Alternation of derivatives is no criterion for choosing an entropy function

Abstract

Of the infinity of state functions of a gas given by integrating functionals of the one-particle distribution function over velocity space, it has been conjectured that only one has the property that all successive time derivatives alternate in sign at all times during free thermal relaxation. A function possessing this property in a certain interval is said to be completely monotonic in that interval. Monotonicity of only the first time derivative is known as an H theorem, and is necessary for a state function to be identifiable as entropy. The conjecture continues with a sufficient condition for this identification that the one completely monotonic state function be taken as the entropy. These conjectures are shown to be false: by choosing the initial distribution function and the differential collision cross-section appropriately in velocity space, the second derivative of any state function which satisfies an H theorem (and decreases) can be shown to change sign during the relaxation. The rate of decrease of such a state function under these circumstances will initially increase, but then level off as thermal equilibrium is approached: the state function evolves with an inflection. Not only must the criterion therefore be invalid, but worse, it cannot serve to distinguish between the various decreasing state functions and pick one out uniquely.