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Fourth Law of Thermodynamics
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0:00
There is a beautiful lecture by Feynman that you can see online it's a historical lecture on the great conservation principles and from that one I learned something important that what we call laws of Nature are essentially general rules that we have been accustomed to consider essential for modeling of anything related to of course natural phenomena. So this is what I want to talk about. So its rules of modeling that are possible candidates for laws of Nature and since my time is limited I'm
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going to start and focus on what I'm trying to suggest that could be considered the fourth law of thermodynamics having to do nonequilibrium. So let me start from an example. Consider a heat transfer problem on a material that is non-isotropic for example layers of a highly conductive material on like an epoxy of low conduction. We know that in this case if you apply a gradient in one direction you get the heat flux that doesn't follow.. that doesn't have the
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same direction and the two are related linearly of course by the conductivity tensor which by the way is symmetric because of Onsager principle. The interpretation of a this is that the material, this particular material which is non-isotropic, feels different directions of heat flux differently and when I say feels it is as if the material really had to decide which direction to pass things along and because of the non-isotropy for this material it is easier to pass
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things along the highly conductive direction than the low conductive direction. So among all these possibilities, again if you put yourself in the shoes of the material the material has to decide where to move or what to do at a given equal cost so somehow this ellipse represents the cost for the material to transfer heat in this various directions. If you look at it this way, then you notice that the local entropy production is the scalar product of the flux and the gradient and so the
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material will choose among those equally costly fluxes the one that maximizes the scalar produc therefore maximizes the local entropy production. This is for a steady state situation. The same idea applies also for nonequilibrium dynamics. If you consider here a plane where say the energy and all the constants of the motion are fixed and these are the contour levels of the entropy so this is the maximum entropy state for those given values of the energy and the other conserved properties
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and suppose you are here.. now if your material is isotropic it feels equal cost to move in all directions so this is like a circle, if your material is anisotropic it's easier for it to move in the horizontal direction for example here because that's the layer of higher conductivity than in the vertical and so these two will evolve in time by searching.. see.. where it is tangent.. this is where among the possible equal cost directions you get the higher increase
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in the entropy, and so on, and then you follow this way the trajectory till you get to equilibrium. Now if you start from the same nonequilibrium state but for a different material that has a different local perspective on what is the cost of moving, this material will follow a different trajectory and here I also consider the possibility that as you move.. as you change in state the metric with which the material decides on how to follow steepest entropy ascent can
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change with state and then it will get again to equilibrium along a different path. Onsager theory belongs to the near-equilibrium region and so this metric when you get near equilibrium is the Onsager generalized conductivity tensor. Now this is a particularly simple material problem with only the transport of energy but as you know if you have also transport of charge and diffusion of particles and so on you get the cross effects and all that. So that's the idea..
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the geometric idea that I call steepest entropy ascent. Now regarding.. I'll go back to that in a moment.. now regarding more basic concepts having to do with the other three laws, energy is the result of the first law so the fact that the system has energy is a typical thing that we want out of our models of physical reality. We also want entropy and some people still argue whether entropy is defined or not in equilibrium, I don't want to argue that because I think it has been settled
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enough in papers by ourselves and also well-known physicists that picked up more recently so all I want to remind you is that you can have for a system, for example a fluid element, this is energy density, entropy density, and concentrations, with the hat means the density, nergy per unit volume, so you can draw an energy versus entropy diagram which is also the cover of our book in 1991 and on this diagram we can show for example the fact that for a given
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energy the system has many states, only one of them is a stable equilibrium state, the others are nonequilibrium and you can be far or near equilibrium. This is the realm of Onsager reciprocity, this is a more open field research. When you want to do modeling of a system, if you constrain yourself just to the equilibrium.. stable equilibrium states then you have to have a model in which you have a fundamental relation, this is a very standard. If you want to model
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nonequilibrium states you need additional variables and so the entropy, the energy, and the concentrations are functions of these variables, I'll call them gamma for today and of course your theory must be compatible with the fact that for given values of the constants of the motion like the energy, there must be one and only one stable equilibrium state so you have a fundamental relation in which these additional variables then take.. are precise functions of those energies
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and other conserved properties. So what characterizes the different theories of Nature that we have, is what we choose for gamma. For example here's the list of state variables of a whole lot of different fields that we adopt and each one is a formula for entropy.. I don't have time to go into details here.. but you probably recognize your own favorite theory here. The only step I want to do is that whenever there is a probability I suggest.. I prefer that we think
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of the square root of the probability so that if you work with the square root then it's easier in your model to keep the non-negativity of the probabilities because you just have to square a number and that's always positive. Apart from that, when you write a dynamical law for your model there is always -- so to speak -- a left-hand side having to do with rate of change of the state or fluxes.. so, rate of exchange of properties with neighboring fluid elements or neighboring systems
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and there is an internal dynamics that is responsible for the irreversibility.. the dissipation. And I like to focus on this one. For example in the quantum.. if I have a D-level system a system with D energy levels in a very particular subset of states where the density operator commutes with the Hamiltonian, just to simplify, you can think of probabilities as being the population of the various energy levels of your system and here's the energy versus entropy diagram for
