Thermal Macroeconomics: A Thermodynamic Approach

• Thermal Macroeconomics is a framework that reinterprets economic aggregates using thermodynamic concepts like entropy and temperature.
• It replaces microfoundations with aggregate axioms, deriving market prices and trade flows from energy-like state variables.
• The approach bridges statistical physics, game theory, and macroeconomics to predict equilibrium, fluctuations, and crises.

Thermal Macroeconomics (TM) is a theoretical framework that applies the mathematical structure and conceptual apparatus of thermodynamics—especially the concepts of temperature, entropy, free energy, and equilibrium—to the aggregate behavior of economic systems. TM provides a bridge between statistical physics, game theory, and macroeconomics, enabling the derivation of robust macro-level laws and predictions without relying on detailed microeconomic foundations or strong rationality assumptions. Rather than focusing on specific agents or optimization, TM is built around aggregate state variables (such as money, goods, aggregate utility, and entropy) and their dynamics, often using analogues of key thermodynamic principles: conservation laws, equations of state, and the second law of thermodynamics.

1. Foundations: Macroscopic Analogues and Axiomatic Formulation

At its core, TM replaces microfoundations by positing macro-level axioms—such as extensivity, scaling symmetry, and accessibility relations among aggregate economic states—mirroring the axiomatization of thermodynamics. States of the economy are represented by aggregate quantities of resources (goods, money, population), and a binary relation of accessibility (“≼”) orders these states according to whether one can be reached from the other by permissible (reversible or irreversible) economic operations (Chater et al., 1 Dec 2024).

An extensive entropy function $S$ is constructed on the economic state space, possessing crucial properties:

• Accessibility Law: $X \preceq X'$ if and only if $S(X) \leq S(X')$.
• Additivity/Extensivity: For two independent economies, $S(X_1 + X_2) = S(X_1) + S(X_2)$.
• Uniqueness up to Affinity: Any other entropy function is an affine transformation $S' = aS + b$, $a>0$.

Economic entropy ($S$) quantifies the aggregate utility (in cardinal terms) or aggregate welfare of the economy.

Economic temperature ($T$) is defined via the marginal change in entropy with respect to total money, i.e.

$\frac{1}{T} = \frac{\partial S}{\partial M},$

which yields a macro-level definition of the marginal utility of money. Importantly, the value of money, market prices, and other traditional macroeconomic variables are extracted directly as thermodynamic derivatives, e.g.,

$$\mu = \frac{\nu}{\beta} = \frac{\partial S/\partial G}{\partial S/\partial M},$$

where $\nu$ is the entropy derivative with respect to the good $G$, and $\beta = 1/T$ is the coolness.

This approach allows TM to derive aggregate-level economic relations—such as the existence of market prices, the meaning of inflation, the compensation effects (macro-Slutsky matrix symmetry and negative-definiteness), and the Le Chatelier–Samuelson principle—without requiring strong assumptions about agent-level rationality or optimization (Chater et al., 1 Dec 2024).

2. Statistical Mechanics and Effective Free Energy

TM generalizes the use of free energy from statistical physics to the analysis of economic systems with individual (possibly noisy or selfish) decision-making (Grauwin et al., 2011). For systems of agents who follow stochastic choice rules parameterized by a “temperature” $T$ (modeling deviations from strict utility maximization), the effective equilibrium distribution is given by

$$\Pi(\vec{q}) = \frac{1}{Z} \exp\left[\frac{L(\vec{q})}{T}\right],$$

where $L(\vec{q})$ is a linking function (or potential, akin to the negative of the energy), and $Z$ is a normalization constant.

At the mesoscopic scale, the stationary distribution of coarse-grained variables (such as the density $\rho_q$ of agents choosing $q$) is governed by the effective free energy:

$F = L + T S,$

with $S$ representing the entropy of configurations. Equilibria and collective patterns emerge as solutions to the variational principle (free energy minimization or entropy maximization), directly linking individual utility functions with macro-level behavior. Examples include the derivation of phase separation (segregation) and congestion phenomena as economic analogues of phase transitions (Grauwin et al., 2011).

