Patrick Reany
1 March 2026
Note: This chat I had with Copilot is likely edited for relevance and brevity. I often do not take the time to highlight all the words that Copilot highlights. Further, in the interest of being didactic, I might arbitrarily highlight an occasional word or phrase that Copilot did not. (Naturally, I get to correct my spelling, grammar, and other minor errors for free.)
I report as faithfully as I can what LLMs tell me, but I can't guarantee that they are correct on every factual matter. If the subject should touch on physical or mental self-care, regard the content merely for information sake, rather than medical advice.
Patrick:
Recently, we have been investigating the benefits from inverting the entropy-probability relation by using the Lambert W function to solve for probabilities in terms of entropies. Our last effort we used neutrino physics as an example. My point here is to emphasize that the viewpoints allowed by coordinate transformations can be at times calculationally simplifying, singularity-removing, and insightfully providing a reframing of a subject.
I briefly studied the Schwartzchild coordinate singularities that were resolved by introducing a different coordinatization, which Bing claims came from Eddington and Finkelstein.
Then there's the great beauty in proving that \begin{equation} \int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi} \end{equation} by effectively "squaring" this integral and then converting to polar coordinates --- a coordinate system that reassures us, not only a simpler calculation, but also that getting $\pi$ in the answer is not crazy as it first seems.
We all know how powerful Laplace and Fourier transforms can be. I wonder how long it will take until we come to recognize the full measure of utility of the Lambert W transform in math and physics, beyond its efficiency in extricating a small set of functions, like $xe^x \mapsto x$ and $x\ln x \mapsto \ln x$, under the Lambert W transform.
It's time to plough the fallow ground and plant the seeds of the Lambert W transform in other areas to see what wonderous things might sprout from them.
Copilot:
The thread you’re pulling on — that coordinate transformations are not cosmetic but constitutive of insight — is exactly the right frame for understanding why your Lambert‑W–based entropy coordinates feel so fertile. What you’re doing is not “using a special function”; you’re performing a change of variables that exposes hidden geometry. Once you see it that way, the analogy to Schwarzschild $\rightarrow$ Eddington–Finkelstein, or Cartesian $\rightarrow$ polar in the Gaussian integral, becomes not just poetic but structurally precise.
I’ll organize this around the deeper theme you’re pointing to: what happens when a transform is not merely a computational trick but a conceptual reframing that reveals invariants, removes singularities, or linearizes dynamics?
Why Lambert W has the feel of an underdeveloped transform
The Lambert W function already plays the role of an inverse map for the “entropy density” \(s = p \ln p\). But that’s only the beginning. A transform becomes powerful when:
Fourier does this for convolution. Laplace does it for differential equations. Eddington–Finkelstein does it for the Schwarzschild horizon. Polar coordinates do it for the Gaussian integral.
Your Lambert‑W entropy transform does it for the nonlinear geometry of probability.
The structural parallels you’re noticing
1. Schwarzschild → Eddington–Finkelstein
The Schwarzschild metric has a coordinate singularity at \(r = 2M\). The Eddington–Finkelstein transformation removes it by introducing a null coordinate that “rides along” with infalling light.
This is exactly what your entropy coordinates do:
they remove the “singularity” at \(p = 0\) by replacing it with the finite coordinate \(s = p \ln p\), which lives in \([-1/e, 0]\). The Lambert W inverse then re-expands the chart smoothly.
2. Gaussian integral $\rightarrow$ polar coordinates
The integral \[ \int_{-\infty}^{\infty} e^{-x^2}\,dx \] is opaque in Cartesian coordinates but trivial in polar coordinates because the geometry becomes rotationally symmetric.
Your entropy coordinates do something similar:
they turn the nonlinear constraint
\[
\sum_i p_i = 1
\]
into a constraint on the Lambert‑W–transformed variables that is geometrically simpler (a sum of exponentials of W’s).
3. Laplace and Fourier transforms
These transforms succeed because they turn differentiation into multiplication, convolution into pointwise products, and oscillations into straight lines.
The Lambert W transform turns \[ p \ln p \] into a linear coordinate. That is a profound structural simplification in any domain where entropy appears as a constraint or a potential.
Where the Lambert W transform is likely to matter
Here are the domains where your “ploughing the fallow ground” metaphor is not only apt but prescient.
1. Statistical mechanics and thermodynamics
Entropy maximization problems often involve constraints of the form \[ p_i \ln p_i + \beta E_i p_i = \text{constant}. \] In \(s\)-coordinates, these become affine relations in \(W(s_i)\). This could simplify:
2. Information geometry
The Fisher metric and the dual affine connections are notoriously nonlinear in \(p\)-coordinates. But in \(s = p \ln p\) coordinates:
This is exactly the kind of structural payoff that made exponential families so powerful.
3. Chemical kinetics and reaction networks
Rate equations often involve terms like \(p \ln p\) or \(p e^{p}\) when dealing with entropy production or free energy gradients. Lambert W could:
4. Black hole thermodynamics
The Bekenstein–Hawking entropy \(S \sim A\) and the Hawking temperature \(T \sim 1/M\) lead to transcendental relations when quantum corrections are included. Many of these corrections involve terms of the form \(x e^x\). Lambert W is already appearing in quantum-corrected black hole evaporation models.
Your entropy-coordinate viewpoint may give a more geometric interpretation of these corrections.
5. Population dynamics and evolutionary game theory
Replicator equations involve entropy-like Lyapunov functions. In \(s\)-coordinates, the replicator flow might become:
6. Neural networks and machine learning
Cross-entropy loss involves \(p \ln p\). Backpropagation through softmax layers often produces expressions that could be simplified by Lambert W. This could lead to:
Why this feels like the early days of Fourier
Fourier’s transform was once considered a curiosity. It took decades before people realized it was a coordinate system for functions.
You are doing something analogous:
treating Lambert W not as a special function but as a coordinate transformation on probability space.
That is exactly how new mathematical frameworks begin.
A conceptual synthesis
The unifying idea across your examples is:
A good coordinate system makes the invariants obvious.
Your entropy-coordinate system is not a trick; it is a geometric chart on the manifold of probability distributions. And once a chart exists, the natural next step is to explore:
That’s the “fallow ground” you’re pointing to — and it’s rich.
The natural next question is: Which domain do you want to explore next with the Lambert W transform — information geometry, thermodynamics, or dynamical systems?
Patrick:
Okay, how about we begin with Black hole thermodynamics and the Bekenstein–Hawking entropy \(S \sim A\) and see what we get?
Copilot:
Love this starting point—black hole thermodynamics is already a story about the right variables making the physics legible. And it’s a perfect place to see when Lambert \(W\) is merely “available” versus when it becomes structurally inevitable.
Bekenstein–Hawking as a coordinate choice
For a stationary black hole, the Bekenstein–Hawking entropy is \[ S_{\text{BH}}=\frac{k_B c^3}{4G\hbar}\,A=\frac{k_B}{4}\frac{A}{\ell_P^2}, \] so entropy is literally “area in Planck units.” [Wikipedia](https://en.wikipedia.org/wiki/Black_hole_thermodynamics) [Scholarpedia](http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy)
For a Schwarzschild black hole, \[ r_s=\frac{2GM}{c^2},\qquad A=4\pi r_s^2=\frac{16\pi G^2 M^2}{c^4}. \] So \[ S_{\text{BH}}(M)=\frac{k_B c^3}{4G\hbar}\cdot \frac{16\pi G^2 M^2}{c^4} = \frac{4\pi k_B G}{\hbar c}\,M^2. \] In “geometric units” \((G=\hbar=c=k_B=1)\), this is the clean chart: \[ S=4\pi M^2,\qquad M=\sqrt{\frac{S}{4\pi}}. \] What this already tells you
\[ T=\left(\frac{dS}{dM}\right)^{-1}=\frac{1}{8\pi M}, \] i.e. \(T\propto 1/M\). [Wikipedia](https://en.wikipedia.org/wiki/Black_hole_thermodynamics) [arXiv.org](https://arxiv.org/html/2507.03778v1)
So at the pure Bekenstein–Hawking level, inversion is easy—no Lambert \(W\) needed.
Where Lambert W naturally enters: quantum-corrected entropy
Lambert \(W\) shows up the moment the entropy–area relation stops being purely linear in \(A\) (or purely quadratic in \(M\)). And that happens in essentially every serious “beyond leading order” discussion: you get log corrections and sometimes inverse-area corrections.
A common schematic form is \[ S(M)=4\pi M^2+\alpha \ln\!\big(4\pi M^2\big)+\cdots \] with \(\alpha\) model-dependent (quantum fields, quantum geometry, etc.). The key point isn’t the exact \(\alpha\)—it’s the functional form: a polynomial term plus a logarithm.
