Could anyone give some definite proof for this? ∫ Q r e v T = Δ ( k B ln ⁡ Ω ) = Δ S

Question

Could anyone give some definite proof for this?  ∫            Q              r        e        v              T    =  Δ  (      k    B    ln  ⁡  Ω  )  =  Δ  S

Claire Love · Accepted Answer

There are many resources out there on microcanonical ensemble. After you come to the conclusion that  β  =            (                        ∂          ln          ⁡          Ω                          ∂          U                    )              N      ,      V      fully characterizes thermal equilibrium, you know that it must be a function of thermodynamic temperature, and thermodynamic temperature alone, by virtue of zeroth law of thermodynamics. So   β  =  f  (  T  )Compare this to      1    T    =            (                        ∂          S                          ∂          U                    )              N      ,      V      and you get that S must be the function of   ln  ⁡  Ω. The remaining question is the exact form of this function, and you already derive it from special cases.

Could anyone give some definite proof for this? int Q/T=/_\(k_b ln omega)=/_\S

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