 00:07:38.150 --> 00:07:52.779 Mark Kushner: and those all then, can participate in the energy conversion, transport as well. So these types of models are used all over the place in plasma physics. So we talked about plasma and 37 00:07:52.910 --> 00:07:56.069 Mark Kushner: plasma waves or instabilities 38 00:07:56.680 --> 00:08:16.270 Mark Kushner: for shocks and reconnection and turbulence. If you think of you know, the basic model for shocks is the ranking agonio analysis, which is a fluid analysis for reconnection. The most basic description of it is the Street Parker analysis, which is a fluid analysis. Turbulence coma Gorov theory. That's a fluid analysis. Right? So 39 00:08:16.700 --> 00:08:23.070 Mark Kushner: is good. It really does a lot of things really well, which is great. 40 00:08:24.240 --> 00:08:30.220 Mark Kushner: taking this a little step further, making a little more getting some equations up here. We like equations. 41 00:08:30.350 --> 00:08:56.649 Mark Kushner: Equation. Sorry so we can write down the equations for the different forms of energy to see how they evolve in time. So this is the bulk, flow, energy, density. You! Here is the bulk, flow velocity. And so this equation tells you how the bulk flow. Energy evolves in time. In a weekly collisional. 42 00:08:56.810 --> 00:08:57.900 Mark Kushner: plasma 43 00:08:58.040 --> 00:09:14.430 Mark Kushner: I'm won't go into too much detail here. But there's also electromagnetic energy density, and that you just get from Maxwell's equation. So that's pointing theorem here. And then you can write an equation for the internal or the thermal energy density which has this equation here? 44 00:09:14.590 --> 00:09:26.699 Mark Kushner: so all of these are telling you something about how energy converts and what the physical mechanisms are for them, for that to happen. One thing I wanted to focus on in this 45 00:09:26.700 --> 00:09:56.070 Mark Kushner: equation here, for the internal energy is, it's really equivalent to the first law of thermodynamics. And that's a little weird, I think, in plasma physics. We don't really talk about it that much. We often just call it the energy equation but in like neutral fluid theory, people recognize all the time that this really is effectively the first law of thermodynamics. So to see where this comes from. This is telling you something about the evolution of the internal energy. Right? And so you can do 46 00:09:56.070 --> 00:10:15.950 Mark Kushner: little bit of math here. There's some subtleties here. These are partial derivatives, and this is the total derivative so effectively. What you're doing is you're moving into the reference frame of the plasma but once you do that, this is telling you about how the internal energy changes in time. This term here is related to the divergence of you. So if you have 47 00:10:16.010 --> 00:10:23.670 Mark Kushner: velocity that's diverging that's expanding so that will cool your plasma. If it's converging, that'll hit your plasma. 48 00:10:23.880 --> 00:10:44.000 Mark Kushner: And so this is really the work that's being done. And so that's this term here. And then you can have heat from various sources, and that would show up here. And so really, again, this is really just the first off thermodynamics just written in a way we're not used to when we take undergraduate physics classes. First off thermodynamics. 49 00:10:44.310 --> 00:10:47.050 Mark Kushner: okay. So first off, thermodynamics is 50 00:10:47.420 --> 00:11:02.040 Mark Kushner: great. It follows essentially from fluid theory, which is great. So when is it that we can use this thing. So the idea is, we said earlier that we're talking about gases or plasmas that are in local thermodynamic equilibrium. 51 00:11:02.400 --> 00:11:05.559 Mark Kushner: And we said, that that means that there's a well defined temperature. 52 00:11:05.780 --> 00:11:25.190 Mark Kushner: So then we can ask, what does it mean to have a well defined temperature? So the way we think about it is you talk about what's called the distribution function, or sometimes the phase space density, and the idea is, it's telling you like. If you picture the gas in this room, there's particles all over the place moving with random velocities. And so you 53 00:11:25.270 --> 00:11:45.399 Mark Kushner: you figure out how the velocities of those particles are distributed. And so this is a very famous distribution function. It's called. So. This here is 0 velocity. So this is, there's a lot of particles moving at 0 velocity, and there's very few particles moving at high velocities. And so 54 00:11:45.750 --> 00:12:13.570 Mark Kushner: this characteristic shape it's called the Gaussian or Maxwell Boltzmann. Distribution bell curve, right all those things. And so, for example, for the gas in this room the particles are distributed with velocities that kind of look like this, and this is a really special one. Because if you have this distribution and you quantify it in this form here, and you can see one of these quantities here is feed for temperature. 55 00:12:13.580 --> 00:12:26.149 Mark Kushner: right? So the temperature in this case is related to the spread of the the distribution of particles. So a higher temperature plasma would be more spread out and a colder plasma would be more 56 00:12:26.510 --> 00:12:29.050 Mark Kushner: localized at smaller velocities. 57 00:12:29.060 --> 00:12:38.040 Mark Kushner: Okay, so that's really what it means to say that you have a well-defined temperature, and that your plasma is in local thermodynamic equilibrium. 58 00:12:38.280 --> 00:12:41.160 Mark Kushner: So this is what the maximum Boltzmann. 59 00:12:43.200 --> 00:12:52.010 Mark Kushner: This is what the Maxwell Boltzmann distribution looks like in oned so this is what it would look like in 2 dimensions looking at it kind of from the top 60 00:12:52.520 --> 00:12:57.679 Mark Kushner: and for this distribution the density is effectively the area under the curve. 61 00:12:57.760 --> 00:13:04.999 Mark Kushner: So the bigger the curve is, the more particles you have and the temperature, like, I said, is related to the spread 62 00:13:05.190 --> 00:13:06.530 Mark Kushner: and 63 00:13:06.880 --> 00:13:16.789 Mark Kushner: for when you're in local thermodynamic equilibrium you can say, well, what's the the net? Random energy you have with these particles that are kind of moving around in all directions. 64 00:13:17.110 --> 00:13:29.420 Mark Kushner: And when you do this calculation, using kinda basic statistics stuff. You find that the internal energy per particle is just 3 halves. Kbt, which is what we know from thermodynamics. Right? 65 00:13:30.150 --> 00:13:34.930 Mark Kushner: By the way, if you want to interrupt with questions that's totally fine. I don't have to wait till the end. 66 00:13:35.330 --> 00:13:44.059 Mark Kushner: So this is. That's what it means to have a plasma that's in local thermodynamic equilibrium. So 67 00:13:44.290 --> 00:14:00.920 Mark Kushner: what happens if you're not in local thermodynamic equilibrium? So it just means that your distribution of the velocities of particles is anything different than this Maxwell Boltzmann distribution, literally anything. So here are some examples from plasma science. 68 00:14:01.450 --> 00:14:12.850 Mark Kushner: I like to start with this one. This is data from the solar wind satellite data. And you can see this distribution right here. Kind of looks like this distribution right here. 69 00:14:12.850 --> 00:14:34.480 Mark Kushner: right? So this one kind of looks like it might be, you know. Hand wavy, anyway, it might be in local thermodynamic equilibrium. But you can see these other 5, which are just a different locations are totally different, if nothing, nothing very little in common with this distribution. And so the point is, they're definitely not in local thermodynamic equilibrium. 70 00:14:34.480 --> 00:15:07.270 Mark Kushner: This is another example. More recently from the magnetospheric multi scale mission Mms. So this is like the highest time, resolution, distribution functions measured in space ever. And these are. This is kind of a famous example where these distributions have these crescent shapes. So this is really important in the early days of Mms. For them to identify magnetic reconnection, which is what their goal was. But again, the the key point for us is just, it's not Maxwellian. So therefore it's not in local thermodynamic equilibrium. 