Thermodynamics extremum principle

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. 

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. 

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. 

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... 

Now we introduce a different extremum principle, one that predicts the distributions of outcomes in statistical systems, such as coin flips or die rolls. This wall lead to the concept of entropy and the Second Law of thermodynamics. 

To use extremum principles, which can identify states of equilibrium, we find points at which derivatives are zero. In mathematics, a critical point is w here a first derivative equals zero. It could be a maximum, a minimum, or a saddle point. In statistical thermodynamics, critical point has a different meaning, but in this chapter critical point is used only in its mathematical sense. 

Equations (7.4) and (7.5) are completely general statements that there are some functional dependences S U, V,N) and U(S, V, N), and that these dependences define T, p, and p. Equations (7.4) and (7.5) are fundamental because they completely specify all the changes that can occur in a simple thermodynamic system, and they are the bases for extremum principles that predict states of equilibria. An important difference between these fundamental equations and others that we will write later is that S and U are functions of only the extensiv e variables. Beginning on page 111 we will show how to use Equations (7.4) and (7.5) to identify states of equilibrium. 

For processes in test tubes in laboratory heat baths, or processes open to the air, or processes in biological systems, it is not the work or heat flow that you control at the boundaries. It is the temperature and the pressure. This apparently slight change in conditions actually requires new thermodynamic quantities, the free energy and the enthalpy, and new extremum principles. S> stems held at constant temperature do not tend toward their states of maximum entropy. They tend toward their states of minimum free energy. 

The Legendre transformation could be formulated as an extremum-task as well [i ]. The Onsager-Machlup function (OM-function) play a central role in the extremum principles of the irreversible thermodynamics based on generalized Onsager constitutive theory. The OM-function could be introduced by the spontaneous entropy production and one of dissipation potentials using the Legendre transformation. 

Equilibrium thermodynamics too has its extremum principles. In this chapter we will see that the approach to equilibrium under different conditions is such that a thermodynamic potential is extremized. Following this, in preparation for the applications of thermodynamics in the subsequent chapters, we will derive general thermodynamic relations. 

We have already seen that all isolated systems evolve to the state of equilibrium in which the entropy reaches its maximum value. This is the basic extremum principle of thermodynamics. But we don t always deal with isolated systems. In many practical situations, the physical or chemical system under consideration is subject to constant pressure or temperature or both. In these situations the positivity of entropy change due to irreversible processes, diS > 0, can also be expressed as the evolution of certain thermodynamic functions to their... 

In thermodynamics the existence of extremum principles has an important consequence for the behavior of microscopic fluctuations. Since all macroscopic... 

When a system is far from thermodynamic equilibrium, the state may not be characterized by an extremum principle and the irreversible processes do not always keep the system stable. The consequent instability within a far-firom-equilibrium system drives it to states with a high level of organization, such as concentration oscillations and spontaneous formation of spatial patterns. We shall discuss far-from-equilibrium in-stability and the consequent selforganization in Chapters 18 and 19. 

Indeed a characteristic feature of equilibrium thermodynamics is the existence of extremum principles. For isolated systems entropy increases and is therefore maximum at equilibrium. In other situations (such as constant temperature) there exist functions called thermodynamic potentials which are also extrema (maxima or minima) at equilibrium. This has important consequences. A fluctuation which leads to a deviation from equilibrium is followed by a response which brings back the system to the extremum of the thermodynamic potential. The equilibrium world is also a stable world. This is no longer so in far-from-equilibrium situations. Here fluctuations may be amplified by irreversible dissipative processes and lead to new space-time structures which... 

The main novelty is that in far-from-equilibrium situations which correspond to the third stages of thermodynamics, an extremum principle seldom exists (Chapters 18-19). As a result, any fluctuations may no longer be damped. Stability... 

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