Laminar-turbulent reactive flows in porous media

We state some recent results concerning liquid-vapor phase transitions for a fluid flow through a porousmedium. The focus is on the friction exerted by the porous medium, which is modeled in such a way to include both laminar and turbulent flows. In this way we obtain a hyperbolic system of three balance laws with a forcing term that is discontinuous in the state variables. Existence, uniqueness and qualitative behavior of traveling waves is proved by a novel regularization technique.     

Insulator-metal transition

A theory of the insulator-metal transition in transition-metal compounds is developed in terms of the collapse of the effective energy gap which is a function of the thermally excited electron-hole pairs. This dependence is shown to arise from the hole-lattice interaction. The reaction of the lattice is found to be equivalent to generating an internal positive pressure (strain). Estimates show that the observed typical behaviour of the conductivity jump and the change of volume at the transition temperature can be explained by the present theory.     

Markov transition probabilities

Summary A continuous-parameter Markov process on a general state space has transition function P _t (x,E). The theory of regenerative phenomena is applied to the question: what functions of t can arise in this way? Particular attention is paid to processes of purely discontinuous type, to which are extended known results for processes with a countable state space.     

Solutions with transition layer and spike in an inhomogeneous phase transition model

We consider the following singularly perturbed elliptic problem

     

Column continuous transition functions

A column continuous transition function is by definition a standard transition function P(t) whose every column is continuous for t?0 in the norm topology of bounded sequence space l_∞. We will prove that it has a stable q-matrix and that there exists a one-to-one relationship between column continuous transition functions and increasing integrated semigroups on l_∞. Using the theory of integrated semigroups, we give some necessary and sufficient conditions under which the minimal q-function is column continuous, in terms of its generator (of the Markov semigroup) as well as its q-matrix. Furthermore, we will construct all column continuous Q-functions for a conservative, single-exit and column bounded q-matrix Q. As applications, we find that many interesting continuous-time Markov chains (CTMCs), say Feller-Reuter-Riley processes, monotone processes, birth-death processes and branching processes, etc., have column continuity.     

Non-equilibrium gas-liquid transition model

A new rigorous mathematical model for evaporation/condensation, including boiling, has been proposed. A problem of phase transition and in particular evaporation/condensation is one of the most acute problems of modern technology with numerous applications in industry, such as: in refrigeration, distillation in chemical industry. It is very common to use equilibrium evaporation model, which assumes that concentrations of species in the gas phase is always at saturated condition. Such kind of approach can lead to significant errors, resulting in negative concentrations in complex computer simulations. In this work two analytical solution of simplified differential-algebraic system have been obtained. One of them was deduced using assumption that the process is isothermal and gas volume fraction is constant. In the second solution the assumption about gas volume fraction has been removed. The code for numerical solution of differential-algebraic system, using conservative scheme, has been developed. It was designed to solve both systems of equations with boiling and without. Numerical calculations of ammonia-water system with various initial conditions, which correspond to evaporation and/or condensation of both components, have been performed. It has been shown that, although system quickly evolves to quasi equilibrium state (the differences between current and equilibrium concentrations are small) it is necessary to use non-equilibrium evaporation model, to calculate accurately evaporation/condensation rates, and consequently all other dependent variables. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Standard tri-point transition function

It is usually difficult to express a family of tri-point transition function (TTF) by a transition matrix as Markov processes with one parameter. In this paper, we define three kinds of connection matrixes on the states of standard tri-point transition function (STTF) and study their essential character, give a constructive method on the constant-value standard tri-point transition function and a general expression of the state-symmetric standard tri-point transition function by a sequence of the transition matrixes of special and simple Markov processes with one parameter.     

Dual and Feller-Reuter-Riley transition functions

In this paper, we investigate duality and Feller-Reuter-Riley (FRR) property of continuous-time Markov chains (CTMCs). A criterion of dual q-functions is given in terms of their q-matrices. For a dual q-matrix Q, a necessary and sufficient conditions for the minimal Q-function to be a FRR transition function are also given. Finally, by using dual technique, we give a criterion of FRR Q-functions when Q is monotone.     

Continuity of non-stationary transition matrices

Conditions are given implying the continuity of the transition functions in a standard Markov transition matrix. An example of a standard transition matrix with discontinuous transition functions is presented and its relation to the Kolmogorov differential equations is discussed.     

Crystal-liquid transition in metallic hydrogen

Using the functional approach to the theory of crystals, we show that, with an increase in pressure, the zero-point vibrations of the crystal lattice break it down. From comparison of the pressure functionals for the crystalline and liquid phase of metallic hydrogen, we obtain a lower estimate for the critical density in the low-temperature limit.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 12–17, 1988.     

Phase transition in loop percolation

We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in Le Jan (C R Math Acad Sci Paris 350(13–14):643–646, 2012, Ill J Math 57(2):525–558, 2013). It is a model with long range correlations with two parameters \(\alpha \) and \(\kappa \). The non-negative parameter \(\alpha \) measures the amount of loops, and \(\kappa \) plays the role of killing on vertices penalizing (\(\kappa >0\)) or favoring (\(\kappa <0\)) appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525–558, 2013) that for any fixed \(\kappa \) and large enough \(\alpha \), there exists an infinite cluster in the loop percolation on \({\mathbb {Z}}^d\). In the present article, we show a non-trivial phase transition on the integer lattice \({\mathbb {Z}}^d\) (\(d\ge 3\)) for \(\kappa =0\). More precisely, we show that there is no loop percolation for \(\kappa =0\) and \(\alpha \) small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of \(\alpha \), namely, for \(\kappa =0\) and any sub-critical value of \(\alpha \), the probability of one-arm event decays at most polynomially. For \(d\ge 5\), we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For \(\alpha \) below this threshold, we calculate, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For \(d=3\) or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, we show that the one-arm exponent in dimension 3 depends on the intensity \(\alpha \).     

On a phase transition model

An Allen–Cahn phase transition model with a periodic nonautonomous term is presented for which an infinite number of transition states is shown to exist. A constrained minimization argument and the analysis of a limit problem are employed to get states having a finite number of transitions. A priori bounds and an approximation procedure give the general case. Decay properties are also studied and a sharp transition result with an arbitrary interface is proved.     

Phase transition in multicomponent systems

Phase transition in a system of several components is studied including chemical reactions by means of an approach based on non-equilibrium thermodynamics. A non-linear constitutive relation with discontinuities across the interphase is considered. Existence of at least one weak solution is proved, using an a priori estimate based on the second principle of thermodynamics. The above formulation is compared with Fourier's and Fick's equations and transformation formulas are given.     

Criteria for Feller transition functions

This paper presents some conditions for the minimal Q-function to be a Feller transition function, for a given q-matrix Q. We derive a sufficient condition that is stated explicitly in terms of the transition rates. Furthermore, some necessary and sufficient conditions are derived of a more implicit nature, namely in terms of properties of a system of equations (or inequalities) and in terms of the operator induced by the q-matrix. The criteria lead to some perturbation results. These results are applied to birth-death processes with killing, yielding some sufficient and some necessary conditions for the Feller property directly in terms of the rates. An essential step in the analysis is the idea of associating the Feller property with individual states.