71 00:15:07.520 --> 00:15:16.710 Mark Kushner: There's some examples here. No temperature, plasma physics, energy density plasma experiments. And I wanted to sort of highlight one 72 00:15:16.900 --> 00:15:43.339 Mark Kushner: some of you may know, or I'll see me. he's a West Virginia University. Couple of doors down for me. He's got this experiment where you take 2 flex ropes and they basically smash together and you get magnetically connection, and you can measure distribution functions in the lab and you can see here again, these are not only in distributions, and these are kind of space, relevant, relevant plasma parameters as well. 73 00:15:43.340 --> 00:15:50.770 Mark Kushner: So you can see it's really common for plasmas to not be in local thermodynamic equilibrium kind of almost 74 00:15:51.120 --> 00:16:08.899 Mark Kushner: most plasmids, you might say, are not right. And the reason is that plasmas are often hot. They're often not very dense. So that means. There's not very many collisions and collisions, or what you need to get to drive your system down to the local thermodynamic equilibrium right? So the gas in this room. 75 00:16:08.900 --> 00:16:32.909 Mark Kushner: Collisions happen really fast. And so you you very quickly get back to local thermodynamic equilibrium. Plasma is not so much right. And low temperature plasmas. It's really interesting. That what's important there, you might think well, low temperatures. Maybe it would be in local thermodynamic equilibrium. But the fact that the electron mass and the iron mass are so different is that when you get collisions 76 00:16:32.910 --> 00:16:44.290 Mark Kushner: the electrons and the ions don't equilibrate as rapidly as you might think they would. So, even though temperature plasmas are often not any in thermodynamic equilibrium. 77 00:16:44.700 --> 00:17:11.320 Mark Kushner: just to make this a little more vivid. We have these nice animations that the folks at NASA have made. So this top one is of magnetic reconnections. So here we have magnetic field lines that are going in opposite directions, and they kind of come together. And this is gonna illustrate why you might get a plasma that's not in local thermodynamic equilibrium. So what you're gonna see when I animate it is particles representing ion. let's 78 00:17:11.695 --> 00:17:34.210 Mark Kushner: say, protons and electrons that are coming in towards the center and just watch their dynamics as they come in. You can see the electrons are doing completely in red are doing completely different things than the protons in that light blue color. And so you get effects that are different, depending on the Larmor radius of the particles. 79 00:17:34.370 --> 00:17:41.509 Mark Kushner: And so that is one thing that can get you out of level thermodynamic equilibrium. Just kind of neat. 80 00:17:42.620 --> 00:17:55.919 Mark Kushner: Another example of that is here which this is representing a collision with shock and what you're gonna see when I hit play is there's a bunch of particles here and here of different signs. 81 00:17:55.920 --> 00:18:24.390 Mark Kushner: And so representing again protons and electrons. And so here these are the electrons with the smaller gyro radius, and here ions with larger, and you can see when they pass through this shock they just start doing completely different things. Even amongst ions they're doing different things. And here amongst electrons are doing different things. And these are all particles that started at almost the exact same location. Yet they have completely different dynamics because they're finite gyro radius effects. 82 00:18:24.810 --> 00:18:26.020 Mark Kushner: Okay? 83 00:18:26.270 --> 00:18:52.019 Mark Kushner: So then we can ask the question, how do we describe the evolution of plasmas when they're not in local thermodynamic equilibrium, because it seems that they're kind of almost never in local thermodynamic equilibrium. And the answer is, we need kinetic theory. Which is also called non-equilibrium statistical mechanics. Right? So it's specifically says we're treating a plasma that's not in thermodynamic equilibrium 84 00:18:53.220 --> 00:18:57.110 Mark Kushner: skip the turbulence fun. So 85 00:18:57.140 --> 00:19:00.060 have folks studied kinetic theory at all. 86 00:19:00.190 --> 00:19:06.049 Mark Kushner: Yeah. So not everyone. Okay, good. So kinetic theory is 87 00:19:06.170 --> 00:19:35.150 Mark Kushner: Part of it is really describing once you know your distribution of particles, but you know that they're not in local thermodynamic equilibrium. How does it evolve in time. And so the way you describe this is, you have position and velocity, and you can think of it as position on one axis and velocity, and then the other axis. So it's phase space. If you're familiar with that and then you ask, Well, how does the distribution function and face density evolve in time. 88 00:19:35.620 --> 00:19:40.870 Mark Kushner: okay, so yeah. 89 00:19:41.070 --> 00:19:43.040 Mark Kushner: Oops. Okay. 90 00:19:44.330 --> 00:19:57.480 Mark Kushner: and the answer goes back a hundred 52 years to Boltzmann. So Boltzmann. in this paper, back in 1872, I wrote this 91 00:19:57.700 --> 00:20:16.989 Mark Kushner: really ugly equation. So this is actually, before vector symbols are around. Even. So, it looks a little different in the old notation, but this is what it looks like in the modern notation. So you see, this tells you how the distribution function is locally changing in time, in phase, space. 92 00:20:16.990 --> 00:20:22.650 Mark Kushner: And this says that if there's a gradient of the distribution function in position. 93 00:20:22.650 --> 00:20:47.520 Mark Kushner: then that can give you a change in your face, in your distribution function. If there's a gradient in the velocity direction. And there's a force then that can lead to another way that you can change your distribution function. And then the C here is Boltzmann's version of C, but what see means is that it's a basically an operator that describes collisions. 94 00:20:47.630 --> 00:20:57.990 Mark Kushner: And Boltzmann had his particular collision operator. Here, I'm just gonna leave it unspecified. It's whatever collision operator. You need to describe your system. 95 00:20:58.280 --> 00:21:21.979 Mark Kushner: Okay, so this is the equation for kinetic theory. You can see in Olson's case this equation here at the bottom, right? This collision operator has integrals in it. So if you've never seen this before as ugly as this is, it's actually. And this is a differential equation. But this term has integrals in it. So it's an integral differential equation. 96 00:21:21.980 --> 00:21:28.489 Mark Kushner: as you probably figure out, it's really hard to solve analytical solutions are 97 00:21:28.540 --> 00:21:32.150 Mark Kushner: not. They're are few and far between. 98 00:21:32.920 --> 00:21:35.480 Mark Kushner: okay, so that's the equation for 99 00:21:35.800 --> 00:21:50.540 Mark Kushner: the distribution function. From this equation. If you take essentially average it over all articles with all velocities. Then, what you get from this are 100 00:21:50.960 --> 00:22:10.619 Mark Kushner: basically what's happening in bulk. And so the way we think of this is just like, if you have a distribution of mass you can think of moments of the mass the moments of the mass distribution. So how much mass there are? The center of mass is into the left or right. Those kinds of things you can do the same kind of tricks with 101 00:22:10.620 --> 00:22:23.470 Mark Kushner: the with the distribution function. So if you average up the distribution function over all velocities that just tells you how much stuff there is. So that's the number density. And we call that the zeroth moment. 102 00:22:24.200 --> 00:22:37.079 Mark Kushner: If we first multiply by the velocity and then integrate we get the first moment. And this is just telling you if there's more particles, or essentially more motion to the left versus the right. So this is the bulk, flow velocity. 103 00:22:37.260 --> 00:22:59.989 Mark Kushner: And you can keep going if you multiply by 2 powers of the velocity and then integrate, you know, average up over velocity. Turns out, this is related to the pressure. But in kinetic theory, in principle, this is a tensor second rank tensor and again, you can keep going. So you can take 3 powers here, and that's related to the heat flux or the tensor heat flux. 104 00:22:59.990 --> 00:23:10.649 Mark Kushner: and when we think about things like moment. Moments of mass distributions right? The total mass, the center of mass moment of inertia. Usually. That's it. We stop at the second 105 00:23:10.650 --> 00:23:34.439 Mark Kushner: moment. But there's no reason to stop, and actually, in principle, if you have some strange distribution, and you want to describe it, you effectively need all you need every moment which would mean an infinite number of moments to be able to specifically distinguish that this is what the shape of your distribution function is. So in principle you get an infinite number of moments. 106 00:23:35.180 --> 00:23:47.749 Mark Kushner: it's hard to solve an infinite number of equations. So that's not necessarily a good thing. But let's put that aside for a minute? And go back to this question about energy conversion. 107 00:23:48.220 --> 00:24:02.969 Mark Kushner: So when you're doing kinetic theory, when you have distributions that are not in local thermodynamic equilibrium, we said. The the definition itself is that you can't define a temperature in a well-defined way. 108 00:24:03.320 --> 00:24:33.169 Mark Kushner: But what you can do is define an internal energy or an internal energy per particle, because you can still just say, Well, what's the average random velocity? Before we would have had the Maxwell Boltzmann distribution here. But if you have some different distribution, you just put that distribution in here, and that tells you what the average energy is, and that's totally fine, right? And so what people do is, you can say, well, we know, if it was in equilibrium, the internal energy would just be 3 halves. Kt. 109 00:24:33.180 --> 00:24:51.300 Mark Kushner: so if for something not in thermodynamic equilibrium, we can just define a lot of people just call it temperature. But I call it an effective temperature, because it's not in thermodynamic equilibrium, and it's what the temperature would be if it wasn't thermodynamic equilibrium, but had with the same amount of internal energy. 