Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

ADVANCES ON PHASE DIAGRAMS FOR

ROEGENIAN ECONOMICS: THEORY AND

EMPIRICAL VALIDATION^∗

Massimiliano Ferrara^†

Communicated by S. Trean¸t˘a

DOI

10.56082/annalsarscimath.2026.2.161

Abstract

This paper extends the foundational work on Roegenian Economics

by developing a rigorous mathematical framework for the thermo-

dynamic-economic correspondence initiated by Georgescu-Roegen. We

provide an enhanced dictionary between thermodynamic and economic

state variables. The main contributions include: (i) a formal proof of

the existence and uniqueness of the economic triple point; (ii) charac-

terization theorems for critical phenomena; (iii) derivation of Maxwell-

type relations for economic potentials; and (iv) stability analysis via

Legendre transforms. We present empirical validation using World

Bank Governance Indicators and IMF inﬂation data for 2022-2023,

demonstrating that the phase diagram captures the relationship be-

tween institutional stability and price dynamics. The analysis iden-

tiﬁes candidates for triple point behavior (Venezuela, Zimbabwe) and

supercritical regimes (Switzerland, Singapore).

Keywords: Roegenian economics, econophysics, phase diagrams, Gibbs-

Pfaﬀ equation, thermodynamic-economic correspondence, triple point.

MSC: 91B02, 91B55, 80A10, 37N40.

^∗Accepted for publication on February 02, 2026

^†massimiliano.ferrara@unirc.it, Universita` degli Studi Mediterranea di Reggio Cal-

abria, Dipartimento DiGiES, Decisions LAB; ICRIOS - Bocconi University, Milan, Italy

161

Phase diagrams for Roegenian economics

162

1 Introduction

The interdisciplinary dialogue between thermodynamics and economics has

a distinguished intellectual history, tracing back to the pioneering work of

Nicholas Georgescu-Roegen [3], who recognized that economic processes

are fundamentally entropic in nature. Unlike the Newtonian mechanical

paradigm that dominated classical economics, thermodynamics provides a

framework that naturally incorporates irreversibility, qualitative change, and

the arrow of time features essential for understanding real economic phenom-

ena.

The present work builds upon and signiﬁcantly extends the systematic

development of what we term Roegenian Economics: an economic theory

structured via formal correspondence with thermodynamics [12,14,18]. Pre-

vious contributions established the fundamental dictionary and explored the

geometric structure of economic state spaces [13, 16, 17]. Here, we deepen

the mathematical foundations by proving new theorems on phase equilib-

ria, developing the economic interpretation more thoroughly, and for the

ﬁrst time in this research program providing empirical validation using real-

world macroeconomic data.

The thermodynamic approach to economics is not merely a formal anal-

ogy but reﬂects deep structural similarities in how complex systems organize

and transform. Just as a thermodynamic system can exist in distinct phases

(solid, liquid, gas) with well-deﬁned transitions between them, an economic

system exhibits qualitatively diﬀerent operational regimes. The identiﬁca-

tion of these regimes and their transition boundaries constitutes a primary

objective of macroeconomic analysis.

Our phase diagram shows how political stability (I) and price level (P)

jointly determine the macroeconomic state. Political stability reﬂects insti-

tutional quality; the price level captures monetary conditions. The three

phases inﬂation, monetary liquidity, and income correspond to observable

macroeconomic regimes.

The existence of a triple point where all three phases coexist and a criti-

cal point beyond which phase distinctions blu provides powerful conceptual

tools for understanding macroeconomic dynamics. Historical episodes such

as the Weimar Republic hyperinﬂation (1923) and recent crises in Venezuela

and Zimbabwe exhibit characteristics consistent with triple point behavior.

The paper proceeds as follows. Section 2 presents the extended thermo-

dynamic-economic dictionary with detailed economic justiﬁcation for each

correspondence. Section 3 develops the mathematical theory of the Gibbs-

Pfaﬀ economic equation, including the contact geometry framework and

M. Ferrara

163

Maxwell relations. Section 4 establishes rigorous results on phase equilibria,

including existence and uniqueness theorems for the triple point. Section 5

analyzes critical phenomena and derives scaling relations. Section 6 presents

the empirical validation using World Bank and IMF data. Section 7 discusses

economic applications and policy implications. Section 8 concludes.

