 Good morning everyone and welcome to Trieste and thank you for being with us and with me in particular this morning. Today I will report on some work that has been done in Sissa over the past few years to combine the Green Cubo theory of linear response for transport coefficients with abynition molecular dynamics with the simulation methods based on density functional theory that until recently were thought to be incompatible with each other. So let me just break the ice and define some terminology and some concepts what he transport is on about it's if you wish the defining feature of the second law thermodynamics which says that heat flows from from the warm to the cool until the thermal equilibrium is established and the key concept of the key quantity you want to compute is the rate at which the thermal equilibrium is established that is the thermal conductivity that is the ratio between the heat flux and the thermal and the temperature gradient. By combining this constitutive law for the density of heat current and the temperature gradient with energy conservation we arrive at the celebrated Fourier law for thermal transport. This is well established it's 19th century science but as we will see shortly there are a few surprises when it comes to combine these well-established concepts with state of the art electronic structure methods. First of all why is thermal transport important? Well there is a whole set of very important technologies that are based on thermal management that include energy saving for instance in civil engineering or heating heat dissipation in devices. Heat management in any device when you want to keep the operating temperature of a device at an optimal value you have to care about how heat is dissipated. Heat shielding in for instance in the aerospace industry or energy conversion in thermoelectric devices all the way up to the heavens all the way up to the thermal history of the earth and planets that crucially depend on how heat is dissipated from the core to the outer space due to thermal transport. More than this it's as we have seen an important conceptual ingredient of modern science and modern physics and still as I was mentioning is poorly understood. Let's let's see why is that. So the the modern theory of thermal transport roots into the linear response theory of Green and Kubo that is summarized in this very beautiful equation that establishes a relation between non-equilibrium property that is the transport coefficient and the equilibrium fluctuation of the conjugated currents. So basically at equilibrium you have the average current the average heat current is zero but fluctuations are not zero and the and the the decay time of spontaneous fluctuations basically determines the magnitude of the heat current. How to compute this from first principles? Well the heat flux is the integral of a microscopic density current. The density current is the heat density current is defined as the non-convective component of the energy current so essentially you you subtract from the energy current the convective component and you have the heat current in in one component fluids and in solids you can because of momentum conservation you can neglect the velocity field and essentially heat and energy currents coincide. The energy current in turn is given by the continuity equation because energy is conserved and by multiplying this continuity equation by r and integrating by parts you end up with with the relation that the current entering the Green-Kubo relation essentially is the first moment of the time derivative of the energy of the energy current. So ever in order to implement this idea you have to have a prescription on how to compute from first principles the energy current, the energy density. This is done since since the early days of molecular dynamics in classical statistical physics basically what you do for the energy current you assign at each atom a local atomic energy and you pretend that the energy is concentrated in lamps localized at every atom and this is this is the resulting expression by using the continuity equation and computing the first moment of the energy of the time derivative of this of this of this density you arrive with some with some algebra to the you arrive at this expression for the energy flux that you can find in many textbooks of computer simulation and statistical physics of liquids. Here comes the first obstacles that have to be overcome in order to combine this Green-Kubo theory with ab initio electronic structure based methods. Basically it was widely believed until recently that the Green-Kubo relation does not serve our purposes to compute the thermal conductivity from first principles for the same for the simple reason that it is impossible to uniquely decompose the total energy into individual contributions and from each atom. The same actually holds for any energy density right because the energy density quantum mechanical is ill-defined as as the the transport coefficient depend on the currents and the currents depend on the density if the density is ill-defined it seems that the transport coefficients are ill-defined as well but here we have problems because transport coefficients can be measured how is it that something can be measured and cannot be computed from theory. More than this if even in classical molecular dynamics where the setting is apparently well-defined and we have a well established recipe to to compute transport coefficients it seems that because of the limited time scales accessible in ab initio molecular dynamics runs thermodynamic fluctuations are dominated at the auto correlation time of the of the current which in turn prevents a reliable and numerically stable assessment of the thermal conductivity. In