Thermo-Credit theory — entropy, CET1 slack, and AI reliability

Plain-English abstract

Thermo-Credit Theory reinterprets the Quantity Theory of Credit (QTC) through the lens of thermodynamics. It treats credit expansion and repayment as energy-like flows within a financial system that has measurable capacity, pressure, and dispersion. This mapping is not physics—it is a structured bookkeeping analogy designed for decision support and early-warning analytics.

QTC extends the classic Quantity Theory of Money (QTM) by adding an explicit capacity term, \(V_C\), for the banking system. Just as physical systems have pressure–volume interactions, credit systems respond to changes in balance-sheet headroom and regulatory constraints. This allows us to separate random liquidity diffusion from intentional policy work, and to test whether data behave like state variables.

The theory introduces entropy-like dispersion, credit potential, and free-energy measures (\(F_C, X_C\)) to monitor stress and policy space. These quantities can be calculated from public balance-sheet and market data, producing falsifiable indicators. Thermo-Credit therefore serves as both a conceptual bridge between money and information, and a practical framework for quantitative supervision.

Contents

Plain-English summary
Overview
Correspondence table
Bank credit creation
Free energy (F_C) & optional exergy (X_C)
Maxwell-like relations
Is it merely a rephrasing?
What the analogy adds (insights & tests)
Limitations & scope
Minimal references
Disclaimer & status
App behavior

Recommended reading order: Plain-English summary → Correspondence table → Bank credit creation → Insights & tests.

Key equations: (1) (2) (3) (4) (5)

Plain-English summary (QTM vs QTC & the first/second laws)

What we compare. Two stories about money/credit: the classic Quantity Theory of Money (QTM) [3,4,6] and Werner’s Quantity Theory of Credit (QTC) [9].

QTM — Quantity Theory of Money (money-first): more money → higher prices, if money moves fast enough.
QTC — Quantity Theory of Credit (credit-first) [9]: banks create deposits when they lend; what credit is used for matters.

First law (bookkeeping idea).

QTM: in the classic quantity-theory view [3, 4, 6], changes are explained via money stock, velocity, and policy shocks, without an explicit “capacity” state variable. Here we restate that as a split between (a) policy work and (b) extra mixing of money across uses.
QTC: same split, but now we also track capacity/headroom on bank balance sheets. Using up headroom shows up like a pressure × volume term (work from constraints).

Second law (one-way tendency).

Both: random mixing tends to raise our entropy-like index. To reverse it (make it more concentrated) you need structure—policy, rules, guidance.
Only in QTC: tightening or relaxing capacity shifts the “pressure” of constraints, which changes how hard it is to concentrate or relax the system.

Why QTC adds value. QTC makes the hidden capacity/pressure channel explicit. That lets us distinguish “more random dispersion” from “intentional work by policy or rules” — which is exactly what you want for audit trails and early-warning dashboards.

Overview

The index we informally described above is made precise as follows.

Definition. Monetary dispersion entropy (entropy-like, extensive): \[ S_M \;=\; k\,M_{\mathrm{in}}\,H(q),\quad H(q)\equiv -\sum_i q_i\,\log q_i \tag{1} \] where M_in is the actual money-in-circulation over a chosen period/system and q are composition shares (MECE, stable).

Entropy form follows Shannon [12]. Economic applications of entropy/dispersion include Theil’s information-theoretic index [7]. We map ideal mixing \(\Delta S_{mix}=k_B N H(x)\) to money shares \(q\) and scale \(M_{in}\).

Correspondence table

Thermodynamics column is literal; QTM/QTC columns are analogy-level correspondences. Heat/Work entries are “-like” bookkeeping splits, not physical identities.

Notes are hidden behind the ⓘ icons; hover (or tap) to read.

