Entry #001: Partition Function as State Space

Praswell

The Macro Picture

One of the most important and useful concepts that creeps out of physics is the partition function. The main idea I will be trying to form here isn’t another summary of thermodynamics, but rather how this phenomena—which is extremely useful for our physical observations—can be seen in our every day lives and, most importantly, in our economies.

The Physical Definition

We start by the usual definition of our probability of the system being in a particular state $i$. For everyone familiar with the concept of normalization, we know that this $Z$ factor has to allow this entire thing to add up to be one:

$$1=\sum_{s}P(s)=\sum_{s}\frac{1}{Z}e^{\frac{-E(s)}{kT}}=\frac{1}{Z}\sum_{s}e^{\frac{-E(s)}{kT}}$$

Which then gives us the Partition Function ($Z$):

$$Z=\sum_{s}e^{\frac{-E(s)}{kT}}$$

In physics, this quantity is a map to how many states an atom has accessible to populate. One thing that is apparent is that while the energy itself is part of the state, the partition function is only a function of temperature.

The Economic Mapping

In our economies, the variables represent different parts of information. Here is how we connect our original expressions to the market:

  • Wealth ($w$): The discrete state of capital accumulation; this is the energy level ($E$) of the acting individual.
  • The State ($s_i$): A discrete level of wealth or capital accumulation.
  • Temperature ($T$): The economic analogue to temperature. In physics, high $T$ means particles move; in our economic example, this relates to the ease of money and capital moving through various states (liquidity, M2 supply, velocity, etc.).
  • Action Scale ($\alpha$): Kept equal to one for now; this functions as the economic equivalent of the Boltzmann constant.
  • Market Friction ($\phi$): The total resistance of the system $(\phi=\frac{1}{\alpha\mathcal{T}})$, representing state influence or intervention by the Central Bank.

The New Expression

Substituting these for proper labeling, we get our “new” expression for the partition function:

$$Z=\sum_{s}e^{-\frac{w_{S}}{\alpha T}}$$

And with the substitution for market friction:

$$Z=\sum_{s}e^{-\phi w_{s}}$$

The Logic of Inelasticity: As the market is “cooled,” the partition function approaches 1. The higher the value for our partition function (Hot), the lower the probability of finding an individual in a particular state. As $T \to \infty$, the distribution flattens, maximizing individual mobility.