2 Extended thermodynamic-economic dictionary

The construction of Roegenian Economics proceeds via a systematic cor-

respondence between thermodynamic and economic concepts. This section

presents an extended and economically grounded version of this dictionary,

with detailed justiﬁcation for each identiﬁcation.

2.1

State variables and their economic meaning

Table 1: Extended Thermodynamic-Economic Dictionary: State Variables

Thermodynamics

Economics

Economic Justiﬁcation

U = Internal energy

G = Growth potential

Latent capacity for transforma-

tion

T = Temperature

I = Political stability

Intensity of systemic activity

S = Entropy

E = Economic entropy Irreversibility, resource degra-

dation

P = Pressure

P = Price level (inﬂa- Purchasing power compression

tion)

V = Volume

Q = Volume, quality

Economic capacity and output

Aggregate economic ﬂow

M = Total energy

µ_k= Chemical pot.

Y = National income

ν_k= Sectoral potential Marginal contribution by sector

N_k

=

Number of N_k= Economic moles

Standardized value units

moles

W

=

Mechanical W = Wealth of system Accumulated productive capac-

work

ity

Q_heat= Heat

q = Stock market ﬂow

Financial energy transfer

2.2

Temperature and internal political stability

In thermodynamics, temperature (T) measures the average kinetic energy of

particles and determines the direction of heat ﬂow between systems. More

fundamentally, temperature is the intensive variable conjugate to entropy—

Phase diagrams for Roegenian economics

164

it governs the spontaneous direction of energy transfer and determines equi-

librium conditions.

In economic systems, internal political stability (I) plays a precisely anal-

ogous role. We deﬁne I as a composite measure reﬂecting: (i) institutional

quality: the eﬀectiveness of governance structures, rule of law, and regu-

latory frameworks; (ii) policy predictability: the degree to which economic

agents can anticipate government actions; (iii) property rights security: the

protection of investments and contractual obligations; (iv) social cohesion:

the absence of politically-motivated violence or terrorism.

The World Bank’s Worldwide Governance Indicators (WGI) provide an

operational measure of I through the “Political Stability and Absence of

Violence/Terrorism” index, which ranges from approximately −2.5 (highly

unstable) to +2.5 (highly stable).

The correspondence T ↔ I is economically meaningful because both are

intensive variables that equilibrate across connected systems. Just as two

bodies in thermal contact reach the same temperature, economically inte-

grated regions tend toward similar institutional quality over time through

competitive pressures, treaty obligations, and demonstration eﬀects. Both

govern the direction of spontaneous ﬂow: heat ﬂows from high to low tem-

perature; capital and skilled labor ﬂow from low to high political stabil-

ity. Both determine system accessibility: high temperature makes more

microstates accessible to a thermodynamic system; high political stability

makes more economic strategies viable for ﬁrms and households. The third

law analog holds: as I → 0 (complete institutional collapse), economic en-

tropy approaches zero not because disorder vanishes, but because the system

becomes frozen incapable of any organized economic activity.

Remark 1 (Measurement of Political Stability). In empirical applications,

I can be measured using several available indices: the World Bank’s WGI

Political Stability indicator (used in Section 6), the Economist Intelligence

Unit’s Political Instability Index, or composite measures from Political Risk

Services. For dimensional consistency in the Gibbs-Pfaﬀ equation, we nor-

malize I to a percentage scale (0–100) based on percentile rank.

2.3

Pressure and price level

Thermodynamic pressure (P) represents the force per unit area exerted

by a system on its boundaries the intensive variable conjugate to volume.

It measures how strongly the system “pushes” against constraints on its

expansion.

M. Ferrara

165

The economic analogue is the price level (P), which represents the pur-

chasing power pressure exerted on economic agents. This identiﬁcation cap-

tures several parallel features: (i) compression eﬀect: high pressure com-

presses volume; high price levels compress real purchasing power and living

standards; (ii) expansion tendency: just as high pressure systems tend to

expand into low pressure regions, high-price economies exert competitive

pressure on low-price economies through trade and labor migration; (iii)

equation of state: the relationship P = P(Q, I) in economics how price

levels depend on output and stability mirrors the thermodynamic equa-

tion of state P = P(V, T); (iv) work conjugacy: thermodynamic work is

dW = PdV ; economic wealth creation involves dW = PdQ, the product of

prices and quantities.

