Statistical Mechanics - Physics at Oregon State University

1.4. AVERAGES. 13
< f >= 1
L
systems
f(system) = 1
L
which reduces to our previous definition.
Reality is not uncorrelated!
Lg(N, M)f(M)2 −N
M
(1.34)
In a real experiment we find average quantities in a very different way. If
there are no external probes which vary as a function of time, one measures f
over a certain time and takes the average:
< f >time= 1
T
f(t)dt (1.35)
T 0
The basic assumption is that if we wait long enough a system will sample all
accessible quantum states, in the correct ratio of times! Nobody has yet been
able to prove that this is true, but it is very likely. The alternative formulation
of our basic assumption tells us that all accessible states will be reached in this
way with equal probability. This is also called the ergodic theorem. Hence we
assume that, if we wait long enough,
< f >ensemble=< f >time
(1.36)
In a measurement over time we construct a sequence of systems which are correlated
due to the time evolution. The ergodic theorem therefore assumes that
such correlations are not important if the time is sufficiently long. Again, beware
of metastable states!
Spatial averages.
If the function f is homogeneous, or the same everywhere in the sample,
it is also possible to define a spatial average. Divide the sample into many
subsystems, each of which is still very large. Measure f in each subsystem and
average over all values:
< f >space= 1
V
f(r)d 3 r (1.37)
The subsystems will also fluctuate randomly over all accessible states, and this
spatial average reduces to the ensemble average too. In this formulation we use
a sequence of systems which are again correlated. We assume that when the
volume is large enough these correlations are not important.
The difference between the three descriptions is a matter of correlation between
subsystems. In the ensemble average all subsystems are strictly uncorrelated.
In the time average the system evolves and the subsystem at time t+dt is