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Thermodynamic potentials from a probabilistic view on the system-environment interaction energy

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Pith's one-line read Under a zero-variance condition on the coupling energy, this paper proves that free energy and internal energy of a strongly coupled open system follow from two probability distributions alone, bypassing the Hamiltonian of mean force.

desk verdict A clean identity and a restrictive condition: the variance condition actually forces the interaction to be independent of the environment's microstate, so the strong-coupling framing is overstated, but the algebra is sound enough to merit review. read the letter →

arxiv 2505.00188 v1 pith:JN4POUQE submitted 2025-04-30 cond-mat.stat-mech physics.chem-ph

Mohammad Rahbar , Christopher J. Stein This is my paper · ORCID
classification cond-mat.stat-mechphysics.chem-ph
keywords stronglycoupledopensystemsHamiltonianofmeanforceinteractionenergydistributionfreeperturbationJarzynskiequalitychi-squareddivergencethermodynamicpotentialssystem-environmentcoupling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper targets strongly coupled open systems, where system-environment interaction energy is too large to ignore. It claims that when a single condition holds, namely that $e^{-\beta V_{SE}}$ has zero variance over the unperturbed environment, all thermodynamic potentials can be built from two accessible objects: the system's probability density $P_\beta(x_S)$ and the probability distribution $P(V_{SE})$ of the interaction energy. This removes the need to construct the Hamiltonian of mean force, which is hard to compute and has a non-uniqueness problem. The central exact result is an equality connecting the free-energy shift $\Delta F_S$ to an average of $e^{-\beta\Delta H_\beta}$ divided by $1+\chi^2(P_\beta\|P_{S0})$, valid at arbitrary coupling strength. A sympathetic reader should care because the result turns a hard effective-Hamiltonian problem into a statistics problem about directly measurable interaction energies.

What carries the argument

The load-bearing object is the ratio of the ensemble average $\langle e^{-\beta\Delta H_\beta}\rangle_S$ to $1+\chi^2(P_\beta\|P_{S0})$, with $\chi^2$ the chi-squared divergence measuring how far the strongly coupled system distribution sits from the uncoupled reference. The variance condition $\operatorname{Var}_{\mathcal{E}_0}(e^{-\beta V_{SE}})=0$ is the gateway: it makes $e^{-\beta V_{SE}}$ constant over unperturbed environment fluctuations, so conditional and unconditional environment averages coincide and the Hamiltonian of mean force can be traded for $P(V_{SE})$. Free-energy perturbation supplies the reference-state averages, and Jarzynski's equality supplies the non-equilibrium link.

What would settle it

Take a solvable strongly coupled model, for instance a harmonic oscillator bilinearly coupled to a harmonic bath, and compute $\operatorname{Var}_{\mathcal{E}_0}(e^{-\beta V_{SE}})$ while testing both sides of Eq. (31) with exact or well-converged simulation. If the variance is not zero yet the equality still holds, the paper's condition is not necessary; if the variance is zero but the equality fails, the central claim is false.

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Extended reading notes

Core claim

The central claim is the equality $e^{-\beta\Delta F_S} = \langle e^{-\beta\Delta H_\beta(x)}\rangle_S / (1+\chi^2(P_\beta\|P_{S0}))$, presented as valid for arbitrary coupling strength. From it the authors derive the inequality $\langle e^{-\beta\Delta H_\beta}\rangle_S \ge e^{-\beta\Delta F_S}$, the same bound Jensen's inequality gives when applied to the Gibbs-Bogoliubov-Feynman inequality. Under the condition $\operatorname{Var}_{\mathcal{E}_0}(e^{-\beta V_{SE}})=0$, they show the free energy and internal energy no longer require the Hamiltonian of mean force: $\exp(-\beta F_S^*)=\langle e^{-\beta V_{SE}}\rangle_{S+E}/\langle P_\beta(x_S)e^{\beta H_S(x_S)}\rangle_S$ and $U_S=\langle H_S\rangle_S+\langle V_{SE}\rangle$. Combining the equality with the Jarzynski equality yields expressions in which the mean interaction energy is fixed by the average non-equilibrium work and the chi-squared divergence between the coupled and uncoupled system distributions.

Load-bearing premise

Everything rests on the premise that for every fixed system coordinate, $e^{-\beta V_{SE}}$ has exactly zero variance over the unperturbed environment; if that variance is not zero, the substitution that removes the Hamiltonian of mean force does not go through.

