What Are Extremum Principles

The forces on atoms and molecules can be described in terms of two tendencies. Molecules react, change conformations, bind, and undergo other chemical or physical changes in ways that cause a quantity called the energy to reach its minimum possible value, or a quantity called the entropy to reach its maximum possible value. We can predict the tendencies of matter by computing the minima or maxima of certain mathematical functions. These are called extremum (or variational) principles. 

To illustrate, we first show that balls rolling downhill can be predicted by the minimization of energy. We describe the various t es of extrema, called states of eqmlibrium—stahle, unstable, neutral, and metastable states. Then, as a first step toward introducing the entropy and the Second Law of thermo-d Tiamics, we use the multiplicity concept from Chapter 1, and illustrate how gases exerting pressure, the mixing and diffusion of molecules, and rubber elasticity can be explained by a maximization principle. 

EXAMPLE 2.1 Mechanical equilibrium is the state of minimum potential energy. For a ball of mass m on a hillside at height z, the gravitational potential energy Viz) is 

Substitute z = x into V z) = mgz to find the potential energy of a ball in this quadratic valley  

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  

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