6 V O L . 3 , D O C . 1 0 a B O L T Z M A N N S P R I N C I P L E total time during which the system is just assuming the state . We shall call the ratio the probability W of the state concerned. If this definition of the probability of a state is taken as a basis, it is generally understandable that a system changes on average from one state such that from this state follows with the greatest probability the neighboring state . I only have to mention this without going into the proof. This is the answer to the second of the questions posed above. It is essential that the definition of the probability of a state be definable inde- pendent of the kinetic picture probability W is a magnitude in principle accessible to observation, even though in most cases direct observation of it is excluded, owing to the brevity of the time at our disposal. If a system in a state substantially differing from thermodynamic equilibrium is left alone, it successively assumes states of ever greater W. A state’s probability W shares this property with the entropy S of a system, and Boltzmann found out that the relation between W and S holds, where k is a universal constant, i.e., independent of the system chosen. This is the important equation that ... the mathematical expression of the Boltzmann conception This Boltzmann equation can be applied in two different ways. There can be a more or less complete picture of molecular theory on the basis of which one can calculate the probability W. The Boltzmann equation then yields the entropy S. This was how Boltzmann’s equation has mostly been applied hitherto. Example.[6] In a volume V, let there be N molecules, i.e., one gram molecule of a particular type. The volume is large enough compared to the eigenvolume of the N molecules and of the other existing matter besides the N molecules—provided such matter is there—distributed evenly over so that the various points of are equivalent for each of the N molecules. This is an incomplete expression of how we visualize an ideal gas or a dilute solution. How large is the probability W that at a randomly chosen instant all N molecules are within the partial volume V of vol- ume ? A simple consideration yields . From this, using the Boltzmann constant equation, we find τ T -- - [p. 6] Za Zb S klgW = [p. 7] V0 V0 V0 W V V0 ----- - N =
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