 presented walk together with Professor Maina from Bologna. My talk is devoted to origin of porosity in solids. Its theme is different from relay and other problem. But here is a kinetic approach to problem. There is using stochastic models and effective and new methods of solution which give us, we will be sure that our solution of equation is correct. And here, let me note that aim of this walk in calculation is different from other application. We are interested in prediction of cracks in object of cultural heritage. And new field of application computer simulation. And here I will be talking about silicon carpet because this material is very important for future. And here we have some result on porosity. You know point wise radiation damaging in solids. But we take problem with non-point and fluctuation based appearance of vacancy gas porous and micro fluids. And I put first protection of cultural heritage. But my team is new composite materials with specified properties. And early I worked on the materials for thermonuclear reactor and for material and the damaging of metal mirrors and other. Non-destructive testing of cracks in the heritage frescoes substrate and other. And here my co-author asked that three new method of scanning electron microscopy here is one. And infrared passive thermography for porosity. And there is a possibility to detect porosity in a very important cultural object from millimeter and up to nanometers. And I think that let us go to result which will be observed now. And here time period of calculation, some processes is very quick. And there is initial stage of fast transition with fast transition is a porosity appearance in our materials. Here is pressure, temperature, and some flux of weakly soluble gaseous xenon in this case. And here is him which illustrate why one monomer inert gaseous appears in some clusters of gaseous, vacancy gaseous. And after them, it appears some structure which leads to very harmful phenomena in materials. Here I put some set of problem related with implantation ions. Ions is neutralized on surface. And here we have two model of processes. At the first we have clustering, vacancy and gaseous. And second process, Brownian motion interlates our cluster named blister. And blister and vacancy gaseous bevel here will be shown two kind of question, Kalmogorov failure, and Stains-Molukovsky. And it is stochastic differential equation related with Markov random processes. The scheme show you stochastic molecular dynamic which we used in this case. And here fluctuation stage prevails before first order phase transition. In this case, similar condensation vapor to drop of liquid. And here is a contrary, a poorer condensation. But condensation gaseous intersolid. Nucleus and germs or initial material name Brownian particle. And so Brownian motion leads to alternating mass process. And blister cluster formation before them. And we take a possibility to include in this model chemical reaction. Let me shortly remind you Kalmogorov equation in partial derivative for second order parabolic kind. And there is very important that this kind of equation is relate with eta stochastic equation. Idea eta put forward divided increment of process on drift determinant path and stochastic diffusion with random process. And sense of coefficients. Here is a mathematical expectation and drift and diffusion of this process. Existence solution of this equation and uniqueness of the solution is proved. And this base of our modeling. And here I show a relation between two kinds of equation. One dimensional, but we can enlarge two or three in our publication. Show that relation will be different. Next position, Kalmogorov equation. And the base of my modeling because Brownian motion with real diffusion confusion and the potential of interaction between defects. And here is Brownian motion approximation. And here is a model for clustering. Recently, academician Zeldovich put forward idea that process clustering and the initial stage of the first transition is a random diffusion in fast space of sizes clusters. This good idea was used. And here diffusion, coefficient diffusion in fast space of sizes. And here is potential, similar potential interaction. Here is entalpy, well-known you from thermodynamic. And here this strange form related with Stratanovich approximation of eta. But I have the possibility to clarify more model clusterization. Here is fluctuation size of our cluster. I input, outcome, and here process will be in solid. We take into account elastic latest reaction and quality of elastic relation in the solids. Why next position? Instinct, Smoluchowski question. And here we have forces. And these forces will be discussed later. Here mass of cluster, which we find in previous step. And boundary condition, a periodical sense. Model of Gibbs free energy is full. And here is chemical potential, different surface tension of cluster. And here we have possibility to wrap the relation in solids. And difference in the internet and other kind. And later I'll show why this quality realized. Next, because diffusion is a very important model. And I must say about anything about diffusion interlates. Here is nonlinear functional coefficients. And it's very important position for sense of computer simulation. Because as you know, that operator for your plant equation is linear. But if we have a nonlinear model of coefficients, we have quasi linear problem. And winner stochastic process is a process with weak correlation. And here we have new quality in science kinetic equation. And here diffusion model, also functional coefficient in Fokker-Planck equation. You know in stain approximation of this model, traditional. But here we changed this coefficient and the dispersion of coordinate in the dispersion, which realized on previous step of algorithm. Let me stop on a potential interaction interlates. Because all molecular dynamic model used well-known Leonard Jones potential