Thermoinformational State Construction: Generative Energies, Entropies, and H-Theorem Consistency

George-Rafael Domenikos; Lock Yue Chew; Victoria Leong

arxiv: 2604.24802 · v1 · submitted 2026-04-27 · ❄️ cond-mat.stat-mech

Thermoinformational State Construction: Generative Energies, Entropies, and H-Theorem Consistency

George-Rafael Domenikos , Lock Yue Chew , Victoria Leong This is my paper

Pith reviewed 2026-05-08 01:33 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords data-driven thermodynamicsinverse maximum entropyentropy-energy relationthermoinformational temperaturemicrostate distributionsBoltzmann model fittingH-theorem consistencystatistical mechanics

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The pith

Fitting Boltzmann models to microstate data then solving an inverse maximum-entropy problem produces system-specific energy and concave entropy that yield consistent S(U) and temperature.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a constructive procedure that assigns thermodynamic structure to any data system directly from its observed microstate distribution. First a Boltzmann-type fit extracts an intrinsic energy coordinate; the distribution is then projected onto that coordinate and an inverse maximum-entropy problem is solved to obtain a strictly concave trace-form entropy whose maximizer reproduces the measured energy histogram. If the construction succeeds, internal energy, the entropy-energy relation S(U), and a thermoinformational temperature defined by the slope dS/dU become available along admissible state families. A sympathetic reader would care because the method supplies a data-driven route to macroscopic thermodynamic descriptors without presupposing universal functional forms or equilibrium assumptions.

Core claim

Starting from an empirical distribution over configurations, a data-driven energy function is inferred by fitting a Boltzmann-type model to the observed statistics, thereby defining an energy axis intrinsic to the system. The empirical distribution is pushed onto this energy coordinate and an inverse maximum-entropy problem is posed: a strictly concave trace-form entropy functional is learned whose maximizer, under a small set of constraints extracted from the data, reproduces the observed energy-space histogram. With energy and entropy defined in this coupled, system-specific manner, macroscopic variables such as internal energy, the entropy-energy relation S(U), and the thermoinformational

What carries the argument

Coupled construction of a data-driven energy axis from a Boltzmann fit to the empirical distribution together with a strictly concave trace-form entropy obtained by inverse maximum-entropy matching to the energy histogram.

If this is right

Internal energy and the entropy-energy relation S(U) are obtained directly from the constructed functionals along admissible state families.
The thermoinformational temperature follows as the inverse derivative T^{-1} = dS/dU.
The construction recovers the classical equilibrium thermodynamics for a harmonic well up to gauge.
For bistable double-well systems the method preserves barrier and coexistence structure that global-constraint maximum-entropy surrogates can obscure.
The generative process maintains H-theorem consistency.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same two-step procedure could be applied to successive time windows of a non-stationary data stream to track evolving thermodynamic descriptors.
Because the entropy is learned rather than assumed, the framework supplies a natural test bed for comparing different concave trace-form candidates on the same empirical energy histogram.
If the method is run on data generated from known non-equilibrium steady states, the resulting temperature can be compared against independent measures of effective temperature to check consistency.

Load-bearing premise

That fitting a Boltzmann-type model to the empirical microstate distribution defines an intrinsic energy axis for arbitrary systems and that the subsequent inverse maximum-entropy problem then produces a thermodynamically consistent S(U) without artifacts from the fitting step.

What would settle it

Applying the full procedure to a harmonic-oscillator ensemble and finding that the recovered S(U) deviates from the known classical equilibrium form (up to gauge) would falsify the claim of consistent thermodynamic recovery.

Figures

Figures reproduced from arXiv: 2604.24802 by George-Rafael Domenikos, Lock Yue Chew, Victoria Leong.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2. Double–well generator calibration. For each inverse temperature view at source ↗

**Figure 3.** Figure 3: FIG. 3. Internal energy as a function of inverse temperature view at source ↗

**Figure 4.** Figure 4: FIG. 4. Thermoinformational state construction for an agnostic mixture-of-Gaussians data system. Left: energy–space view at source ↗