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such a system, this is a seven level system with the blue curve and for a given energy you can have different states for example this is a pure state in which all the energy is just on one level the third level. This is the canonical the Gibbs state, canonically distributed with positive temperature, and that's with -- can I say pseudo -- negative temperature okay -- I'm just joking because of Henning's talk yesterday-- this is a point that represents various possible states that all have
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the same energy the same entropy distributed this way okay if instead of several energy levels we just have a bit.. a qubit.. two-level system then the states map on a unit ball, and so that means this is the Bloch ball, the pure states are on the Bloch sphere and as you see here for example you have constant energy states and these are the entropy contours and you can draw various diagrams for example I can instead of doing X Y Z component of the magnetic moment I could select as
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independent variable the entropy then I have X, Z (which is also energy), and entropy and so I get this paraboloid kind of description and if I project it on the E versus S this is the E versus S diagram and that's.. and this is the constant energy plane. So now my concern is how good to describe a time evolution of a state that would evolve towards equilibrium along the steepest entropy ascent. The idea is that the various laws of evolution can be different depending
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on your model but should have a structure in which you have.. the rate of change of the state is related to a part which accounts for the reversible dynamics and a part which accounts for the dissipative dynamics. And then you apply this equation and then the key is that the system at every equilibrium state must have defined some metric which is the way in which it perceives distances between states. It's the thing that makes the system decide which direction
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to move, like in the example we've seen graphically before. So for a given effort in moving, the system will choose the direction of steepest.. maximal entropy increase.. maximum entropy production. Then the problem becomes just a Lagrange multiplier optimization and simple mathematics.. you can apply.. I am not going to show this of course and don't be worried about the number here I'm not talking about all these slides they are ust for questions later So here's the result of applying that idea
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you get spiraling motion towards equilibrium and this motion can be viewed in various points on this diagram is just moving from here to there a nice aspect of this kind of dynamical model is that you can solve it not only forward in time but also backwards in time in other words it's mathematically reversible though it describes the physical irreversibility differently from other models like the linear Lindblad equation that if you solve it backwards in time at a certain
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stage it will boil out. So my suggestion is that this idea of steepest entropy ascent could be considered as a general principle at the same fashion as uniqueness of stable equilibrium States is the general principle that we call the statement of the second law, the Hatsopoulos-Keenan statement of the second. Now, I'm saying this not because just of our proposal but because essentially the steepest entropy ascent has been a precursor of several modern theories of nonequilibrium that some
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which you've heard and some which you know some of which you are working on and you see here they start.. we all started around 1984 Miroslav Grmela was among them and then we have various other contributions that go in the same direction even gradient flows in mathematics, large deviation theory, you'll hear about it soon and also maximum entropy production from the information theory point of view. So, the last concept I want to talk about is well, why then are people talking about
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minimum entropy production? Is it minimum or maximum? And I want to clarify this because it is a source of confusion sometimes in the literature. Locally the system chooses the direction of maximum entropy increase. The theorem on minimum entropy production which was proven by Glansdorff and Prigogine in 1954 has to do with global, so you have like a control volume of many fluid elements and you look at the global entropy production and here you have a formal proof that under
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very wide sets of boundary conditions the rate of change of the entropy production rate of the overall so the integral of the overall entropy production of the volume is negative, it means that you start from a state, it has a certain value, it goes down in time until it reaches either zero -- and then that would be equilibrium -- or a fixed value which is steady state. Here is the example. Suppose you take just a simple material and you impose an unsteady initial temperature
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distribution like that. You compute the integral of the local entropy production so the local entropy production is maximal.. is steepest entropy ascent but when you integrate you get a certain value. If you let.. if you solve this problem.. let it go in time using simple Fourier law, eventually you get to a steady state, in the steady state the entropy production has halved. Or, for example, start from here where you have imposed this odd distribution of temperature, so you
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have an entropy production then you let it go.. it will reach zero.. so it will go down until you go to zero and that is a equilibrium situation. So there is no contradiction between minimum production theorems and local maximum entropy production. Thank you. [Applause] [Peter Van]Question about relativity theory. I'm not an expert I don't dare saying anything about relativity and.. I suppose that one should use the same kind of logic but I hope you can do it I am not
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able.. sorry, too difficult. [David Jou]Question about how to measure temperature in nonequilibrium. Okay, now, there is a distinction that one should make -- I think I understand the question -- and one has to make a distinction between what is the definition of entropy, which is model independent and it's something model independent, so you should have a definition that applies to all systems, and then the particular dependence that you impose in your model between the
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internal variables and the entropy I mean this formula here so the various models differ in this one then once you have a formula then it's simple to do the maximization and find the Lagrange multiplier the beta and so on and and then you can identify the temperature. Now, why do you call it temperature? Because it comes out when you ask what are the necessary conditions for mutual equilibrium between systems okay thereby the idea of a thermometer and and all that. So you measure on the
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basis of this theorem that says that the partial derivatives of the entropy.. the inverse of the partial derivative of the entropy with respect to energy should be equal between the two systems that are in contact and so that's the basis for the instruments and it is a basis that is nice because being independent of the model you measure something of an object of which you know nothing using the thermometer and that's how you build up the properties of that material that's how you can structure
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the properties of that material [unknown voice]Question on steepest entropy ascent and Onsager reciprocity. The fact that it is a metric.. a metric by definition.. geometry.. has the symmetry properties so automatically that will give you near-equilibrium Onsager reciprocity it's built in. But it also gives you other symmetries that may emerge far from equilibrium which are there to be discovered Thank you
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