Memorial to the Classical Statistics 



By KARL K. D ARROW 



ONE of the most elusive and perplexing, hazy and confusing of the parts 

 of theoretical physics is that which bears the name of "statistical 

 mechanics".* On the principle that a tree is to be judged by its fruits, this 

 must be ranked as high as the tree which bore the golden apples of the 

 Hesperides; for among its fruits are the Maxwell-Boltzmann distribution- 

 law, the black-body radiation law, the value of the chemical constant, the 

 Fermi distribution-law for the electrons in metals, the alternating intensities 

 in band-spectra — and indeed the tree might lay a valid claim to the whole of 

 quantum-theory. The singular thing is that such wonderful fruits should 

 have grown from, or should have been grafted upon, so badly-rooted a tree. 

 To change the metaphor, one frequently feels that the superstructure is 

 sustaining the foundations, and the premises are flowing from the con- 

 sequences, rather than the other way about. Perhaps anyone who feels 

 this way should be disqualified from writing about the subject; but on the 

 present occasion, the attempt is going to be made. 



Statistical mechanics — hereinafter to be called "S.M." at times for short — 

 did not of course arise from any desire to solve the problems suggested above, 

 which came late. It seems to have sprung from attempts to answer older 

 questions, of which the following may serve as an example. Consider a gas 

 in a box, with an electric fan or something of the sort fitted inside to stir 

 it up. The gas having been stirred up, the fan is stopped, leaving it in a 

 state of surging and whirling about within the confines of the box. Very 

 shortly, however, the surging and the whirling cease, the gas having passed 

 of itself into a state of tranquillity and uniformity^ — uniform density, uni- 

 form pressure, uniform temperature. From this state it never departs, un- 

 less stirred up afresh. There is a tendency of the gas to go of itself from the 

 state of surging into the state of uniformity, and no tendency at all for it to 

 go from the state of uniformity back into the state of surging. This is very 

 unlike the behavior of a pendulum, which having fallen from one end of the 

 arc of its sweep to the middle thereof, moves on to the opposite end, re- 

 traces its path and returns to its first situation. Why should the gas behave 

 that other way? 



* I acknowledge with gratitude the incentive given me b}' Smith College to explore 

 this subject, by offering me the opportunity of giving a course on statistical and chemi- 

 cal physics in the spring semester of 1942. 



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