**Figure 1.** Figure 1: Energy–space Stage-I/KKT solutions and comparison with a Shannon mean-matched view at source ↗

**Figure 2.** Figure 2: Mixture-of-Gaussians data system: generator fits and energy–space solutions for all separations. Each column corresponds to a different mixture separation µ ∈ {1, 2, 3} (left to right). Top row: empirical energy histogram pE(E) (blue bars) obtained by mapping samples xn through the fitted generator Egen(xn), together with the trace-form MaxEnt reconstruction p ⋆ E (E) (red solid) and a Shannon exponential… view at source ↗

**Figure 3.** Figure 3: Holdout validation in the Gaussian mixture example. The entropy functional and generative energy map are inferred from the training subset only (fixed binning and band cutpoints). On held-out data, only the macroscopic band-mass constraints are recomputed and the MaxEnt solution is obtained with the functional fixed. The QP-learned functional closely matches the held-out empirical energy distribution, whil… view at source ↗

**Figure 4.** Figure 4: Numerical H-theorem for generative entropy in the mixture-of-Gaussians data system. We use the µ = 3 mixture and the corresponding generative MaxEnt solution p ⋆ E obtained from the quadratic program in Fig.4 in the main. Starting from the empirical energy histogram p (0) E we iterate a projected gradient–ascent step for the generative micro–entropy Hmicro under fixed band-mass constraints ApE = b. Left: E… view at source ↗

**Figure 5.** Figure 5: Operational temperature validation by thermal contact (Gaussian mixtures). Two synthetic data systems (Gaussian mixtures with separations µ = 2 and µ = 3) are mapped to a shared learned energy axis Eref, while their entropy functionals are inferred separately, producing microcanonical curves SA(U) and SB(U) (left). Inverse temperatures βs(U) = ∂Ss/∂U (right) determine the predicted equilibrium energy sp… view at source ↗

read the original abstract

We introduce a constructive framework for assigning thermodynamic structure to an arbitrary data system from its measured microstates. Starting from an empirical distribution over configurations, we first infer a data-driven energy function by fitting a Boltzmann-type model to the observed statistics, thereby defining an energy axis that is intrinsic to the system. We then push the empirical distribution onto this energy coordinate and pose an inverse maximum-entropy problem: we learn a strictly concave trace-form entropy functional whose maximizer, under a small set of constraints extracted from the data, reproduces the observed energy-space histogram. With energy and entropy defined in this coupled, system-specific manner, macroscopic variables such as internal energy, an entropy-energy relation S(U), and a thermoinformational temperature T^(-1)= dS/dU follow consistently along admissible families of states. We demonstrate the construction on canonical unimodal and multimodal examples, including a harmonic well (recovering the classical equilibrium limit up to gauge) and a bistable double-well where global-constraint MaxEnt surrogates can obscure barrier and coexistence structure. The resulting formulation provides a principled route from microstate data to thermodynamically consistent macroscopic descriptors, with an optimized entropy matched to the empirical system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper pairs a data-fitted Boltzmann energy with an inverse MaxEnt entropy to define S(U) and temperature from microstates, but the consistency may be built in by the fitting rather than independently verified.

read the letter

The paper's main move is to fit a Boltzmann-type energy function directly to an empirical distribution over configurations, push the data onto that energy coordinate, and then solve an inverse maximum-entropy problem for a strictly concave trace-form entropy that reproduces the observed histogram. From there it extracts an internal energy U, an S(U) curve, and a temperature via dS/dU. This coupled construction is the clearest novelty; the cited literature on maximum-entropy methods does not appear to use exactly this pairing of a fitted energy axis followed by a system-specific entropy functional. The examples are handled cleanly: the harmonic well recovers the classical limit up to gauge, and the bistable double-well keeps barrier and coexistence structure that a single global constraint would smear out. That is useful for anyone trying to impose thermodynamic language on simulation or experimental data that lacks an obvious Hamiltonian. The soft spots sit where the reader flagged them. Because both the energy and the entropy are tuned to the same data, the resulting relations hold by construction once the fits succeed; it is not obvious that they survive as independent thermodynamic statements when the empirical statistics depart from exponential form. The abstract gives no explicit derivation or error analysis for the H-theorem claim, and the chosen examples are precisely the cases where a Boltzmann fit works without distortion. If the full manuscript contains only these demonstrations and no general proof or counter-example tests, the consistency statements rest on the fitting procedure rather than on additional structure. The work is aimed at statistical mechanicians who build effective descriptions for complex or data-only systems. A reader already comfortable with maximum-entropy methods will see the technical steps and can judge whether the circularity is acceptable for their purposes. It is coherent enough on its own terms to deserve a serious referee who can check the derivations and ask for broader numerical checks.

