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In these lecture notes (http://ocw.mit.edu/courses/physics/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2007/lecture-notes/lec1.pdf [Broken]), they get the equation:

[tex] F_{AC} (A_1, A_2, \ldots; C_2, C_3, \ldots) = F_{BC} (B_1, B_2, \ldots; C_2, C_3, \ldots) [/tex]

Then they claim that the additional contraint

[tex] f_{AB}(A_1, A_2, \ldots; B_1, B_2, \ldots) = 0 [/tex]

means that the first equation is independent of [itex] C_i [/itex], and so there are functions [itex] \Theta_A [/itex] and [itex] \Theta_B [/itex] such that

[tex] \Theta_A(A_1, A_2, \ldots) = \Theta_B (B_1, B_2, \ldots ) [/tex]

Maybe I'm missing something, but the whole thing feels a little hand-wavy to me, and I'm having trouble seeing a more mathematically rigorous justification for this step. Can anyone help me fill in the gaps and justify this step in a little more detail?

[tex] F_{AC} (A_1, A_2, \ldots; C_2, C_3, \ldots) = F_{BC} (B_1, B_2, \ldots; C_2, C_3, \ldots) [/tex]

Then they claim that the additional contraint

[tex] f_{AB}(A_1, A_2, \ldots; B_1, B_2, \ldots) = 0 [/tex]

means that the first equation is independent of [itex] C_i [/itex], and so there are functions [itex] \Theta_A [/itex] and [itex] \Theta_B [/itex] such that

[tex] \Theta_A(A_1, A_2, \ldots) = \Theta_B (B_1, B_2, \ldots ) [/tex]

Maybe I'm missing something, but the whole thing feels a little hand-wavy to me, and I'm having trouble seeing a more mathematically rigorous justification for this step. Can anyone help me fill in the gaps and justify this step in a little more detail?

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