Variable	Thermodynamics	QTM (money-first)	QTC (credit-first)
Mixing entropy	\(\Delta S_{mix}=k_B N H(x)\)	\( S_M = k\,M_{in}\,H(q) \)	\( S_M = k\,M_{in}\,H(q) \)
Temperature	\( T \)	\( T_L \)	\( T_L \)
Internal energy	\( U \)	\( U_M(S_M) \)	\( U(S_M,V_C,\dots) \)
Volume	\( V \)	—	\( V_C \)
Pressure	\( p \)	—	\( p_C \equiv -\,(\partial U/\partial V_C)_{S_M} \)
Heat (heat-like)	\( \delta Q_{\mathrm{rev}} = T\,dS \)	\( Q_M \sim T_0\,\Delta S_M \)	\( Q_C \sim \bar T_L\,\Delta S_M \)
Work (work-like)	\( \delta W = p\,dV \)	\( W_M \equiv W_{\mathrm{policy}} \)	\( W_C \equiv -\,\bar p_C\,\Delta V_C \;+\; W_{\mathrm{policy}} \)
First law	\( \Delta U \equiv T\,\Delta S + W \)	\( \Delta U_M = T_0\,\Delta S_M + W + \varepsilon \)	\( \Delta U = \bar T_L\,\Delta S_M + W + \varepsilon \)
Second law (monotonicity)	\( \Delta S \ge 0 \)	\( \Delta S_M \ge 0 \)	\( \Delta S_M \ge 0 \)
Helmholtz free energy	\( F \)	\( F_M \equiv U_M - T_0 S_M \)	\( F_C \equiv U - T_0 S_M \)
Exergy / availability	\( X \)	\(X_M \approx U_M - T_0 S_M\)	\( X_C = \Delta U + p_0\Delta V_C - T_0 \Delta S_M \)

Note — What does U mean here?

QTM: Bookkeeping potential \(U_M(S_M)\) so that a first‑law‑like split holds: \(\Delta U_M = T_0\,\Delta S_M + W_{policy} + \varepsilon\). Not a standard variable in classic QTM.
QTC: Credit state potential \(U(S_M,V_C,\dots)\) with conjugates \(T_L=(\partial U/\partial S_M)_{V_C}\), \(p_C=-(\partial U/\partial V_C)_{S_M}\).

We use \(U\) as an analytic device; original QTM/QTC do not name \(U\).

Bank credit creation (within this mapping)

Balance-sheet identity (loans create deposits)

When a bank grants a new loan, it books the loan as an asset and simultaneously creates a matching deposit as a liability. On the customer side, the new deposit is an asset and the loan is a liability. This standard identity is the only accounting ingredient we need.

\[ \Delta U \;\approx\; \bar T_L\,\Delta S_M \;- \; \bar p_C\,\Delta V_C \;+ \; W_{\mathrm{policy}} \;+ \; \varepsilon \tag{2} \]

This expresses the credit‑creation energy balance — the “ΔU of credit creation,” decomposing changes into liquidity (T̄_L ΔS_M), capacity (−p̄_C ΔV_C), and policy work (W_policy).

\( \Delta S_M \): with \( S_M = k\,M_{\mathrm{in}}\,H(q) \), credit creation typically raises \( M_{\mathrm{in}} \) (scale term) and, via allocation changes, may alter \( H(q) \), pushing \( S_M \) upward.
\( \Delta V_C \): new lending uses headroom so \( \Delta V_C < 0 \) (capacity declines). Hence the term \( -\bar p_C\, \Delta V_C > 0 \) contributes positively.
\( W_{\mathrm{policy}} \): includes structural work from regulation, capital injections, guarantees, purchases (QE/collateral ops), etc.

Creation vs repayment (sign conventions)

Notation: \(\Delta M_{\mathrm{in}}\) = change in money-in-circulation over the period. Arrows ↑/↓ denote increase/decrease.