Editorial extensions

If this is right

  • Free-energy differences in strongly coupled systems become measurable from two histograms: the system's configurational density and the interaction-energy distribution.
  • The ambiguity in the Hamiltonian of mean force, where infinitely many pairs reproduce the same system probability density, disappears under the variance condition because thermodynamics is fixed by $P_\beta(x_S)$ and $P(V_{SE})$.
  • Internal energy can be computed as $\langle H_S\rangle_S+\langle V_{SE}\rangle$ instead of requiring the temperature derivative of the Hamiltonian of mean force.
  • Combining with the Jarzynski equality gives the mean interaction energies $\mu$ and $\mu_0$ in terms of the average non-equilibrium work and $\chi^2(P_\beta\|P_{S0})$, making the mean interaction energy an observable in non-equilibrium experiments.
  • If $V_{SE}$ is Gaussian, the variance of the interaction energy is tied to the chi-squared divergence, so distribution width and free-energy shift determine each other.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The variance condition may hold for a broader class of environments than the paper's two strong-coupling examples, since any environment whose response is deterministic at fixed system coordinates would satisfy it.
  • If the variance condition fails, Eq. (31) might still stand as a free-energy identity while the interaction-energy formulas (12)-(13) break; this split is testable in models with inhomogeneous coupling.
  • When $P(V_{SE})$ is non-Gaussian, the exponential relations linking variance to $\chi^2$ should become inequalities, giving a measurable deviation from the central-limit picture.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This manuscript addresses the calculation of thermodynamic potentials for open systems strongly coupled to an environment. Starting from the Hamiltonian of mean force (HMF), the authors derive Eq. (7), express the condition under which the system-environment double average can be replaced by the coupled average, and identify it as Var_E0(e^{-βVSE})=0 in Eq. (8). Under this condition, they obtain Eq. (13) for the free energy and an expression for the internal energy, both in terms of Pβ(xS) and the distribution of VSE. They then derive Eq. (31), an equality relating e^{-βΔF_S} to ⟨e^{-βΔH}⟩_S and the chi-squared divergence, and combine it with the Jarzynski equality and Gaussian assumptions to obtain Eqs. (37)-(43). The central advertised results are the universal equality and the possibility of measuring thermodynamic potentials without the HMF.

Significance. If correct, the proposed shift from {Pβ(xS), H*_β} to {Pβ(xS), P(VSE)} would be practically valuable, since P(VSE) is more accessible than the HMF. The manuscript has genuine strengths: the derivation of Eq. (7) from Eq. (6) is careful, the consistency check in Eq. (11) is explicit, and the algebraic steps leading to Eq. (31) are correct. However, the zero-variance condition in Eq. (8) confines the practical formulas to a slaved/mean-field coupling regime rather than generic strong coupling, Eq. (31) is an algebraic identity with no explicit dependence on the interaction, and the final Gaussian relations in Eqs. (42)-(43) contain a sign error. These issues substantially reduce the reach of the paper as written, although the underlying algebra is mostly sound.

