Observables in Statistical Physics
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.. index:: Observables

Observables in Equilibrium State
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In statistics, the average of an observable is

.. math::
   O = Tr(\hat O \rho)

where :math:`\rho` is the "probability matrix". In QM, this is

.. math::
   \rho = \frac{ e^{-\beta H} }{\mathrm{Tr} ( e^{-\beta H} ) }


If a system of QM comes to equilibrium, that means

1. :math:`\rho  = \frac{ e^{-\beta H} }{\mathrm{Tr} e^{-\beta H} }`;
2. :math:`\rho` is diagonal in energy eigen space.