Variable	Credit creation (new lending > repayments)	Credit contraction (repayments > new lending)
\(\Delta M_{\mathrm{in}}\)	> 0	< 0
\(\Delta S_M\)	increase (scale ↑; dispersion may also rise)	decrease (scale ↓; if concentration increases, fall is larger)
\(\Delta V_C\)	\(< 0\) (headroom used)	\(> 0\) (headroom restored)
\(-\,\bar p_C\, \Delta V_C\)	> 0	< 0
\(\Delta F_C = \Delta(U - T_0 S_M)\)	sign depends on \(T_0\), dispersion changes, and \(p_C\)	same (case by case)

Measurement proxies (practical)

Credit flows/stocks: new loans, margin/credit for securities, corporate bonds/CP issuance (use stock series differences if needed).
Shares \(q\): use a stable MECE partition by use/sector/instrument and update monthly.
Capacity \(V_C\): CET1/RWA headroom, LCR/NSFR slack, HQLA-based lending capacity.
Temperature \(T_L\): composite intensity index from spreads, turnover, and order-book depth (z-scored).
Pressure \(p_C\): estimate via \(p_C \approx -\Delta U/\Delta V_C\) regressions or policy-shock differences.

Monthly algorithm (implementation steps)

Aggregate credit flows/stocks \(f_i\); compute shares as \(q_i = f_i/\sum_i f_i\).
Observe \(M_{\mathrm{in}}\) and compute \(S_M = k\,M_{\mathrm{in}}\,H(q)\).
Build \(V_C\) from RWA, LCR/NSFR, and HQLA metrics.
Compute \(T_L\) from market microstructure (spreads/turnover/depth).
Evaluate \(\Delta U = \bar T_L \Delta S_M - \bar p_C \Delta V_C + W_{policy} + \varepsilon\).
Detect events: label months with \(\Delta M_{\mathrm{in}}\) spikes (or net credit inflow if available) as credit-creation periods and decompose contributions.

Free energy \(F_C\) — and optional exergy \(X_C\)

Why this exists. We want a single scalar potential for the QTC side that (i) decreases as dispersion \(S_M\) increases under a fixed environment, and (ii) provides an upper bound on structured work over a cycle. A Helmholtz-style free energy plays exactly this role.

Definition. With a state potential \(U(S_M,V_C,\dots)\) and fixed environment \(T_0\): \[ F_C \equiv U - T_0 S_M,\qquad dF_C = -\,p_C\,dV_C + \delta W_{other} \tag{3} \]

It serves as an early-warning gauge: when \(\Delta F_C \to 0\), policy headroom for structured work dries up.

Exergy \(X_C\) (optional, environment-dependent)

When an ambient pressure-like term matters, use the exergy-like availability: \[ X_C = (U-U_0) + p_0\,(V_C - V_{C0}) - T_0\,(S_M - S_{M0}) \tag{4} \]

It reduces to \(-\Delta F_C\) if \(p_0\) effects are negligible and \(V_C\) is fixed. Because \(X_C\) depends on boundary choices \((T_0, p_0)\), we treat it as an advanced/optional metric. This Version 2 update refines the definition and boundary conditions for \(X_C\) compared to earlier drafts.

Gibbs free energy \(G_C\) (optional, fixed \(p_0\) environment)

When both temperature- and pressure-like environments are treated as fixed, the Gibbs free energy is convenient: \[ G_C \equiv U + p_0\,V_C - T_0\,S_M,\qquad dG_C = V_C\,dp_0 - S_M\,dT_0 + \delta W_{other} \] In practice, we use \(F_C\) for fixed-volume analyses and \(G_C\) when an ambient pressure-like \(p_0\) is the natural control.

Maxwell-like relations (integrability)

Given the state potential \(U(S_M,V_C,\dots)\) implied by the correspondence table and the \(F_C\) definition above, mixed partials must commute, giving a Maxwell-like condition:

\[ T_L = \left(\frac{\partial U}{\partial S_M}\right)_{V_C},\quad p_C = -\left(\frac{\partial U}{\partial V_C}\right)_{S_M} \;\Rightarrow\; \left(\frac{\partial T_L}{\partial V_C}\right)_{S_M} = -\left(\frac{\partial p_C}{\partial S_M}\right)_{V_C} \tag{5} \]

Violations in data falsify the mapping (i.e., chosen variables are not state-like or proxies are inadequate).

Is this merely a rephrasing?

Mathematically it is a change of coordinates that isolates scale and dispersion. It becomes non-trivial because it yields: (i) falsifiable integrability constraints (Maxwell-like), (ii) a clean separation of policy work vs dispersion, and (iii) an exergy/free-energy lens for early-warning ceilings via \(X_C\) or \(F_C\).

