Potential extremum value

Prior to the advent of high-speed computers, methods of optimization were limited primarily to analytical methods, that is, methods of calculating a potential extremum were based on using the necessary conditions and analytical derivatives as well as values of the objective function. Modem computers have made possible iterative, or numerical, methods that search for an extremum by using function and sometimes derivative values of fix) at a sequence of trial points x1, x2,. 

A thermodynamic potential reaches an extremum value toward equilibrium under various conditions. The Helmholtz free energy A is particularly useful for systems at constant volume and temperature. Combining Eq. (1.76) and Eq. (1.244) at constant temperature yields... 

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 ... 

Potentials are of particular interest in their extremum values, that is maxima or minima, since this indicates a sort of limit on the system concerned. In the case of a ball rolling (without friction) in a parabolic trough that we considered in Chapter 3 (Figure 5.1),... 

Unless demanded by symmetry, the value of will not be an extremum at a critical point in p. Thus, the critical points in V and p will not, in general, coincide and the distribution of eleetronie eharge, even in a one-electron system, is not determined entirely by the external foree — V V t). Bohm (1952) ascribed the stability of a stationary state in a quantum system to the balance between the classical force —W and the quantum mechanical force which is given by the gradient of the quantum potential . For a one-electron system at a critical point in p(r), Bohm s quantum mechanical force is just the right-hand side of eqn (3.8). The Laplacian of the charge density, the quantity appearing in the quantum potential, is an important local property of a system and is the subject of Chapter 7. 

In the equilibrium statistical mechanics, the unknown probabilities of microstates p, are found from the second part of the second law of thermodynamics, i.e., from the constrained extremum of the thermodynamic potential (Eq. (29)) as a function of the variables (pv pw) under the condition that the variables (pv. .., pw) satisfy Eq. (27). Moreover, it is supposed that the value of the entropy in the i th microstate of the system is a function of the probability pt of this microstate, i.e., =Sf=Sf(pf). Then to determine the unknown probabilities [pt] at... 

This is the criterion for material or diffusional stability for a binary mixture to be differentially stable, the mixture must have Q > 0, Kt- > 0, and (at fixed T and P) the chemical potential of component 1 must always increase in response to any increase in Nj. This means that if an isothermal-isobaric plot of the chemical potential (or fugacity) passes through an extremum with Xj, then the mixture is unstable for some Xj-values. The result (8.3.13) confirms (3.7.29) in which we claimed that the chemical potential of a pure component is always greater than its value in any mixture at the same T and P. 

So far, we use the model that assumes the fact that critical nuclei of aU phases, allowed by the phase diagram, appear at once (the unlimited nucleation model). It is known that the growth of a new phase from the nucleus is energetically favorable only in the case of nucleus size exceeding some critical value la, determined from the extremum condition of Gibbs thermodynamic potential. In a one-component substance, the extremum condition is expressed simply by the derivative of G with respect to the nucleus size being equal to zero ... 

The extremum principle says that the ball will roll to wherever the potential energy is a minimum. To find the minimum, determine what value, x = x, causes the derivative to be zero ... 

Due to the Lagrangian of the functional (99) is the sum of the dissipation potentials, which is equal to the entropy production in case of every real steady-state physical processes, this extremum theorem involves the minimum principle of global entropy production (MPGEP). The physical meaning of MPGEP needs a clarification. Consider the variations of the fluxes and of the intensive parameters as fluctuations of the system around their stationer state values. When these fluctuations are small, the fluctuation of the global entropy production of the system is equal to its first approximations and it has a form... 

As since the necessary condition of the extremum of functional is the zero value of its first variation. Due to the variations of the potential are arbitrary and independent of the variations of the flux, than we get with S

[Pg.267]

To satisfy this statement, the expression in parentheses describing the potential energy is required to assume a stationary value. Furthermore, it can be shown that this extremum has to be the minimum of the potential energy, see Sokolnikoff [167] or Knothe and Wessels [113]. Thus, Dirichlet s principle of minimum potential energy can be extended to electromechanically coupled materials ... 

They both have a single minimum, an inflexion point for values of X attained at the explosion time, and no further extremum in the region of high values of x. The only difference between them is that in the thermal case the minimum is at 2 = 0 (closed system, complete combustion) whereas in the chemical case it is located at a finite value of x (open system) corresponding to the unique stable state of the rate equation in the region outside the curves (a ) and ( ) in Fig. 3. In a typical situation the system is started in the region of x values for which the potential is flat. A slow evolution, reflected by this flatness, first take place but when the vicinity of the inflexion point is reached, X becomes quickly depleted and finally evolves to the unique final state. 

Before taking up the specific discussion on our results, let us highlight that the existence of extremum points in the SC is a consequence of the confinement effects brought about by the presence of external fields. This is in line with the recent finding of Patil et al. [25], who show for some specific constrained Coulomb potentials that the simplest composite uncertainty measure, the Heisenberg uncertainty product, of the electron density presents an extremum located at a critical position which scales as the reciprocal value of the potential strength. 

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