2.4

Entropy and economic entropy

The concept of entropy has the most profound implications when transferred

to economics. Thermodynamic entropy (S) measures the number of micro-

scopic conﬁgurations compatible with a macroscopic state, the degree of

disorder or “spread” of energy across available states, and the irreversibility

of natural processes (second law).

Georgescu-Roegen’s fundamental insight was that economic processes

necessarily increase entropy: “Matter-energy enters the economic process

in a state of low entropy and comes out in a state of high entropy” [3].

Economic entropy (E) therefore measures: (i) resource degradation: the

transformation of concentrated, high-quality resources (low entropy) into

dispersed, low-quality waste (high entropy); (ii) process irreversibility: un-

like the reversible exchanges of neoclassical equilibrium theory, real economic

processes are fundamentally irreversible commodities are consumed, capi-

tal depreciates, and knowledge becomes obsolete; (iii) information disper-

sion: market aggregation destroys information about individual preferences

and endowments, increasing the eﬀective entropy of the economic system;

(iv) structural complexity: as economies develop, they tend toward greater

structural complexity and specialization, which can be viewed as increasing

conﬁgurational entropy.

Remark 2 (The Second Law in Economics). The economic second law

states that dq ≤ IdE, where q represents ﬁnancial market activity. In

reversible (ideal) economic processes, dq = IdE; in irreversible (real) pro-

cesses, dq < IdE. This implies that ﬁnancial activity generates less “useful”

economic transformation than the entropy increase would suggest there are

Phase diagrams for Roegenian economics

166

always transaction costs, information asymmetries, and coordination fail-

ures.

2.5

The economic phases

The three phases in our economic phase diagram correspond to qualitatively

distinct macroeconomic regimes:

Inﬂation Phase. The inﬂation phase is characterized by sustained

increase in the general price level eroding purchasing power, typically asso-

ciated with low political stability (weak institutional constraints on money

creation), high velocity of money as agents ﬂee from depreciating currency,

and redistribution from creditors to debtors, from ﬁxed-income earners to as-

set holders. Historical examples include Weimar Germany 1923, Zimbabwe

2008, and Venezuela 2018–present.

Monetary Policy of Liquidity Phase. This intermediate phase is

characterized by active central bank intervention in credit markets, eﬀective

transmission of monetary policy to real economic variables, price stability

as a policy target (neither deﬂation nor high inﬂation), and liquidity provi-

sion as the primary tool of economic stabilization. Examples include most

developed economies under inﬂation-targeting regimes.

Income Phase. The income phase is characterized by stable income

growth and accumulation, high political stability enabling long-term invest-

ment and planning, low and stable inﬂation, and eﬀective property rights

and contract enforcement. Examples include Switzerland, Singapore, and

Nordic countries.

3 The Gibbs-Pfaﬀ economic equation

3.1

Fundamental formulation

The core mathematical object in Roegenian Economics is the Gibbs-Pfaﬀ

equation, which encodes the constraints on economic state transitions.

Deﬁnition 1 (Gibbs-Pfaﬀ Economic Equation). The fundamental economic

equation is the Pfaﬃan diﬀerential equation:

ω = dG − IdE + PdQ = 0,

(1)

where (G, I, E, P, Q) ∈ R⁵are the economic state variables.

This equation combines the ﬁrst and second laws of economics: the ﬁrst

law dW = PdQ (elementary wealth creation through production) and the

M. Ferrara

167

second law dq = IdE (reversible) or dq < IdE (irreversible ﬁnancial activ-

ity).

Deﬁnition 2 (Third Law of Economics). The economic analogue of the

third law of thermodynamics states:

lim E = 0.

(2)

I→0

This limit has a precise economic interpretation: as internal political sta-

bility approaches zero (complete institutional collapse), the economic system

loses its capacity for organized activity.

3.2

Contact geometry framework

The geometric structure underlying the Gibbs-Pfaﬀ equation is that of a

contact manifold, which provides powerful tools for analyzing economic dy-

namics.

Proposition 1 (Contact Structure). The 1-form ω = dG − IdE + PdQ is

a contact form on R⁵= {(G, I, E, P, Q)}, i.e., ω ∧ (dω)²= 0.

Proof. We compute the exterior derivative:

dω = −dI ∧ dE + dP ∧ dQ.

The square of this 2-form is:

(dω)²= (−dI ∧ dE + dP ∧ dQ) ∧ (−dI ∧ dE + dP ∧ dQ)

= −2 dI ∧ dE ∧ dP ∧ dQ.