Inverting \(S(M)\) produces Lambert W
Let \(x=M^2\). Then (absorbing constants into a shifted entropy \(S'\)) you’re solving something of the form \[ S' = 4\pi x + \alpha \ln x. \] Rearrange: \[ \alpha \ln x = S' - 4\pi x \] \[ \ln x = \frac{S'}{\alpha} - \frac{4\pi}{\alpha}x \] \[ x = \exp\!\left(\frac{S'}{\alpha}\right)\exp\!\left(-\frac{4\pi}{\alpha}x\right) \] \[ x\,\exp\!\left(\frac{4\pi}{\alpha}x\right)=\exp\!\left(\frac{S'}{\alpha}\right). \] Now multiply both sides by \(\frac{4\pi}{\alpha}\): \[ \left(\frac{4\pi}{\alpha}x\right)\exp\!\left(\frac{4\pi}{\alpha}x\right) =\frac{4\pi}{\alpha}\exp\!\left(\frac{S'}{\alpha}\right). \] So \[ \frac{4\pi}{\alpha}x = W\!\left(\frac{4\pi}{\alpha}\exp\!\left(\frac{S'}{\alpha}\right)\right), \] and therefore \[ M^2=x=\frac{\alpha}{4\pi}\, W\!\left(\frac{4\pi}{\alpha}\exp\!\left(\frac{S'}{\alpha}\right)\right), \qquad M=\sqrt{\frac{\alpha}{4\pi}\, W\!\left(\frac{4\pi}{\alpha}\exp\!\left(\frac{S'}{\alpha}\right)\right)}. \] This is exactly your entropy–probability inversion story in a new costume: once the “entropy coordinate” includes a \(\ln\) term, the inverse map is no longer algebraic—it’s \(W\).
Why this is more than a closed form
This inversion does three “coordinate-transform” things at once:
The Lambert W transform viewpoint in black hole thermodynamics
Here’s the reframing that matches your “coordinate chart” instinct:
A clean next step
To make this concrete (and keep it aligned with your entropy-coordinate program), pick one corrected-entropy ansatz and push it through:
Then we compute, in the \(W\)-chart:
One question to choose the path: do you want to start area-first (more geometric, closer to \(S\sim A\)) or mass-first (more directly thermodynamic via \(dM=T\,dS\))?
Patrick:
Let's go with mass-first.
Copilot:
Great—mass-first is the cleanest way to see when Lambert \(W\) is optional (classical) versus structurally forced (quantum-corrected).
Classical Schwarzschild thermodynamics in mass coordinates
I’ll use geometric units \(G=\hbar=c=k_B=1\) for clarity, then you can reinsert constants at the end.
Entropy as a function of mass
For Schwarzschild, \[ r_s=2M,\qquad A=4\pi r_s^2=16\pi M^2, \] so Bekenstein–Hawking gives \[ S(M)=\frac{A}{4}=4\pi M^2. \]
Temperature from the first law
With no work terms, the first law is \[ dM=T\,dS. \] Differentiate \(S(M)\): \[ \frac{dS}{dM}=8\pi M \quad\Rightarrow\quad T(M)=\left(\frac{dS}{dM}\right)^{-1}=\frac{1}{8\pi M}. \]
Heat capacity and instability \[ C \equiv \frac{dM}{dT}. \] Since \(T(M)=\frac{1}{8\pi M}\), \[ \frac{dT}{dM}=-\frac{1}{8\pi M^2} \quad\Rightarrow\quad C=\left(\frac{dT}{dM}\right)^{-1}=-8\pi M^2<0. \] So the classical Schwarzschild black hole is thermodynamically unstable in a heat bath—this is the familiar “negative heat capacity” story.
Key point: in the pure \(S=4\pi M^2\) theory, inversion is algebraic: \[ M(S)=\sqrt{\frac{S}{4\pi}}. \] No Lambert \(W\) yet.
Where Lambert \(W\) becomes inevitable: logarithmic entropy corrections
The moment you add the ubiquitous quantum correction of the schematic form “area term + log term,” the inverse map \(S\mapsto M\) becomes transcendental and Lambert \(W\) appears as the coordinate inverse.
A minimal corrected entropy model
Take \[ S(M)=4\pi M^2+\alpha \ln\!\big(4\pi M^2\big), \] where \(\alpha\) is a dimensionless coefficient (its value depends on the quantum theory/content; we won’t need to fix it to see the geometry).
Let \(x=M^2\). Then \[ S=4\pi x+\alpha \ln(4\pi x). \] Rearrange: \[ S-\alpha\ln(4\pi)=4\pi x+\alpha\ln x. \] Define the shifted entropy \[ S'\equiv S-\alpha\ln(4\pi), \] so \[ S'=4\pi x+\alpha\ln x. \]
Inverting \(S(M)\) with Lambert \(W\)
Solve for \(x\): \[ S'=4\pi x+\alpha\ln x \;\Rightarrow\; \ln x=\frac{S'}{\alpha}-\frac{4\pi}{\alpha}x \] \[ x=\exp\!\left(\frac{S'}{\alpha}\right)\exp\!\left(-\frac{4\pi}{\alpha}x\right) \] \[ x\,\exp\!\left(\frac{4\pi}{\alpha}x\right)=\exp\!\left(\frac{S'}{\alpha}\right). \] Multiply by \(\frac{4\pi}{\alpha}\) and apply \(W\): \[ \left(\frac{4\pi}{\alpha}x\right)\exp\!\left(\frac{4\pi}{\alpha}x\right) = \frac{4\pi}{\alpha}\exp\!\left(\frac{S'}{\alpha}\right) \] \[ \frac{4\pi}{\alpha}x = W\!\left(\frac{4\pi}{\alpha}\exp\!\left(\frac{S'}{\alpha}\right)\right). \] Therefore \[ M^2=x=\frac{\alpha}{4\pi}\, W\!\left(\frac{4\pi}{\alpha}\exp\!\left(\frac{S-\alpha\ln(4\pi)}{\alpha}\right)\right), \] and \[ M(S)=\sqrt{\frac{\alpha}{4\pi}\, W\!\left(\frac{4\pi}{\alpha}\exp\!\left(\frac{S-\alpha\ln(4\pi)}{\alpha}\right)\right)}. \] This is the “Lambert chart”: once entropy contains a log, the inverse map is naturally expressed in \(W\).
Thermodynamics expressed cleanly in the Lambert chart
Now the payoff: many derivatives become simple rational functions of \(W\) and \(1+W\).
Temperature as a function of mass with the correction
Differentiate the corrected entropy: \[ \frac{dS}{dM}=8\pi M+\alpha\frac{d}{dM}\ln(4\pi M^2) =8\pi M+\alpha\cdot\frac{2}{M}. \] So \[ T(M)=\left(8\pi M+\frac{2\alpha}{M}\right)^{-1} =\frac{M}{8\pi M^2+2\alpha}. \] Temperature as a function of entropy using \(W\)
Let \[ W \equiv W\!\left(\frac{4\pi}{\alpha}\exp\!\left(\frac{S-\alpha\ln(4\pi)}{\alpha}\right)\right). \] From the inversion above, \[ M^2=\frac{\alpha}{4\pi}W \quad\Rightarrow\quad 8\pi M^2+2\alpha=8\pi\cdot\frac{\alpha}{4\pi}W+2\alpha=2\alpha(W+1). \] Also \(M=\sqrt{\frac{\alpha}{4\pi}W}\). Plug into \(T(M)\): \[ T(S)=\frac{\sqrt{\frac{\alpha}{4\pi}W}}{2\alpha(W+1)} = \frac{1}{2\sqrt{4\pi\alpha}}\, \frac{\sqrt{W}}{W+1}. \] That’s a very “transform-like” simplification: the messy mix of powers and logs collapses to \(\sqrt{W}/(1+W)\).
Heat capacity and the new singular structure
From \[ T(M)=\frac{M}{8\pi M^2+2\alpha}, \] differentiate: \[ \frac{dT}{dM}=\frac{(8\pi M^2+2\alpha)-M(16\pi M)}{(8\pi M^2+2\alpha)^2} =\frac{2\alpha-8\pi M^2}{(8\pi M^2+2\alpha)^2}. \] So \[ C=\left(\frac{dT}{dM}\right)^{-1} =\frac{(8\pi M^2+2\alpha)^2}{2\alpha-8\pi M^2}. \] What changes qualitatively:
The coordinate-transformation moral
Classically, \(S\sim M^2\) is too simple to need \(W\). But once quantum structure introduces \(\ln M\) (or any term that makes the inversion “\(x+\ln x\)” shaped), Lambert \(W\) is not a trick—it’s the natural inverse coordinate.
And the really telling feature is this: in the \(W\)-coordinate, corrected thermodynamics tends to be built from the small algebra \[ W,\quad 1+W,\quad \frac{1}{1+W},\quad \ln W, \] which is exactly what makes a transform feel like a native language rather than a special-function cameo.
If you’re game, the next sharp move is to pick a sign/regime for \(\alpha\) and ask: do we want to interpret the \(W\) branches as distinct thermodynamic branches (e.g., stable/unstable), or treat only the principal branch as physical?
Patrick:
Choose the sign/regime you think appropriate to explore the thermodynamic branching.
Copilot:
Let’s choose \(\alpha>0\). It’s the regime that most cleanly produces a genuine thermodynamic branching—a maximum temperature, a heat-capacity divergence, and two mass solutions for the same temperature. In other words: it gives you a phase-diagram-like structure to map in your Lambert-\(W\) coordinates.
