Page 175 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Statistics
N
three spatial components. With particles this forms a 6N
-dimensional
i
phase space to represent the different implementations . We can now
construct a hypersurface in the 6N -dimensional space that corresponds
to the energy E .
i
In an interval δE that is small in comparison to the total energy of the
system, an external observer cannot distinguish between the different
i
implementations . Even if there was a slight difference in the energy it
would not be resolvable by the accuracy limit of possible measurements.
i
Every implementation therefore has an equal probability P to be real-
i
ized, because from a macroscopical point of view the implementations
are indistinguishable.

 c for E < E < E + δE)
(
P =  i (5.1)
i
 0 otherwise
c
Here is a constant which comes from the normalization that requires
that ΣP = 1 , where the sum counts every state that A is allowed to
i
assume in the given energy interval. This means that in an ensemble that
has much more members than the maximum number of implementations
each implementation is realized by the same number of member systems.

The volume Γ E() that the system occupies in the 6N -dimensional
phase space, i.e., the number of possible implementations in the small
energy interval E <( E < E + δE) , is given by
i
∫
Γ E() = d 3N qd 3N p (5.2)
( E < E i < E + δE)

which evaluates to
(
Γ E() = VE + δE) VE() (5.3)
–
where VE() is the volume of the system occupying in the energy range
( 0 < E < E) . For the case where δE « E , we expand (5.3) into a linear
i
Taylor series to obtain

172 Semiconductors for Micro and Nanosystem Technology