New Approach to Modeling Non-equilibrium Processes

Abstract

AbstractFor a long time the main mathematical apparatus for modeling dynamic processes in physics and mechanics was the formalism of nonlinear differential equations. However, when high-rate and short-duration processes began to be used in modern technique and technology, the insufficiency of such modeling became more and more evident. Numerous attempts to modify this apparatus to describe such processes have failed because the processes far from local thermodynamic equilibrium are accompanied by such effects that make the commonly used models unsuitable. In previous chapters these effects were considered in details. Back in the second half of the last century in the framework of non-equilibrium statistical mechanics (Richardson in J Math Anal Appl 1:12–60, 1960; Piccirelli in Phys Rev 175:77–98, 1968; Zubarev in Non-equilibrium statistical thermodynamics Springer, 1974) it has been proven that far from local equilibrium the correct mathematical description should be based on integral–differential equations which include nonlocal and memory effects. The constitutive relationships between thermodynamic forces \({\mathbf{G}}\) and dissipative fluxes \({\mathbf{J}}\) closing the macroscopic transport equations are integral both in space and time. The original structure of the constitutive relationships is presented in Chap. 3. The closure problem of macroscopic transport equations out of equilibrium is considered in Chaps. 1 and 2. In this Chapter we present a new self-consistent approach to describe processes far from local equilibrium based on the nonlocal thermodynamic relationships with memory (Richardson in J Math Anal Appl 1:12–60, 1960; Piccirelli in Phys Rev 175:77–98, 1968; Zubarev in Non-equilibrium statistical thermodynamics. Springer, 1974) and one of the methods of control theory of adaptive systems presented in Chap. 6. Instead of the distribution function the nonlocal equations contain more rough characteristics of non-equilibrium processes, the correlation functions, that can be expressed in the form of functionals of macroscopic fields and the history of the system loading. In Sect. 5.9 the model correlation function with parameters having physical meaning is constructed. Modeling such parameters as functionals of the process allows us to pass from macroscale to mesoscale where most non-equilibrium effects occur. This opens principally new opportunities to describe processes far from thermodynamic equilibrium. Identification of the interrelationships between the concepts of spatiotemporal correlations and turbulent structures accompanying high-rate momentum transport is a new step into the study of non-equilibrium phenomena. Non-equilibrium distribution function by means of correlation functions has been found to generate vectors connected to rotational modes making the medium polarized. In Sect. 5.8 we show that this is a special feature of any high-rate process. In Sect. 5.11 we consider self-organization of discrete-size dynamic structures which size is determined by boundary and loading conditions imposed on the system. Far from local equilibrium, such structures are unstable and evolve in the direction that approaches the system state as close to local equilibrium as the imposed constraints permit. Description of temporal evolution of the system is based on principle of maximum entropy presented in Chap. 4. Evolution of dynamic structures resulting from self-organization on the mesoscale defines macroscopic behavior of the system far from local equilibrium. In order to describe the structural evolution we use the algorithm of the Speed Gradient principle determining the fastest way to the goal. The temporal evolution of the dynamic structures and its influence on macroscopic properties of the system will be considered in Chap. 6 by using methods of control theory of adaptive systems. In the general case, the self-organized dynamic structures are turbulent structures. Therefore, the developed approach proposes a fresh look at the theory of turbulence. On the other hand, the idea of describing the evolution of a system far from local thermodynamic equilibrium through the evolution of its turbulent structure is principally new in non-equilibrium thermodynamics. The relations of the proposed approach with statistical and quantum mechanics, thermodynamics and control theory make it multi-disciplinary. It turns out that only relying on this entire set of methods can one move further from local thermodynamic equilibrium. In Sect. 5.12, we present the formalism of special type nonlinear operators that can help to solve boundary problems for the processes far from local equilibrium. We show that non-equilibrium processes on the mesoscale have a lot in common with quantum mechanics. In the last Sect. 5.13, we list all the special features of the developed approach that distinguish it from others and make it fundamentally new. In Chaps. 7–10 we apply this approach to modeling shock-induced processes in real systems.KeywordsNon-equilibriumNonlocal and memory effectsCorrelation functionSelf-consistent modelBoundary problemTurbulent structuresSize spectrum