110 00:24:51.410 --> 00:24:55.930 Mark Kushner: So this is essentially the kinetic version of temperature. 111 00:24:56.720 --> 00:25:21.139 Mark Kushner: And so for any distribution, right? What you can do is find the density, find the bulk flow, find this effective temperature, and say, Well, if it was in local thermodynamic equilibrium, it would be this Maxwell Boltzmann distribution. With these this temperature. And this you know, the bulk flow and the density that you have. So that is called the Maxwellianized distribution. 112 00:25:21.360 --> 00:25:32.789 Mark Kushner: a lot of people have been using this lately. But it turns out because all the way back to 1965 so it's been around for a long time. This will come up later. That's why I'm telling stuff right now. 113 00:25:33.700 --> 00:25:47.270 Mark Kushner: So then what you can do is say, well, if I wanna look at energy conversion, in a plasma that's not in local thermodynamic equilibrium. You take moments of the 114 00:25:47.530 --> 00:25:54.029 Mark Kushner: the Boltzmann equation itself. And that tells you how the moments of 115 00:25:54.050 --> 00:26:09.909 Mark Kushner: the the the distribution function evolve in time. So the zeroth moment. If you just take the whole Boltzmann equation and integrate it over velocity space you find the continuity equation. It just says that the number of particles is conserved, which is great. 116 00:26:09.910 --> 00:26:26.579 Mark Kushner: If you first multiply by the velocity and then integrate. You get the momentum equation, which looks a lot like that mo momentum equation. I showed you earlier in the fluid picture. But now the pressure is a tensor, instead of being a scalar like it was before. 117 00:26:26.580 --> 00:26:43.950 Mark Kushner: Things like that. And then, similarly, you can find the internal energy evolution equation related to this effective temperature thing. And it looks like this. And again, it's very similar to what it was before. But this is a little more 118 00:26:44.040 --> 00:26:50.450 Mark Kushner: involved, because there's sensors and all this stuff, and there's some heat fluxes and things like that that show up 119 00:26:51.070 --> 00:27:17.129 Mark Kushner: alright. So like, I said in principle. When you're not in local thermodynamic equilibrium, you need an infinite number of equations. You can just keep taking as many moments as you want and actually you have to keep taking moments, because here you have the density in order to tell how the density evolves in time. You need the bulk flow velocity. So you need an equation for the bulk flow of velocity. But that depends on the pressure. 120 00:27:17.130 --> 00:27:43.909 Mark Kushner: So you need an equation for the pressure, the internal energy that's here. But this depends on heat should be the third moment. And so in principle, you have to just keep going, and what you get is an infinite set of equations, which, of course, you can't solve an infinite set of equations. So that's bad. And this is called the closure problem. Because without some sort of assumption somewhere, you would need an infinite number of equations to describe your plasma. 121 00:27:44.220 --> 00:27:49.040 Mark Kushner: which is bad. alright. So 122 00:27:49.610 --> 00:28:13.810 Mark Kushner: again, we can write down the equations for the evolution of the kinetic energy density. And this is the equation for the internal energy density. And this is again pointing through, and we can do all the same stuff that we did before. Where we looked at the evolution of the energies. But now these are not required to be in local thermodynamic equilibrium. Right? So that's great. 123 00:28:14.130 --> 00:28:19.180 Mark Kushner: and I think I won't go into too much detail here. Actually. 124 00:28:19.200 --> 00:28:22.820 Mark Kushner: yeah, I think I'm gonna skip this stuff because it's a little 125 00:28:23.670 --> 00:28:53.519 Mark Kushner: little off topic. So I'm gonna skip those things. But the important thing I wanted to mention is that if you look at the internal energy equation just like very similar to what we did before, right in in the fluid model, we said that the internal energy equation is related to the first law of thermodynamics, right? And so this equation now is valid even outside of the fluid model. It's valid for distributions that are not plasmas that are not in local thermodynamic equilibrium 126 00:28:53.690 --> 00:29:10.649 Mark Kushner: right? And so just like it was before these 2 terms are related to the internal energy. And so you can write it like this. This term, right here can be broken up into these 2 terms, and we see something that kind of looks like that that compression or expansion term we saw before. 127 00:29:10.660 --> 00:29:18.840 Mark Kushner: And then this is some extra stuff, and the queues have to do with. He's you might expect right? So, sure enough. This really is. 128 00:29:19.170 --> 00:29:27.880 Mark Kushner: Well, the way I like to think of it is, it's really a non local thermodynamic equilibrium generalization of the first law of thermodynamics, because 129 00:29:28.230 --> 00:29:43.140 Mark Kushner: it no longer requires for you to be in the from a dynamic equilibrium. But it's still telling you how there's this, what the interplay is between changing the internal energy, doing work on it, checking heat, those kinds of things 130 00:29:43.490 --> 00:29:54.389 Mark Kushner: and so like, I said, the important thing is, this is rigorous. We've not made any assumptions anywhere here, and it's really just telling you how things work, even when you're not in local thermodynamic equilibrium. 131 00:29:55.030 --> 00:30:19.399 Mark Kushner: So that's great. And you might think that's the end of the story. But the rest of the story is this so the energy is essentially the second moment of the distribution function, which is great. And in in this equation here, that also depends on. It's related to work and work is, remember changing the density. 132 00:30:19.400 --> 00:30:31.230 Mark Kushner: So the density itself is the 0 moment of the distribution function. So the first law of thermodynamics, or it's generalization relates how, when you change the second moment 133 00:30:31.310 --> 00:30:49.060 Mark Kushner: you know, how does the 0 moment change when you also change the second moment. Right? But we said that in when you're not in local thermodynamic equilibrium, these fluid variables aren't the fundamental variables. It's the distribution function, right and the distribution function can have infinite number of moments. 134 00:30:49.290 --> 00:31:00.249 Mark Kushner: Right? And so if you're not in local thermodynamic equilibrium, okay, the second moment is energy. We like that. Because if there's a name for it, that's great. That's what we're used to thinking about when we do. 135 00:31:00.250 --> 00:31:24.909 Mark Kushner: Fluid theory. But why, what makes the second moment so special like. Why not any other moment? Why not the 40 seventh moment? Right? If you have this system and it's evolving, and it's not in local thermodynamic equilibrium. If the 40 seventh moment changes, it's telling you your systems evolving. And that's maybe that's important, too. And so only thinking about energy kind of leaves out all these other moments of the distribution. 136 00:31:25.360 --> 00:31:27.510 Mark Kushner: And 137 00:31:28.930 --> 00:31:53.199 Mark Kushner: that's not necessarily a good thing, right? And actually, I'll go back. And so yeah, in the work I'm about to show you is we started thinking more about this? Because, essentially, you know, people in the field are really focused on energy. Because why not? That's what the fluid model tells us we should be thinking about. So we started thinking about what happens if we start thinking about these other moments again. 138 00:31:53.570 --> 00:32:03.110 Mark Kushner: And so what that kind of gets us to think about is what happens really holistically, you might say, when you're not in local thermodynamic equilibrium. 139 00:32:03.180 --> 00:32:08.040 Mark Kushner: And so the inspiration for what we were kind of thinking about is 140 00:32:08.720 --> 00:32:32.309 Mark Kushner: what happens to the second law of thermodynamics when you're not in the local red dynamic equilibrium. And the answer, I'll skip ahead to the answer. But here's Boltzmann again. He figured this out in 1,877. So we're coming up on the Ces Cesco centennial for that one but he basically first. So the second law of thermodynamics is that the entropy is non decreasing in time. 141 00:32:32.630 --> 00:32:46.840 Mark Kushner: And so, if you want to try to describe this, so this is valid in local thermodynamic equilibrium, you can see the usual way, right? This has a temperature in it, right? So if you're not in local thermodynamic equilibrium. 