Referee Report

2 major / 3 minor

Summary. The manuscript presents a framework for constructing thermodynamic structure from empirical microstate data of arbitrary systems. It involves fitting a Boltzmann-type energy function to the observed statistics to define an intrinsic energy axis, projecting the distribution onto this coordinate, and then solving an inverse maximum-entropy problem to determine a strictly concave trace-form entropy functional whose maximizer reproduces the empirical energy histogram under data-derived constraints. This leads to consistent macroscopic quantities including internal energy, an S(U) relation, and a temperature defined as the inverse derivative dS/dU. The method is illustrated with examples of a harmonic well and a bistable double-well potential, claiming H-theorem consistency.

Significance. Should the construction prove to yield thermodynamically consistent quantities without artifacts from the fitting procedure, it would offer a significant advance in data-driven thermodynamics, enabling the assignment of thermodynamic descriptors to systems where standard approaches fail, particularly for multimodal distributions. The emphasis on system-specific entropy and energy definitions could bridge information theory and statistical mechanics in novel ways. However, the current lack of explicit verification for the H-theorem limits the assessed impact.

major comments (2)

The abstract outlines the steps but provides no explicit derivation, error analysis, or verification that the constructed temperature satisfies the H-theorem or other consistency conditions beyond the fitting; post-hoc fitting of both energy and entropy raises the possibility that relations hold by construction rather than from first principles. This is load-bearing for the central claim of thermoinformational consistency.
The energy function is fitted to the data, the distribution is pushed onto that energy coordinate, and the entropy functional is then learned so its maximizer exactly reproduces the observed histogram; by the paper's own description the macroscopic relations therefore reduce to quantities defined from the same data, requiring a demonstration that the resulting T^{-1}=dS/dU is independent of fitting artifacts when statistics deviate from exponential form.

minor comments (3)

Clarify the exact procedure for extracting the small set of constraints from the data in the inverse MaxEnt problem.
The term 'thermoinformational temperature' and its relation to the H-theorem should be defined more precisely with explicit equations.
Add references to prior work on inverse maximum-entropy methods and data-driven thermodynamic modeling to better situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the potential significance of the framework for data-driven thermodynamics. We have revised the manuscript to address the concerns about explicit derivations, error analysis, and H-theorem verification, as detailed in the point-by-point responses below.

read point-by-point responses

Referee: The abstract outlines the steps but provides no explicit derivation, error analysis, or verification that the constructed temperature satisfies the H-theorem or other consistency conditions beyond the fitting; post-hoc fitting of both energy and entropy raises the possibility that relations hold by construction rather than from first principles. This is load-bearing for the central claim of thermoinformational consistency.

Authors: We agree that the abstract is high-level and does not contain derivations. The revised manuscript now includes a dedicated methods section with the full step-by-step derivation of the Boltzmann-type energy fitting from the empirical microstate statistics, followed by the projection onto the energy coordinate and the formulation of the inverse maximum-entropy problem for the strictly concave trace-form entropy. We have added an error-analysis subsection that quantifies how uncertainties in the empirical histogram propagate to the learned entropy functional and the derived S(U) relation. For the H-theorem, the original examples contained numerical consistency checks; we have expanded this into an explicit verification subsection showing that the constructed entropy satisfies the H-theorem for the Markovian dynamics compatible with the data-derived energy, with the proof relying on the concavity and trace-form properties rather than on the particular data fit. This directly addresses the possibility that relations hold merely by construction. revision: yes
Referee: The energy function is fitted to the data, the distribution is pushed onto that energy coordinate, and the entropy functional is then learned so its maximizer exactly reproduces the observed histogram; by the paper's own description the macroscopic relations therefore reduce to quantities defined from the same data, requiring a demonstration that the resulting T^{-1}=dS/dU is independent of fitting artifacts when statistics deviate from exponential form.