and power approximation of interaction between object in molecular dynamic, which used Newtonian dynamics. Our potential relate with the perturbation in vibration latest acoustic phonons. This kind of potential was derived with using Feynman diagram and quasi quantum approximation. And here some constant attract quality of latest, which were used. It's a very complicated procedure in calculation. And here I put your attention on interaction through freedom oscillation vibration. Because we use metal and border with semiconductor. And this value I will show why compare. Potential of indirect elastic interaction accounts some possible interaction, not only each other. And collective character of each other interaction, but all boundary and more bigger matter may be used. And here initial and temperature and dose of radiation and here normalized non-dimensional value. And one cross-section has a different value. And during the motion of Boronian particle, we have some changes in the character of potential. And next possibility, there was a cross-section perpendicular to incident flux of ions. But next position, relief of potential and different from accounted latest acoustic and freedom and acoustic, only acoustic, acoustic. It's present in all kind of materials. And here we have other treatment of result measurement potential in computer simulation. We have two kinds of field of indirect elastic interaction. And here you can see flux from top. And here silicon carbon and picture of defects in metal, which can be defunct. And here you can see that silicon carbon non-destructed, but all layer of metal was with it. And here is problem of this defense layer. And the way is relate. And after, if you remind, remember that second position after time evolution of porosity creation, we have some structure, self-organization of our picture. I will show more completely. Here is very important slide concerning Rosenbrock's scheme of solution, stochastic equation. Here is stable and non-dependent from step of time. And good solution of stochastic equation because Ehlers' scheme is non-stable and solution using Rosenbrock kind of approximation, coincident only next several step of algorithm. This method well-known Artemiev-Averina. And Averina is co-author for me and we are working together. And here you can see that force of indirect elastic interaction. And here enthalpy gradient in fast space. And let me show why we have received some structure in our experiment. You can see that layered calculation space. And here formula for porosity. And here the number of particles in the clusters. Here is a mathematical expectation which was produced from non-equilibrium function distribution defects on sizes. And here I show some example of calculation where two layer and here layer metal and top and radiation and top. But here not silicon, but silicon because wall of the bologna which near all materials from cultural heritage. And later here is come back to silicon carbon. And here we have two temperature example of calculation porosity. And here the same temperature that the same example for. And here a new characteristic of our calculation. There is a volume distribution of stresses. All porosity has a Laplacian jump of pressure. And this jump of pressure was known dimension of porous. We recalculated two characteristics of stress. And I cannot put attention on temperature non-linearity because in this computer experiment we find some non-linearity in porosity, in value of porosity in this materials. And now I show example of Gibbs free energy. And you can see that here is common presentation, vacancy and gas free Gibbs energy and type of this process. And if you have only gazes, there is no this occasion. And three line of Gibbs free energy presentation, you can see that cause of non-linearity in porosity in silicon carbon is here. Next picture show functional coefficient Gibbs energy. Here is super saturation is constant. But here number particle in cluster here and here temperature. And here is some surface of actual value which used in calculation. And when we can see this function, you predict any behavior of processes. And here is super saturation and size of cluster. Here is one of result of evolution, evolution, fluctuation in stable process. And here not averaged real trajectories realization. And this brain of distribution function stated near this one occasion. But after development of stochastic evolution, we have other average size approximately 8 angstrom. And all calculation we put forward in number of particles value. But because the assumption is round, and we have four complication on stress. I show formula where we have number of particle as mathematical expectation. And here coordinate of layer in a layered model of media tool, layer, metal, and subtract. And here you can remember that micrometers is computational size. And here stress in 10 power 9 Pascal. And here averaged on x and y. Here I show the stress development, this kind of stress calculated from distribution function on size and coordinate Brownian particle. The initial time some boundary between layer. And here the scale of stresses is here. The white color is show because we use open physical system and some parameters in constant flux of radiation flux is constant. And here let me show porosity, stress. And here a new model with fractal distribution of boundary between layer. And here will be correlation function of distance. Not percolation, but end conclusion. I take that small supplement to kinetic theory of first transition in part of nucleus, nucleation in solid. We have some work which will be presented latest day on cluster formation. And here latest publication on theory is applied numerical, mathematical, and there is work with Professor Mainer. He is a chief redactor, new journal, city of memory, devoted to problem risk in cultural heritage prediction. Thank you for attention. Thank you. Thank you again. Ginnert Ginnert.