142 00:32:46.840 --> 00:33:11.809 Mark Kushner: you don't even have temperature. So you can. This. This doesn't mean anything. When you're out of local thermodynamic equilibrium. So Boltzmann said, what is entropy when you're not in local thermodynamic equilibrium and you've undoubtedly seen this in like a thermodynamics class statistical mechanics. So let's say you have a box, and you have a bunch of particles in there. Right the entropy, as Boltzmann defined it, as 143 00:33:11.810 --> 00:33:17.449 Mark Kushner: of the number of ways you can exchange particles and get the same system back. 144 00:33:17.450 --> 00:33:35.800 Mark Kushner: And so here, you have 4 particles on the left and 4 particles on the right, and so you can exchange them. I didn't put the numbers here, but there's like 70 different ways, that you can arrange these particles in different ways to still have 4 on the left and 4 on the right. 145 00:33:35.850 --> 00:34:03.090 Mark Kushner: And you contrast that with, let's say, all the particles were on the left. So there's really only one way to put all the particles on the left. All the particles are there. And so Boltzmann said, that this is re, the number of different it's called microstates. That gives you a given macro state, which is just related to the number of particles, or the number of permutations of these particles is related to entropy 146 00:34:03.660 --> 00:34:29.199 Mark Kushner: with the idea that the more disordered States are the ones that will are more common, right? And so, for example, for the gas in this room, we never have to worry about all the gas being up in a square centimeter in the corner of the room, and us not being able to breathe right. The gas is everywhere right, and so the equation that puts them right down, for this is that the entropy is K log, W. It's up here on is 147 00:34:29.210 --> 00:34:40.059 Mark Kushner: this grave? Some there? And the idea, like we said, is that systems tend to evolve towards the more disordered state or the higher entropy states. 148 00:34:40.409 --> 00:35:04.679 Mark Kushner: So what Boltzmann did is, he showed that in kinetic theory you can write this W in terms of the distribution function, because the distribution function is telling you where the particles are. So if you know the distribution function, you can figure out how many ways you can exchange the particles. And so it turns out that the entropy and kinetic theory is the integral of an entropy density here. And the entropy density is related to F log. F, 149 00:35:04.980 --> 00:35:08.459 Mark Kushner: okay, so that's kind of this famous form going back to Boltzmann. 150 00:35:09.120 --> 00:35:31.190 Mark Kushner: and what Boltzmann did, which is absolutely amazing and crazy, and we don't have huge amount of time to go into it. But he said. Well, what if you take the time derivative of S, because this is related to the second law of thermodynamics? How the entropy changes in time, and what he was able to show is when you take a time derivative. Here you end up with terms that are proportional to Df Dt 151 00:35:31.340 --> 00:35:55.260 Mark Kushner: right, and he invented the Boltzmann equation. So he he knew exactly how F. Evolves in time. So you plug in for Dfdt. And what you get is the Dsdt is some function of the collision operator, and he invented a collision operator. So he put that in and amazingly was able to show that the entropy was not decreasing. 152 00:35:55.640 --> 00:36:14.310 Mark Kushner: Okay, and all of this is in completely in terms of the distribution function. And it's valid for anything whether or not. You're in local thermodynamic equilibrium. And so in many ways. I would say, this generalizes the second on thermodynamics but when you're not in local thermodynamic equilibrium. 153 00:36:14.680 --> 00:36:23.060 Mark Kushner: Alright, there's a lot of philosophical issues, if those of you that are historians of physics probably know 154 00:36:23.140 --> 00:36:35.489 Mark Kushner: what happened to Boltzmann. But anyway, there's a lot of fighting about it. And 155 00:36:35.580 --> 00:36:46.449 Mark Kushner: yeah, let me bring it up. Just so we have it here on the screen. So what he did is he essentially assumed, like, you know. Back then they were thinking about like the guests in this room. Right? So 156 00:36:46.800 --> 00:37:13.670 Mark Kushner: What Boltzmann said is just pretend that every collision is essentially a binary collision, the particles 2 particles with different velocities. And so that depends on their distribution function. So that that's why you see a product of 2 factors of F here, and so they they come in with some velocity, and they come out with some velocity. And so that's why there's 2 factors here and 2 factors here. And so this sigma. 157 00:37:13.780 --> 00:37:40.499 Mark Kushner: Okay, he calls it B is related to the impact parameter. So it's related to the Cross section for a particular collision to happen. So you're sort of integrating over that cross section. And this contains the relative velocity which says that if 2 particles are moving at the same velocity they're never going to collide right? They need to be moving relative to each other. And so it's really just kind of bookkeeping of what binary collisions should look like. 158 00:37:41.010 --> 00:37:42.379 Mark Kushner: Yeah, does that make sense? 159 00:37:47.110 --> 00:37:48.770 Mark Kushner: Okay? So 160 00:37:49.410 --> 00:37:57.820 Mark Kushner: so the point is, for 150 years, we've known how to generalize the second law of thermodynamics for non-equilibrium on lte systems. 161 00:37:58.500 --> 00:38:28.070 Mark Kushner: So this was kind of a pet project over during the lockdown pandemic. We had lots of time to think and try things. And so I'm just gonna essentially tell you the results of what we were thinking about. So the idea is, what if? And this is really weird? But let's just go with it for a bit. What if you start with this kinetic entropy, density, this F. Log f thing that Boltzmann wrote down. And what if you take the time? Evolution of this quantity? 162 00:38:28.090 --> 00:38:38.970 Mark Kushner: Right? So dt ddt of it. So from that you can derive an evan evolution equation for the kinetic entropy, density. And of course, again, Boltzmann did all this stuff. 163 00:38:38.990 --> 00:39:07.039 Mark Kushner: Fighting this paper from 2,018. But Boltzmann did it 150 years ago. And the answer is this equation, which looks like a continuity equation, right? And there's this source term over here for collisions. And this J here represents a flux like an entropy flux. So this basically says, if you don't have collisions, entropy basically is conserved to just kind of convex with your plasma and get compression and stuff like that. But otherwise that's basically it? 164 00:39:07.820 --> 00:39:20.929 Mark Kushner: So we did 2 things right. The first thing is, we said, well. it's okay to have a continuity equation. Right? It tells you how things are changing locally. But 165 00:39:20.930 --> 00:39:44.849 Mark Kushner: if you're interested in like, you know, changes in energy and things like that, if you're looking at something locally like I'd say, put a probe here that measures the thermometer. You measure the temperature. And you say, Oh, the temperature just went up right. Is that heating? Well, it can be, but it also could just be that your plasma is flowing or your gas is flowing and hotter. Plasma over here just moves in 166 00:39:44.950 --> 00:39:56.999 Mark Kushner: to where your thermometer is, and then your thermometer would say, it's heating, but it's really not. It's just moving right? So a better way to describe. If you have heating is to go into the reference frame of the plasma. 167 00:39:57.000 --> 00:40:21.939 Mark Kushner: and in that reference frame, if you see the temperature go, that, then you know this is heating right? So we did the same thing with the entropy. As we start. We went into the co-moving or the Lagrangian reference frame and the other thing we did is there is this. And again, you're just gonna have to trust me on. Why, we did this thing cause I'll come back to it. Is. You can break up the entropy into 2 pieces. 168 00:40:21.940 --> 00:40:28.200 Mark Kushner: One that's called the position space entropy, and one that's called the velocity space entropy. 169 00:40:28.200 --> 00:40:44.329 Mark Kushner: And again, just kind of work with me on this for a second, and just trust me as we do this. So if you take this equation. Make this decomposition for the entropy and then move it into the co-moving Lagrangian reference frame. Then you get this equation right here. 170 00:40:45.000 --> 00:40:51.770 Mark Kushner: Okay, so it looks kinda different than this one, but it's got the same information, and there's no approximations going on anywhere. 