Boltzmann Probability

Thermodynamics is the study of energy and its transformations. Before the atomic theory of matter was generally accepted a considerable progress was made in thermodynamics. Classical Thermodynamics encompasses a very powerful set of laws, but they offer zero molecular insight. With the development of the atomic and molecular understanding of matter, statistical thermodynamics was developed to connect microscopic properties to already well established macroscopic behavior.

Statistical thermodynamics relates the averages of molecular properties to bulk thermodynamic properties like pressure, temperature, and enthalpy.

Boltzmann equation of statistical thermodynamics

$$S={{k}_{B}}\operatorname{lnW}$$

Entropy S and variable W that can represent number of micro-states, disorder, degeneracy, likelihood or probability are related by the Boltzmann constant k_B which is equal to:

$${{k}_{B}}=1.38\cdot {{10}^{-23}}J{{K}^{-1}}$$
$$R={{k}_{B}}{{N}_{A}}$$

The very large water cooler (Ensemble at fixed temperature T)

Imagine a very large water cooler (infinitely very large), arbitrarily large but not necessarily infinite. Water cooler held at constant temperature. There are about 10²⁵ molecules (N) in a 330 mL volume (V) with the molecules interacting with one another in each bottle. We know from quantum mechanics that there is an enormous set of allowed macroscopic energies. This set of energies will be a function of both N and V: 

$$\left\{ {{E}_{i}}\left( N,V \right) \right\}~-\text{ }Allowed\text{ }set\text{ }of\text{ }energies.$$

So the question is: “What is the probability that your water will be in state i with energy E_i?”

Number of states – We will use a_i to indicate the number of bottles having energy E_i(N,V) for the (fixed) number of particles N and volume V. The ratio of the number of bottles in states j and k will be given by some function of energy f such that: 

$$\frac{{{a}_{j}}}{{{a}_{k}}}=f\left( {{E}_{j}},{{E}_{k}} \right)$$

This ratio should not depend on the arbitrary choice of zero for E, so we have:

$$\frac{{{a}_{j}}}{{{a}_{k}}}=f\left( {{E}_{j}}-{{E}_{k}} \right)$$

Consider another system having energy E_l. The ratio of a_j to a_i and of a_k to al must be respectively:

$$\begin{align} & \frac{{{a}_{j}}}{{{a}_{l}}}=f\left( {{E}_{j}}-{{E}_{l}} \right) \\ & \frac{{{a}_{k}}}{{{a}_{l}}}=f\left( {{E}_{k}}-{{E}_{l}} \right) \\ & \frac{{{a}_{j}}}{{{a}_{l}}}=\frac{{{a}_{j}}}{{{a}_{k}}}\frac{{{a}_{k}}}{{{a}_{l}}}=f\left( {{E}_{j}}-{{E}_{k}} \right)f\left( {{E}_{k}}-{{E}_{l}} \right) \\ & f\left( {{E}_{j}}-{{E}_{l}} \right)=f\left( {{E}_{j}}-{{E}_{k}} \right)f\left( {{E}_{k}}-{{E}_{l}} \right) \\ \end{align}$$

The exponential is the function that has the property:

$$f\left( x+y \right)=f\left( x \right)f\left( y \right)$$

This implies that our original number of states for a given energy E should be determined from:

$${{a}_{i}}=C{{e}^{-\beta {{E}_{i}}}}$$

Where C and β are arbitrary positive constants yet to be determined. B must be positive or our water cooler would be infinitely large. Energy must be bounded from below for the same reason.