1. D 6 Summary and Conclusions a) Entropy as defined from a microscopic point of view is a measure of randomness in a system b)The entropy is related to the probabilities pi of the individual quantum states of the system by ∑P2lmP where k, the Boltzmann constant is given by R/NAvogadre c) For a system in which there are Q2 quantum states, all of which are equally probable(for which the probability is P=1/ 2), the entropy is given by s=kIne The more quantum states, the more the randomness and uncertainty that a system is in a particular quantum state d ) From the statistical point of view there is a finite, but exceedingly small possibility that a system that is well mixed could suddenly"unmix"and that all the air molecules in the room could suddenly come to the front half of the room. The unlikelihood of this is well described by Denbigh [Principles of Chemical equilibrium, 1981]in a discussion of the behavior of an isolated system In the case of systems containing an appreciable number of atoms, it becomes increasingly improbably that we shall ever observe the system in a non-uniform condition. For example, it is calculated that the probability of a relative change of density, Ap/p, of only 0.001% in 1 cm'of air is smaller than 10 and would not be observed in trillions of years. Thus, according to the statistical interpretation the discovery of an appreciable and spontaneous decrease in the entropy of an isolated system, if it is separated into two parts, is not impossible, but exceedingly improbable. We repeat, however, that it is an absolute impossibility to know when it will take place. 1D-71D-7 1.D.6 Summary and Conclusions a) Entropy as defined from a microscopic point of view is a measure of randomness in a system. b) The entropy is related to the probabilities pi of the individual quantum states of the system by S kpp i i = − ∑ i ln where k, the Boltzmann constant is given by R/ NAvogadro . c) For a system in which there are Ω quantum states, all of which are equally probable (for which the probability is pi = 1/Ω ), the entropy is given by S k = lnΩ . The more quantum states, the more the randomness and uncertainty that a system is in a particular quantum state. d) From the statistical point of view there is a finite, but exceedingly small possibility that a system that is well mixed could suddenly "unmix" and that all the air molecules in the room could suddenly come to the front half of the room. The unlikelihood of this is well described by Denbigh [Principles of Chemical Equilibrium, 1981] in a discussion of the behavior of an isolated system: "In the case of systems containing an appreciable number of atoms, it becomes increasingly improbably that we shall ever observe the system in a non-uniform condition. For example, it is calculated that the probability of a relative change of density, ∆ρ ρ, of only 0.001% in 1 cm3 of air is smaller than 10 108 − and would not be observed in trillions of years. Thus, according to the statistical interpretation the discovery of an appreciable and spontaneous decrease in the entropy of an isolated system, if it is separated into two parts, is not impossible, but exceedingly improbable. We repeat , however, that it is an absolute impossibility to know when it will take place