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**Additional resources for On the Gravitational Field of a Sphere of Incompressible Fluid according to Einstein’s Theory**

**Example text**

Then each integer point in the multidimensional space composed of 3N quantum numbers represents a microscopic state of the system. 13) where positive and negative integers nb n2, n3, ... , n3N are quantum numbers. The total numbers j(E) of quantum states with energy less than E is given by the volume of the 3N-dimensional sphere of radius P = -I2mE divided by the unit (hIL)3. This volume is 11:3N 12 r(3N12 + I)P 3N "" (~)3N 12 3N12 3N ,.... 14) is the asymptotic value for N» 1. 15) but this is not correct as can be seen from the following consideration.

Then the series will begin with the second power, since S must increase irrespective of the sign of a. Thus dSjdt oc a2 or dSjda oc a for a small value of a. Therefore, when the rate of change a approaches 0, the change in S due to the change in a, or dSjda, also approaches 0. We see that in the limit of small a, the process (quasi static process in thermodynamics) gives no change in entropy and is shown to be reversible. In this consideration, the system was assumed to have received no influence other than the change in the external variable.

In other 46 2. Outlines of Statistical Mechanics words, the entropy S = k lnj of the system does not change during an adiabatic change. To show the above relation, we rewrite it in a differential form as Aj = :~ AE + g~ Aa = 0 . 22) The second term is the increase in the number of states below the energy surface :If = E when the external variable is changed by the amount Aa. In other words, the number of states below E is decreased every time an energy eigenvalue exceeds E. To calculate this, we note that D is the number of states in a unit energy range just below :If = E, and let r be the numbering of these states arranged in order according to the magnitude of oEr/oa.