Computing the wedge product with ω:

ω ∧ (dω)²= (dG − IdE + PdQ) ∧ (−2 dI ∧ dE ∧ dP ∧ dQ)

= −2 dG ∧ dI ∧ dE ∧ dP ∧ dQ = 0.

Since this 5-form is non-vanishing, ω is a contact form.

Corollary 1. The economic state space (R⁵, ω) is a contact manifold of

dimension 5, with contact distribution ker(ω) of rank 4.

Phase diagrams for Roegenian economics

168

3.3

Integral manifolds and economic systems

Theorem 1 (Classiﬁcation of Integral Manifolds). The integral manifolds

of ω = 0 are of dimension at most 2. Speciﬁcally:

(i) Integral curves (economic paths): One-parameter families

(G(t), I(t), E(t), P(t), Q(t))

˙

satisfying G − IE + PQ = 0. These represent time-evolution paths of

economic systems.

(ii) Integral surfaces (simple economic systems): Two-parameter fami-

lies satisfying

∂G

∂E

∂Q

− I

+ P

= 0,

(3)

(4)

∂x

∂G

∂x

∂E

∂x

∂Q

− I

+ P

= 0.

∂y

Proof. By the Darboux theorem for contact forms, the contact distribution

ker(ω) is maximally non-integrable, meaning no integral manifold of dimen-

sion greater than (5 − 1)/2 = 2 exists.

3.4

Maxwell relations for economics

The integrability conditions for integral surfaces yield the economic analogue

of Maxwell relations.

Theorem 2 (Economic Maxwell Relations). On any integral surface of the

Gibbs-Pfaﬀ equation, the following relation holds:

∂I ∂E

∂x ∂y

∂E ∂I

∂x ∂y

∂P ∂Q

∂x ∂y

∂Q ∂P

∂x ∂y

−

=

−

.

(5)

(6)

Equivalently, in Jacobian notation:

∂(I, E)

∂(P, Q)

.

∂(x, y)

Proof. Diﬀerentiating (3 ) with respect to y and (4) with respect to x:

∂²G

∂x∂y

∂²G

∂I ∂E

∂y ∂x

∂I ∂E

∂x ∂y

∂²E

∂x∂y

∂²E

∂P ∂Q

∂y ∂x

∂P ∂Q

∂x ∂y

∂²Q

∂x∂y

∂²Q

−

− I

+

+ P

= 0,

− I

+ P

= 0.

∂y∂x

Subtracting and using the symmetry of mixed partials yields (5).

M. Ferrara

169

3.5

Economic potentials via Legendre transforms

Following the thermodynamic pattern, we deﬁne economic potentials through

Legendre transformations.

Deﬁnition 3 (Economic Potentials). The four fundamental economic po-

tentials are:

(i) Growth potential: G = G(E, Q) natural variables are entropy and

quantity

(ii) Free economic energy: F = G − IE natural variables are stability

and quantity

(iii) Wealth function (Enthalpy): H = G + PQ natural variables are

entropy and price

(iv) Economic potential (Gibbs function): Φ = G − IE + PQ natural

variables are stability and price

Theorem 3 (Potential Representations). Each economic potential provides

a complete description of a simple economic system:

∂G

∂E

∂G

(i) From G(E, Q):

I =

,

P = −_∂Q

∂F

(ii) From F(I, Q):

E = − _∂I

∂H

∂E

,

P = −_∂Q

∂H

∂P

(iii) From H(E, P):

I =

,

Q =

∂Φ

∂P

(iv) From Φ(I, P):

E = − _∂I

,

Q =

4 Phase equilibria: triple point theory

4.1 Economic phase space

Deﬁnition 4 (Economic Phases). In the Roegenian framework, an economic

system can exist in three distinct phases:

(i) Inﬂation phase (I): Characterized by sustained price increases, typ-

ically at low political stability.

(ii) Monetary policy of liquidity phase (M): An intermediate regime

where monetary interventions eﬀectively modulate economic activity.

Phase diagrams for Roegenian economics

170

(iii) Income phase (Y): Characterized by stable income growth and accu-

mulation.

Deﬁnition 5 (Phase Boundary). A phase boundary is a curve γ : [0, 1] →

R²in the (I, P) plane along which two phases coexist in equilibrium.

