Thermodynamic Lyapunov function

Conditions (66) and (67) ensure the existence of Lyapunov s convex function for eqns. (17) GGjdNi = fit. With a known type of the potentials /i, for which condition (1) is fulfilled, one can obtain Lyapunov s thermodynamic functions for various (including non-isothermal) conditions. Thus, for an ideal gas and the law of mass action [16]... 

Studies of linear systems and systems without "intermediate interactions show that a positive steady state is unique and stable not only in the "thermodynamic case (closed systems). Horn and Jackson [50] suggested one more class of chemical kinetic equations possessing "quasi-ther-modynamic properties, implying that a positive steady state is unique and stable in a reaction polyhedron and there exist a global (throughout a given polyhedron) Lyapunov function. This class contains equations for closed systems, linear mechanisms, and intersects with a class of equations for "no intermediate interactions reactions, but does not exhaust it. Let us describe the Horn and Jackson approach. 

The first of the principal Horn and Jackson results is as follows. If the system obeys the law of mass action (or acting surfaces), then if it has a positive PCB it demonstrates a "quasi-thermodynamic behaviour, i.e. its positive steady state is unique and stable and a global Lyapunov function exists. 

Theory of electric circuits (Kirchhoff), 7 Thermal nitrogen oxides, formation of, 55 Thermodynamic Lyapunov functions, 3... 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

The Lyapunov function resembles the thermodynamic entropy production function and the asymptotic stability principle. If the eigenvalues of the coefficient matrix of the quadratic form of the entropy production are very large, then the convergence to equilibrium state will be rapid. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

The simplest case is to find the Lyapunov function for systems with an arbitrary set of the intermediate linear chemical transformations. In prox imity to thermodynamic equilibrium, this function becomes identical to the Rayleigh Onsager functional and thus relates to the principle of the minimal rate of entropy production. 

If the positively defined Lyapunov function exists for a particular kinetic scheme, this scheme has the stationary state that is stable in respect to the concentrations of the intermediates, whether they are close to or far from thermodynamic equilibrium. 

From analyzing this expression, the symmetry of coefficients M p in respect to the permutation of indices a and P occurs necessarily only in the case of independent rushes fij of internal parameters (concentrations of intermediates involved in the reaction groups). It was shown previ ously that these are namely the systems allowing the construction of the Lyapunov function to describe their evolution through the entire region of applicability of the thermodynamic approach. It is usually impossible to find the Lyapunov functional for the intermediate nonlinear schemes. 

In terms of thermodynamics, the energy dissipation P (or the positively defined Lyapunov function O) has its local minimums in the stable stationary points, and spontaneous jumping between stable stationary states in the system is only allowed when the identical "input" parameters are inherent in two states this may be, for example, common affinity ArE which is given from outside and provides the process. Therefore, we can consider these transitions as related to overcoming some dynamic "potential" barrier (see following). 

The stationary states that faU into fragment 1 of the curve (Figure 3.5) are stable at minor deviations of a from (1q in virtue of the theorem on the minimal rates of entropy production in these states. On further run ning away from the point a = aq, we may faU outside the region of applicability of nonlinear thermodynamics while remaining in the thermodynamic branch, which is described, for example, by a station ary state functional as a kind of positively defined Lyapunov function... 

However, the stationary states of the catalyst may be stable far from thermodynamic equilibrium—for example, due to the existence of the positively defined Lyapunov function for the given catalytic process (see Section 3.4). In particular, there are always stable stationary states of cata lytic systems with an arbitrary set of monomolecular transformations of catalytic intermediates ( the reactant-active center complexes) or any other set of these transformations, when they are linear in respect to the intermediate concentration or its thermodynamic rush. 

The first and second conditions ensure the existence of the thermodynamic Gibbs free energy function or, using the mathematical term, the convex Lyapunov function for kinetic equations. The Lyapunov function is a strictly positive function with a nonpositive derivative. The one exception to this definition is that at the equilibrium point, the Lyapunov function equals zero. In physicochemical sciences, the Gibbs free energy is an extremely important Lyapunov function for understanding the stability of equilibria. 

Special attention has to be paid to thermodynamic Lyapunov functions (Gorban, 1984). Basic variables describing a chemical state are amounts of components /, which can be expressed as amount of substance, or concentration, c,-. If the chemical system is homogeneous, the conditions are fixed and the equation of state is known, all system characteristics can be expressed using the vector of amounts of substance, n, and some constant parameters, for example... 

For the regimes defined in Section 7.2.1, a thermodynamic Lyapunov function, L, can be constructed, for example... 

The corresponding thermodynamic Lyapunov function LJji,U,T) at constant V and T can be found from... 

For any classical conditions, the derivative of the thermodynamic Lyapunov function will be presented as follows ... 

The second and, apparently, the more general idea is the idea of formation of a new scientific discipline - "Model Engineering" (Gorban and Karlin, 2005 Gorban et al., 2007). The subject of the discipline is the choice of an outset statement of the solved problem which is the most suitable (optimal) both for conceptual analysis and for computations. The transfer of kinetic description into the space of thermodynamic variables became a main method for this discipline. In the method the solved problem can be represented as one-criterion problem of search for extremum of the function that has the properties of the Lyapunov functions (monotonously moving to fixed points). 

Passive systems can be proven to be asymptotically stable because the dissipative storage function is a Lyapunov function. So if an arbitrary dynamic system can be made passive, the system can be stabilized. Ydstie and Alonso [14] have proven that for process systems a storage function exists that leads to a passive system. We call this storage function the Ydstie function. This Ydstie function is constructed from variables already available in the thermodynamic description. It is given by ... 

What are the suitable thermodynamic Lyapunov functions (evolution criteria) ... 

The Lyapunov function < ), (2.13), is both the thermodynamic driving force toward a stable stationary state and determines the stationary probability distribution of the master equation. The stationary distributions (2.33, 2.34) are nonequilibrium analogs of the Einstein relations at equilibrium, which give fluctuations around equilibrium. 

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