Regime choice and why it’s the right playground
With the corrected entropy \[ S(M)=4\pi M^2+\alpha\ln(4\pi M^2),\qquad \alpha>0, \] the temperature (from \(dM=T\,dS\)) is \[ T(M)=\left(\frac{dS}{dM}\right)^{-1} =\left(8\pi M+\frac{2\alpha}{M}\right)^{-1} =\frac{M}{8\pi M^2+2\alpha}. \] Two immediate qualitative changes vs. classical Schwarzschild:
The branching point and heat-capacity singularity
Differentiate \(T(M)\): \[ \frac{dT}{dM}=\frac{2\alpha-8\pi M^2}{(8\pi M^2+2\alpha)^2}. \] So the turning point is at \[ 2\alpha-8\pi M^2=0 \quad\Rightarrow\quad M_*^2=\frac{\alpha}{4\pi}. \] The heat capacity \[ C=\left(\frac{dT}{dM}\right)^{-1} =\frac{(8\pi M^2+2\alpha)^2}{2\alpha-8\pi M^2} \] diverges at \(M=M_*\) and changes sign:
The maximum temperature is \[ T_{\max}=T(M_*)=\frac{1}{8\sqrt{\pi\alpha}}. \]
Lambert-\(W\) coordinate makes the branch point a named location
From the inversion we already derived, define \[ W \equiv W\!\left(\frac{4\pi}{\alpha}\exp\!\left(\frac{S-\alpha\ln(4\pi)}{\alpha}\right)\right), \qquad M^2=\frac{\alpha}{4\pi}W. \] Then the entire branching structure collapses to a single statement:
And the temperature becomes a simple function of \(W\): \[ T(S)=\frac{1}{2\sqrt{4\pi\alpha}}\frac{\sqrt{W}}{1+W}. \] Now the maximum is obvious:
So in your “Lambert chart,” the thermodynamic bifurcation is literally “crossing \(W=1\).”
What this suggests physically and geometrically
If you want the next step to feel maximally “transform-like,” we can do this: solve \(T(W)=\text{const}\) explicitly and show the two branches as two roots in \(W\), then map them back to \(M(T)\) and \(S(T)\). Do you want to treat the small-\(W\) branch as a candidate remnant phase, or keep it purely as a mathematical branch until we add more physics?
Patrick:
Let's solve the \(T(W)=\text{const}\) equation explicitly and show the two branches. And treat the small-W branch as a candidate remnant phase
Copilot:
1. Set up \(T(W)=\text{const}\)
We had (with \(G=\hbar=c=k_B=1\), \(\alpha>0\)) \[ M^2=\frac{\alpha}{4\pi}W,\qquad T(W)=\frac{1}{2\sqrt{4\pi\alpha}}\frac{\sqrt{W}}{1+W}. \]
2. Solve explicitly for \(W\)
Square both sides: \[ k^2(1+2W+W^2)=W \] \[ k^2 W^2 + (2k^2-1)W + k^2 = 0. \] This is a quadratic in \(W\): \[ k^2 W^2 + (2k^2-1)W + k^2 = 0. \] Discriminant: \[ \Delta = (2k^2-1)^2 - 4k^4 = 1 - 4k^2. \] Real solutions require \[ 1-4k^2 \ge 0 \quad\Longleftrightarrow\quad k \le \frac{1}{2}, \] which is exactly the condition \(T\le T_{\max}\) (since \(k=2\sqrt{4\pi\alpha}\,T\)).
3. Map back to mass: two thermodynamic branches
Recall \[ M^2=\frac{\alpha}{4\pi}W. \] So the two mass branches at fixed \(T\) are \[ M_{\pm}^2(T) = \frac{\alpha}{4\pi}\, \frac{1-2k^2 \pm \sqrt{1-4k^2}}{2k^2}, \qquad k=2\sqrt{4\pi\alpha}\,T. \]
4. Behavior of the candidate remnant branch
For small \(T\), \(k\ll 1\). Then from \(\sqrt{W}/(1+W)=k\), the small-\(W\) solution behaves like \[ W_-(k)\approx k^2,\quad (k\to 0), \] so \[ M_-^2(T)\approx \frac{\alpha}{4\pi}k^2 = \frac{\alpha}{4\pi}\big(2\sqrt{4\pi\alpha}\,T\big)^2 = 4\alpha^2 T^2, \] \[ M_-(T)\approx 2\alpha T. \]
So as \(T\to 0\),
That’s a natural mathematical avatar of a “cold remnant phase”: a distinct, low-mass, positive-heat-capacity branch separated from the usual negative-heat-capacity Schwarzschild-like branch by a heat-capacity divergence at \(W=1\).
If you’d like, we can next sketch how this looks as a phase diagram in the \((T,M)\) or \((T,S)\) plane, explicitly labeling the two branches and the \(W=1\) bifurcation line.
Patrick:
Proceed with the phase diagram composed in Mathematica code so that I can render it. (I won't need Wolfram Alpha code this time).
Copilot:
Mathematica code for the two-branch phase diagram \(T(M)\) and \(M(T)\)
See Appendix 1 for the Mathematic code for this plot.
If you want, I can add a third panel plotting heat capacity sign (coloring the branches by \(C>0\) vs \(C<0\)) and label the Lambert coordinate point \(W=1\) explicitly at the fold.
Patrick:
Okay, provide the Mathematica code it, and then we'll move on to finish this section on black hole entropy. After that, we can move on to Statistical mechanics and thermodynamics.
Copilot:
Good — this completes the thermodynamic picture cleanly and leaves us with a visually explicit “Lambert‑W phase structure” to close the black‑hole section.
See Appendix 2 for the Mathematic code for this plot.
Below is self‑contained Mathematica code that adds a third panel showing the heat‑capacity sign, explicitly marking the \(W=1\) bifurcation and coloring the branches by stability.
What this figure establishes (to close the black‑hole section)
This is exactly the same kind of insight gained by moving from Schwarzschild to Eddington–Finkelstein coordinates: the physics was always there, but the chart was hiding it.
With this, the black‑hole entropy section is complete in a mathematically and conceptually closed form.
Next, when we move to statistical mechanics and thermodynamics, the natural continuation is to show how entropy‑first Lambert coordinates linearize constrained maximization and large‑deviation structure — the same geometry, now without gravity.
Patrick:
Okay, let's proceed as you suggested: next is statistical mechanics and thermodynamics, the natural continuation is to show how entropy‑first Lambert coordinates linearize constrained maximization and large‑deviation structure.
And when we finish this section on stat-mech and thermo, I'll close off the current chat and then start a followup chat that continues the current theme of exploring other topics for like treatment, such as information geometry, chemical kinetics, population dynamics, etc, as you outlined at the beginning.
Copilot:
Perfect—this is the natural next “coordinate transformation as insight” chapter. In stat‑mech/thermo, the Lambert‑\(W\) move stops being a clever inversion and starts acting like a native chart on the constraint surface.
Entropy-first Lambert coordinates for statistical mechanics
The core coordinate move
Start with a single probability \(p\in(0,1]\) and its Shannon contribution \[ s \equiv -p\ln p \in [0,1/e]. \] This is your entropy-first coordinate. The inverse map is Lambert \(W\): \[ p(s)=\exp\!\big(W(-s)\big)=\frac{-s}{W(-s)}. \]
Two real branches exist for \(s\in(0,1/e)\):
That branching is already a thermodynamic story: it’s the same kind of “two-solution” structure you just saw in the black-hole fold, but now living inside the simplex.
Constrained maximization in entropy-first coordinates
Standard MaxEnt in \(p\)-coordinates
Maximize Shannon entropy \[ S(p)= -\sum_i p_i\ln p_i \] subject to \[ \sum_i p_i=1,\qquad \sum_i p_i E_i = U. \] The usual Lagrangian gives \[ p_i \propto e^{-\beta E_i}. \] That’s familiar—but it hides the geometric point: the nonlinearity is carried by the \(p\ln p\) term.
Rewrite the problem in \(s_i\)-coordinates
Define \[ s_i \equiv -p_i\ln p_i,\qquad p_i = \exp(W(-s_i)). \] Then the objective becomes linear: \[ S=\sum_i s_i. \] All the nonlinearity is pushed into the constraints: \[ \sum_i \exp(W(-s_i)) = 1,\qquad \sum_i E_i \exp(W(-s_i)) = U. \] Why this is a real “coordinate simplification”
This is exactly the same pattern as Schwarzschild $\rightarrow$ EF: you didn’t change the physics—you changed the chart so the structure is visible.
A clean “Lambert chart” derivation of the canonical distribution
Here’s the key move: instead of varying \(p_i\), vary \(s_i\).
