Thermodynamic theory of voting and EU elections

Thermodynamic Theory of Voting Explains Election Vote Distributions

Researchers have found that the way votes are distributed among political parties follows the same mathematical rules as energy in physical systems or wealth in an economy. By treating votes like energy, they can accurately predict how much power a few top parties hold compared to many smaller ones.

In political science, modeling how voters choose between dozens of competing parties is notoriously difficult. Most existing mathematical theories treat electors as independent, non-interacting agents. They essentially treat a crowd of voters like a collection of isolated particles that never bump into one another. While these models work for simple two-party contests, they often struggle to capture modern multi-party systems. In these systems, a massive number of small parties compete alongside a few dominant giants.

The central question remains: why do election results consistently exhibit such extreme inequality? Why do a tiny fraction of parties capture the vast majority of the vote? A new study from Klaus M. Frahm and Dima L. Shepelyansky suggests the answer lies in the fundamental laws of statistical mechanics (the branch of physics dealing with large-scale systems).

Beyond Independent Agents

Traditional voting theories often rely on the assumption that voters act in isolation. However, the authors argue that this ignores the complex, nonlinear interactions between electors in a society. In the real world, opinions are shaped by social coupling. This is similar to how the movement of one molecule in a gas affects its neighbors.

The authors point out that previous attempts to model wealth inequality using the Boltzmann-Gibbs approach often fall short. This approach is a statistical method used to describe energy distribution among particles. Specifically, the authors note that the Boltzmann-Gibbs approach produces a Lorenz curve (a graph showing cumulative inequality) that fails to account for the extreme "tail" of the distribution. In economic terms, it struggles to explain why a tiny "oligarchic" class holds such a disproportionate share of total wealth.

To address this, the researchers propose the Thermodynamic Theory of Voting (TTV). Instead of looking at individual voter psychology, the TTV treats the total pool of votes as a conserved quantity. This is similar to the total energy in a closed physical system.

The Mechanism of Rayleigh-Jeans Condensation

The core of the TTV approach is the application of Rayleigh-Jeans (RJ) thermalization. In physics, thermalization is the process by which a system reaches a steady state of equilibrium. The authors suggest that interactions between voters lead to a "thermal" distribution of votes among participating parties.

The mechanism works through several key steps:

1. Energy Mapping: The authors map vote fractions ($V_m$) to energy states ($E_m$). Just as a physical system distributes energy among available modes, a political system distributes votes among available parties.
2. Conservation Constraints: The theory assumes two primary constraints. These are the conservation of total energy (total votes) and the conservation of the probability norm (total number of parties).
3. Nonlinear Coupling: The authors posit that nonlinear interactions between electors drive the system toward an RJ distribution. This distribution is defined by the expression $\rho_m = T / (E_m - \mu)$. Here, $T$ is the system temperature and $\mu$ is the chemical potential (a parameter describing particle density).
4. Condensation: Crucially, the authors describe a phenomenon called Rayleigh-Jeans condensation (RJC). When the "temperature" of the political system is sufficiently low, a macroscopic portion of the total votes "condenses" into the lowest energy states.

In a political context, this condensation manifests as a few dominant parties capturing the lion's share of the electorate. This leaves a "poverty phase" where many minor parties receive negligible support. This pattern is visible in the Lorenz curves for German and French EU elections .

Evidence from EU and French Elections

The authors validate the TTV by comparing its predictions against decades of election data. They focus on two primary metrics: the Lorenz curve and the Pareto curve (a plot showing the fraction of parties with a vote share above a certain threshold).

The paper reports that the Rayleigh-Jeans Standard (RJS) model provides an accurate fit for recent election data. For the 2024 EU elections in Germany and France, the theoretical RJS curves align closely with the actual recorded vote distributions .

In Germany (DE), the model reflects a scenario where the top 10% of parties capture approximately 57% of the total votes. In France (FR), that figure rises to 68%.

The study further demonstrates the universality of this mechanism across different scales. The authors analyze the 2024 EU elections for ten of the most populated EU countries. They use a density color plot to visualize how vote concentration varies by population size .

They find that the Gini coefficient—a standard metric for inequality—varies significantly between nations. Values range from roughly 0.55 in Belgium to 0.85 in Spain .

Even in systems with fewer participants, the theory holds. The authors consider the first round of French presidential elections. The RJS model successfully recovers the dispersion of votes between individual candidates .

Limits of the Thermodynamic Approach

While the TTV offers a powerful descriptive tool, it has clear boundaries. First, the authors explicitly state that the theory is a statistical description of distributions. It is not a predictive tool for specific outcomes. The TTV can tell you how unequal the vote distribution is likely to be. However, it cannot tell you which specific party will win.

Second, the model relies on a significant number of competitors. The authors note that the theory is designed for environments where the number of parties ($N_p$) is relatively high. While they applied it to French presidential elections, the number of candidates there is smaller. The statistical strength of the "condensation" effect depends on the number of available modes in the system.

Finally, the theory assumes a level of "thermalization." This means it assumes that voter interactions have reached a statistical steady state. It does not account for sudden, external shocks. A mid-campaign scandal might temporarily disrupt the underlying statistical equilibrium.

The Verdict

The Thermodynamic Theory of Voting is a compelling piece of "sociophysics." It bridges the gap between statistical mechanics and political science. By moving away from the assumption of independent agents, the authors provide a framework for understanding political concentration.

If you are looking for a tool to predict the winner of the next election, this is not it. However, if you want to understand why multi-party systems tend toward "winner-takes-most" outcomes, the TTV provides a rigorous foundation. The theory serves as a diagnostic tool to quantify the "political temperature" and inequality of different electoral systems.

Figures from the paper