M.9 Ergodicity & Statistical Mechanics

Basic Prerequisites: Probability & Measure Theory (expectations, laws of large numbers, martingales) and Calculus & Optimization (maximising $\mathbb{E}[\ln W]$, concavity). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Almost all of textbook finance computes ensemble averages: the expected value of a portfolio across many parallel copies of the world at one instant. But you do not get to live across parallel copies - you live along one path through time. When wealth compounds multiplicatively, the ensemble average and the time average are not the same number, and the difference is not a rounding error - it is the whole game. A strategy whose ensemble expectation is $+5\%$ per round can bankrupt a single investor with probability one. That is non-ergodicity, and it is the single most important correction to the “expected return” reflex in the Atlas.

This folder is the ergodicity & statistical-mechanics toolbox for the Atlas. Everything reduces to five primitive ideas:

1. Ensemble average $\ne$ time average unless a system is ergodic - and compounding wealth is not ergodic. → 02 · Ensemble vs Time Averages.
2. Multiplicative growth adds in the log, so the growth rate is $\mathbb{E}[\ln (1+R)]$, not $\mathbb{E}[R]$. The gap is the volatility drag $\frac{1}{2}{\sigma }^2$. → 03 · Multiplicative Growth.
3. Maximising $\mathbb{E}[\ln W]$ is the growth-optimal objective - the Kelly criterion - and it has a unique optimum $f^{\ast }$. → 04 · The Kelly Criterion.
4. Beyond $f^{\ast }$ there is a critical fraction $f_c$ past which ruin is certain, and even full Kelly has a fixed, large drawdown law. → 05 · Ruin & Drawdown.
5. In the real world the parameters are estimated, so overbetting is the dominant risk - which is why practice uses fractional Kelly. → 06 · Advanced Extensions.

This page is the hub: it gives the fast formula and growth-law lookup below, then routes you to six sub-pages built from first principles, with working code and failure modes.

The one-sentence essence. “An investor does not hold an average of paths, they hold one path; when wealth multiplies, the typical path drifts toward extinction even as the ensemble mean explodes - so the correct objective is the growth rate of the typical path, $\mathbb{E}[\ln W]$, and the correct bet size is Kelly’s $f^{\ast }$.“

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). The growth/ruin formulas are transcribed from Thorp (2006, Ch. 9) and Kelly (1956); the ensemble/time-average distinction follows Peters (2019); the fat-tail and estimation caveats are cross-checked against Tsay (Ch. 1) and Glasserman (Ch. 1). The numbers in the check column were re-executed and reproduced exactly by the scripts in §3 (see the per-page verified outputs).

Notation: $W_t$ wealth, $R_t$ simple return, $r_t=\ln (1+R_t)$ log return, $p$ win probability, $q=1-p$, $b$ win payout per unit staked, $a$ loss per unit staked. Continuous limit: $m$ instantaneous drift, $s^2$ variance rate, $r$ riskless rate, $f$ invested fraction.

The scaling warning that makes this folder “foundations.” Every number above is a growth-rate quantity. A Sharpe ratio $S$ does not translate into wealth at $S$; it translates into a growth rate $S^2/2+r$. A $+5\%$ arithmetic edge can be a $-5.27\%$ log-growth decel. Confusing additive means with multiplicative growth is the single most common error this folder exists to kill.

3. Computational Implementation - the three laws in one screen

Stdlib only. It (a) shows ensemble vs time average diverging, (b) computes the growth-optimal Kelly fraction, and (c) checks the drawdown law against Thorp’s formula.

The ensemble mean ($131.5$) and the typical outcome ($0.005$) are four orders of magnitude apart - that gap is non-ergodicity, and it is what the §3 experiment of each relevant sub-page quantifies.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure analysis lives in the sub-pages. In one line each:

1. The arithmetic-mean trap. Ranking strategies by $\mathbb{E}[R]$ instead of $\mathbb{E}[\ln (1+R)]$ rewards volatility; high-mean/high-variance strategies look best right before they go extinct (pages 02–03).
2. Full-Kelly overbetting on estimated parameters. $f^{\ast }$ computed from an estimated edge is systematically too large because $\mathbb{E}[\hat{m}]>m$ under mean reversion; production funds run $0.25f^{\ast }$–$0.5f^{\ast }$ (pages 04, 06).
3. The certain-ruin boundary $f_c$. Betting $f\ge 2f^{\ast }$ (and always past $f_c$) makes ruin almost sure even with a positive edge - a positive Kelly edge is not a licence to lever freely (page 04).
4. Drawdowns are first-class, not noise. Full Kelly’s median max-drawdown over 30 years is $\sim 71\%$; investors quit, and quits are indistinguishable from ruin (page 05).
5. Fat tails break the Gaussian growth law. Real returns have excess kurtosis (Tsay Table 1.2); the log-diffusion model behind these formulas understates jump risk, so the practical hedge is fractional Kelly (pages 05–06).

5. References

• Kelly, J. L. jr.: A New Interpretation of Information Rate, Bell System Technical Journal 35(4):917–926 (1956)
• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market, in Handbook of Asset and Liability Management Vol. 1 (Zenios & Ziemba eds., 2006)
• MacLean, Thorp & Ziemba (eds.): The Kelly Capital Growth Investment Criterion: Theory and Practice (World Scientific, 2011)
• Peters, Ole: The Ergodicity Problem in Economics, Nature Physics 15, 1216–1221 (2019)
• Peters & Gell-Mann: Evaluating Gambles Using Dynamics, Chaos 26, 023103 (2016)
• Tsay, Ruey S.: Analysis of Financial Time Series (3rd ed.)
• Glasserman, Paul: Monte Carlo Methods in Financial Engineering

6. Connected Graph Bridges

• Foundational base: Probability & Measure Theory · Calculus & Optimization (concave $\mathbb{E}\ln$, interior optimum)
• Legacy page (archived seed): Ergodicity & Statistical Mechanics (legacy note)
• Applied destination - risk & tails: Extreme Value Theory & Fat Tails · Quantitative Risk
• Applied destination - allocation: Mean–Variance Optimization · Portfolio Optimization (Kelly as the growth-optimal allocation)
• Applied destination - sizing in execution/market-making: Inventory Management & Quote Skewing · Almgren–Chriss
• Related foundation folders: Stochastic Calculus & Itô (GBM, the diffusion model behind $g_{\infty }$) · Econometrics & Time Series (estimation error feeding page 06) · Numerical Methods (Monte Carlo)
• Sub-pages (in-folder): 01 From Zero · 02 Ensemble vs Time Averages · 03 Multiplicative Growth · 04 Kelly Criterion · 05 Ruin & Drawdown · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05