{"id":"853c6d7d-c5c0-43d9-abe9-38b73dd246bc","arxiv_id":"2607.06265","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Thermoelectric voltage generation is irreversible whenever the Thomson coefficient is nonzero, with the voltage described by the Gouy-Stodola equation for classical heat engines.","lead":"The paper argues that thermoelectric voltage generation is inherently irreversible whenever the Thomson coefficient is nonzero, even at open circuit. It extends Thomson's classical thermodynamics argument to finite temperature differences and shows the voltage follows the Gouy-Stodola equation for heat engines.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The τ-dependent entropy generation in Eq. 9 has no counterpart in Onsager irreversible thermodynamics, where open-circuit entropy production is independent of the Thomson coefficient, suggesting the claimed irreversibility is an artifact of the two-reservoir model rather than a physical feature.","rationale":"The reader correctly identified the reservoir collapse as the weakest point, but the concern is more fundamental than 'the collapse might be invalid.' The specific issue is that the entropy generation in Eq. 9, which is the paper's central quantitative result, has no physical counterpart in the standard irreversible thermodynamics framework. At open circuit (J_e = 0), the Onsager entropy production depends only on thermal conductivity and is independent of the Thomson coefficient τ. The paper claims losses 'additional to' Onsager irreversibility but provides no mechanism or independent evidence for this additional entropy generation. The 'lost work' in the Gouy-Stodola decomposition (Eq. 10) arises from comparing the actual voltage to a Carnot reference (π(Th)·(1−Tc/Th)) that assumes a two-reservoir model inapplicable to a continuous-gradient system. The infinitesimal argument (Eqs. 3–4) correctly shows reversibility at each point; the finite-temperature irreversibility is introduced solely by the modeling choice of collapsing intermediate reservoirs. Since the paper's central claim — that open-circuit thermoelectric voltage generation is irreversible when τ ≠ 0 — depends entirely on this artifact, and since no independent physical or experimental evidence is provided to confirm the entropy generation is real, the claim is not adequately supported. The mathematical derivation is internally consistent, but the physical interpretation does not hold up. Moving from CONDITIONAL to REJECT because the load-bearing concern directly undermines the paper's central claim rather than merely weakening it.","tokens_in":3853,"tokens_out":9593,"duration_ms":379718,"concrete_test":"Compute the Onsager entropy production rate σ for a thermoelectric with nonzero τ at open circuit (J_e = 0). The standard result is σ = κ(∇T)²/T², independent of τ. If this holds — i.e., if varying τ (by choosing materials with different dε/dT) does not change σ at J_e = 0 — then Eq. 9's τ-dependent ΔS_i has no physical counterpart and the central irreversibility claim is an artifact of the two-reservoir model. This can be checked analytically from the Onsager-de Groot equations or numerically by solving the coupled heat/charge transport equations with J_e = 0 for two materials with identical κ but different τ.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central claim is that nonzero Thomson coefficient τ implies entropy generation at open circuit (Eq. 9: ΔS_i = ∫_{Tc}^{Th} (1/Tc − 1/T) τ dT). This term arises from forcing a two-reservoir heat-engine model onto a system that physically spans a continuous temperature gradient. The author shows that the infinitesimal process is reversible at each point (Eqs. 3–4: δs = δs'), and irreversibility only appears when intermediate reservoirs are collapsed into a single cold reservoir at Tc. The resulting ΔS_i is proportional to τ, but in the Onsager framework the local entropy production rate at open circuit (J_e = 0) is σ = κ(∇T)²/T², which depends only on thermal conductivity κ and is entirely independent of τ. The author acknowledges the losses are 'additional to' Onsager irreversibility but does not explain what physical mechanism produces this extra entropy or why the Onsager framework fails to capture it. The 'heat rejected to the cold reservoir' (Eq. 8: Q = π(Th) − π(Tc) − qV) is a hypothetical quantity from the heat-engine analogy — at open circuit no electron transport occurs, so no Peltier/Thomson heat is physically transported. The 'Carnot work' reference π(Th)·(1 − Tc/Th) used in the Gouy-Stodola decomposition (Eq. 10) assumes all heat is absorbed at Th and rejected at Tc, which does not describe the thermoelectric's actual operation across a continuous gradient. The 'lost work' Tc·ΔS_i is therefore the difference between an inapplicable Carnot reference and the actual voltage, not a measure of physical dissipation. Without an independent physical mechanism for τ-dependent entropy production at zero current, Eq. 9 appears to be a mathematical artifact of the modeling choice rather than a physical irreversibility.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The manuscript extends Thomson's classical thermodynamic treatment of thermoelectric conversion from an infinitesimal temperature difference to a finite temperature difference. The author argues that when the Thomson coefficient τ is nonzero, open-circuit voltage generation in a thermoelectric material produces entropy and is therefore irreversible. The voltage is expressed via a Gouy-Stodola-type decomposition (Eq. 10) into a reversible (Carnot) contribution and an irreversible lost-work term proportional to entropy generation (Eq. 9). The infinitesimal derivation (Eqs. 1-6) correctly reproduces Kelvin's relations, and the finite-difference extension (Eqs. 7-10) follows by integration and entropy balance.","tokens_in":4159,"tokens_out":1024,"duration_ms":162486,"significance":"The paper addresses a long-standing question in thermoelectric theory: whether open-circuit voltage generation is fundamentally reversible. The derivation is parameter-free and self-contained, using only standard thermoelectric coefficients