171 00:40:51.910 --> 00:41:12.080 Mark Kushner: so now, what is really interesting is, we have this position, space entropy, density, and it depends on density here. So it's n log, N, and here we have a Ddt of it, right? So Ddt of the position. Space entropy divided by N is basically just Ddt of Ln, N. 172 00:41:12.200 --> 00:41:18.850 Mark Kushner: Okay, so the point is, it's all related to Ddt of N. And Ddt of N is compression 173 00:41:18.870 --> 00:41:30.289 Mark Kushner: right? If you have compression, your density goes up. If you have expansion, your density goes down so as we skip to this next slide. Here we find that this 174 00:41:31.780 --> 00:41:39.469 Mark Kushner: okay, I didn't write it here, but that first term is really going to be related to compression. So it actually turns out that it's exactly this work term 175 00:41:39.780 --> 00:41:42.460 Mark Kushner: that you would get from the first law of thermodynamics. 176 00:41:42.710 --> 00:41:54.120 Mark Kushner: which is weird because we started with the entropy. So let's go with it a little bit. Now we have the second term. This velocity, space, entropy, density. and again 177 00:41:54.560 --> 00:42:09.509 Mark Kushner: the the wonders of having free time during pandemic lockdown. If you break this into these 2 terms, you get this one term here, and if you take the time derivative of this, you find that this is related to the internal energy 178 00:42:09.790 --> 00:42:23.179 Mark Kushner: the same. The regular internal energy from non equilibrium is not in local thermodynamic equilibrium, but it's still the internal energy density. And then the then you get this extra term. 179 00:42:23.280 --> 00:42:28.840 Mark Kushner: And so this notation wise. What what you can see is that if we kind of 180 00:42:28.840 --> 00:42:53.829 Mark Kushner: couple of these together, we can think of this is this term is really just the internal energy. So this term has the same unit, so it has to have the units of energy as well. And so, if we just call this the relative energy, and this will call the relative entropy, because it turns out that God did this in 1965 also. Then this whole term, the S. Delta SV. Term is related to the internal energy plus this external 181 00:42:53.830 --> 00:42:54.809 extra piece. 182 00:42:55.150 --> 00:43:08.560 Mark Kushner: Okay? And then, when you do the Entropy flux term, you can do a similar kind of thing where you get the regular old heat part that we had before, plus a relative part, basically the rest of it 183 00:43:08.610 --> 00:43:27.519 Mark Kushner: right? And so when you put these all together. These 4 terms end up being where you can write them like this. And this looks like the first law of thermodynamics. Right? And first of all, this is crazy because the first of thermodynamics is conservation of energy right? And 184 00:43:27.590 --> 00:43:49.550 Mark Kushner: the second law of thermodynamics has to do with entropy. But what we just did is we started with entropy, and we get something that looks like the first law of thermodynamics. And in particular, if you're in local thermodynamic equilibrium. I'll go over this in the next slide. But this relative term over here goes away. And so this goes away, and this goes away, and you literally get the first of thermodynamics back 185 00:43:50.050 --> 00:44:14.389 Mark Kushner: from entropy. not energy, which is again crazy. So we got excited about this stuff. we decided to be brash. So we call this the first law of kinetic theory. And and in a sense, we're basically arguing that it's a non-equilibrium thermodynam or non thermodynamic equilibrium generalization, if you will, of the first law of thermodynamics. 186 00:44:14.420 --> 00:44:19.460 Mark Kushner: which is again crazy because it's coming from entropy. Yeah. 187 00:44:19.490 --> 00:44:35.340 Mark Kushner: But to evaluate that, you still need to know the distribution function. Right? So how is it a generalization? Because if you know the distribution function, there's not a culture problem you couldn't solve. 188 00:44:36.170 --> 00:44:57.230 Mark Kushner: Yeah, it's just so. Yeah, there's in a sense, there's no real. There's not. There's no real new information relative to the Boltzmann equation. It's it's all in there. It's still the same stuff. We didn't make any approximations. It's it all comes from Boltzmann. So it's just another way to think of it. Another way to to write it, and in what I'll show you in a little bit. 189 00:44:57.290 --> 00:44:59.319 Mark Kushner: let's see how we doing on time. 190 00:45:00.380 --> 00:45:25.370 Mark Kushner: 10 min. Okay, well, I'll show you in a little bit is how we might be able to use this to think of things a little differently than we might normally think of it, and we get some insights out of it. But the idea is just as you were saying, Scott, this effectively avoids the closure problem because you don't need an infinite number of equations to get each moment they're all in there. Right? So let's talk about that a little bit. 191 00:45:25.520 --> 00:45:48.060 Mark Kushner: So the key of all of that is this relative entropy term. And again in the pandemic, I thought we invented this, and we went back in literature and found that grad did everything long before any of us did it? But anyway, so this relative entropy term it actually comes from information theory. If you're into that. It's related to the callback libler divergence 192 00:45:48.060 --> 00:46:17.030 Mark Kushner: which tells you the statistical difference between 2 different distribution functions. So here our reference. Distribution is a Maxwell Boltzmann distribution, and then F is just whatever the distribution is right. And so, if you have a distribution, that is Maxwellian, then F over FM is one. And so the natural log is 0, and the relevant key is 0. So it's 0 if you're in the Maxwellian, a Maxwellian state, and it turns out that if f is anything else. 193 00:46:17.210 --> 00:46:20.179 Mark Kushner: then the relative entropy is negative. 194 00:46:20.260 --> 00:46:33.300 Mark Kushner: right, and the more different your distribution is from being Maxwellian, the bigger magnitude the relative entropy is. So it's really a measure of how different, how far away from local thermodynamic equilibrium you are! 195 00:46:34.070 --> 00:46:36.660 Mark Kushner: And so with 196 00:46:36.660 --> 00:46:43.999 Mark Kushner: what when we did this derivation. And we get this term that's related to the time derivative of the relative entropy that 197 00:46:44.000 --> 00:47:08.309 Mark Kushner: we grab didn't do that as far as we know but that you know there was. There's physics in there, right? So we said, Well, what does it mean if the relative entropy is changing in time, right? And what it means is that your distribution function? It's telling you that you're not any it. Well, that's the local at a local time. It's telling you how different your distribution function is. So it's time derivative is telling you how that's changing. 198 00:47:08.550 --> 00:47:20.379 Mark Kushner: Right? So as you if you have a system that's not in local thermodynamic equilibrium, and it's evolving in time. This is telling you how rapidly the the non lte part of it is changing in time. 199 00:47:21.340 --> 00:47:26.460 Mark Kushner: And so that's really what this means, physically. And 200 00:47:26.740 --> 00:47:42.899 Mark Kushner: yeah. So we've decided to, since this is related to something that has units of energy. We decided to write this in the form of something that's a power density. So that rate of energy change per, you know, volume 201 00:47:42.900 --> 00:48:07.880 Mark Kushner: and so the reason that we would do that is because, we wanna be able to compare it to actual power density. So things like j dot e convert other energy conversion metrics. So now we can compare these things to this quantity here, and be able to say, like, for example, the gas in this room. It really doesn't depart very far from local thermodynamic equilibrium. 