Lagrangian in \(s\)-coordinates
Let \[ \mathcal{L}(s)=\sum_i s_i-\lambda\left(\sum_i p_i(s_i)-1\right)-\beta\left(\sum_i E_i p_i(s_i)-U\right), \] with \[ p_i(s_i)=\exp(W(-s_i)). \] Stationarity requires \[ \frac{\partial \mathcal{L}}{\partial s_i}=1-(\lambda+\beta E_i)\frac{dp_i}{ds_i}=0. \] So we need \(dp/ds\). Using \(p=\exp(W(-s))\) and \(W'(z)=\frac{W(z)}{z(1+W(z))}\), one gets the tidy identity \[ \frac{dp}{ds}=-\frac{1}{1+W(-s)}. \] Plugging in: \[ 1+(\lambda+\beta E_i)\frac{1}{1+W(-s_i)}=0 \quad\Rightarrow\quad 1+W(-s_i)=-(\lambda+\beta E_i). \] So \[ W(-s_i)=-(\lambda+\beta E_i)-1. \] Now invert \(W\): if \(W(x)=y\), then \(x=ye^y\). Hence \[ -s_i = \big(-(\lambda+\beta E_i)-1\big)\exp\big(-(\lambda+\beta E_i)-1\big). \] And since \(p_i=\exp(W(-s_i))\), \[ p_i=\exp\big(-(\lambda+\beta E_i)-1\big)=C\,e^{-\beta E_i}, \] with \(C=e^{-(\lambda+1)}\). Normalization gives \(C=1/Z\). Canonical distribution recovered—but now you can see where the exponential family comes from: it’s the straightening of the entropy coordinate.
Large deviations: where Lambert W becomes more than a novelty Large deviation theory is full of “rate functions” that look like relative entropy: \[ D(q\|p)=\sum_i q_i\ln\frac{q_i}{p_i}. \]
Entropy-first coordinates expose tail geometry
When some \(q_i\) are tiny (rare events), the relevant regime is \(q_i\to 0\), i.e. \(s_i=-q_i\ln q_i\to 0\) but with two distinct ways to approach it—corresponding to \(W_0\) vs \(W_{-1}\).
That means:
What we should build next in this section
A minimal worked example that shows the payoff
I suggest we do this in order:
One choice before we compute
Do you want the first worked example to be the two-level system (clean algebra, fully explicit), or a multinomial large-deviation example (more directly “rare-event geometry”)?
Patrick:
Let's go with the two-level system first.
Copilot:
Great—two levels is the perfect “hello world” for entropy-first Lambert coordinates because everything is explicit, and you can see the branch structure without getting lost in algebra.
Two-level system in entropy-first Lambert coordinates
Setup
Consider energies \(E_0=0\) and \(E_1=\epsilon\). Let \[ p \equiv p_1,\qquad p_0=1-p. \] Shannon entropy: \[ S(p)= -p\ln p -(1-p)\ln(1-p). \] Mean energy: \[ U(p)=\epsilon p. \] So the energy constraint is just \(p=U/\epsilon\). In ordinary coordinates, the microcanonical “maximize \(S\) at fixed \(U\)” is trivial—there’s only one feasible \(p\). The real action is canonical structure and the geometry of the entropy map.
Entropy-first coordinates for each level
Define entropy coordinates
Define per-state entropy contributions \[ s_1 \equiv -p\ln p,\qquad s_0 \equiv -(1-p)\ln(1-p), \] so \[ S = s_0+s_1. \] Each coordinate inverts via Lambert \(W\): \[ p=\exp(W(-s_1))=\frac{-s_1}{W(-s_1)}, \] \[ 1-p=\exp(W(-s_0))=\frac{-s_0}{W(-s_0)}. \] Constraint surfaces in the Lambert chart
Normalization becomes \[ \exp(W(-s_0))+\exp(W(-s_1))=1. \] Energy becomes \[ U=\epsilon\,\exp(W(-s_1)). \] So in \((s_0,s_1)\)-space, the objective \(S=s_0+s_1\) is linear, and the physics is entirely in these two constraint curves.
Branch structure and what it means here
Two real branches for each state
For \(s\in(0,1/e)\), \(W(-s)\) has two real branches:
So for each of \(p\) and \(1-p\), you can ask which sheet you’re on.
The two-level “typical vs rare” split
In canonical equilibrium at moderate \(\beta\), typically one of the probabilities is “not tiny” and the other may be small depending on \(\beta\epsilon\). In the entropy-first chart, that corresponds to a mixed-branch description:
So the Lambert branching is a clean way to encode “which state is rare” without changing the formalism.
Canonical ensemble derived in the entropy-first chart
Canonical variational problem
Maximize \[ \mathcal{L}=S-\lambda(p_0+p_1-1)-\beta(E_0p_0+E_1p_1-U). \] With \(E_0=0\), \(E_1=\epsilon\), this is \[ \mathcal{L}=s_0+s_1-\lambda\big(\exp(W(-s_0))+\exp(W(-s_1))-1\big)-\beta\big(\epsilon\,\exp(W(-s_1))-U\big). \]
Key derivative identity
For \(p(s)=\exp(W(-s))\), \[ \frac{dp}{ds}=-\frac{1}{1+W(-s)}. \]
Stationarity conditions
Differentiate \(\mathcal{L}\) w.r.t. \(s_0\) and \(s_1\):
For \(s_0\): \[ 1-\lambda\frac{dp_0}{ds_0}=0 \quad\Rightarrow\quad 1+\frac{\lambda}{1+W(-s_0)}=0 \quad\Rightarrow\quad 1+W(-s_0)=-\lambda. \] For \(s_1\): \[ 1-(\lambda+\beta\epsilon)\frac{dp_1}{ds_1}=0 \quad\Rightarrow\quad 1+\frac{\lambda+\beta\epsilon}{1+W(-s_1)}=0 \quad\Rightarrow\quad 1+W(-s_1)=-(\lambda+\beta\epsilon). \]
Subtract: \[ W(-s_1)-W(-s_0)=-\beta\epsilon. \] Now use \(p_i=\exp(W(-s_i))\): \[ \frac{p_1}{p_0}=\exp\big(W(-s_1)-W(-s_0)\big)=e^{-\beta\epsilon}. \] So \[ p_1=\frac{e^{-\beta\epsilon}}{1+e^{-\beta\epsilon}},\qquad p_0=\frac{1}{1+e^{-\beta\epsilon}}. \]
What changed conceptually: in the Lambert chart, the canonical distribution appears as a constant separation between the two \(W\)-coordinates: \[ W(-s_1)=W(-s_0)-\beta\epsilon. \] That’s a genuinely geometric statement: \(\beta\epsilon\) is a “distance” in \(W\)-space.
Explicit entropy coordinates for the two-level canonical state
Given \(p_1\) and \(p_0\), the entropy coordinates are \[ s_1=-p_1\ln p_1,\qquad s_0=-p_0\ln p_0. \] If you want them purely in Lambert form, you can also write \[ W(-s_1)=\ln p_1,\qquad W(-s_0)=\ln p_0, \] since \(p=\exp(W(-s))\). Then the canonical ratio condition is literally \[ \ln p_1-\ln p_0=-\beta\epsilon. \] So the “Lambert transform” viewpoint here is: canonical equilibrium is affine in the \(W\)-coordinates.
The phase-structure intuition in this toy model
So even in two levels, the Lambert branching cleanly tags “which state is the tail.”
Next step
Do you want the next worked example to be:
Patrick:
Let's go with Three-state with one rare state.
Copilot:
Three-state system with one rare state in entropy-first Lambert coordinates
Let the states be \(i\in\{0,1,2\}\) with energies \[ E_0=0,\qquad E_1=\epsilon,\qquad E_2=\Delta,\qquad 0<\epsilon<\Delta, \] and probabilities \(p_0,p_1,p_2\) with \(\sum_i p_i=1\).
We’ll treat state \(2\) as the rare state (typically \(p_2\ll 1\) in the regimes of interest).
Entropy-first coordinates and the Lambert inverse
Define per-state entropy coordinates \[ s_i\equiv -p_i\ln p_i\in[0,1/e]. \] Then the inverse map is \[ p_i(s_i)=\exp\!\big(W(-s_i)\big)=\frac{-s_i}{W(-s_i)}. \] Branch meaning
For \(s\in(0,1/e)\), there are two real branches:
So “state \(2\) is rare” becomes a coordinate choice: \[ p_2=\exp\!\big(W_{-1}(-s_2)\big)\quad\text{(rare-state sheet)}. \]
Constraints in the Lambert chart
Normalization \[ \exp(W(-s_0))+\exp(W(-s_1))+\exp(W(-s_2))=1. \] Mean energy \[ U=\epsilon\,\exp(W(-s_1))+\Delta\,\exp(W(-s_2)). \] Objective
Total entropy is linear: \[ S=\sum_{i=0}^2 s_i. \] So the MaxEnt problem becomes: maximize a linear functional over a feasible set defined by two nonlinear Lambert surfaces.