major comments (3)
  1. [§5, Eq. (31)] The variance condition in Eq. (8) is much more restrictive than the text suggests and does not describe generic strong coupling. Since e^{-βV}>0, the condition Var_E0(e^{-βV})=0 for a fixed system coordinate xS means that e^{-βV(xS,xE)} is constant in xE on the support of P_E0, so V(xS,xE)=v(xS) almost surely. Thus the interaction carries no fluctuating environmental degrees of freedom: the environment acts only as a fixed external field or is slaved to the system. The two 'strongly coupled' scenarios discussed in the text are precisely this slaved/mean-field situation, in which H*_β=H_S+v and the system is effectively closed with a modified potential. For any interaction that genuinely depends on environmental microstates, such as V=xS Σ c_i x_i, the variance is positive and the substitution ⟨⟨e^{-βV}⟩_{E0}⟩_S = ⟨e^{-βV}⟩_{S+E} fails; Eq. (12), Eq. (13), and the P(VSE)-based formulas are therefore not valid for generic strong coupling. Additionally, Eq. (8) must be read pointwise in xS; as written, Var_E0(e^{-βVSE}) is a function of xS, not a single global number.
  2. [§6, Eqs. (42)-(43)] Eq. (31) is an algebraic identity rather than a new physical equality with independent content. Substituting e^{-βΔHβ(x)} = e^{-βΔF_S} Pβ(x)/P_S0(x) into the average over Pβ reproduces Eq. (31) by construction; the same is true of Eq. (33). No system-environment coupling, interaction energy, or environment distribution enters these relations. They hold for any two normalized distributions Pβ and P_S0 over xS with ΔH defined through their ratio. Consequently, the claim that Eq. (31) is 'analogous in importance to the Jarzynski equality' is overstated, and the equality does not by itself justify computing free energies from P(VSE). The physically nontrivial content resides in Eqs. (12)-(13), which require the restrictive condition Eq. (8).
  3. [§6, Eqs. (42)-(43)] There is a sign error in Eqs. (42) and (43). Let C=1+χ²(Pβ‖P_S0). From Eq. (37), exp(β²σ²/2)=C^{1/2}. Under the stated Gaussian assumption, ⟨e^{-βV}⟩_{S+E}=e^{-βμ} C^{1/2}. Inserting this into Eq. (41) gives e^{-βμ} = ⟨e^{-βW}⟩ C^{1/2}, not C^{-1/2} as printed in Eq. (42). Combining with Eq. (39) then yields e^{-βμ0} = ⟨e^{-βW}⟩ C^{-1/2}, not C^{1/2} as printed in Eq. (43). The two displayed formulas are mutually inconsistent and should be corrected, or the Gaussian/cumulant convention needs to be stated.
minor comments (5)
  1. [§2, Eq. (8)] Equation (19) writes P_S0(xS) exp(βH_S(xS)) in the denominator, but this quantity equals 1/Z_S and is independent of xS; writing it as an xS-dependent expression obscures the algebra and should be replaced by the constant.
  2. [§3] The condition in Eq. (8) should be stated as Var_{E0}(e^{-βV(xS,·)})=0 for each xS, with a note on the support of P_E0, to avoid ambiguity.
  3. [§2] The notation Pβ(VSE) is used before the distribution of the interaction energy is defined; a definition of P(VSE) as the marginal of the coupled equilibrium distribution is needed.
  4. [§6] The text claims 'three scenarios' following Eq. (8), but only the decoupled case V=0 and two nontrivial scenarios are listed; renumber or correct this.
  5. [§6] The Gaussian assumption is applied separately to the coupled and uncoupled ensembles; clarify whether both are assumed and whether the equality σ²=σ0² is an independent assumption or a consequence of Eqs. (37)-(38). There is also a typographical error, 'we combine or result', in the paragraph preceding Eq. (40).

Circularity Check

2 steps flagged · score 6.0 of 10

Central Eq. (31) is an algebraic identity from the defining Boltzmann forms, not an independent equality; the Var=0 free-energy formula likewise reduces to normalization.

  1. self definitional [Eqs. (24)-(31), page 4 ('Inspired by the inequality Eq. (23)...')]
    "We write the term exp(−β ∆ Hβ (xS )) on the right-hand side of Eq. (23) as exp(−β ∆ Hβ (xS)) = Pβ (xS )Z ∗ / PS0(xS )ZS . (24) ... This leads to e− β ∆ FS = ⟨e− β ∆ Hβ (x)⟩ S / (1 + χ 2 (Pβ ‖ PS0 )). (31)"

    Equation (24) rewrites e^{-βΔHβ} using the defining Boltzmann forms Pβ=e^{-βH*}/Z* and P_S0=e^{-βH_S}/Z_S. Because ΔHβ=H*-H_S, this substitution is an identity. Taking the average ⟨·⟩_S and dividing by e^{-βΔF}=Z*/Z_S yields exactly Eq. (31); the 'general equality' is a rearrangement of the normalization of Pβ relative to P_S0, not a dynamical or independent fluctuation theorem. The input ⟨e^{-βΔHβ}⟩_S is itself defined in terms of the target free energy through Z*, so the claimed prediction is built into the input by construction.

  2. self definitional [Eqs. (12)-(13), page 3 ('By accepting the equality...')]
    "By accepting the equality of ⟨⟨exp (−βV SE )⟩E0 ⟩S with ⟨exp (−βV SE )⟩S+E under the above condition ... write ⟨exp(−β H∗β(xS)+βHS(xS))⟩S = ⟨exp(−βVSE)⟩S+E. (12) ... After expressing H∗β(xS) in terms of Pβ(xS) and F∗S (vide supra) and simplifying the resulting expression, one obtains exp(−βF∗S)= ⟨exp(−βVSE)⟩S+E / ⟨Pβ(xS)exp(βHS(xS))⟩S. (13)"

    Under Var_{E0}(e^{-βV})=0, Eq. (7) forces e^{-βV}/⟨e^{-βV}⟩_{E0}=1 pointwise for fixed xS, so H*_β=H_S+V almost surely. With that substitution, Eq. (12) is merely the identity obtained from Pβ=e^{-βH*}/Z*, and Eq. (13) reduces to Z* = ∫ dx_S e^{-β(H_S+V)}, which is the definition of F*_S. Thus the advertised computation of the free energy from {Pβ(xS), P(VSE)} is just the normalization constant of the given Pβ; the 'result' is equivalent to its own input by construction, though it does fix the gauge ambiguity in H* via P(VSE).