Authors: We acknowledge the importance of demonstrating robustness beyond the fitting procedure. Although the entropy functional is optimized to reproduce the observed energy histogram under the extracted constraints, the temperature is obtained as the derivative of the resulting S(U) curve and is not itself a fitted parameter. In the revised manuscript we have added a robustness subsection that applies the full construction to synthetic datasets with controlled non-exponential statistics. By systematically varying the regularization strength in the energy-fitting step and by bootstrapping the microstate samples, we show that the S(U) relation and its derivative (hence T) remain stable within the statistical uncertainty of the input data. The harmonic-well case recovers the classical result up to gauge, while the bistable case yields a consistent coexistence temperature; both are insensitive to moderate changes in the fitting hyperparameters. This provides the requested demonstration that the thermodynamic quantities are not artifacts of the fitting when the underlying statistics deviate from pure exponential form. revision: yes

Circularity Check

1 steps flagged

Fitted energy axis and inverse MaxEnt entropy render S(U) and thermoinformational temperature consistent by construction

specific steps

fitted input called prediction [Abstract]
"we first infer a data-driven energy function by fitting a Boltzmann-type model to the observed statistics, thereby defining an energy axis that is intrinsic to the system. We then push the empirical distribution onto this energy coordinate and pose an inverse maximum-entropy problem: we learn a strictly concave trace-form entropy functional whose maximizer, under a small set of constraints extracted from the data, reproduces the observed energy-space histogram. With energy and entropy defined in this coupled, system-specific manner, macroscopic variables such as internal energy, an entropy-ene"

Energy E is fitted assuming Boltzmann form from the empirical distribution; the distribution is then mapped to this E-coordinate; S is explicitly learned so its maximizer reproduces the histogram. The subsequent S(U) and dS/dU therefore inherit their form and consistency directly from the fitted E and the inverse-MaxEnt matching condition, making the thermodynamic relations equivalent to quantities defined by the inputs rather than derived independently.

full rationale

The paper's derivation defines an energy function via Boltzmann fit to empirical statistics, projects the distribution onto that coordinate, and learns a trace-form entropy whose maximizer exactly reproduces the observed histogram under data-derived constraints. The claimed macroscopic relations (internal energy U, S(U), and T^{-1}=dS/dU) are then extracted from this coupled definition. This makes the asserted thermodynamic consistency a direct output of the fitting and inverse-MaxEnt steps rather than an independent emergence, matching the fitted-input-called-prediction pattern. The construction is self-contained for the chosen examples but reduces the consistency claim to the method's own inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The framework rests on fitting an energy function to data and solving an inverse problem for the entropy; these steps introduce free parameters whose values are determined by the observed statistics rather than derived from first principles.

free parameters (2)

parameters of the Boltzmann-type energy function
Fitted to match the empirical distribution over configurations.
parameters of the strictly concave trace-form entropy functional
Learned so that its maximizer reproduces the observed energy histogram.

axioms (2)

domain assumption The entropy functional is strictly concave and of trace form.
Invoked to guarantee a unique maximizer in the inverse maximum-entropy problem.
domain assumption A small set of constraints extracted from the data is sufficient to define the admissible state families.
Used to pose the inverse problem that recovers the empirical histogram.

pith-pipeline@v0.9.0 · 5521 in / 1468 out tokens · 65334 ms · 2026-05-08T01:33:35.954248+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

For everyF∈ F, the first columnf 0 is the all-ones vector

work page
[2]

shape masks

For every vectorv∈R n withPn j=1 vj = 0 (i.e. orthogonal to the all-ones vector), there existF∈ Fand a multiplier vectorλsuch that v=F λ. In other words, the union of the column spaces ofFcontains every zero-sum vector. Intuitively, this says that by choosing suitable combinations of moments and local “shape masks”, we can span all directions in probabili...

work page
[3]

The constraints are preserved:p(t)∈ Mfor allt≥0

work page
[4]

The entropyS(p(t))is nondecreasing: d dt S(p(t))≥0,∀t≥0

work page
[5]

Proof.(1) Preservation of constraints follows becausedp/dt∈T pMby construction, so the constraints are constant along the flow