4.2

Coexistence conditions

Theorem 4 (Coexistence Criteria). Two economic phases α and β coexist

in equilibrium if and only if:

I_α= I_β, _{α β α}

P = P ,

Φ (I, P) = Φ_β(I, P),

(7)

where Φ is the economic potential (Gibbs function).

Theorem 5 (Economic Clausius-Clapeyron Equation). Along a phase bound-

ary between phases α and β, the slope satisﬁes:

E_β− E_α

dP

∆E

=

,

(8)

dI

Q_β− Q_α

∆Q

where ∆E and ∆Q are the entropy and volume diﬀerences between phases.

Proof. Along the coexistence curve, Φ_α(I, P) = Φ_β(I, P). Taking the to-

tal diﬀerential along the curve: dΦ_α= dΦ_β. From Theorem 3(iv), dΦ =

−EdI + QdP, so: −E_αdI + Q_αdP = −E_βdI + Q_βdP. Rearranging yields

(8).

4.3

Existence and uniqueness of the triple point

Theorem 6 (Existence of Triple Point). Under the following regularity con-

ditions on the economic potentials:

(11) Φ_I, Φ_M, Φ_Y∈ C²(R²) (smooth potentials),

(22) The phase boundaries are non-degenerate: ∇Φ_α= ∇Φ_βfor α = β,

(33) The potentials satisfy the crossing condition: Φ_I(0, 0) > Φ_M(0, 0) >

Φ_Y(0, 0) and Φ_I(I₀, P₀) < Φ_M(I₀, P₀) < Φ_Y(I₀, P₀) for some (I₀, P₀) ∈

R²₊₊

,

there exists a point (I_tr, P_tr) where all three phases coexist.

M. Ferrara

171

Proof. Deﬁne the functions measuring potential diﬀerences:

f₁(I, P) = Φ_I(I, P) − Φ_M(I, P),

f₂(I, P) = Φ_M(I, P) − Φ_Y(I, P).

By condition (A2), ∇f₁= 0 on f⁻¹(0) and ∇f₂= 0 on f⁻¹(0). By the

1

2

implicit function theorem, γ_IM= f₁⁻¹(0) and γ_MY= f₂⁻¹(0) are smooth

1-dimensional curves.

By condition (A3), f₁and f₂change sign. By the intermediate value

theorem, there exists (I_tr, P_tr) where f₁= f₂= 0, implying Φ_I= Φ_M

Φ_Y.

=

Theorem 7 (Uniqueness of Triple Point). Under conditions (A1)–(A3) and

the additional transversality assumption:

(A4) The gradients ∇f₁and ∇f₂are linearly independent at every point

where f₁= f₂= 0,

the triple point is locally unique.

Proof. By condition (A4), det(DF) = 0 at the triple point. By the inverse

function theorem, F⁻¹(0, 0) is a single point in a neighborhood of (I_tr, P_tr).

Deﬁnition 6 (Triple Point). The unique point (I_tr, P_tr) satisfying

Φ_I(I_tr, P_tr) = Φ_M(I_tr, P_tr) = Φ_Y(I_tr, P_tr)

is called the economic triple point. At this point, all three macroeconomic

phases coexist in equilibrium.

4.4

Gibbs phase rule for economics

Theorem 8 (Economic Phase Rule). For a Roegenian economic system

with n components (economic sectors) and π coexisting phases, the number

of degrees of freedom is:

f = n + 2 − π.

(9)

Corollary 2 (Single-Component System). For a single-component economic

system (n = 1): single phase (π = 1) gives f = 2; two phases (π = 2) gives

f = 1; three phases (π = 3) gives f = 0.

Phase diagrams for Roegenian economics

172

5 Critical phenomena in economic systems

Deﬁnition 7 (Economic Critical Point). A point (I_c, P_c) is a critical point

if:

(i) It lies on a phase boundary (speciﬁcally, the M-Y boundary).

(ii) For I > I_cand P > P_c, the phases M and Y become indistinguishable.

Theorem 9 (Critical Point Conditions). At a critical point, the following

conditions hold for the equation of state P = P(Q, I):

ꢀ

2

∂P

∂ P

= 0,

= 0.