a nutshell it is a common practice in the classical molecular dynamics that it takes molecular dynamics run as long as several nanoseconds to extract accurate thermal accurate accurate transport coefficients and it seems that even though the first obstacle is overcoming a way or the other the time scales necessary to to compute transport properties are so long as to be hardly attainable using ab initio molecular dynamics. So let us face the two hurdles in turn let us start from the first that is the most challenging in my opinion and and to see how this conundrum can be can be solved by inspecting the very concepts that lead that lead to the formulation of a transport theory in extended systems. So this is a summary of the relations that we have already seen before the energy is the total energy is a sum of atomic energies in classical molecular dynamics which is a constant and and using the continuity equation we arrive at this seemingly well defined expression for the current but is it is it really well defined actually this unpack unpacking of the energy into local contribution is to a large extent arbitrary bear with me and instead of unpacking the energy in the blackway with the black recipe supposed to multiply each interatomic interaction between atom i and atom j with times a factor that I define rather arbitrarily as one plus an antisymmetric matrix gamma ij is any matrix that is anti-symmetric for the exchange of the i and j indices because gamma is anti-symmetric and v is symmetric the sum over all the pairs is zero so the sum of all the epsilons is the same good bonafide the total energy independent of the gammas you do the same algebra the same that leads from the black atomic energies to the black current and you end up with a different current so the total energy does not depend on gamma but the total current does if you do a simulation here is a simulation for liquid argon and you can use a different recipes for gamma the blue recipe is the obvious one is a zero and we have two different recipes in purple and yellow that I don't even remember and they have a hard time reading in in the slide but they are different recipes arbitrary choices of these gamma matrix the autocorrelation function of the current is different is different even in quantum mechanics even in classical mechanics where things seem to well defined that they are not the current and and and the current autocorrelation function depend on an arbitrary choice of hours nevertheless when you integrate these autocorrelation function up to a large time so as to obtain the transport coefficient according to the green cubo formula apparently this is the integral as a function of the upper limit of integration all the recipes converge at the same time at the same value this is good value good news right because this is what you measure the it is the value at the large time that is measured so how is this so a hint a hint is that the arbitrary part of the of the of the current if you inspect carefully it turns out to be to be the total derivative of a bounded vector what what are the implications of this well you have two currents that differ one from the other by the total derivative of a of of a vector let us try to compute the autocorrelation time so the integral from zero to infinity of the primed and the non-prime the currents they differ by some by some terms and then you exploit the stationarity is to exploit the time micro micro reversive the reversibility exploit a few a few properties that I have not the time to describe but if you do your homework carefully all the extra terms will cancel as a consequence of the fact that the two currents differ by total time derivative then the two thermal transport coefficients coincide how what is the real reason for that well this is a consequence of a property that we like to call the gauge invariance of of thermal transport that basically comes from energy conservation so the energy conservation energy conservation and the extensivity so all you when you have a system that is described by two subsystems the energy of the system is equal to the energy of the two subsystems plus some interaction energy whose magnitude scales as the interface between the two subsystems right but when when you treat when you treat the energy as extensive which is in which is essential to define an energy density the very fact that you define a density is a consequence of extensivity so that when you implement extensivity you have to make a decision on how to partition the interaction energy between two subsystems for macroscopic systems this doesn't matter because the interaction energy scales as the interface the interfacial energy that in the thermodynamic limit that goes to zero but when it comes to defining a density it does matter and the two different energy densities should be considered equivalent if their integral in the thermodynamic limit coincide but the integral of two scalar fields coincide in the thermodynamic limit if they differ by the divergence of a bounded field you you insert this equivalence into the continuity equation and you see that if two energy densities are equivalent in the in the sense that they they yield the same total energy then the corresponding currents must differ by a total time derivative and the total flux must also differ by a total time derivative that is the condition that we found for the two transport coefficients that to coincide to summarize any any two energy densities that differ by the divergence of a bounded vector field are physically equivalent and the corresponding energy fluxes differ by a total time derivative so that the heat transport coefficients coincide here is the conundrum salt so there is indeed an ill-defined ms of the ingredients