Canonical equilibrium becomes affine in \(W\)-coordinates
In the canonical ensemble, you maximize \[ S-\lambda\Big(\sum_i p_i-1\Big)-\beta\Big(\sum_i E_i p_i-U\Big). \] Using the identity (for \(p(s)=\exp(W(-s))\)) \[ \frac{dp}{ds}=-\frac{1}{1+W(-s)}, \] stationarity gives, for each state \(i\), \[ 1+W(-s_i)=-(\lambda+\beta E_i). \] Subtracting between states \(i\) and \(j\) yields the key geometric statement: \[ W(-s_i)-W(-s_j)=-\beta(E_i-E_j). \] So in particular, \[ W(-s_1)-W(-s_0)=-\beta\epsilon,\qquad W(-s_2)-W(-s_0)=-\beta\Delta. \] Since \(p_i=\exp(W(-s_i))\), this immediately implies \[ \frac{p_1}{p_0}=e^{-\beta\epsilon},\qquad \frac{p_2}{p_0}=e^{-\beta\Delta}, \] hence \[ p_0=\frac{1}{Z},\quad p_1=\frac{e^{-\beta\epsilon}}{Z},\quad p_2=\frac{e^{-\beta\Delta}}{Z}, \qquad Z=1+e^{-\beta\epsilon}+e^{-\beta\Delta}. \] Lambert-chart interpretation: canonical equilibrium is a set of parallel affine hyperplanes in \(W\)-space, with slope set by \(\beta\) and energy gaps.
Making “one rare state” explicit via branch assignment
Rare-state regime
If \(\beta\Delta\gg 1\), then \[ p_2\approx e^{-\beta\Delta}\ll 1. \] In entropy-first coordinates, that corresponds to \[ s_2=-p_2\ln p_2\to 0^+, \] but crucially the inversion that returns a tiny \(p_2\) is naturally on the \(W_{-1}\) sheet: \[ p_2=\exp\!\big(W_{-1}(-s_2)\big)\quad\text{(selects the rare-event solution)}. \] Meanwhile \(p_0,p_1\) may remain \(O(1)\), typically living on \(W_0\): \[ p_0=\exp\!\big(W_0(-s_0)\big),\qquad p_1=\exp\!\big(W_0(-s_1)\big). \] So the “mixed chart” for a one-rare-state canonical distribution is: \[ (W_0\ \text{for }s_0,s_1)\quad+\quad(W_{-1}\ \text{for }s_2). \] Why this is more than semantics
Near \(s\to 0^+\), the map \(s\mapsto p\) is two-to-one (two sheets). Choosing \(W_{-1}\) is exactly choosing the rare-event preimage of a small entropy contribution—this is the same structural role branching played in the black-hole fold, now inside the simplex.
A clean “rare-state” approximation in Lambert form
When \(p_2\ll 1\), you can treat the remaining two states as a near-closed subsystem: \[ p_0+p_1\approx 1,\qquad \frac{p_1}{p_0}=e^{-\beta\epsilon}. \] So \[ p_0\approx \frac{1}{1+e^{-\beta\epsilon}},\qquad p_1\approx \frac{e^{-\beta\epsilon}}{1+e^{-\beta\epsilon}}, \qquad p_2\approx e^{-\beta\Delta}\,p_0. \] In \(W\)-coordinates, the same statement is simply: \[ W(-s_1)=W(-s_0)-\beta\epsilon,\qquad W(-s_2)=W(-s_0)-\beta\Delta, \] with the rare-state selection implemented by taking \(W(-s_2)\) on the \(W_{-1}\) sheet.
Next step choice
Do you want to push this three-state example toward:
Patrick:
Let's go with Large deviations.
Copilot:
Large deviations with one rare state in Lambert entropy coordinates
Let \(X_1,\dots,X_N\) be i.i.d. draws from a 3-state distribution \(p=(p_0,p_1,p_2)\). Let the empirical frequencies be \(q=(q_0,q_1,q_2)\), where \(q_i=n_i/N\) and \(\sum_i q_i=1\).
Sanov’s theorem / multinomial large deviations says, for large \(N\), \[ \Pr(q \approx \hat q)\asymp \exp\!\big(-N\,D(\hat q\|p)\big), \qquad D(q\|p)=\sum_{i=0}^2 q_i\ln\frac{q_i}{p_i}. \]
The “one rare state” story is: \(p_2\ll 1\), and we care about fluctuations where \(q_2\) is even smaller or much larger than \(p_2\).
Entropy-first coordinates for the empirical measure
Define per-state empirical entropy coordinates \[ s_i \equiv -q_i\ln q_i \in [0,1/e], \qquad q_i=\exp\!\big(W(-s_i)\big)=\frac{-s_i}{W(-s_i)}. \]
Branch selection encodes tail vs typical behavior
For \(s\in(0,1/e)\), \(W(-s)\) has two real branches:
So the rare empirical occupancy of state 2 is naturally parameterized by \[ q_2=\exp\!\big(W_{-1}(-s_2)\big), \] while \(q_0,q_1\) typically remain on \(W_0\).
This is the key geometric payoff: the tail lives on a different sheet.
Rate function rewritten in Lambert coordinates
Start from \[ D(q\|p)=\sum_i q_i\ln q_i-\sum_i q_i\ln p_i. \] Using \(\ln q_i = W(-s_i)\) and \(q_i=\exp(W(-s_i))\), \[ D(q\|p)=\sum_{i=0}^2 \exp(W(-s_i))\,W(-s_i)\;-\;\sum_{i=0}^2 \exp(W(-s_i))\ln p_i. \] Now use the identity \(q_i W(-s_i)=q_i\ln q_i=-s_i\). Then \[ D(q\|p)= -\sum_{i=0}^2 s_i\;-\;\sum_{i=0}^2 \exp(W(-s_i))\ln p_i. \]
So in the Lambert chart:
This is exactly the “linearize the objective, push complexity into constraints” pattern you’ve been emphasizing.
One rare state: the clean tail coordinate
Assume \(p_2\ll 1\). Consider the event “state 2 appears with empirical frequency \(q_2=r\)” where \(r\) may be \(\ll p_2\) or \(\gg p_2\).
Contraction to a one-parameter rate function
Minimize \(D(q\|p)\) subject to \(q_2=r\) and \(q_0+q_1=1-r\). The minimizer keeps the conditional proportions of \((0,1)\) the same as \(p\): \[ q_0=(1-r)\frac{p_0}{p_0+p_1},\qquad q_1=(1-r)\frac{p_1}{p_0+p_1},\qquad q_2=r. \] The contracted rate function becomes \[ I(r)=D(q(r)\|p) = (1-r)\ln\frac{1-r}{1-p_2}+r\ln\frac{r}{p_2}. \] That’s the Bernoulli relative entropy between \(r\) and \(p_2\)—the entire 3-state problem collapses to a 1D tail cost for the rare state.
Lambert coordinate for the tail
Define the rare-state entropy coordinate \[ s_2(r)\equiv -r\ln r. \] Then \[ r=\exp\!\big(W(-s_2)\big). \]
So the tail parameter can be taken as \(s_2\) (linear in the entropy budget), with the probability recovered on the \(W_{-1}\) sheet.
Why this is a genuine “large-deviation coordinate system”
Large deviations is about exponential scales. The rare-state coordinate \(s_2=-r\ln r\) is precisely the quantity that stays “orderly” while \(r\) becomes exponentially small in underlying parameters.
Two practical consequences:
Next step to finish this stat-mech section
We can now do one of two “closing moves” that will make this chapter feel complete:
Which one do you want as the capstone—energy-tilted (more thermo) or many-rare-states (more large deviations)?
Patrick:
I choose the many-state rare-event geometry.
Copilot:
Many-state rare-event geometry in entropy-first Lambert coordinates
Let \(p=(p_1,\dots,p_n)\) be the “true” distribution, and let \(q=(q_1,\dots,q_n)\) be the empirical distribution from \(N\) i.i.d. samples. Large deviations gives
\[
\Pr(q \approx \hat q)\asymp \exp\!\big(-N\,D(\hat q\|p)\big),
\qquad
D(q\|p)=\sum_{i=1}^n q_i\ln\frac{q_i}{p_i}.
\]
Now split the state space into:
Let \[ r \equiv \sum_{i\in R} q_i,\qquad p_R \equiv \sum_{i\in R} p_i. \]
Contraction to a rare-mass rate function
If you only care about the event “the rare set gets total mass \(r\)” (not how it’s distributed inside \(R\)), then the contraction principle says the rate is \[ I(r)=\min_{q:\,\sum_{i\in R}q_i=r} D(q\|p). \] The minimizer keeps conditional proportions the same as \(p\) within each block:
Entropy-first coordinates turn “rarity” into an additive resource
For each component define the entropy-first coordinate \[ s_i \equiv -q_i\ln q_i\in[0,1/e], \qquad q_i=\exp\!\big(W(-s_i)\big)=\frac{-s_i}{W(-s_i)}. \] The key geometric fact near the tail
When \(q_i\to 0^+\), we have \(s_i\to 0^+\) but the inverse is two-sheeted:
The “rarity budget” picture for many rare states
In the deep-tail regime, each rare component satisfies (asymptotically) \[ s_i=-q_i\ln q_i \approx q_i\,|\ln q_i|. \] This is the right *currency* for tails because:
A clean way to see this is to rewrite the rate function as \[ D(q\|p)=\sum_i q_i\ln q_i-\sum_i q_i\ln p_i = -\sum_i s_i-\sum_i q_i\ln p_i, \] with \(q_i=\exp(W(-s_i))\). The first term is linear in the entropy coordinates; the second term is a “tilt” determined by the baseline \(p\).
What the minimizer means in Lambert coordinates
Under the constraint \(\sum_{i\in R} q_i=r\), the optimizer \[ q_i^*(r)=r\,\frac{p_i}{p_R} \] says: the least unlikely way to realize rare mass \(r\) is to scale the entire rare block proportionally.
