Statistical Physics

30 2 Entropy2.3.2 Equilibrium Condition when Molecules are ExchangedNext we consider the case in which two systems are connected by a small hole,and molecules can be exchanged between them. Since the molecules carryenergy, energy is also exchanged. The energy and the number of molecules ineach system are not conserved, but the total number of molecules N = N I +N IIand the total energy E = E I +E II are conserved. We already know the densityof states of the total system when each of the two systems has a fixed numberof molecules. The density of states in the present case is obtained by summingthe density of states for a fixed number over the possible distributions of themolecules, i.e. for 0 ≤ N I ≤ N. Thus, the density of states of the total system isN∑∫ EΩ(E,N)= Ω I (E I ,N I )Ω II (E − E I ,N − N I )dE I . (2.17)N I=00The probability for system I to have an energy between E I and E I +dE I andN I molecules isf(E I ,N I )dE I = Ω I(E I ,N I )Ω II (E − E I ,N − N I )dE I.Ω(E,N)(2.18)Thermal equilibrium is attained when this probability reaches its maximum.Since the logarithm of the numerator is the entropy of the total systemS(E,N) =S I (E I ,N I )+S II (E II ,N II ), the condition is that the total entropyis maximized under the constraint of fixed E and N:( ∂S(E,N)∂E I) ( ∂SI (E I ,N I )∂E I) ( )∂SII (E − E I ,N − N I )∂Eand( )∂S(E,N)∂N IE,N,N I=E,N,E I=N I−= 0 (2.19)( )∂SI (E I ,N I )∂N IE I−( )∂SII (E − E I ,N − N I )∂N=0. (2.20)N IE IFig. 2.3. Two systems connected by a small hole