Generalizing Statistical Mechanics

Gibbs’ approach to thermodynamics used information theory to derive its laws directly from the canonical distribution. Gibbs’ formalism is much more general in its application than simply describing classical or quantum mechanical systems. It is not mechanic in nature, it is information theoretic in nature. Its generality allows the description of the time evolution of our knowledge of a system.

To demonstrate the generality the paper begins with the log-normal distribution. This distribution appears throughout nature and human systems. It can be used to express our knowledge of species distribution, income distributions, crypto currency transaction distributions, particle size distribution, etc. Because of its ubiquitous nature, taking a closer look through Gibbs’ lens can provide some profound insights into the world around us.

This paper will derive the second law expression for a multivariate log-normal distribution using an adaptation of Gibbs’ method. From here, we will derive the equation of state that is the expression of the multivariate log-normal distribution. Next, we will contrast conventional macro-economic theory with this expression of a system of people and show that entropy is the missing factor in macro-economics. To test this, we will apply this to the Bitcoin blockchain and show how to practically estimate an equation of state in a complex economic system. We will see that entropy is not disorder as is conventionally described and communicated. It is, instead, simply a measure of a system’s complexity.

Keywords: entropy, information theory, statistical mechanics, economics, log-normal, bitcoin, income inequality, Cobb-Douglas production function, Keynes’ General Theory