(10)

∂Q

∂Q²

I_c

5.1

Van der Waals equation for economics

Deﬁnition 8 (Economic Van der Waals Equation). The economic van der

Waals equation relates price level, volume, and stability:

ꢁ

ꢂ

a

Q²

P +

(Q − b) = RI,

(11)

where a, b, R > 0 are economic parameters: a measures “cohesive pressure”

from market interactions, b represents minimum irreducible economic vol-

ume (subsistence level), and R is the economic gas constant.

Theorem 10 (Critical Parameters). For the economic van der Waals equa-

tion, the critical point parameters are:

8a

a

27b²

I_c=

,

P_c=

,

Q_c= 3b.

(12)

27Rb

Theorem 11 (Law of Corresponding States). In reduced variables π =

P/P_c, φ = Q/Q_c, ι = I/I_c, the van der Waals equation takes the universal

form:

ꢁ

ꢂ ꢁ

ꢂ

3

φ²

1

8ι

π +

φ −

=

.

(13)

3

Theorem 12 (Mean-Field Critical Exponents). For the economic van der

Waals equation, the critical exponents are:

1/2

Order parameter:

Compressibility:

Critical isotherm:

Q_Y− Q_{M c}

∼ |I − I|

(β = 1/2)

(γ = 1)

(δ = 3)

(14)

(15)

(16)

κ_I∼ |I − I_c|⁻¹

3

|P − P_c| ∼ |Q − Q_c|

M. Ferrara

173

6 Empirical validation

This section presents an empirical investigation of the Roegenian phase di-

agram using real-world macroeconomic data.

6.1

Data sources

We use two primary data sources: (i) Political Stability Index: World

Bank Worldwide Governance Indicators (WGI), speciﬁcally the “Political

Stability and Absence of Violence/Terrorism” index, ranging from approxi-

mately −2.5 (highly unstable) to +2.5 (highly stable), using 2023 data; (ii)

Inﬂation Rate: IMF World Economic Outlook and World Bank data on

consumer price inﬂation (annual percentage change), using 2022–2023 data.

6.2

Sample selection

We select a diverse sample of 20 countries spanning the full range of stability

and inﬂation conditions:

Table 2: Sample Countries: Political Stability and Inﬂation Data (2022-

2023)

Country

Stab. Index Stab. %

Inﬂ. (%)

High Stability, Low Inﬂation (Income Phase)

Switzerland

Singapore

Norway

New Zealand

Denmark

1.25

1.42

1.15

1.22

0.86

95.8

97.2

93.4

95.3

85.4

2.8

6.1

5.8

7.2

7.7

Medium Stability, Moderate Inﬂation (Monetary Liquidity)

United States

Germany

France

Italy

Japan

0.19

0.55

0.34

0.52

0.98

63.2

76.4

68.4

74.5

89.2

8.0

8.7

5.9

8.7

2.5

Lower Stability, Higher Inﬂation (Transitional)

Brazil

−0.32

−0.58

−0.18

−0.72

−1.04

42.0

32.5

47.2

27.8

15.6

9.3

7.9

7.0

6.7

Mexico

South Africa

India

Turkey

72.3

Low Stability, High Inﬂation (Inﬂation Phase)

Argentina

Venezuela

Zimbabwe

Sudan

−0.08

−1.28

−1.04

−2.45

−2.75

51.4

8.5

15.6

1.4

72.4

189.8

285.0

154.9

139.0

Syria

0.5

Sources: World Bank WGI (2023); IMF WEO (2022-2023).

Phase diagrams for Roegenian economics

174

6.3

Phase diagram construction

Figure 1 presents the empirical phase diagram with countries plotted ac-

cording to their stability-inﬂation coordinates.

log₁₀(P)

ZWE

200%

VEN

SDN

SYR

100%

Inﬂat_Tio_URn

ARG

50%

Triple Pt.

Critical Pt.

25%

Monetary

Liquidity

BRA

10%

DEU

5%

USA

DNK

NOR

ITA

FRA

SGP

NZL

MEX

IND

ZAF

Income

JPN

2.5%

CHE

1%

I (Stability %)

0

20

40

60

80

100

Figure 1: Empirical Phase Diagram: Political Stability vs. Inﬂation (2022–

2023). The x-axis shows World Bank Political Stability percentile rank (0-

100); the y-axis shows log₁₀of annual inﬂation rate. Country codes: CHE

(Switzerland), SGP (Singapore), NOR (Norway), NZL (New Zealand), DNK

(Denmark), USA (United States), DEU (Germany), FRA (France), ITA

(Italy), JPN (Japan), BRA (Brazil), MEX (Mexico), ZAF (South Africa),

IND (India), TUR (Turkey), ARG (Argentina), VEN (Venezuela), ZWE

(Zimbabwe), SDN (Sudan), SYR (Syria).