of the green Kubo theory at the microscopic level but what you can measure that is the transport coefficients of themselves is insensitive of such a ill-defined ms which is what we call a gauge invariance how much time do I have Roberto really oh five minutes okay so this gives us a way of solving the problem in density function theory all you have to do is to break the total energy into contributions to which you associate densities you plug those that expression in the continuity equation you work hard and you arrive at a rather awful expression for the total flux which however is ill-defined I will not comment on any of these terms but the third but the the first that implies matrix elements of the position operator that as we know very well in our community are kind of black beasts because they they are ill-defined in periodic boundary conditions but because those matrix elements are among are calculated between occupy states and linear combination of unoccupied states they are actually well defined and can be computed using density function perturbation theory this is the results that we get for water let me go very fast through illustrating what the problem here is it is because the the power spectrum of the current has high frequency components that are associated with the internal degrees of freedom the signal oscillates very fast even at very large integration time so that it is very difficult to get the signal you see this is the integral and seems to never converge because of these high energy components of of of the noise you can convert this green Kubo expression into an Einstein's relation so that the transport coefficient can be computed as the slope of of the integral of the energy displacements and by doing careful and painful error analysis you get the result that is in fair agreement with experiment but still but still in spite of the fact that the first hurdle has been seemingly overcome the second is there to stay at least apparently because of these long long time tail of of the noise that are difficult to get rid of very quickly our recipe to cope with this problem is to express the thermal transport coefficient as the zero frequency component of the power spectrum that via the Wiener-Kinchin theorem is estimated as the expectation value of the square modulus of the time series of the current that is a very very noisy the spectrum the major spectrum is very noisy essentially because it is the product of the theoretical spectrum that you want to measure times uncorrelated the stochastic variables that are uncorrelated a monster neighboring neighboring frequencies as the frequencies are as the longer the longer the trajectory and this is a very gives this noise that makes it very difficult to extract the spectrum we are interested in the low frequency value here on the spectrum so what we do in order to cope with this problem is to work rather than with the spectrum itself with the logarithm of the spectrum that turns this multiplicative noise into an additive noise and by concentrating on the low frequency portion of the spectrum what we can do we can define the Fourier coefficients of the logarithm of the spectrum and if you know or if you suppose or if you postulate that the number of these Fourier coefficients is small then you can reduce the noise by limiting the number of Fourier coefficients up to a pre-scribed number p star so the black the gray value here the gray line is the original spectrum then we filter this spectrum by retaining 1000 Fourier coefficients or 500 or 100 or 10 until eventually the noise a few a few seconds the noise is under is under control and these are the results of classical molecular dynamics runs performed over different systems and we see that we can achieve an accuracy of 10% of the order of 10% with trajectories of 1,100 picoseconds or a few hundreds at most the second the second hurdle seems to be overcome as well still the workflow for doing this is very heavy because at each time step or every given number of time steps you have to solve this you have to compute this current using density function perturbation theory so that the estimator the valuation of the estimator is as heavy as a configuration sampling so either you post process the the trajectory which is expensive or you integrate the estimator evaluation in ab initia molecular dynamics codes which is the opposite of what the Carlos suggested we should do to keep our codes clean and the third way is to share big data amongst executable through persistent memory very briefly what persistent memory is I learned about this a few days ago from Carlos actually this is not something I am very knowledgeable about but basically persistent memory combines the virtues of mass storage to be persistent with the virtues of memory to be faster and by addressable so that the different different applications that easily communicate through persistent memory that as as said is byte addressable and so easy to to share so heat currents are intrinsically defined at the atomic scale gauge invariance makes a heat transport coefficients independent of such an in the in the terminacy and computable within ab initia molecular dynamics the statistical theory of time series can be leveraged to estimate and significantly improve the accuracy of the transport coefficients computed from molecular dynamics and when stretched to the extreme scale to extreme scale simulations the resulting computational workflow sets challenges to information technology persistent memory may be an asset that to win the challenge these are the guys who contributed to this work loris and now at CISA Aris previously at CISA now at EPFL and IBM Zurich Federico and Ricardo and CISA and this is it thank you very much for your attention