**Maximum Entropy Distributions**

Entropy is an important topic in many fields; it has very well known uses in statistical mechanics, thermodynamics, and information theory. The classical formula for entropy is Σ_{i}(p_{i} log p_{i}), where p=p(x) is a probability density function describing the likelihood of a possible microstate of the system, *i*, being assumed. But what is this probability density function? How must the likelihood of states be configured so that we observe the appropriate macrostates?

In accordance with the second law of thermodynamics, we wish for the entropy to be maximized. If we take the entropy in the limit of large N, we can treat it with calculus as S[φ]=∫dx φ ln φ. Here, S is called a functional (which is, essentially, a function that takes another function as its argument). How can we maximize S? We will proceed using the methods of calculus of variations and Lagrange multipliers.

First we introduce three constraints. We require **normalization**, so that ∫dx φ = 1. This is a condition that any probability distribution must satisfy, so that the total probability over the domain of possible values is unity (since we’re asking for the probability of *any possible event* occurring). We require **symmetry**, so that the expected value of *x* is zero (it is equally likely to be in microstates to the left of the mean as it is to be in microstates to the right — note that this derivation is treating the one-dimensional case for simplicity). Then our constraint is ∫dx x·φ = 0. Finally, we will explicitly declare our **variance** to be σ², so that ∫dx x²·φ = σ².

Using Lagrange multipliers, we will instead maximize the augmented functional S[φ]=∫(φ ln φ + λ_{0}φ + λ_{1}xφ + λ_{2}x²φ dx). Here, the integrand is just the sum of the integrands above, adjusted by Lagrange multipliers λ_{k} for which we’ll be solving.

Applying the Euler-Lagrange equations and solving for φ gives φ = 1/exp(1+λ_{0}+xλ_{1}+x²λ_{2}). From here, our symmetry condition forces λ_{1}=0, and evaluating the other integral conditions gives our other λ’s such that q = (1/2πσ²)^{½}·exp(-x² / 2σ²), which is just the Normal (or Gaussian) distribution with mean 0 and variance σ². This remarkable distribution appears in many descriptions of nature, in no small part due to the Central Limit Theorem.