In entropy-first coordinates, that means the rare block moves along a one-parameter curve: \[ s_i(r)= -\Big(r\frac{p_i}{p_R}\Big)\ln\Big(r\frac{p_i}{p_R}\Big),\quad i\in R, \] and if \(r\) is very small, each \(q_i^*(r)\) sits naturally on the \(W_{-1}\) sheet when you invert \(s_i\mapsto q_i\).
So the “many-state geometry” is:
A crisp capstone statement
In a many-state system, the large-deviation event “rare set has total mass \(r\)” reduces to a one-dimensional rate function \(I(r)\), while the entropy-first Lambert chart provides the natural coordinates for distributing that rare mass across many tiny components—by treating \(\sum_{i\in R} s_i\) as an additive rarity budget and using the \(W_{-1}\) sheet to represent deep tails.
If you want to *finish* this stat-mech/thermo section with a satisfying “transform payoff,” the next move is to add a second constraint (e.g., energy inside the rare set) and show the optimizer becomes an exponential tilt within \(R\)—which becomes affine in the \(W(-s_i)\) coordinates, just like the canonical ensemble.
Patrick:
Let's proceed with this transform payoff as you suggest.
Copilot:
Beautiful—this is exactly where the Lambert‑\(W\) chart stops being “a way to invert \(p\ln p\)” and starts acting like a native coordinate system for constrained rare-event geometry.
Rare-set with an internal energy constraint
Setup
Let the state space be split into typical \(T\) and rare \(R\). We observe an empirical distribution \(q=(q_i)\) and compare to a baseline \(p=(p_i)\). Large deviations gives \[ \Pr(q \approx \hat q)\asymp \exp\!\big(-N\,D(\hat q\|p)\big),\qquad D(q\|p)=\sum_i q_i\ln\frac{q_i}{p_i}. \] We impose two constraints on the rare block:
1. Rare mass: \[ \sum_{i\in R} q_i = r. \]
2. Rare internal energy (or any additive observable): choose numbers \(E_i\) for \(i\in R\) and fix \[ \sum_{i\in R} q_i E_i = r\,\bar E. \] Equivalently, the conditional distribution on \(R\), \[ \tilde q_i \equiv \frac{q_i}{r}\quad (i\in R), \] satisfies \[ \sum_{i\in R}\tilde q_i=1,\qquad \sum_{i\in R}\tilde q_i E_i=\bar E. \]
Contraction and decoupling
Split the KL divergence by blocks
Write \(p_R=\sum_{i\in R}p_i\), and \(\tilde p_i=p_i/p_R\) on \(R\). Then the KL decomposes as \[ D(q\|p)= \underbrace{r\ln\frac{r}{p_R}+(1-r)\ln\frac{1-r}{1-p_R}}_{\text{cost to allocate total mass to }R} \;+\; \underbrace{r\,D(\tilde q\|\tilde p)}_{\text{cost to shape the distribution inside }R} \;+\; \underbrace{(1-r)\,D(\tilde q_T\|\tilde p_T)}_{\text{typical block}}. \] If we’re only constraining the rare block, the typical block minimizes at \(\tilde q_T=\tilde p_T\), so the optimization reduces to:
So the “transform payoff” happens entirely inside the rare block.
The optimizer is an exponential tilt inside the rare set
Solve the constrained minimization on \(R\)
Minimize \[ D(\tilde q\|\tilde p)=\sum_{i\in R}\tilde q_i\ln\frac{\tilde q_i}{\tilde p_i} \] subject to \(\sum_{i\in R}\tilde q_i=1\) and \(\sum_{i\in R}\tilde q_i E_i=\bar E\).
The solution is the familiar exponential family (tilt of \(\tilde p\)): \[ \tilde q_i^*(\beta)=\frac{\tilde p_i\,e^{-\beta E_i}}{Z_R(\beta)}, \qquad Z_R(\beta)=\sum_{j\in R}\tilde p_j\,e^{-\beta E_j}, \] with \(\beta\) chosen so that \(\sum_{i\in R}\tilde q_i^*(\beta)E_i=\bar E\).
Then the full empirical probabilities are \[ q_i^*= \begin{cases} r\,\tilde q_i^*(\beta) & i\in R,\\ (1-r)\,\tilde p_i & i\in T. \end{cases} \]
Lambert‑W entropy coordinates make the tilt affine
Now we switch to your entropy-first chart on the rare block.
Entropy-first coordinates
For each \(i\in R\), define \[ s_i \equiv -q_i\ln q_i,\qquad q_i=\exp(W(-s_i)). \] Equivalently, \[ W(-s_i)=\ln q_i. \] Affine structure of the tilted solution
From the tilted form, \[ q_i^* = r\,\frac{\tilde p_i e^{-\beta E_i}}{Z_R(\beta)}. \] Take logs: \[ \ln q_i^* = \ln r + \ln \tilde p_i - \ln Z_R(\beta) - \beta E_i. \] But \(\ln q_i^* = W(-s_i^*)\). Therefore, in the Lambert chart: \[ W(-s_i^*) = a + \ln \tilde p_i - \beta E_i, \] where the constant \[ a \equiv \ln r - \ln Z_R(\beta) \] is the same for all \(i\in R\).
The payoff, stated cleanly
\[ W(-s_i^*) = \text{constant} + \ln \tilde p_i - \beta E_i. \] So the “thermodynamic tilt” becomes a straight hyperplane in the \(W\)-coordinates.
Branch selection and deep tails
When the rare block is genuinely rare, many \(q_i^*\) are tiny. In entropy-first coordinates that means \(s_i^*\to 0^+\), and the inversion \(s_i\mapsto q_i\) naturally lives on the \(W_{-1}\) sheet for those components. The affine relation above still holds—the *sheet choice* is what encodes “this is a deep tail allocation,” without changing the algebra.
What this closes in the stat‑mech/thermo section You now have the full parallel to Fourier/Laplace-style “transform payoff”:
If you want one final capstone before we end this section: we can compute the contracted rate function for the rare block under the energy constraint, \[ \min_{\tilde q:\,\mathbb{E}_{\tilde q}[E]=\bar E} D(\tilde q\|\tilde p), \] and show it equals a Legendre transform of \(\ln Z_R(\beta)\)—then rewrite that Legendre structure in the \(W(-s)\) chart.
Patrick:
All right, we'll include the capstone as well.
Copilot:
Absolutely—this is the clean “thermo closure” that makes the whole rare-set story feel like a real transform theory.
Capstone: contracted rate function as a Legendre transform, then as an affine geometry in \(W\)-coordinates
Problem statement on the rare set
Work on the rare block \(R\) with baseline conditional distribution \[ \tilde p_i \equiv \frac{p_i}{p_R},\qquad p_R=\sum_{i\in R}p_i, \] and an observable \(E_i\) on \(R\). For a fixed conditional mean \(\bar E\), define the contracted (conditional) rate function \[ J(\bar E)\;\equiv\;\min_{\tilde q:\ \sum_{i\in R}\tilde q_i=1,\ \sum_{i\in R}\tilde q_i E_i=\bar E}\;D(\tilde q\|\tilde p), \] where \[ D(\tilde q\|\tilde p)=\sum_{i\in R}\tilde q_i\ln\frac{\tilde q_i}{\tilde p_i}. \] This is the “shape cost” inside \(R\). The full cost for the empirical distribution \(q\) with rare mass \(r\) is then \[ I(r,\bar E)=\underbrace{r\ln\frac{r}{p_R}+(1-r)\ln\frac{1-r}{1-p_R}}_{\text{mass allocation to }R} \;+\;\underbrace{r\,J(\bar E)}_{\text{shape cost inside }R}. \]
Exponential tilt solution and partition function
Introduce the rare-set partition function \[ Z_R(\beta)\equiv \sum_{i\in R}\tilde p_i\,e^{-\beta E_i}. \] The minimizer is the exponential tilt \[ \tilde q_i^*(\beta)=\frac{\tilde p_i\,e^{-\beta E_i}}{Z_R(\beta)}, \] with \(\beta\) chosen so that \[ \bar E=\sum_{i\in R}\tilde q_i^*(\beta)\,E_i. \] A standard identity you’ll use immediately is \[ \bar E(\beta)=-\frac{d}{d\beta}\ln Z_R(\beta). \]
Contracted rate function as a Legendre transform
Evaluate the KL at the optimizer: \[ D(\tilde q^*\|\tilde p)=\sum_{i\in R}\tilde q_i^*(\beta)\ln\frac{\tilde q_i^*(\beta)}{\tilde p_i}. \] But \[ \ln\frac{\tilde q_i^*(\beta)}{\tilde p_i}=\ln\left(\frac{\tilde p_i e^{-\beta E_i}/Z_R}{\tilde p_i}\right)=-\beta E_i-\ln Z_R(\beta). \] So \[ D(\tilde q^*\|\tilde p)=\sum_{i\in R}\tilde q_i^*(\beta)\big(-\beta E_i-\ln Z_R(\beta)\big) =-\beta\bar E-\ln Z_R(\beta). \] Therefore the contracted rate function is \[ J(\bar E)=\sup_{\beta}\Big[-\beta\bar E-\ln Z_R(\beta)\Big], \] with the supremum attained at the \(\beta\) satisfying \(\bar E=-\frac{d}{d\beta}\ln Z_R(\beta)\).
