thermodynamics - Why does the Boltzmann factor $e^{-E/kT}$ seem to imply that lower energies are more likely?

I'm looking for an intuitive understanding of the factor $$e^{-E/kT}$$ so often discussed. If we interpret this as a kind of probability distribution of phase space, so that $$\rho(E) = \frac{e^{-E/kT}}{\int_{0}^{\infty}e^{-E'/kT}dE'}$$ then what, precisely, does this probability correspond to? Is its physical significance obvious? Specifically, why is it largest for energies close to zero?

Edit: I'd like to know how it is possible that for increasing $T$ the lower energy states grow more populated. The naive thing to think is "If I stick my hand in an oven and turn up the temperature, I'm assuredly more likely to be burned." This raises the question, exactly what does it mean for the lower energy states to grow more populated relative to the higher energy ones?

Answer

The Maxwell-Boltzmann distributes $N$ particles in energy levels $E_i$ such that the entropy is maximized for a fixed total energy $E=\sum E_i N_i$.

The probability that a particle is in the energy level $E_i$ is proportional to the number of particles in the energy level $E_i$ in this particular arrangement of particles in which entropy is maximized (the Maxwell-Boltzmann distribution), which is $N_i$. It so happens that when we distribute the particles such that entropy is maximized, more particles populate the lower energy levels.

I don't follow the above argument @Ben Crowell - we want to show that more particles are distributed in the lowest energy level. In the above, we write $\sum E_i = N_0 E_0 + E_R$ conclude that probability is maximized if the energies of the particles in the lowest energy state, $N_0 E_0$, is minimized, which occurs for $N_0 = 0$ - the opposite of what was desired.

I'm not sure how to intuitively explain the solution to distributing the particles such that entropy is maximized. If we agree that by distributing the particles as evenly as possible in the energy levels, we will maximize the entropy, we can try:

Suppose we require $\sum N_i E_i \equiv \hat{E}$ and that we have plenty of particles $N$, and that $E_i$ increases with $i$:

1. Starting from $E_0$, put one particle in each energy level whilst $\sum E_i < \hat{E}$.


2. We need to distribute the remaining particles. Starting from $E_0$, again put one particle in each energy level whilst $\sum N_i E_i < \hat{E}$. The lowest energy level, $E_0$, now has 2 particles


3. Repeat 2., until all particles are distributed.

We can see that the lowest energy level will be most populated, and that $N_i$, and hence the probability that a particle is in state $E_i$, decreases with $i$. This algorithm won't exactly reproduce the Maxwell-Boltzmann distribution of particles in the energy levels, but it might help with an intuitive feel of why the lower energy levels are more probable.