Caption: Exponential decay illustrated
for several INVERSE
efolding constants: 25, 5, 1, 1/5, 1/25.
The exponential decay formula is:
F(x) = e^{x/x_e} ,
where
e = 2.71828 ...
(the famous irrational number
e)
and
x_e is the efolding constant.
Features:
 Here we are interested in the
the exponential decay formula's
use in thermodynamics
in its microscopic formulation
statistical mechanics.

Systems (or particles) obeying the
Boltzmann distribution
are distributed among microscopic states of energy E
according the Boltzmann factor:
F(E) ∝ e^{E/(kT)} ,
where
E is energy with zeropoint set at the
ground state,
k = 8.6173324(78)*10**(5) ≅ 10**(4) eV/K is
Boltzmann's contant,
T is Kelvin temperature,
and kT is an energy
parameter in units of
electronvolts (eV).
 As you can see, kT is the
efolding constant of
the Boltzmann factor.
Explicitly,
kT = 8.6173324(78)*10^{5} eV/K * T ≅ 10^{4} eV/K * T .
 KT controls the distribution of systems among the microscopic states:
 The smaller kT (or T), the more rapid the
exponential decay with E
and systems are more concentrated in the lower energy states (lower excitation states).
 If kT = 0 (i.e., the overall system is at
absolute zero),
all the systems will be in their
ground states.
 The larger kT (or T), the more slow the
exponential decay with E
and systems are spread out more to higher energy states (higher excitation states).
 If T < 0, then you have a funny topheavy overall system in which the individual systems
are concentrated in the highest energy states.
There must be an upper bound on E for such overall systems.
Negativetemperature overall systems
can be made in the laboratory and must occur in somewhere
in nature, but probably only rarely and fleetingly.
 Now the energy of
microscopic systems have a vast range.
However, it turns out the electronvolt (eV)
is the best overall
natural unit.
That's why we've written
Boltzmann's contant in terms of eV's.
 The quantized
energy levels
of
atoms
and molecules
are often separated by energies of order an eV.
Thus, unless T ≥ ∼ 10**(4) K,
the ground state
(which has E=0) has overwhelmingly the
highest
occupation number and
most atoms
and molecules
are sitting in the ground state.
This is the case in the ordinary terrestrial environment and
in most
stellar photospheres.
Credit/Permission: ©
User:PeterQ,
2010
(uploaded to
Wikimedia Commons
by User:Autopilot,
2010) /
CC BYSA 3.0.
Image link: Wikimedia Commons.
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