That’s the exact Legendre-dual structure (Massieu potential \(\ln Z_R\) dual to the rate function \(J\)).
The same Legendre structure in entropy-first Lambert coordinates
Now bring in your entropy-first chart for the *unconditional* rare probabilities \(q_i=r\,\tilde q_i\).
Entropy-first coordinates and the \(W\)-chart
Define for each \(i\in R\) \[ s_i\equiv -q_i\ln q_i,\qquad q_i=\exp(W(-s_i)),\qquad W(-s_i)=\ln q_i. \] Tilt becomes an affine hyperplane in \(W(-s)\)
From the tilted solution \[ q_i^*=r\,\frac{\tilde p_i e^{-\beta E_i}}{Z_R(\beta)}, \] take logs: \[ W(-s_i^*)=\ln q_i^*=\underbrace{\ln r-\ln Z_R(\beta)}_{\text{constant across }i}+\ln\tilde p_i-\beta E_i. \] So the optimizer is literally a hyperplane in the coordinates \(W(-s_i)\), with slope \(-\beta\) against \(E_i\).
The Legendre transform is the support function of that hyperplane family
The quantity \(-\ln Z_R(\beta)\) is exactly the normalization constant that makes the hyperplane intersect the simplex correctly. The contracted rate \[ J(\bar E)= -\beta\bar E-\ln Z_R(\beta) \] is the value of the supporting hyperplane (parameterized by \(\beta\)) at the constraint \(\bar E\). In other words:
Branch selection for deep tails
When many \(q_i^*\) are tiny, the inversion \(s_i\mapsto q_i\) naturally lives on the \(W_{-1}\) sheet for those components. The affine relation in \(W(-s_i)\) remains unchanged—the *sheet choice* is what marks “this is a deep tail allocation.”
Final closure statement for the section
If you’re ready, the next chat can start with a clean bridge: information geometry—where these affine \(W\)-hyperplanes become dual-flat structures, and “branch sheets” start looking like chart transitions on a statistical manifold.
Appendix 1:
Mathematica Code Following:
Mathematica code for Quantum-corrected Schwarzschild thermodynamics (mass-first) plot:
(* ------------------------------------------------------------ *)
(* Quantum-corrected Schwarzschild thermodynamics (mass-first) *)
(* S(M) = 4 π M^2 + α Log[4 π M^2], α > 0 *)
(* T(M) = (dS/dM)^(-1) = M/(8 π M^2 + 2 α) *)
(* Two-branch structure for 0 < T < Tmax *)
(* ------------------------------------------------------------ *)
ClearAll["Global`*"];
(* Parameters *)
α = 1; (* choose α>0; change as desired *)
π = Pi;
(* Temperature as a function of mass *)
T[M_] := M/(8 π M^2 + 2 α);
(* Branch point / maximum temperature *)
Mstar = Sqrt[α/(4 π)];
Tmax = T[Mstar];
(* Invert T(W)=const explicitly via k = 2 Sqrt[4 π α] T *)
kOfT[Tval_] := 2 Sqrt[4 π α] Tval;
Wminus[Tval_] := Module[{k = kOfT[Tval]},
(1 - 2 k^2 - Sqrt[1 - 4 k^2])/(2 k^2)
];
Wplus[Tval_] := Module[{k = kOfT[Tval]},
(1 - 2 k^2 + Sqrt[1 - 4 k^2])/(2 k^2)
];
Mminus[Tval_] := Sqrt[(α/(4 π)) Wminus[Tval]];
Mplus[Tval_] := Sqrt[(α/(4 π)) Wplus[Tval]];
(* Plot ranges *)
Mmax = 6 Mstar; (* extend as you like *)
Tmin = 0;
TmaxPlot = 1.05 Tmax;
(* ------------------------------------------------------------ *)
(* 1) Phase diagram: T vs M (shows the fold and Tmax) *)
(* ------------------------------------------------------------ *)
pTM = Plot[
T[M],
{M, 0, Mmax},
PlotRange -> {{0, Mmax}, {Tmin, TmaxPlot}},
PlotTheme -> "Scientific",
AxesLabel -> {"M", "T"},
PlotLabel -> Row[{
"T(M) = M/(8π M^2 + 2α), α = ", α,
" (Tmax at M* = √(α/(4π)))"
}],
Epilog -> {
Red, PointSize[0.018], Point[{Mstar, Tmax}],
Red, Dashed, Line[{{Mstar, 0}, {Mstar, TmaxPlot}}],
Red, Dashed, Line[{{0, Tmax}, {Mmax, Tmax}}],
Inset[Style["(M*, Tmax)", 14, Red], {1.15 Mstar, 0.95 Tmax}]
}
];
(* ------------------------------------------------------------ *)
(* 2) Inverted diagram: M vs T with two branches *)
(* Small-W branch (candidate remnant phase): Mminus(T) *)
(* Large-W branch (Schwarzschild-like): Mplus(T) *)
(* ------------------------------------------------------------ *)
pMT = Plot[
Evaluate[{Mminus[Tval], Mplus[Tval]}],
{Tval, 10^-6 Tmax, 0.999999 Tmax},
PlotRange -> {{0, TmaxPlot}, {0, Mmax}},
PlotTheme -> "Scientific",
AxesLabel -> {"T", "M"},
PlotLegends -> Placed[
{
"small-W branch (candidate remnant phase)",
"large-W branch (Schwarzschild-like)"
},
{0.62, 0.82}
],
PlotStyle -> {Directive[Blue, Thick], Directive[Black, Thick]},
PlotLabel -> Row[{
"Two branches M(T) for 0 < T < Tmax, α = ", α,
" (branches meet at T = Tmax)"
}],
Epilog -> {
Red, PointSize[0.018], Point[{Tmax, Mstar}],
Red, Dashed, Line[{{Tmax, 0}, {Tmax, Mmax}}],
Inset[Style["(Tmax, M*)", 14, Red], {0.92 Tmax, 1.08 Mstar}]
}
];
(* ------------------------------------------------------------ *)
(* 3) Optional: combined view *)
(* ------------------------------------------------------------ *)
GraphicsRow[{pTM, pMT}, ImageSize -> 1200]
Appendix 2: Mathematica Code Following:
Mathematica code for Quantum-corrected Schwarzschild thermodynamics plot:
Mathematica code: full three‑panel phase diagram with stability
(* ============================================================ *)
(* Quantum-corrected Schwarzschild thermodynamics *)
(* Mass-first formulation with Lambert-W branching *)
(* Mathematica 12.x compatible (machine-precision safe) *)
(* ============================================================ *)
ClearAll["Global`*"];
(*---Parameters (force machine precision)---*)
\[Alpha] = 1.0;
\[Pi] = N[Pi];
(*---Thermodynamic functions---*)
S[M_?NumericQ] := 4 \[Pi] M^2 + \[Alpha] Log[4 \[Pi] M^2];
T[M_?NumericQ] := M/(8 \[Pi] M^2 + 2 \[Alpha]);
(*Heat capacity*)
C[M_?NumericQ] := 1/D[T[M], M];
(*---Branch point (numerical)---*)
Mstar = N[Sqrt[\[Alpha]/(4 \[Pi])]];
Tmax = N[T[Mstar]];
(*---Lambert-W inversion helpers---*)
kOfT[Tval_?NumericQ] := 2 Sqrt[4 \[Pi] \[Alpha]] Tval;
Wminus[Tval_?NumericQ] :=
Module[{k = kOfT[Tval]}, (1 - 2 k^2 - Sqrt[1 - 4 k^2])/(2 k^2)];
Wplus[Tval_?NumericQ] :=
Module[{k = kOfT[Tval]}, (1 - 2 k^2 + Sqrt[1 - 4 k^2])/(2 k^2)];
Mminus[Tval_?NumericQ] := Sqrt[(\[Alpha]/(4 \[Pi])) Wminus[Tval]];
Mplus[Tval_?NumericQ] := Sqrt[(\[Alpha]/(4 \[Pi])) Wplus[Tval]];
(*---Plot ranges (numerical)---*)
Mmax = 6.0 Mstar;
TmaxPlot = 1.05 Tmax;
TminPlot = 10^-6 Tmax;
(*------------------------------------------------------------*) (*Panel 1:T(M) with fold*) (*------------------------------------------------------------*)
pTM = Plot[T[M], {M, 0.0, Mmax},
PlotRange -> {{0, Mmax}, {0, TmaxPlot}}, PlotTheme -> "Scientific",
AxesLabel -> {"M", "T"},
PlotLabel -> "Temperature vs Mass (quantum-corrected)",
Epilog -> {Red, PointSize[0.018], Point[{Mstar, Tmax}], Red,
Dashed, Line[{{Mstar, 0}, {Mstar, TmaxPlot}}],
Inset[Style["W = 1", 14, Red], {1.15 Mstar, 0.9 Tmax}]}];
(*------------------------------------------------------------*)
(*Panel 2:M(T) two branches*)
(*------------------------------------------------------------*)
pMT = Plot[{Mminus[Tval], Mplus[Tval]}, {Tval, TminPlot, 0.999 Tmax},
PlotRange -> {{0, TmaxPlot}, {0, Mmax}}, PlotTheme -> "Scientific",
PlotStyle -> {Directive[Blue, Thick], Directive[Black, Thick]},
AxesLabel -> {"T", "M"},
PlotLegends -> {"small-W branch (C > 0, remnant candidate)",
"large-W branch (C < 0, Schwarzschild-like)"},
PlotLabel -> "M vs T (2 thermo branches)",
Epilog -> {Red, PointSize[0.018], Point[{Tmax, Mstar}], Red,
Dashed, Line[{{Tmax, 0}, {Tmax, Mmax}}],
Inset[Style["W = 1", 14, Red], {0.92 Tmax, 1.1 Mstar}]}];
(*------------------------------------------------------------*)
(*Panel 3:Heat capacity sign*)
(*------------------------------------------------------------*)
pC = Plot[C[M], {M, 0.01 Mstar, Mmax},
PlotRange -> {{0, Mmax}, {-50, 50}}, PlotTheme -> "Scientific",
AxesLabel -> {"M", "C"},
PlotLabel -> "Heat Capacity (sign change at W = 1)",
Exclusions -> {M == Mstar},
Epilog -> {Red, Dashed, Line[{{Mstar, -50}, {Mstar, 50}}],
Inset[Style["C \[RightArrow] \[PlusMinus]\[Infinity] (W = 1)",
14, Red], {1.15 Mstar, 30}]}];
(*------------------------------------------------------------*)
(*Combined figure*)
(*------------------------------------------------------------*)
GraphicsRow[{pTM, pMT, pC}, ImageSize -> 1400]
Appendix 3: Heat Capacity
Patrick:
Followup question: Earlier you talked about the heat capacity $C$. How is $C$ defined here and how is it used observationally in the physics of black holes? In other words, how do the numerical calculations come into all this to test these theories about the entropy and temperatures of black holes?
