Which relation E ( β ) is required to ensure that he micro-canonical distribution and the canonical distribution have the same expected energy?

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Which relation   E  (      β    ) is required to ensure that he micro-canonical distribution and the canonical distribution have the same expected energy?

Aidyn Mccarthy · Accepted Answer

The relation between energy and temperature (or its inverse   β) is highly system-dependent. The energy E, in the microcanonical ensemble, is a fixed external parameter, while       β          m      i      c      r      o       is             ∂                        S                      m            i            c            r            o                                      ∂              E            , within a constant factor (the inverse Boltzmann constant). In the canonical ensemble,       β          c      a      n       is an independent fixed parameter, while the average internal energy can be obtained as      U          c      a      n        =            ∂              (                  β                      c            a            n                                    F                      c            a            n                          )                    ∂                        β                      c            a            n                                ,      F          c      a      n        =      U          c      a      n        −      T          c      a      n            S          c      a      n       being the Helmholtz free energy.The possibility that      β          c      a      n        =      β          m      i      c      r      o          ⟺        U          c      a      n        =  Eis the so-called equivalence of the ensemble.It holds under quite general conditions on the hamiltonian (existence of the thermodynamic limit) as an asymptotic result in the limit of large systems. It does not depend on a specific   E  (  β  ) relation.

Which relation E(beta) is required to ensure that he micro-canonical distribution and the canonical distribution have the same expected energy?

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