Statistics of single systems
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While probabilities of single events are meaningless (see the FAQ entry
on this), it is meaningful to do statistics on single systems, if
the statistics of interest is that of the system's behavior in time.
Indeed, there are many single systems with a sound statistical
interpretation. Once one has a time series of a single system that
empirically looks fluctuating, one can do valid statistics with it.
Sunspot activity data or El Nino data (both time series for single
physical systems) are traditional test data for statistical procedures.
Single systems data are routinely explained and forecast in terms of a
stochastic process whose associated ensemble is a mathematical fiction,
not a physical reality.
The association of a ficticious ensemble to single thermal systems goes
back to Gibbs 1900 (or even earlier), as one can see by reading his book
on statistical mechanics. He was very aware that thermodynamics and
hence statistical mechanics applies to single physical systems.
His arguments are today as cogent as when he introduced them.
His statistical mechanics formalism survived the quantum revolution
almost without change.
''We may imagine a great number of systems of the same nature, but
differing in the configurations and velocities which they have at a
given instant, and differing not merely infinitesimally, but it may
be so as to embrace every conceivable combination of configuration
and velocities.'' (from the preface, p. vii)
''Let us imagine a great number of independent systems, identical in
nature, but differing in phase, that is, in their condition with
respect to configuration and velocity.'' (p.5)
J.W. Gibbs,
Elementary Principles in Statistical Mechanics