6.4

Empirical ﬁndings

The data reveal patterns consistent with the theoretical phase diagram:

Income Phase Identiﬁcation. Countries with high political stability

(percentile > 85) and low inﬂation (< 10%) cluster in the predicted “income

phase” region: Switzerland (95.8%, 2.8% inﬂation), Singapore (97.2%, 6.1%

inﬂation), Norway (93.4%, 5.8% inﬂation), Japan (89.2%, 2.5% inﬂation).

Inﬂation Phase Identiﬁcation. Countries with low political stability

(percentile < 20) and high inﬂation (> 100%) cluster in the predicted “inﬂa-

tion phase” region: Venezuela (8.5%, 189.8% inﬂation), Zimbabwe (15.6%,

M. Ferrara

175

285.0% inﬂation), Sudan (1.4%, 154.9% inﬂation), Syria (0.5%, 139.0% in-

ﬂation).

Triple Point Candidates. Historical and contemporary candidates

include Weimar Germany (1923), Venezuela (2018-present), and Zimbabwe

(2008, 2020s).

6.5

Statistical analysis

We test the relationship between political stability (I) and inﬂation (P)

using log-linear regression: log(P_i) = α + βI_i+ ε_i.

Table 3: Regression Results: Stability-Inﬂation Relationship

Variable

Coeﬃcient Std. Error t-statistic p-value

Constant (α)

4.82

0.45

10.7

< 0.001

Stability (β)

−0.048

0.67

0.008

−6.0

< 0.001

R²

N

20

The negative coeﬃcient on stability (β = −0.048, p < 0.001) conﬁrms

the theoretical prediction: higher political stability is associated with lower

inﬂation. The R²of 0.67 indicates that stability alone explains a substantial

fraction of cross-country inﬂation variation.

7 Economic applications and policy implications

Proposition 2 (Stability Near Triple Point). Near the triple point, small

policy interventions can produce large and unpredictable regime changes. Op-

timal policy requires maintaining (I, P) well away from (I_tr, P_tr).

Policy implication: Countries experiencing simultaneous institutional

weakness and inﬂationary pressure should prioritize fundamental institu-

tional reform over marginal monetary adjustments.

Proposition 3 (Supercritical Regime Management). For (I, P) ꢀ (I_c, P_c),

traditional distinctions between monetary and real economic policy become

less meaningful.

Policy implication: During periods of unconventional monetary policy,

central banks should coordinate closely with ﬁscal authorities.

Phase diagrams for Roegenian economics

176

8 Conclusions

This paper has developed an extended mathematical framework for Roege-

nian Economics, establishing rigorous foundations for the thermodynamic-

economic correspondence initiated by Georgescu-Roegen and systematized

in earlier collaborative work.

The main contributions are: (i) Enhanced Theoretical Framework:

Rigorous proofs for the existence and uniqueness of the economic triple

point, the economic Clausius-Clapeyron equation, Maxwell relations for eco-

nomic potentials, and the economic phase rule. (ii) Deep Economic Inter-

pretation: Extended dictionary with detailed justiﬁcation for each thermo-

dynamic-economic correspondence. (iii) Critical Phenomena Analysis:

Characterization of critical points, derivation of the economic van der Waals

equation, and calculation of mean-ﬁeld critical exponents. (iv) Empirical

Validation: First systematic empirical test of the Roegenian framework

using World Bank and IMF data.

The thermodynamic approach to economics provides a powerful concep-

tual framework that naturally incorporates irreversibility, phase transitions,

and critical phenomena features absent from traditional mechanical models

but essential for understanding real economic dynamics.

Acknowledgments. I am deeply grateful to the late Professor Con-

stantin Udri¸ste (University Politehnica of Bucharest), whose pioneering math-

ematical vision created the foundations of this research program. I also

honor the memory of Professor Dorel Zugr˘avescu (Institute of Geodynam-

ics, Bucharest). I thank Professor Ionel T¸evy (University Politehnica of

Bucharest) and Dr. Florin Munteanu (Institute of Geodynamics) for their

continuing collaboration. This research was supported by the Decisions LAB

at Universit`a Mediterranea di Reggio Calabria.

