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I confess that I can't parse defn 4 ....
.... I can't make any sense of Def 4 either.
4. I'm for the standard definition of heat given in statistical
mechanics as the integral of the infinitesimal contributions to the
differential change in the macroscopic energy expectation due to a
change in the probability distribution for the system occupying its
various microscopic states. Such a change in the macroscopic energy
expectation will typically be associated with a change in the
system's entropy resulting from a change in the system's
distribution of microstates which are accessible to the system's
microscopic dynamics over the time interval that the changes occur.