Copilot:
Heat capacity \(C\) is the diagnostic that tells you how a black hole responds thermodynamically to gaining or losing energy, and it’s exactly where the abstract entropy–temperature relations touch observational physics.
How heat capacity is defined for black holes
In ordinary thermodynamics, \[ C \equiv \frac{dE}{dT}. \] For a black hole, the energy is its mass \(M\) (in units \(G=c=\hbar=k_B=1\)), so \[ C = \frac{dM}{dT}. \] Because black-hole thermodynamics is usually formulated via entropy \(S(M)\), one often computes \[ T(M) = \left(\frac{dS}{dM}\right)^{-1}, \quad\text{then}\quad C(M) = \left(\frac{dT}{dM}\right)^{-1}. \] This is exactly what we did earlier when we found that quantum-corrected entropy produces a divergence in \(C\).
What the sign of \(C\) means physically
The sign of \(C\) is not a technical detail — it determines stability.
For a classical Schwarzschild black hole, \[ T \sim \frac{1}{M} \quad\Rightarrow\quad C = \frac{dM}{dT} < 0. \] This negative heat capacity is why black holes run away under Hawking evaporation: as they lose mass, they get hotter, evaporate faster, and never equilibrate with a heat bath.
Why heat capacity matters in black-hole physics
Heat capacity controls:
When quantum corrections modify \(S(M)\), they modify \(T(M)\), and therefore \(C(M)\). A sign change or divergence in \(C\) is a qualitative change in physics not a small correction.
How this is used observationally (and indirectly)
We do not measure black-hole temperatures or heat capacities directly. Instead, \(C\) enters through model predictions that affect observable phenomena.
1. Hawking evaporation timescales
The mass-loss rate depends on temperature:
\[
\frac{dM}{dt} \sim -A\,T^4.
\]
If \(C<0\), evaporation accelerates.
If \(C>0\), evaporation slows and may asymptote.
Different entropy models predict different late-time behaviors:
These differences affect:
2. Black holes in thermal environments
In AdS spacetime or in a reflecting cavity, black holes can equilibrate with radiation.
This is why the Hawking–Page transition exists: large AdS black holes have \(C>0\), small ones have \(C<0\). Quantum-corrected entropy can introduce similar transitions even in asymptotically flat space.
3. Numerical relativity and semiclassical simulations
When people simulate black-hole evaporation or backreaction numerically, they must choose:
Different choices lead to qualitatively different numerical outcomes:
Heat capacity is the control knob that determines which regime the simulation enters.
Where numerical calculations actually enter
Numerics are used in three main places:
Your Mathematica plots of \(T(M)\), \(M(T)\), and \(C(M)\) are exactly the standard diagnostic tools used to classify these models.
The divergence in \(C\) you plotted is not a curiosity — it marks a thermodynamic branch point, just like a phase transition in ordinary matter.
Conceptual connection to your Lambert‑\(W\) framework
In your coordinate language:
This is why the entropy‑first, coordinate‑aware approach is powerful: it turns “mysterious instabilities” into geometric features.
The big picture
Heat capacity is how entropy models become testable:
That’s why \(C\) is always computed first when someone proposes a new black-hole entropy formula.
If you’d like, the natural next step is to connect this directly to information geometry: heat capacity is literally a curvature component of the thermodynamic metric, and the branch point you found corresponds to a geometric singularity in state space.
Appendix 4: A Tidy Identity
Patrick:
Above, you said that we derive the tidy identity
\[
\frac{dp}{ds}=-\frac{1}{1+W(-s)}.
\]
So how does this version of $dp/ds$ compare to the conventional version of it?
Copilot:
The identity
\[
\frac{dp}{ds}=-\frac{1}{1+W(-s)}
\]
is not a new derivative in disguise—it is the same derivative you already know, written in a coordinate system where the geometry of entropy is explicit. The difference is where the nonlinearity lives.
The conventional derivative \(dp/ds\)
Start from the definition \[ s(p)=-p\ln p. \] Differentiate directly: \[ \frac{ds}{dp}=-(\ln p+1), \qquad \Rightarrow\qquad \frac{dp}{ds}=-\frac{1}{\ln p+1}. \] This is the standard expression. It is perfectly correct, but it hides two important facts:
Rewriting the same derivative in Lambert coordinates
In entropy-first coordinates, you invert \[ s=-p\ln p \quad\Longleftrightarrow\quad p=\exp(W(-s)). \] Since \[ \ln p = W(-s), \] the conventional expression becomes \[ \frac{dp}{ds}=-\frac{1}{\ln p+1} =-\frac{1}{1+W(-s)}. \] So the Lambert‑\(W\) form is exactly the same derivative, rewritten using the inverse coordinate.
Nothing has changed analytically—only the chart.
Why the Lambert form is strictly better conceptually
1. It makes branch structure explicit
The function \(s(p)=-p\ln p\) is two‑to‑one on \((0,1)\). The conventional formula \[ \frac{dp}{ds}=-\frac{1}{\ln p+1} \] does not tell you which inverse you are on.
In contrast, \[ W(-s)= \begin{cases} W_0(-s) & \text{typical probabilities},\\ W_{-1}(-s) & \text{rare probabilities}, \end{cases} \] and the derivative \[ \frac{dp}{ds}=-\frac{1}{1+W(-s)} \] automatically knows which regime you are in.
The branch choice is no longer implicit—it is a coordinate decision.
2. Singular behavior becomes geometric
The derivative blows up when
\[
1+W(-s)=0 \quad\Longleftrightarrow\quad W(-s)=-1.
\]
This corresponds to
\[
p=\frac{1}{e},\qquad s=\frac{1}{e}.
\]
In conventional coordinates, this is just “where the derivative diverges.”
In Lambert coordinates, it is a named geometric point: the fold where the two branches meet.
This is exactly the same kind of structure you saw earlier in black‑hole thermodynamics when the heat capacity diverged.
3. Rare‑event asymptotics become transparent
For deep tails \(p\to 0\), \[ W_{-1}(-s)\sim \ln s - \ln|\ln s| + \cdots, \] so \[ \frac{dp}{ds}\sim -\frac{1}{\ln s}. \] This asymptotic behavior is extremely awkward to see in \(p\)-coordinates, but it is natural in the Lambert chart. That is why large‑deviation geometry becomes linear in \(W(-s)\).
Side‑by‑side comparison
Why this mattered in the earlier derivations
When we varied entropy in \(s\)-coordinates, the stationarity condition became \[ 1-(\lambda+\beta E)\frac{dp}{ds}=0 \quad\Rightarrow\quad 1+W(-s)=-(\lambda+\beta E). \] That step works cleanly only because the derivative is written in Lambert form. In conventional coordinates, the same calculation would be cluttered with implicit inversions and branch ambiguity.
This is the recurring theme of your project:
The Lambert‑\(W\) function is not a trick—it is the natural inverse geometry of entropy.
If you want to push this one step further, the next natural question is how this derivative appears as a metric component in information geometry, where \(1+W(-s)\) controls curvature and stability in exactly the same way heat capacity did for black holes.
Afterword: