Caption: Exponential decay illustrated for several INVERSE e-folding 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 e-folding constant.

Features:

1. Here we are interested in the the exponential decay formula's use in thermodynamics in its microscopic formulation statistical mechanics.

2. 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 zero-point 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 electron-volts (eV).

3. As you can see, kT is the e-folding constant of the Boltzmann factor.

   Explicitly,

   kT = 8.6173324(78)*10^-5 eV/K * T ≅ 10^-4 eV/K * T .

4. KT controls the distribution of systems among the microscopic states:
   1. 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).

   2. If kT = 0 (i.e., the overall system is at absolute zero), all the systems will be in their ground states.

   3. 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).

   4. If T < 0, then you have a funny top-heavy 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.

      Negative-temperature overall systems can be made in the laboratory and must occur in somewhere in nature, but probably only rarely and fleetingly.

5. Now the energy of microscopic systems have a vast range. However, it turns out the electron-volt (eV) is the best overall natural unit.

   That's why we've written Boltzmann's contant in terms of eV's.

6. 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.