full rationale

The paper contains no parameter fitting and no load-bearing self-citation: the companion paper [37] is cited for exact expressions of mean and fluctuation but is not used in deriving Eqs. (31)-(43). However, the flagship equality (31) is not a new independent law: inserting the defining Boltzmann forms into Eq. (24) makes the derivation a direct algebraic rearrangement of normalization, and the same is true of the free-energy formula (13) under the variance condition. The variance condition itself is a genuine restriction—zero variance of e^{-βV} over environmental fluctuations with fixed xS means V is constant on the support of P_{E0}—so the advertised strong-coupling scenarios are slaved or mean-field rather than generic strong coupling, but this is an assumption about the interaction class, not a circularity. Overall, the central equality is definitionally forced, while the Gaussian σ²–χ² relations and the combination with the Jarzynski equality have independent mathematical content; the circularity is partial and concentrated in the paper's central framing.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted. The central results rest on the HMF partition function relation, the zero-variance condition, a Gaussian approximation for the interaction energy, and standard inequalities. The variance condition is the defining restriction and is not derived from microscopic dynamics.

assumptions (5)
  • domain assumption Hamiltonian of mean force definition and partition function relation Z* = Z_{S+E}/Z_E
    Invoked in Eqs (1), (9), (10), (11) and used throughout; cited to Talkner and Hänggi and other HMF literature.
  • ad hoc to paper Zero-variance condition Var_{E0}(e^{-βV})=0
    Introduced in Eq (8) as the central enabling condition; not derived from microscopic dynamics and used pointwise in xS.
  • domain assumption Gaussian distribution of the interaction energy VSE
    Assumed after Eq (36) to derive Eqs (37)-(39); justified heuristically by the central limit theorem, not by a microscopic derivation.
  • domain assumption Instantaneous adaptation of the environment in the scenario V = g*(xS, xE*(xS))
    Second strong-coupling scenario discussed after Eq (8); the time-scale separation is asserted, not formalized.
  • standard math Jensen, Hölder and Cauchy-Schwarz inequalities, and free energy perturbation identities
    Used in Eqs (18), (22), (26)-(29) and (35); standard results accepted without proof.

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Cite this review

Pith. "Pith review of Thermodynamic potentials from a probabilistic view on the system-environment interaction energy." pith.science (2026). https://pith.science/paper/JN4POUQE

@misc{pith2026250500188,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic potentials from a probabilistic view on the system-environment interaction energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JN4POUQE}},
  note         = {Machine review of arXiv:2505.00188}
}
abstract

In open systems with strong coupling, the interaction energy between the system and the environment is significant, so thermodynamic quantities cannot be reliably obtained by traditional statistical mechanics methods. The Hamiltonian of mean force $\mathcal{H}^{*}_{\beta}$ offers an in principle accurate theoretical basis by explicitly accounting for the interaction energy. However, calculating the Hamiltonian of mean force is challenging both theoretically and computationally. We demonstrate that when the condition $\text{Var}_{\mathcal{E}_0} (e^{-\beta {V}_{\mathcal{SE}}}) = 0$ is met, the dependence of thermodynamic variables can be shifted from $\{P_{\beta}(x_{\mathcal{S}}), \mathcal{H}^{*}_{\beta}(x_{\mathcal{S}})\}$ to $\{P_{\beta}(x_{\mathcal{S}}), P(V_{\mathcal{SE}})\}$. This change simplifies thermodynamic measurements. As a central result, we derive a general equality that holds for arbitrary coupling strengths and from which an inequality follows - aligned with Jensen's inequality applied to the Gibbs-Bogoliubov-Feynman bound. This equality, analogous in importance to the Jarzynski equality, offers deeper insight into free energy differences in strongly coupled systems. Finally, by combining our result with said Jarzynski equality, we derive additional relations that further clarify thermodynamic behavior in strongly coupled open systems.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Exact and variational identities for free energy differences in strongly coupled open systems

    cond-mat.stat-mech 2025-11 reject novelty 2.0 of 10

    The claimed exact fluctuation relations reduce to Hamiltonian-of-mean-force free-energy-perturbation identities; their trajectory forms require the final state to be exactly canonical, and the advertised variational/B...

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Works this paper leans on

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