3. Core Macroeconomic Laws: Equilibrium, Dynamics, and Fluctuations

TM extends the analogy to define macroeconomic variables analogous to thermal systems:

Thermodynamic	Economic	Macroeconomic Formulation
Internal energy	Total money $E$	$E = \sum_i \epsilon_i$
Volume	Total goods $V$	$V$ (aggregate stock or flow)
Pressure	Price $p$	$dW = p\,dV$
Temperature	Money temperature	$1/T = \partial S / \partial E$
Chemical potential	Financial potential $u$	$dE = u\,dN$
Entropy	Economic entropy $S$	Aggregate welfare

The first and second laws in TM take the forms:

$$\delta Q = dE + p\,dV - u\,dN \ T\,dS = dE + p\,dV - u\,dN$$

where $Q$ is injected “money,” $E$ is the money balance, $V$ is the goods volume, $N$ is the population, and $u$ is the financial potential (Rashkovskiy, 2020).

Statistical mechanics yields partition functions for economic ensembles, with grand-canonical or canonical forms, leading to the macro-distribution of quantities (wealth, price, goods) and explicit formulas for fluctuations (e.g., price volatility scales inversely with number of agents) (Rashkovskiy, 2020, Zheng, 2015). The equilibrium theorem states that when two open economic systems reach equilibrium, they share a common economic temperature (marginal utility of money) and chemical/economic potential (migration/inclusion incentive) (Zheng, 2015).

4. Thermodynamic Interpretation of Macroeconomic Phenomena

• Money Flows and Trade: When two economies are put in contact (via trade or financial integration), money flows from the economy with higher temperature (lower marginal utility of money) to the one with lower temperature, mirroring heat flow from hot to cold. The aggregate entropy of the joint system never decreases, enforcing an “economic second law” for transitions (Chater et al., 1 Dec 2024, Luo et al., 27 Oct 2024).
• Price Formation and Market Prices: Market prices arise naturally as the ratio of entropy derivatives, describing the equilibrium exchange rate between money and goods. Arbitrage and the direction of trade obey the condition that aggregate entropy not decrease (thus, profitable trades are only feasible in the presence of temperature/value differentials).
• Carnot and Onsager Analogues: TM predicts an upper bound on the efficiency of profit extraction via trade between economies at different temperatures, directly analogous to the Carnot efficiency $1-T_C/T_H$ (Luo et al., 27 Oct 2024). Onsager symmetry and reciprocity relations are observed for the macro-flows in response matrices.
• Economic Potential and Firm Creation: In models of firm growth, the chemical potential $\mu$ regulates creation or extinction regimes, with the observed Pareto exponent of the firm size distribution linked to the macroeconomic equilibrium (Zipf law when $\mu=0$) (Zambrano et al., 2015).
• Aggregate Fluctuations: Partition functions provide explicit formulas for macroeconomic variances, capturing aggregate market fluctuations and allowing for rigorous analysis of stability (Rashkovskiy, 2020).

5. Application to Complex, Non-Equilibrium, and Social Systems

TM has been generalized to contexts such as:

• Complex Production Economies: Statistical mechanics of complex economies admits phase transitions (e.g., collapse of production sector when a critical complexity or resource threshold is breached), rationalized by vanishing feasible volumes in high-dimensional constraint spaces (Bardoscia et al., 2015).
• Stock Markets and Financial Networks: Phase transition and renormalization group methods distinguish between market regimes (e.g., bull/bear transitions) and critical points (e.g., volatility spikes), with temperature serving as a marker for financial instability (Subias, 2015).
• Agent-Based and Opinion Systems: Temperature in agent-based models corresponds to randomness or volatility in decision-making, directly measurable as a function of state surplus and response to external signals (Börner et al., 11 Jul 2025).
• Social Thermodynamics: Thermodynamic variables are mapped to societal structures, with entropy interpreted as social freedom and the second law governing irreversible societal evolution and phases (e.g., Marxist society as a critical point) (Tsekov, 2023).
• Resource and Energy Constraints: Thermodynamic potential models embed economic flows in a physical stock–flow consistent framework, allowing explicit calculation of optimal intensities, friction losses, and entropy production under integrated ecological-economic constraints (Herbert et al., 2022).
• Carbon Footprint and Input-Output Accounting: Nonequilibrium thermodynamic concepts (eco-majorization, entropy, flow directionality) clarify the structural biases induced by standard input-output models, partitioning between structure-driven and artifact-driven emissions and wealth flows (Loomis et al., 2021).

6. Implications and Theoretical Payoffs

The TM framework yields several fundamental advances:

• It provides a macroeconomic derivation of key results (existence and formation of market prices, the nature of inflation, the regulation of aggregate demand, and the structural role of money) with minimal behavioral assumptions (Chater et al., 1 Dec 2024).
• It identifies new macro-level variables and laws (economic entropy, temperature, money capacity) with direct interpretations and measurable consequences.
• It bridges to nonequilibrium thermodynamics and dynamical systems, allowing for the rigorous paper of transitions, instabilities, and irreversibility in economic development, resource usage, and crises.
• It supports robust, empirically confirmed predictions for simulated and real economies—whether agent-based, statistical, or field-theoretic in nature—independent of detailed agent behavior (Luo et al., 27 Oct 2024).
• It generalizes traditional equilibrium concepts (e.g., Slutsky matrix, Le Chatelier principle, phase equilibria) to the macroeconomic domain.

7. Methodological Extensions and Empirical Status

The validity of TM’s predictions has been confirmed in agent-based simulations, controlled microeconomies, and empirical studies of firm growth, financial networks, and complex input-output systems (Luo et al., 27 Oct 2024, Zambrano et al., 2015, Bardoscia et al., 2015). Key steps for implementation and further development include:

• Measuring aggregate entropy and temperature empirically in real economies, often via observed macro-variables and fitting to derived equations of state.
• Applying maximum entropy and grand canonical ensemble methods to derive macroscopic distributions (wealth, firm size, price, etc.) under specified constraints.
• Using response function and Onsager formalism to predict flows between sectors, economies, or subsystems subject to small perturbations or policy changes.
• Embedding TM modules within broader macroeconomic, ecological, or social models, including explicit resource, energy, and waste flows as dictated by thermodynamic reasoning (Herbert et al., 2022).

Summary Table of Core Quantities and Their Macroeconomic Roles

Macroeconomic Quantity	Thermodynamic Analogue	Formula/Role
Economic Entropy $S$	Entropy	$$S = N \ln\left[\left(G/N\right)^{\alpha}\left(M/N\right)^{\eta}\right]$$ (Chater et al., 1 Dec 2024)
Money Temperature $T$	Temperature	$T = \left( \partial S/\partial M \right)^{-1}$
Value $\nu$	Chemical/Economic Potential	$\nu = \partial S/\partial G$
Market Price $\mu$	Dual variable (pressure)	$$\mu = \nu / \beta = \frac{\partial S/\partial G}{\partial S/\partial M}$$
Money Capacity $C$	Heat Capacity	$C = \partial M / \partial T$
Free Energy $F$	Free Energy	$F = L + T S$
Fluctuations	Response functions	$\text{Var}(T) \propto T^2/N$

Thermal Macroeconomics, by systematically importing key theoretical and mathematical structures from thermodynamics, enables a consistent, general, and empirically testable approach to understanding aggregate economic phenomena—encompassing equilibrium and nonequilibrium dynamics, trade, value formation, wealth distribution, resource constraints, and even social evolution—on a purely macro-level foundation.