## Section 15.1: Exploring Functions: *g*(ε), *f*(ε), and *n*(ε)

*Please wait for the animation to completely load.*

A statistical system (whether classical or quantum mechanical)
can be described in terms of the number of particles with a given energy, *n*(ε). Restart. In general, this function is itself the product of two
functions: *n*(ε) = *g*(ε)*f*(ε). *g*(ε) counts the number of states of a given energy and is also
called the density of states. *f*(ε) is the probable distribution of
the particles through the states and is also called the occupancy. The form of
these functions varies depending on the system of interest: whether governed by
classical statistics or quantum statistics.

The animation shows you these distribution functions
for a system of fermions (since these are fermions, the correct statistics is Fermi-Dirac) at a particular temperature. You can evaluate the
integral of each function from any starting point and end point. For example, in order to find the total
number of particles, you must integrate *n*(ε) over the entire energy range.
How many particles are in this system? Over what energy range are states
filled (a probability of 1)? Over what range are they empty (a
probability of 0)? For this system, is the degeneracy constant as a
function of energy?