The only stationary point of(44)inMis the MaxEnt statep ⋆, and every trajectoryp(t) converges top ⋆ ast→ ∞. Proof.(1) Preservation of constraints follows becausedp/dt∈T pMby construction, so the constraints are constant along the flow. (2) The time derivative ofSalong the flow is d dt S(p(t)) =∇S(p(t))· dp dt =∇S(p(t))·Π p(t)∇S(p(t)). Since Π p(t) is the ...

work page
[6]

heat- like

The temperatures are defined by 1 TA(UA) := dSA dUA (UA), 1 TB(UB) := dSB dUB (UB), and are strictly decreasing functions ofU A andU B, respectively. 6.3.2 Total entropy maximisation under fixed total energy Let the total internal energyU tot of the composite system be fixed: Utot =U A +U B. Assuming weak coupling (no interaction term in the entropy), the...

work page
[7]

Ifp AB is independent, i.e.p ij =p A i pB j for alli, j, thenΦ corr(pAB) = 0and S(A∪B) =S(A) +S(B) + Φ prod(pA, pB)

work page
[8]

31 Proof.By adding and subtracting P i,j G(pA i pB j ), we write S(A∪B) = X i,j G(pij) = X i,j G(pA i pB j ) + X i,j G(pij)−G(p A i pB j )

IfG(u) =−ulogu(Shannon generator), thenΦ prod ≡0and S(A∪B) =S(A) +S(B) + Φ corr(pAB), withΦ corr reducing to the negative mutual information:Φ corr(pAB) =−I(A;B). 31 Proof.By adding and subtracting P i,j G(pA i pB j ), we write S(A∪B) = X i,j G(pij) = X i,j G(pA i pB j ) + X i,j G(pij)−G(p A i pB j ) . The second sum is precisely Φ corr(pAB). Adding and s...

work page
[9]

thermodynamic compatibility

there exists a constantκ >0 and a continuous functionρ: (0,1]→Rwith lim u↓0 ρ(u) = 0 such that, for all sufficiently smallu, G(u) =κ ulog 1 u +ρ(u)u logu .(52) 34 Equivalently, lim u↓0 G(u) ulog(1/u) =κ. The constantκsets the entropy units; choosing units so thatκ= 1 recovers the standard normalisation. Remark7.9.Why (52) is the right notion of “thermodyn...

work page
[10]

essential smoothness

For each (β,λ) in an open domainB ⊂R m+1, there exists a unique (U ⋆(β,λ),X ⋆(β,λ))∈ Dachieving the supremum in (61). Remark8.6.Assumption 8.5 is a standard “essential smoothness” condition ensuring thatS and Φ are Legendre duals in the classical sense: smooth, strictly concave/convex, and with one-to-one gradient mapping between (U,X) and (β,λ). 8.3.2 Co...

work page
[11]

, m.(62)

The supremum in(61)is attained at the unique point(U ⋆,X ⋆)satisfying the first-order conditions β= ∂S ∂U (U ⋆,X ⋆), λ i = ∂S ∂Xi (U ⋆,X ⋆), i= 1, . . . , m.(62)

work page
[12]

At this point, S(U ⋆,X ⋆) =βU ⋆ + mX i=1 λiX ⋆ i −Φ(β,λ).(63)

work page
[13]

Massieu potential

The dual potentialΦis differentiable onBand ∂Φ ∂β (β,λ) =U ⋆(β,λ), ∂Φ ∂λi (β,λ) =X ⋆ i (β,λ).(64) Proof.(1) SinceS(U,X) is strictly concave and differentiable, the function Ψ(U,X;β,λ) :=βU+ mX i=1 λiXi −S(U,X) is strictly convex in (U,X) for fixed (β,λ), and the supremum in (61) is attained at a unique point (U ⋆,X ⋆) where the gradient with respect to (U...

work page 2015

[1] [1]

For everyF∈ F, the first columnf 0 is the all-ones vector

work page

[2] [2]

shape masks

For every vectorv∈R n withPn j=1 vj = 0 (i.e. orthogonal to the all-ones vector), there existF∈ Fand a multiplier vectorλsuch that v=F λ. In other words, the union of the column spaces ofFcontains every zero-sum vector. Intuitively, this says that by choosing suitable combinations of moments and local “shape masks”, we can span all directions in probabili...

work page

[3] [3]