What the analogy adds (insights & tests)

Integrability test (Maxwell-like): Estimate \(T_L(S_M,V_C)\), \(p_C(S_M,V_C)\); check \((\partial T_L/\partial V_C)_{S_M} \approx - (\partial p_C/\partial S_M)_{V_C}\). Persistent failure ⇒ mapping/proxies are wrong.
Work vs dispersion: Decompose \(\Delta U = \bar T_L\,\Delta S_M - \bar p_C\,\Delta V_C + W_{policy}\) to separate random mixing from structured/policy effects.
Free-energy/exergy ceilings: Use \(F_C\) or \(X_C\) as early-warning gauges when they approach zero under chosen boundaries.
Loop area (hysteresis): Non-zero loop area in \((p_C,V_C)\) over policy cycles measures dissipative stress.

Limitations & scope

Category dependence: \(S_M\) depends on how uses/sectors are binned; require robustness checks.
Proxy noise: \(T_L, V_C, p_C\) are noisy proxies; mis-measurement can break Maxwell-like relations.
Quasi-static only: Fast crises and regime shifts violate the smooth state-variable picture.
Non-physical: No microscopic “money particles” are assumed; this is structured bookkeeping, not physics.
Identification: Policy and shocks move terms jointly; causal claims need proper empirical design.
Scaling conventions: Parameters like \(k\) are conventional; report normalized, sensitivity-tested metrics.
No guarantee of physical laws. Nothing here asserts or guarantees that macro-financial data obey the physical first or second laws; all mappings are analogy-level and subject only to empirical testing.

Minimal references (selected)

[1] Kocherlakota, N. R. (1998). “Money is Memory.” Journal of Economic Theory 81(2): 232–251.
[2] Kocherlakota, N. R. (2002). “Money is Memory.” Minneapolis Fed Quarterly Review 26(1): 2–10.
[3] Fisher, I. (1911). The Purchasing Power of Money. Macmillan.
[4] Friedman, M. (1956). “The Quantity Theory of Money—A Restatement.” In Studies in the Quantity Theory of Money (ed. Friedman). University of Chicago Press.
[5] Friedman, M., & Schwartz, A. J. (1963). A Monetary History of the United States, 1867–1960. Princeton University Press.
[6] Laidler, D. (1985, 3rd ed.). The Demand for Money: Theories, Evidence, and Problems. Harper & Row.
[7] Theil, H. (1967). Economics and Information Theory. North-Holland.
[8] Bank of England (McLeay, Radia, Thomas) (2014). “Money creation in the modern economy.” Quarterly Bulletin.
[9] Werner, Richard A. (2011 keynote; 2012 journal; 2014 article). “Quantity Theory of Credit” program and bank money creation evidence. International Review of Financial Analysis.
[10] Borio & White (2004). “Whither monetary and financial stability?” BIS Working Paper 147.
[11] Fontana, G. (2004). “Rethinking endogenous money.” Metroeconomica.
[12] Shannon, C. E. (1948). “A Mathematical Theory of Communication.” Bell System Technical Journal.
[13] Jaynes, E. T. (1957). “Information Theory and Statistical Mechanics.” Physical Review.
[14] Callen, H. B. (1985, 2nd ed.). Thermodynamics and an Introduction to Thermostatistics. Wiley.
[15] Zemansky, M. W., & Dittman, R. H. (1997, 7th ed.). Heat and Thermodynamics. McGraw-Hill.
[16] Foley, D. K. (1994). “A statistical equilibrium theory of markets.” Journal of Economic Theory.
[17] Drăgulescu, A., & Yakovenko, V. M. (2000). “Statistical mechanics of money.” European Physical Journal B.
[18] Wall, G. (1977). “Exergy—a useful concept.” Chalmers University preprint; later textbooks.

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App behavior

Monthly build computes \(S_M\), \(T_L\), loop_area (PLD), and \(X_C\) from public data.
Interactive report (with PNG fallbacks): Open report
Repository: GitHub repo

Thermo-Credit Theory (QTC ↔ Thermodynamics)

Key links