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S1m. When is an object macroscopic?
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One says that thermodynamics and statistical mechanics apply to
macroscopic objects. But when is an object macroscopic?
Thermodynamics and statistical mechanics are approximate, asymptotic
descriptions valid for 'sufficiently large' objects.
The approximations made are better and better the larger the object.
One can place the barrier anywhere; if one puts it too low, the
approximate description will be poor, if one puts it too high it
won't apply to the system of interest.
Thus the loose language accommodates the freedom in modeling the
user has when choosing the description level and the accuracy level.
It is only in the same sense subjective as is the choice of a
system of interest. What is interesting for one person or investigation
may be different from what is interesting for another person or
investigation; nevertheless, both may employ objective tools.
The mathematical meaning underlying this loose language is called the
thermodynamic limit. It makes the term 'macroscopic'
precise in a similar way as the mathematical notion of a limit N->inf
makes the term 'N sufficiently large' precise.
If one accepts the vague terminology to avoid talking always about
limits, one can give the following definition (which reflects the
subjectivity in the qualification about the modeling accuracy):
In statistical mechanics, all macroscopic observables are ensemble
averages. Thus, formally, a "macroscopic observable" is the expectation
of a space-time dependent field operator which remains constant
within the modeling accuracy under changes in space and time
smaller than the modeling accuracy.