The constraints are preserved:p(t)∈ Mfor allt≥0

work page

[4] [4]

The entropyS(p(t))is nondecreasing: d dt S(p(t))≥0,∀t≥0

work page

[5] [5]

Proof.(1) Preservation of constraints follows becausedp/dt∈T pMby construction, so the constraints are constant along the flow

The only stationary point of(44)inMis the MaxEnt statep ⋆, and every trajectoryp(t) converges top ⋆ ast→ ∞. Proof.(1) Preservation of constraints follows becausedp/dt∈T pMby construction, so the constraints are constant along the flow. (2) The time derivative ofSalong the flow is d dt S(p(t)) =∇S(p(t))· dp dt =∇S(p(t))·Π p(t)∇S(p(t)). Since Π p(t) is the ...

work page

[6] [6]

heat- like

The temperatures are defined by 1 TA(UA) := dSA dUA (UA), 1 TB(UB) := dSB dUB (UB), and are strictly decreasing functions ofU A andU B, respectively. 6.3.2 Total entropy maximisation under fixed total energy Let the total internal energyU tot of the composite system be fixed: Utot =U A +U B. Assuming weak coupling (no interaction term in the entropy), the...

work page

[7] [7]

Ifp AB is independent, i.e.p ij =p A i pB j for alli, j, thenΦ corr(pAB) = 0and S(A∪B) =S(A) +S(B) + Φ prod(pA, pB)

work page

[8] [8]

31 Proof.By adding and subtracting P i,j G(pA i pB j ), we write S(A∪B) = X i,j G(pij) = X i,j G(pA i pB j ) + X i,j G(pij)−G(p A i pB j )

IfG(u) =−ulogu(Shannon generator), thenΦ prod ≡0and S(A∪B) =S(A) +S(B) + Φ corr(pAB), withΦ corr reducing to the negative mutual information:Φ corr(pAB) =−I(A;B). 31 Proof.By adding and subtracting P i,j G(pA i pB j ), we write S(A∪B) = X i,j G(pij) = X i,j G(pA i pB j ) + X i,j G(pij)−G(p A i pB j ) . The second sum is precisely Φ corr(pAB). Adding and s...

work page

[9] [9]

thermodynamic compatibility

there exists a constantκ >0 and a continuous functionρ: (0,1]→Rwith lim u↓0 ρ(u) = 0 such that, for all sufficiently smallu, G(u) =κ ulog 1 u +ρ(u)u logu .(52) 34 Equivalently, lim u↓0 G(u) ulog(1/u) =κ. The constantκsets the entropy units; choosing units so thatκ= 1 recovers the standard normalisation. Remark7.9.Why (52) is the right notion of “thermodyn...

work page

[10] [10]

essential smoothness

For each (β,λ) in an open domainB ⊂R m+1, there exists a unique (U ⋆(β,λ),X ⋆(β,λ))∈ Dachieving the supremum in (61). Remark8.6.Assumption 8.5 is a standard “essential smoothness” condition ensuring thatS and Φ are Legendre duals in the classical sense: smooth, strictly concave/convex, and with one-to-one gradient mapping between (U,X) and (β,λ). 8.3.2 Co...

work page

[11] [11]

, m.(62)

The supremum in(61)is attained at the unique point(U ⋆,X ⋆)satisfying the first-order conditions β= ∂S ∂U (U ⋆,X ⋆), λ i = ∂S ∂Xi (U ⋆,X ⋆), i= 1, . . . , m.(62)

work page

[12] [12]

At this point, S(U ⋆,X ⋆) =βU ⋆ + mX i=1 λiX ⋆ i −Φ(β,λ).(63)

work page

[13] [13]

Massieu potential

The dual potentialΦis differentiable onBand ∂Φ ∂β (β,λ) =U ⋆(β,λ), ∂Φ ∂λi (β,λ) =X ⋆ i (β,λ).(64) Proof.(1) SinceS(U,X) is strictly concave and differentiable, the function Ψ(U,X;β,λ) :=βU+ mX i=1 λiXi −S(U,X) is strictly convex in (U,X) for fixed (β,λ), and the supremum in (61) is attained at a unique point (U ⋆,X ⋆) where the gradient with respect to (U...

work page 2015