The Physics: Energy, Probability, and Temperature
Imagine a physical system that can exist in several possible states. Each state has an associated energy. Some states are cheap, requiring very little energy, while others are expensive. One fundamental question statistical physics asks is deceptively simple:
Given all these possible states, how likely is the system to occupy each one?
At first, it might seem tempting to use the energies themselves as probabilities. But energies aren’t probabilities. A state can have energy 5, -3, or 127—none of which directly indicates likelihood. Probabilities obey rules that energies do not: they can never be negative, and together they must sum to exactly one.
The solution is to stop thinking about probabilities altogether. Instead, each state’s energy is first converted into a weight:
weight = e^(-E_i / kT)
The exponential doesn’t produce probabilities—it produces relative importance. States with low energy receive large weights, while high-energy states receive exponentially smaller ones. At this stage, nothing is normalized; the numbers simply express how strongly each state competes with the others.
Turning those weights into probabilities is almost trivial: divide each state’s weight by the sum of all weights:
P(state) = e^(-E_i / kT) / Σ_j e^(-E_j / kT)
That’s the Boltzmann distribution. Despite its intimidating appearance, the equation performs only two operations. The exponential translates energy into relative preference. The denominator rescales those preferences so they form a valid probability distribution. What initially looks like a complicated formula is really a weighting step followed by a normalization step.
Now, let's turn to the temperature variable. Note that temperature never changes the energies themselves. Instead, it changes how much those energy differences matter.
Consider three possible states with energies 0, 1, and 2. At a very low temperature, say T = 0.1, even these modest energy differences become enormous inside the exponential. The lowest-energy state captures almost all the probability, while the other two become practically impossible. Increase the temperature to T = 10, and the opposite happens. The same energy differences are now heavily compressed, producing nearly identical weights and a distribution that’s close to uniform.
Consider the following two tables to visualize the concept (assuming k = 1 for simplicity):
| Energy (E) | 0 | 1 | 2 |
| Weight at T = 0.1 | e^0 = 1 | e^(-10) ≈ 0.000045 | e^(-20) ≈ 2.06e-9 |
| Probability at T = 0.1 | ≈ 1.0 | ≈ 4.5e-5 | ≈ 2.1e-9 |
| Weight at T = 10 | e^0 = 1 | e^(-0.1) ≈ 0.905 | e^(-0.2) ≈ 0.819 |
| Probability at T = 10 | ≈ 0.37 | ≈ 0.33 | ≈ 0.30 |
As the table shows, low temperatures sharpen the distribution toward the most favorable (lowest-energy) states, while high temperatures flatten it, making all states more equally likely.