References

[1] C. Carath´eodory, Untersuchungen u¨ber die Grundlagen der Thermody-

namik, Math. Ann. 67 (1909), 355-386.

[2] D.S. Chernavski˘ı, N.I. Starkov and A.V. Shcherbakov, On some prob-

lems of physical economics, Phys.-Usp. 45 (2002), 977-997.

[3] N. Georgescu-Roegen, The Entropy Law and the Economic Process,

Harvard University Press, Cambridge, MA, 1971.

M. Ferrara

177

[4] N. Georgescu-Roegen, Matter matters, too. In: K.D. Wilson (Ed.),

Prospects for Growth: Changing Expectations for the Future, pp. 293-

313, Praeger, 1977.

[5] D. Kaufmann and A. Kraay, The Worldwide Governance Indicators:

Methodology and 2024 Update, Policy Research Working Paper, World

Bank Group, Washington, DC, 2024.

[6] S. London and F. Tohm´e, Thermodynamics and economic theory: A

conceptual discussion. In: Proc. XXX Reuni´on Anual AAEP, Univer-

sidad Nacional de R´ıo Cuarto, 2007.

[7] R.N. Mantegna and H.E. Stanley, An Introduction to Econophysics:

Correlations and Complexity in Finance, Cambridge University Press,

2000.

[8] M. Ruth, Insights from thermodynamics for the analysis of economic

processes, In: A. Kleidon, R.D. Lorenz (Eds.), Non-equilibrium Ther-

modynamics and the Production of Entropy, pp. 243-254, Springer,

Berlin, 2005.

[9] W.M. Saslow, An economic analogy to thermodynamics, Am. J. Phys.

67 (1999), 1239-1247.

[10] B.G. Sharma, S. Agrawal, M. Sharma, D.P. Bisen and R. Sharma,

Econophysics: A brief review of historical development, present status

and future trends, arXiv:1108.0977, 2011.

[11] E. Smith and D.K. Foley, Classical thermodynamics and economic gen-

eral equilibrium theory, J. Econ. Dyn. Control 32 (2008), 7-65.

[12] C. Udri¸ste, Thermodynamics versus economics, U.P.B. Sci. Bull., Ser.

A 69 (2007), 89-91.

[13] C. Udri¸ste, I. T¸evy and M. Ferrara, Nonholonomic economic systems.

In: C. Udri¸ste, O. Dogaru, I. T¸evy (Eds.), Extrema with Nonholonomic

Constraints, Vol. 4, pp. 139-150, Geometry Balkan Press, Bucharest,

2002.

[14] C. Udri¸ste, M. Ferrara and D. Opri¸s, Economic Geometric Dynamics,

Monographs and Textbooks 6, Geometry Balkan Press, 2004.

[15] C. Udri¸ste, V. B˘adescu, V. Ciancio, F. Ghionea, D. Isvoranu and I.

T¸evy, Black hole geometric thermodynamics. In: Proc. 4th Int. Colloq.

MENP, Bucharest, pp. 186-194, 2007.

Phase diagrams for Roegenian economics

178

[16] C. Udri¸ste, M. Ferrara, D. Zugr˘avescu and F. Munteanu, Geobiody-

namics and Roegen type economy, Far East J. Math. Sci. 28 (2008),

681-693.

[17] C. Udri¸ste, M. Ferrara, D. Zugr˘avescu and F. Munteanu, Nonholonomic

geometry of economic systems. In: Proc. 4th European Computing Con-

ference, Bucharest, ECC’10, pp. 170-177, WSEAS, 2010.

[18] C. Udri¸ste, M. Ferrara, D. Zugr˘avescu and F. Munteanu, Controllability

of a nonholonomic macroeconomic system, J. Optim. Theory Appl. 154

(2012), 1036-1054.

[19] C. Udri¸ste, M. Ferrara, I. T¸evy, D. Zugr˘avescu and F. Munteanu, Phase

diagram for Roegenian economics, Atti Accad. Pelorit. Pericol. Cl. Sci.

Fis. Mat. Nat. 97 (2019), A3.

[20] J. Vollmer, Out of “thin air”: Exploring phase changes, J. Chem. Educ.

77 (2000), 488A.

[21] World Bank, The Worldwide Governance Indicators: Revised Method-

ology, World Bank Group, Washington, DC, 2025.