Equilibrium, global stability

The second law for isolated systems shows that the excess entropy, A.V S SKI < 0, increases monotonically in time, d(AS)/dt > 0. Therefore, it plays the role of a Lyapunov function, and defines a global stability. So, dfi/dt is a Lyapunov function that guarantees the global stability of stationary states that are close to global equilibrium. 

These equations have immediate implications for the signs of the /indices. Namely, by the global stability criterion [31, 49] q = 8p/0N > 0, so that any addition of electrons to the system, dN > 0, raises the global chemical potential dp(dN > 0) > 0. Assume that both initial and final states correspond to the global equilibrium, so that dji = djia = djip =. .. > 0. Moreover in a stable molecular system all hardness eigenvalues are positive h > 0, a = 1,..., m. Hence it immediately follows from the EE equations (23) that, for stable systems,... 

The method will give us a quantitative recipe only for local stability, i.e. where the perturbed solution is sufficiently close to the equilibrium singular point. Global stability cannot be guaranteed without more detailed investigation. 

Thus, it should be recognized that AASf/TO LRFD (AASHTO 2012) includes overall or global stability under the service Umit states design due to complexities involved in defining load and resistance factors when soil acts to exert both load and resistance simultaneously. Slope stability analysis for LRFD is performed in the same manner as for WSD, except that resistance factor, as stated above, is taken as the inverse of the FS determined based on the limit equilibrium methods. 

An immediate consequence is that the closed system will be in stable equilibrium if and only if the entropy of that system is maximum under all admissible variations of its configuration. The stability can be local or global, depending on whether the variations of state used to probe stability are infinitesimal or arbitrary in magnitude. This is the form of the original statement on equilibrium and stability of material systems Gibbs (1876). ... 

The local equilibrium assumption was the basis on which the Brussels school developed a global thermodynamic theory. Use of this assumption makes possible the macroscopic evaluation of entropy production and entropy flow terms with macroscopic thermodynamic methods. The assumption states that "there exists within each small mass element of the medium a state of local equilibrium for which the local entropy, s, is the same function of the local macroscopic variables as at equilibrium state" (Glansdorff and Prigogine, 1971, p. 14). In other words, each small element of a system may be treated as a state near equilibrium but need not necessarily be at equilibrium. This does not mean that the system as a whole need be near equilibrium thus, neighboring local elements may differ in parameters (temperatures, chemical affinities, etc.) which are reflected in the function describing their local entropy. The additional assumption is made that the sum of the criteria of local stability for each element corresponds to the global stability criterion for the whole system. 

The key question we want to answer is what are the intrinsic sequence dependent factors tliat not only detennine tire folding rates but also tire stability of tire native state It turns out tliat many of tire global aspects of tire folding kinetics of proteins can be understood in tenns of tire equilibrium transition temperatures. In particular, we will show tliat tire key factor tliat governs tire foldability of sequences is tire single parameter... 

At one end of the spectrum are first-principles methods where the only input requirements are the atomic numbers Za, Zb,. .. the relevant mole fractions and a specified crystal structure. This is a simple extension to the methods used to determine the lattice stability of the elements themselves. Having specified the atomic numbers, and some specific approximation for the interaction of the relevant wave functions, there is no need for any further specification of attractive and repulsive terms. Other properties, such as the equilibrium atomic volumes, elastic moduli and charge transfer, result automatically from the global minimisation of... 

The completely reliable computational technique that we have developed is based on interval analysis. The interval Newton/generalized bisection technique can guarantee the identification of a global optimum of a nonlinear objective function, or can identify all solutions to a set of nonlinear equations. Since the phase equilibrium problem (i.e., particularly the phase stability problem) can be formulated in either fashion, we can guarantee the correct solution to the high-pressure flash calculation. A detailed description of the interval Newton/generalized bisection technique and its application to thermodynamic systems described by cubic equations of state can be found... 

Let us assume internal equilibrium in Zf, which corresponds to the mutually open subsystems, Zfe=(Z/> Zf 2), with equalized chemical potentials, nf = nf = P.r = dE/dN, at the global chemical potential level. The internal stability refers to intra-5 (hypothetical) charge displacements, dA/y(A) = (A, — A), that preserve N. The corresponding quadratic energy change due to this polarizational displacement from the initial internal equilibrium state ... 

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. 

Equation (12.74) can be used in Eq. (12.69) for stability analysis at near global equilibrium. 

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. 

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