Extremum principles

By analogy with Hamilton s principle of least action, the simplest proposition that could solve the thermodynamic problem is that equilibrium also depends on an extremum principle. In other words, the extensive parameters in the equilibrium state either maximize or minimize some function. 

If the amount of heat transferred is infinitesimally small, the extremum principle dictates that the entropy of the system in virtual equilibrium does not change, dS = 0. Since dU0) 0, it is implied that, at equilibrium... 

The fact that each of the two variables, S and U may be expressed as a function of the other, indicates that the extremum principle could likewise be stated in terms of either entropy or energy. The alternative to the maximum entropy principle is a minimum energy principle, valid at constant entropy, as graphically illustrated for a one-component system in figure 1, below. 

As for U it follows without further analysis that the extremum principle for thermodynamic equilibrium at constant entropy should also apply to the other potentials under suitable conditions, i.e. constant T for minimum F, constant pressure for minimum H, and constant temperature and pressure for minimum G. 

In earlier publications we have shown that the principle of selection can be deduced from the premises of a self-replicating system as an extremum principle. It states that inherent linear autocatalysis causes the relative population numbers to take on values that correspond to the highest reproductive efficiency of the system as a whole. The distribution of relative concentrations in the stationary population is, after a short induction period, independent of changes of the system as a whole. The population consists of a uniquely defined wild-type (or several equivalent,... 

The consequences of this extremum principle, valid for Darwinian systems, are... 

Prior to deriving conditions for equilibrium of a system, mathematics on the extremum principle is revisited in die following ... 

Chemistry remains a mystery without understanding three basic attributes of matter cohesion, structure and affinity, each of them shaped by an extremum principle. The principle of minimum energy regulates chemical cohesion, which results from the interaction between atoms, also known as chemical bonding. Minimization of angular momentum dictates the three-dimensional arrangement of atoms in chemical substances, which defines their... 

An extremum principle minimizes or maximizes a fundamental equation subject to certain constraints. For example, the principle of maximum entropy (dS)v = 0 and, (d2S)rj < 0, and the principle of minimum internal energy (dU)s = 0 and (d2U)s>0, are the fundamental principles of equilibrium, and can be associated with thermodynamic stability. The conditions of equilibrium can be established in terms of extensive parameters U and. S, or in terms of intensive parameters. Consider a composite system with two simple subsystems of A and B having a single species. Then the condition of equilibrium is... 

Equilibrium thermodynamics has various extremum principles. At various conditions, a thermodynamic potential will approach an extremum value as the system approaches equilibrium. For an isolated or closed system, we may consider the following extremum principles ... 

A system reaches the thermodynamic equilibrium state when it is left for a long time with no external disturbances. At equilibrium the internal properties are fully determined by the external properties. This makes it easy to describe such systems for example, if the temperature is not uniform within the system, heat is exchanged with the immediate surrounding until the system reaches a thermal equilibrium, at which the total internal eneigy U and entropy S are completely specified by the temperature, volume, and number of moles. Therefore, at equilibrium there are no thermodynamic forces operating within the system (Figure 2.1). Equilibrium systems are stable. For small deviations, the system can spontaneously return to the state of equilibrium. Equilibrium correlations result from short-range intermolecular interactions. Existence of the extremum principles is a characteristic property of equilibrium thermodynamics. 

Irreversible processes may promote disorder at near equilibrium, and promote order at far from equilibrium known as the nonlinear region. For systems at far from global equilibrium, flows are no longer linear functions of the forces, and there are no general extremum principles to predict the final state. Chemical reactions may reach the nonlinear region easily, since the affinities of such systems are in the range of 10-100 kJ/mol. However, transport processes mainly take place in the linear region of the thermodynamic branch. 

Hill, R. and Power, G., Extremum principles for slow viscous flow and approximate calculation of drag, Quarterly J. Mech. Appl. Math., Vol. 9, No. 3, pp. 313-319, 1956. 

Consider as an illustrative example a single component case. As in the ordinary thermodynamics of open systems [146] the entropy extremum principle of equation (118) requires the constraint of the fixed number of electrons, N p = N°. Moreover, in order to introduce a temperature parameter T, associated with the constraint of the fixed average energy as the inverse of the condition Lagrange multiplier, one... 

Prestipino, S., Giaquinta, P.V. The concavity of entropy and extremum principles in thermodynamics. J. Stat. Phys. 111(1-2), 479 93 (2003). URL http //www.springerlink.com/ content/n411802313285077/fuUtext.pdf... 

Sieniutycz, S., Parkas, H. (eds.) Variational and Extremum Principles in Macroscopic Systems. Elsevier, Amsterdam (2005)... 

Schapery [100] has obtained bonds for the effective for both isotropic and anisotropic composites consisting of isotropic phases by employing extremum principles of thermoplasticity. The exact solutions were derived in particular cases. For a unidirectional composite, the following expressions are obtained. The a (ay) for a fiber-reinforced composite in the longitudinal direction is given by the following equation... 

The introduction and use of parsimony methods in phylogenetic estimation is described and discussed against the background of Ockham s razor and extremum principles in science generally. It is argued that the method of maximum likelihood applied to realistic models of evolutionary divergence is the optimum procedure, to which parsimony methods are often an acceptable approximation. 

These outstandingly successful developments involving extremum principles contributed to the idea that one can elevate into a principle the minimization of some quantity or other — in our case a quantity of evolution — in order to infer an evolutionary tree. But the situations are quite different, and an amount of evolution is analogous neither to the potentials of physics nor the errors of statistics. 

The Reconstruction of Evolution The Principle of Minimum Evolution The Influence of Traditional Procedures Maximum Tikelihood Extremum Principles in Science Epilogue References... 

---