202 00:48:07.880 --> 00:48:22.239 Mark Kushner: So this S relative, the relative entropy, is almost 0. And so this really wouldn't be significant for the gas in this room. And you could say, Okay, this is a lot smaller than all these other things going on. And we could just ignore it right. But in a plasma, we said. 203 00:48:22.520 --> 00:48:49.830 Mark Kushner: plasmas are often really far from local thermodynamic equilibrium. So this might be important, and in in particular, you can quantify. You can say it's half as important or twice as important, or a millionth as important. Because we have this quantity to to calculate it. My grad student Hassanberg. We gave it this funny name, higher order, non-equilibrium terms or hornets. I get to put up this ugly picture. That's I didn't take that 204 00:48:50.640 --> 00:48:52.880 Mark Kushner: And so. 205 00:48:52.940 --> 00:49:16.769 Mark Kushner: coming back to this, how do we think of this in terms of the first law of thermodynamics? Right. So the first law of thermodynamics is still correct. I'm not saying it's not correct. It's still correct. It's still conservation of energy. Nothing will change that but one way to think about what this equation is sort of generalized thing is telling you is that there's these other pieces that also come into it. So you can sort of 206 00:49:16.770 --> 00:49:27.149 Mark Kushner: Victoria Victorialy say the first law of thermodynamics is just relating the work and the heat and the internal energy. So black arrows are saying, there's conversion going on. 207 00:49:27.260 --> 00:49:41.089 Mark Kushner: And so, for the first time, kinetic theory, you still have work. You still have heat, you still have internal energy. You still have these black arrows going between them. Nothing's changed any of that. The only thing that's changed is, you have this other box. 208 00:49:41.180 --> 00:49:54.749 Mark Kushner: Related to this relative part, which is basically quantifying everything that's not in local thermodynamic equilibrium and a similar quantity for heat. And so these red arrows denote that you can get this exchange of 209 00:49:55.300 --> 00:50:15.970 Mark Kushner: it's not energy, right? But it's kind of this, this non lte enous of of your system. And then there's coupling between the 2, because, the internal energy can be affected by the heat, and the heat is part of the non lte stuff right? And Lte, the heat flux is 0. So the non lte part, that's how it couples in. 210 00:50:17.710 --> 00:50:18.640 Mark Kushner: Okay. 211 00:50:20.400 --> 00:50:45.350 Mark Kushner: alright. So I think, since I'm almost out of time I'll just show you little snippets of what we've been trying to do with this, to make sure our interpretation makes sense. Maybe try to learn some physics out of it. So we did some simulations of magnetic reconnection. So here, oppositely directed magnetic fields. I'll skip a lot of the details just because we're running a little short on time. But this here is the part related to the non lte stuff. 212 00:50:45.690 --> 00:50:50.499 Mark Kushner: And so out here. It's white, and that means 0. 213 00:50:50.950 --> 00:51:07.599 Mark Kushner: So in this case, out here, your distributions and this one helps to your distributions are effectively Maxwellian. Everything's a local thermodynamic equilibrium, not changing in time, so nothing exciting is going on. As you get closer to the point where reconnection happens. 214 00:51:07.600 --> 00:51:31.970 Mark Kushner: You can see blue so lights up in blue, and that means these distributions are moving away from local thermodynamic equilibrium in the case of reconnection, this stuff has been studied for 15 years, so we know exactly physically why these distributions are not Maxwellian anymore? So this is just kind of a proof of principle that says, Hey, we can actually see this right, and if that it agrees with this interpretation, I just gave you 215 00:51:31.970 --> 00:51:52.779 Mark Kushner: and then in here you can see a little bit of red. So that means these distributions are coming back towards Maxwellian distributions. And again in reconnection. We understand all this stuff. It's the plasma thermalizing as it moves out away from the reconnection site and things like that. So it really seems to be consistent with the interpretation we have. 216 00:51:52.860 --> 00:52:09.239 Mark Kushner: And then here you can compare the relative part to, let's say, the change in the internal energy. And so this is their ratio. Log scales and stuff like that. But the key is that so way out here in the blue? 217 00:52:09.240 --> 00:52:34.039 Mark Kushner: basically, the relative entropy stuff, the non lte stuff just doesn't matter. So this dark blue represents the relative stuff not mattering. And you can basically ignore it. But anywhere like this light blue or even into these reds, is where, dynamically. You're seeing significant changes of the non lte part compared to the regular energy part and because they have the same units, we can really make that comparison. 218 00:52:36.590 --> 00:52:53.320 Mark Kushner: You can keep going. And do. We did stuff with reconnections. So we use this hornet quantity and compared it to these other energy conversion metrics. We didn't do Jd, in this plot, actually. But we did like compression. For example, here heat flux. 219 00:52:53.320 --> 00:53:09.760 Mark Kushner: We did this for reconnection, and here's for turbulence. And again, one of the nice things about this is you can quantify it. So here hornet goes up to Point 7. And whatever units we're using. And compression in this case goes up to O, 3, 8. 220 00:53:09.890 --> 00:53:26.049 Mark Kushner: Okay, so it's maybe you know, one fifth or so. Okay, so not huge, but not small. In the turbulent simulation. If you average over the entire domain, it turns out at least for this one simulation. The non lte stuff can be like two-thirds 221 00:53:26.070 --> 00:53:37.979 Mark Kushner: of the energy conversion of the amount of energy conversion. So in that case it's actually pretty significant. And these are the types of things. Now, you can just make a plot and look at it and average it up and and get these numbers out. 222 00:53:39.220 --> 00:53:53.870 Mark Kushner: Just really quick. Been working with some colleagues Matt Argo in particular New Hampshire. So he uses this magnificale mission, which is measuring these distribution functions faster than any previous satellite. 223 00:53:54.190 --> 00:54:18.350 Mark Kushner: and he's been able to calculate these entropy related quantities over here, and we did some particle and cell simulations of the same event. So compare, you know, are we getting entropies that make sense compared to what they're getting in the data? I'm not gonna lie. It's not perfect. But there's, you know, some reasonable agreement, and there's some trends that are essentially pretty similar. So this gives us 224 00:54:18.350 --> 00:54:30.959 Mark Kushner: some suggestion that it it might be possible to use Mms data, and be able to, you know, apply what we've learned in our simulation studies to what's going on in space. So that's 225 00:54:30.970 --> 00:54:34.399 Mark Kushner: This is active research. They're all 226 00:54:34.560 --> 00:54:36.630 Mark Kushner: hopefully get some good stuff out of. 227 00:54:36.870 --> 00:54:52.320 Mark Kushner: I think maybe I'll skip, land out and think, throw almost at a time and just go to potential applications for this. So like I said, the hope is, I mean, it was really fun to do this project, but the hope is it might be useful for something. So 228 00:54:52.320 --> 00:55:09.289 Mark Kushner: you know, going back you know any plasma simulation where you're getting distribution functions out of it. You might be able to do these kinds of calculations so particle and cell for hybrid simulations. And as it shows satellite observations. So 229 00:55:09.290 --> 00:55:31.640 Mark Kushner: you know, we're trying to apply it to reconnection and turbulence. We started collaborating on some collisionless shock work, and we've been doing wave particle interactions with land out damping, for example. But I haven't done much of, and I'm trying to work on the theory a little bit to make it more applicable as a low temperature plasmas and things like that. So that could be interesting. 230 00:55:31.640 --> 00:55:38.129 Mark Kushner: And then systems outside of plasma physics that also use the Boltzmann equation. 231 00:55:38.130 --> 00:56:04.120 Mark Kushner: This stuff might be useful for them. So, like neutrino physics, dark matter physics, they start from the Boltzmann equation as well. So that could be really interesting. Also, outside of plasma, physics is like molecular dynamics, simulations, chemistry and biology applications. So I'm not really an expert on these kinds of things, but it's something that I think would be interesting to pursue. And it turns out 232 00:56:04.120 --> 00:56:18.939 Mark Kushner: there's also a quantum mechanical analog to what we did. We didn't know about it when we did our work, but we found that someone doing quantum statistical mechanics for quantum entanglement. Did essentially the the analogous 233 00:56:18.940 --> 00:56:42.500 Mark Kushner: study that we did and when I read their paper I'm not. I don't do quantum mechanics anymore. But I found it really hard to interpret the physics of what they were talking about. So I think, being able to do this classically in a plasma and being able to really pick out the physics might help us. Tell them, hey, this is the physics of what you're doing, and you have these equations. But here's physically what's going on 234 00:56:43.310 --> 00:56:50.359 Mark Kushner: and I think I will just end there. So thank you for your time and attention. 235 00:56:50.750 --> 00:56:51.760 Mark Kushner: Okay? 236 00:57:04.260 --> 00:57:06.430 Mark Kushner: Questions, comments. 237 00:57:10.460 --> 00:57:22.929 Mark Kushner: Yes. So you talked about this essentially, the metric is totally dependent on the whole equation. You try doing any averaging so that you could express it purely in terms of fluid variables. Be able to tell like or 238 00:57:23.080 --> 00:57:26.780 Mark Kushner: Oh, that's that's a really good question. 