Thermo-Credit: from measurement to a candidate state model

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

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 \(M_{in}\).

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.

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.

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)

1. Aggregate credit flows/stocks \(f_i\); compute shares as \(q_i = f_i/\sum_i f_i\).
2. Observe \(M_{\mathrm{in}}\) and compute \(S_M = k\,M_{\mathrm{in}}\,H(q)\).
3. Build \(V_C\) from RWA, LCR/NSFR, and HQLA metrics.
4. Compute \(T_L\) from market microstructure (spreads/turnover/depth).
5. Evaluate \(\Delta U = \bar T_L \Delta S_M - \bar p_C \Delta V_C + W_{policy} + \varepsilon\).
6. 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.

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:

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.

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