RECENT STATISTICAL THEORIES 745 



so the value predicted for the rate of generation of heat per unit 

 volume amounts to this: 



,J'd( dT\ ,3kdT ^ ,.,_, 



The first term is obviously the Joule heat; the second is not directly 

 a consequence of this current-flow, as it would occur whatever the 

 agency which set up the temperature-distribution in question. It is 

 the third term which concerns us; this is a "reversible heat," pro- 

 portional to the first power of the current, so that when the current 

 flows in one sense heat is absorbed and when it is reversed heat is 

 evolved. The sign is such, that heat is absorbed when the electrons 

 are flowing towards the hotter part of the metal; the magnitude is 

 such, that as the electrons move onward they acquire just enough 

 energy to raise their temperature to that of the regions which they 

 enter. The coefiicient of this term therefore represents the specific 

 heat of the electron-gas, which is the same as that of any other mona- 

 tomic gas when referred to equal numbers of particles. 



Adopting instead the Fermi distribution, and inserting into (142) 

 and (143) the values of v and v~^ and v^ prevailing at absolute zero, 

 we find on making the substitutions in the expression (138) for r that 

 the terms containing the first power of J balance one another out. 

 This might have been expected; for we have just seen that these 

 terms form a sum which is proportional to the specific heat of the 

 electron-gas; and if this result may be extended to an electron-gas 

 conforming to the Fermi distribution, then since the specific heat 

 vanishes at zero so also must this "reversible heat." Working 

 through the second approximation, Sommerfeld found that the net 

 coefiicient of the term in / in the expression for r is in fact proportional 

 to the specific heat of the electron-gas, being therefore proportional 

 to T, and given by the formula: 



I^^llfY'r. (146) 



3 elr \ 3n / 



Now it is a fact of experience that when an electric current flows 

 along a uniform wire of uneven temperature, heat is generated at a 

 rate which involves a term proportional to the current and which 

 changes sign when the current changes sense. This "Thomson heat," 

 like the maximum value which experiments allow us to admit for the 

 specific heat of the electron-gas, has always been much smaller than 

 the value which the classical statistics requires provided that the free 