239 00:57:26.940 --> 00:57:38.749 Mark Kushner: I mean, in the fluid sense, you might argue, it's essentially 0, right, at least in you know. Certainly in Lte would be 0. So one of the things I kind of skipped over here is 240 00:57:38.820 --> 00:57:45.050 Mark Kushner: one of the things that I hope might, this might be useful, for as well is people doing 241 00:57:45.080 --> 00:58:07.450 Mark Kushner: like closures? Right? So in a plasma, you, you might say. Well, let's say for the gas in this room. The way you close the system of equations is with the collisions. And you, you say, Okay, if it's a small enough departure from Lte. You you can end up showing that you get things like viscosity and resistivity and 242 00:58:07.770 --> 00:58:26.879 Mark Kushner: thermal conductivity and all that. And once you do that, you then close your equation so you can run a fluid. You can use the fluid model with those corrections, non lte corrections, right? But if you're really far from thermodynamics equilibrium, it's really questionable about whether that would work right. So I think the short answer to your question is 243 00:58:26.880 --> 00:58:49.980 Mark Kushner: no cause. It's really hard, right? But I think it's what might be really cool is that some people are doing like even machine learning to try to. You like you run a particle in cell code, and you say that the machine use machine learning to say, Okay, this is the correction you should use in your fluid model, and what they get is well, case what we know. It's a heat flux or things like that. 244 00:58:50.470 --> 00:59:13.480 Mark Kushner: And so one thing that I think would be cool looking forward would be, instead of doing that to approximate the heat flux maybe do that on the entire non lte, you know, to all orders, and that might be able to. Maybe the machine is smarter than we are and can tell you, okay, if you use this closure, it'll effectively capture everything instead of just like a low order corrections. 245 00:59:13.790 --> 00:59:15.260 Mark Kushner: But that's 246 00:59:15.360 --> 00:59:31.059 Mark Kushner: I don't. I don't do much with machine learning. So that's definitely a future work. We have a question from online people. If you could show from Robert Krasny if you could show the London pay 247 00:59:33.880 --> 00:59:36.119 Mark Kushner: and Robert, can he ask you a question. 248 00:59:39.540 --> 00:59:42.080 Mark Kushner: I can't hear you. You're you're muted. 249 00:59:43.820 --> 00:59:48.169 Robert Krasny: So yeah, I just wanted to see Paul. Could you just quickly tell us what 250 00:59:48.450 --> 00:59:50.450 Robert Krasny: we should know about 251 00:59:51.040 --> 00:59:52.269 Robert Krasny: the results here. 252 00:59:56.540 --> 00:59:58.089 Robert Krasny: Now, I can't hear you. 253 00:59:58.650 --> 01:00:06.610 Robert Krasny: Okay, what's on there? 254 01:00:08.440 --> 01:00:10.579 Robert Krasny: What did you find about landau damping? 255 01:00:10.950 --> 01:00:19.570 Mark Kushner: Oh, what do we find about? Okay, thank you. so what we did with is 256 01:00:19.590 --> 01:00:44.389 Mark Kushner: we basically. So what we're trying to do is reinvestigate it, thinking about it through the lens of entropy. And so what we know about landau damping, by the way, real quick, for those don't know what land out damping is in a plasma. If you have like a wave that's propagating at a certain speed. The particles in the plasma that move at the same speed effectively are seeing the wave that stationary frame. So if there's an electric field in that frame. 257 01:00:44.390 --> 01:00:55.240 Mark Kushner: then the particle can either gain or lose a lot of energy, and so depending on where your particles are. You know, how fast they're moving things like that. 258 01:00:55.630 --> 01:01:20.550 Mark Kushner: the wave can damp that can give energy to the particles or it can actually grow as well by taking energy from the particles. So when you have land out damping the wave itself damps away. So the amplitude of the magnet or the electric field goes down, the amplitude of the bulk flow, velocity goes down, all that stuff happens, and in order to conserve energy, the temperature has to go up 259 01:01:20.690 --> 01:01:44.179 Mark Kushner: and so that's been known for a really long time. And these are some simulation results that basically say, Okay, our simulation works. We're seeing conservation of energy, conversion from bulk flow and electric energy into thermal energy, which is what we expect in this bottom plot shows that the temperature agrees with what it showed from conservation of energy. So what we've been doing 260 01:01:44.180 --> 01:02:00.649 Mark Kushner: now is saying, well, what about conservation of entropy right? So we know that the total entropy can be should be conserved. But when we decompose it into these 3 different forms, right? This position space, and then this velocity, space one and this relative entropy. 261 01:02:00.930 --> 01:02:11.090 Mark Kushner: we should be able to use that as a you know, it should be conserved as well. And so when you do that, you can predict what the find, what the change in these entropies 262 01:02:11.090 --> 01:02:31.640 Mark Kushner: are as as you get land out damping. And the short story is that they agree with the theory that I'm not showing you but the key result of all of that is, you get a nonzero change in the relative entropy. So even if you start with a plasma that's in local thermodynamic equilibrium, you'll end up with a plasma that's not. 263 01:02:31.710 --> 01:02:56.429 Mark Kushner: and you can quantify that with the relative entropy. And in particular, we've known for 100 years that when you have land out damping this plot on the left here. That, your your distribution function is non Maxwellian after Max Lando damping happens right? So this gives us a way to effectively quantify using just a single number. How non Maxwellian your distribution function will be 264 01:02:56.610 --> 01:02:58.710 Mark Kushner: after Landout damping has occurred. 265 01:02:59.260 --> 01:03:06.470 Robert Krasny: So that's kind of where we're at with that I hope that answers your question. Thank you. Very interesting. I see there's a paper in preparation about that. 266 01:03:07.680 --> 01:03:13.850 Mark Kushner: Is there a paper on? Not yet, my students working on it. 267 01:03:14.590 --> 01:03:22.590 Mark Kushner: Go ahead. Hi, Do. Various kinds of departures from Lte have different names. 268 01:03:22.900 --> 01:03:33.639 Mark Kushner: Oh, it's a great question, and I'll actually answer a slightly broader question. Some some of them do some departures do, and sometimes the mechanism for 269 01:03:33.680 --> 01:03:55.710 Mark Kushner: taking your plasma out of thermodynamic equilibrium or moving it further away. Those are related to physics. Right? So for example, lando, damping or cyclotron damping or beta trying acceleration or fermi installation. All these things are different ways that particles can gain or lose energy. And that will change your distribution right? In characteristic ways. 270 01:03:56.260 --> 01:04:24.270 Mark Kushner: The bad news about what I'm showing you is that it's just a this relative entropy stuff is just a single quantity that tells you it's further away or closer to local thermodynamic equilibrium. So you can have 2 completely different distribution due to completely different physics that are, you know, equivalently far away from local thermodynamic equilibrium. And this would just say, well, okay, they're far thermodynamic equilibrium, but it doesn't distinguish the physical mechanism that caused it. 271 01:04:24.460 --> 01:04:30.559 Mark Kushner: So that's a real drawback of this but the idea would be that if you can make. 272 01:04:32.390 --> 01:04:34.229 Mark Kushner: if you can make these 273 01:04:34.370 --> 01:05:02.330 Mark Kushner: these plots right like, you can just look at this and say, Okay, well, here, it's not in local thermodynamic equilibrium. And here's something exciting is happening. But we don't know what. But now we at least at least know where to go, where we can look to see the exciting stuff and try to pull apart. Okay, this land out damping or cyclotron damping, or any of these other things. And there's lots of other work that's been done on identifying sort of the physics by looking at the distribution function. 274 01:05:02.390 --> 01:05:07.290 Mark Kushner: What's the metric that you use for your non, Max family. 275 01:05:07.860 --> 01:05:24.290 Mark Kushner: Yeah, it's in this one or showing, I guess, on the second panel. Yeah, yeah, we call this M. Bar. Kp, it turns out, it's a factor of 3 within a factor of one and a half. It's really just the relative entropy. 276 01:05:24.570 --> 01:05:30.970 Mark Kushner: Yeah. So think of within a factor of one just there? As well. 277 01:05:31.560 --> 01:05:32.300 Mark Kushner: Yeah. 278 01:05:32.790 --> 01:05:46.579 Mark Kushner: there was another question. Yeah. In Connecticut. Why do you in principle have to solve an infinite number of equations to satisfy, to solve the system. If there's only a finite number of products, make sure you just describe 279 01:05:46.620 --> 01:05:50.200 Mark Kushner: right? Right? So 280 01:05:51.210 --> 01:06:03.589 Mark Kushner: sure. So in principle, you don't need to solve an infinite number of questions. You just need to solve one. And this is actually related to discuss question earlier. If you know the distribution function, and you know, both from the equation. 