How to derive an expression for entropy in form of S = ln ⁡ Ωfrom the form S = − ∑ i p i ln ⁡ p i ?

Question

How to derive an expression for entropy in form of  S  =  ln  ⁡  Ωfrom the form            S      =      −              ∑        i                          p        i            ln      ⁡              p        i             ?

kunstdansvo · Accepted Answer

If there are   Ω states and       p    i   is constant by the fundamental postulate of statistical mechanics, then you have  1  =      ∑          i      =      1        Ω           p    i    =  Ω         p    i          ⟹            p    i    =      1    Ω    ,and consequently  S  =  −      ∑          i      =      1        Ω           p    i    ln  ⁡      p    i    =  −      ∑          i      =      1        Ω           p    i    ln  ⁡      1    Ω    =      (    −    ln    ⁡          1      Ω        )        ∑          i      =      1        Ω           p    i    =  ln  ⁡  Ω  .

How to derive an expression for entropy in form of S=ln omega from the form S=−sum_i p_i lnp_i ?

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