281 01:06:03.590 --> 01:06:24.450 Mark Kushner: You can just solve it. And that's the end of the story. And that tells you all the information you need to know? The problem is, or maybe not the problem. But what makes this hard is that this is an equation in phase space. And so there's 3. And for system there's 3 position coordinates and 3 velocity coordinates plus time. So F is a 7 dimensional variable 282 01:06:24.560 --> 01:06:30.240 Mark Kushner: right? So if you're doing a simulation. Right? If you're doing a fluid simulation, you tab. 283 01:06:30.250 --> 01:06:49.330 Mark Kushner: let's say 3 spatial coordinates plus time. So it's 4 dimensional. So this is 7 dimensional. So just in terms of. you know, just data, memory, right? It's a lot more expensive to do 70 simulation than a 40 simulation so that's one reason that 40, you know, fluid is kind of 284 01:06:49.390 --> 01:07:10.180 Mark Kushner: quote better in some sense than the fully kinetic one. The problem is that you're losing the physics right? So the physics, when you start to do the fluid equations here. This is so. If you do a simulation, let's say where you're evolving the fluid variables. That's where you say, well, okay, I need an infinite number of them to really do it. 285 01:07:10.210 --> 01:07:12.380 Mark Kushner: So that's the trade-off 286 01:07:14.160 --> 01:07:20.010 Mark Kushner: maximum distribution. 287 01:07:20.350 --> 01:07:26.189 Mark Kushner: Is there any other constraints you need to introduce before. Is that sufficient? 288 01:07:26.300 --> 01:07:48.159 Mark Kushner: That's as far as you know. That's the definition. If you're not, I mean, if I take 289 01:07:48.470 --> 01:08:00.640 Mark Kushner: it depends. Right? I mean, right at the this, the discontinuity. There'd be an issue right? Well, it's going to be in an equilibrium for infinity. 290 01:08:00.940 --> 01:08:08.540 Mark Kushner: Right? Yeah, no, what you're asking is very important, very important questions. What I would say is, 291 01:08:09.080 --> 01:08:34.800 Mark Kushner: like here, when you talk about defining a distribution function, it just means locally at that cell. It's a Maxwell Boltzmann distribution, and the cell next to it would be Maxwell Bolton. But it doesn't have to have the same temperature or the same density. So that sort of one way to think of it is that inside of a particular cell, in phase space, you have enough time to equilibrate. But between cells you have it 292 01:08:35.029 --> 01:09:04.480 Mark Kushner: restriction initially. 293 01:09:04.859 --> 01:09:05.600 Mark Kushner: Yeah. 294 01:09:08.229 --> 01:09:09.580 Mark Kushner: other questions. 295 01:09:11.890 --> 01:09:13.799 Mark Kushner: So I'm sorry 296 01:09:14.060 --> 01:09:23.220 Mark Kushner: it's going. I'm still trying to understand your statement that this avoids the closure problem. When I think about the closure problem, it's usually that 297 01:09:23.279 --> 01:09:33.560 Mark Kushner: have some fluid dynamical variables. I want to relate density, flow velocity and temperature. And I want to system equations where I can. 298 01:09:33.740 --> 01:09:40.910 Mark Kushner: I have a closed system of equations that relate those variables without solving. 299 01:09:41.060 --> 01:09:46.420 Mark Kushner: Okay, do that by preserving the kinetic equation about equilibrium 300 01:09:46.890 --> 01:09:54.289 Mark Kushner: and doing like a moment method or hierarchy method, where it tells you, like a linear constitutive relation for the 301 01:09:54.350 --> 01:10:01.890 Mark Kushner: answer. 302 01:10:02.270 --> 01:10:05.909 Mark Kushner: Then you don't have to solve the kinetic equation 303 01:10:06.120 --> 01:10:13.930 Mark Kushner: so. But your entropy still depends on the solution of the kinetic equation. So it hasn't avoided the closure problem. Right? You still have to solve the kinetic equation. 304 01:10:14.350 --> 01:10:27.009 Mark Kushner: the okay, the okay. So II was totally with you up until that last part. So, yeah. So you have. 305 01:10:27.030 --> 01:10:28.250 Mark Kushner: Okay. So 306 01:10:28.390 --> 01:10:52.560 Mark Kushner: you have a distribution function. You have your fluid equations as essentially in principle, at least, you need an infinite number of food equations. But you use some physics to close them right? So I'm with you on that. And then, so you said for the entropy part. Can you say the last part again? Distribution? Yes. So you have to solve for the distribution function. That's right. 307 01:10:52.840 --> 01:11:00.510 Mark Kushner: Closure probably have to solve the Connecticut. If you have the solution to the connectic equation, you can solve for everyone 308 01:11:00.590 --> 01:11:04.830 Mark Kushner: in these conservation convergence. So yeah, closed. 309 01:11:05.000 --> 01:11:26.619 Mark Kushner: Well, okay. What I would argue is, let's say, for example, you're doing a pick simulation, or a glass of simulation or something like that. Right? So you you're solving. Basically, the Boltzmann equation itself right. And that avoids the closure problem entirely because you don't need the fluid equations right? Any of the moment equations, right? That is true. 310 01:11:26.730 --> 01:11:36.669 Mark Kushner: But people don't do that. At least I mean people do for the first 2 moments, right? But there's been very little people going beyond the first 2 moments, and in particular. It. It's 311 01:11:36.840 --> 01:11:53.680 Mark Kushner: it's not just. I mean, there's logistic purposes there, logistic reasons that people don't do it too. So, for example, with Mms right? The higher the moments you go to the you're getting structure sort of further out in the distribution and just the the resolution run into 312 01:11:53.910 --> 01:12:05.350 Mark Kushner: constraints just from the diagnostics. So you're not getting as good data further out, right? And so it's like, people can measure the heat flux 313 01:12:05.350 --> 01:12:26.979 Mark Kushner: in with Mms data, but I've heard it's hard, and I've never heard of anyone trying anything even higher. Right? So if so, you're totally right, if you have the distribution function. You can go to the 20 seventh moment and say, What's that equation in the fluid sense and have it tell you something. But in practice it's really hard to do these really high order moments. 314 01:12:27.050 --> 01:12:35.640 Mark Kushner: And I would argue, the same is true again, in a simulation like they occur. Maybe flat is probably good enough. But 315 01:12:35.770 --> 01:12:38.349 Mark Kushner: yeah, so does that. I don't know if that helps 316 01:12:54.900 --> 01:12:58.730 Mark Kushner: whatever observable you're interested in. Yes. 317 01:12:58.770 --> 01:13:08.520 Mark Kushner: in principle. Yes, but I would say, it also goes back to the question from earlier, which is that the distribution function is 70, 318 01:13:08.640 --> 01:13:14.870 Mark Kushner: whereas the fluid variable is fourd, and so the entropy is a fourd, you know, space and time, and that's it. 319 01:13:15.020 --> 01:13:38.510 Mark Kushner: Whereas the distribution function is 70. So yes, if you have the full 70 distribution function. You can do whatever you want with it. But typically, it's really hard to extract that kind of information. And so I think one of what I'm the way I'm selling selling this a little bit is that if you use this relative entropy stuff, it's it's now just space and time. 320 01:13:38.510 --> 01:13:50.729 Mark Kushner: You don't have to know what's going on in velocity space explicitly. It's all there, right. It's it's hidden. And once you do that integral but it can tell you what's going on with your distribution function in a fluid variable. 321 01:13:51.000 --> 01:13:58.399 Mark Kushner: That's telling you the non lte stuff. And then, okay, then you can go back and say, Well, alright! What's the whole distribution? What's all the physics going on. 322 01:13:59.090 --> 01:14:01.659 Mark Kushner: Spencer? Next, that I'll see you there 323 01:14:04.020 --> 01:14:06.889 Mark Kushner: when you need us to computer? 324 01:14:07.320 --> 01:14:08.090 Mark Kushner: Yes. 325 01:14:12.830 --> 01:14:17.839 Mark Kushner: maybe you're proposing that, like you run some simulation for a simple 326 01:14:18.060 --> 01:14:25.440 Mark Kushner: or smaller system, and then you should configure transport like to the variables from that, and maybe some not lte 327 01:14:25.610 --> 01:14:33.670 Mark Kushner: relationships. And then you kind of like import that too different. I think that'd be really cool. Yeah. 328 01:14:34.070 --> 01:14:43.059 Mark Kushner: yeah. Cause like, like, Scott was saying you wouldn't be able to solve for app simulation. You're trying to run. So maybe you could. 329 01:14:43.970 --> 01:14:44.640 Mark Kushner: you know. 330 01:14:44.910 --> 01:14:49.929 Mark Kushner: import that information? Yeah, yeah, I think that would be really cool 331 01:14:50.640 --> 01:14:55.009 Mark Kushner: forefront research area. I think it'd be really neat to look into 332 01:14:58.270 --> 01:15:03.740 Mark Kushner: other questions online people. Any questions from there. 333 01:15:07.760 --> 01:15:10.290 Mark Kushner: Yeah, let's thank our speaker again.