and Kelvin's relations. The Gouy-Stodola decomposition (Eq. 10) provides a falsifiable, quantitative prediction linking the Thomson coefficient to irreversibility. If the central claim is correct, it has implications for how thermoelectric efficiency limits are framed. However, the claim's validity depends on resolving the tension between the two-reservoir classical heat-engine model and the Onsager framework, as discussed below.","major_comments":[{"comment":"§3, Eqs. (7)-(9): The central irreversibility claim hinges on collapsing the continuous temperature gradient (with intermediate reservoirs at Th - nδT, as acknowledged by the author) into a two-reservoir heat-engine model. The author explicitly notes this 'contravenes the fundamental premise in the operation of classical heat engines' but proceeds anyway. The resulting entropy generation ΔS_i = ∫_{Tc}^{Th} (1/Tc - 1/T) τ dT (Eq. 9) is proportional to τ, whereas in the Onsager framework the local entropy production rate at open circuit (J_e = 0) is σ = κ(∇T)²/T², which is independent of τ. The manuscript states the losses are 'additional to' Onsager irreversibility but does not identify what physical mechanism produces this extra τ-dependent entropy, nor why the Onsager framework fails to capture it. This is load-bearing for the central claim: if ΔS_i is an artifact of the reservoir model","section":null},{"comment":"§3, Eq. (8): The heat rejected to the cold reservoir Q = π(Th) - π(Tc) - qV is defined within the heat-engine analogy, but at open circuit no electron transport occurs, so no Peltier or Thomson heat is physically transported between reservoirs. The author should clarify what physical quantity Q represents at open circuit and how it relates to measurable heat flow, since Eq. (10) and the Gouy-Stodola decomposition depend on it.","section":null},{"comment":"§3, Eq. (10): The 'Carnot work' reference π(Th)(1 - Tc/Th) assumes all heat is absorbed at Th and rejected at Tc, which does not describe the thermoelectric's operation across a continuous gradient. The 'lost work' Tc·ΔS_i is therefore the difference between this idealized Carnot reference and the actual voltage. The author should address whether this decomposition has physical content beyond a bookkeeping identity, given that the Seebeck voltage qV = ∫ ε dT (Eq. 7) is already well-established and does not require the two-reservoir framing.","section":null}],"minor_comments":[{"comment":"The abstract spells 'Gouy-Stodola' as 'Guy-Stodola'; the body uses both spellings. Standardize.","section":null},{"comment":"Fig. 1 and Fig. 2 labels are small and difficult to read; consider enlarging or simplifying the annotations.","section":null},{"comment":"Eq. (3) and Eq. (4) appear in reverse logical order (Eq. 4 is discussed before Eq. 3 in the text). Consider reordering for clarity.","section":null},{"comment":"The phrase 'effectively infinite number of reservoirs' in §3 could be stated more precisely, e.g., 'a continuum of intermediate reservoirs.'","section":null},{"comment":"Reference 9 (Onsager) is missing the journal name; it should read Phys. Rev. 37, 405 (1931).","section":null}],"recommendation":"major_revision","confidential_remarks":"The core tension is whether the τ-dependent entropy generation is a genuine physical effect or a modeling artifact. The author is extending Thomson's own framework, which has historical legitimacy, but the manuscript needs to engage seriously with the Onsager counterargument rather than deferring it to the discussion. If the author can provide a physical mechanism or a concrete experimental test distinguishing the two frameworks, the paper could make a meaningful contribution. As it stands, the central claim is defensible but under-justified."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and substantive reading of the manuscript. The comments identify a genuine and important tension between the classical two-reservoir heat-engine framework used in the paper and the Onsager irreversible-thermodynamics framework. We address each comment below and indicate where the manuscript will be revised.","responses":[{"response":"The referee correctly identifies the central tension of the paper, and we agree that the manuscript does not currently address it with sufficient clarity. We offer the following clarification, which we will incorporate into the revised manuscript, while being candid about what remains unresolved. The classical thermodynamic analysis in the paper asks a different question from the Onsager framework. The Onsager framework computes the local entropy production rate associated with transport processes (heat conduction, Joule heating, etc.) at a given point in the material. At open circuit (J_e = 0), this local production is indeed σ = κ(∇T)²/T², independent of τ, as the referee states. The classical analysis, by contrast, performs a global entropy balance over the entire device treated as a single heat engine operating between two reservoirs. The entropy generation ΔS_i = ∫_{Tc}^{Th} (1/Tc − 1/T) τ dT arises because, when τ ≠ 0, the Peltier coefficient π(T) = ε(T)·T varies with temperature in a way that prevents the global entropy balance from closing. Physically, the mechanism is the temperature dependence of the Seebeck coefficient: when ε is not constant, the thermoelectric cannot be decomposed into a series of infinitesimal reversible Carnot engines, because the entropy per electron ε(T) changes along the gradient. The 'extra' τ-dependent entropy is therefore not a local transport effect but a consequence of the global thermodynamic accounting when a continuous gradient is mapped onto a two-reservoir model. We acknowledge, however, that the referee raises a legitimate and sharp question: if this entropy generation is physically real, it should be reconcilable with the Onsager framework, which is the standard and complete description of irreversible thermodynamics in the","revision_made":"no","referee_comment":"§3, Eqs. (7)-(9): The central irreversibility claim hinges on collapsing the continuous temperature gradient into a two-reservoir heat-engine model. The resulting entropy generation is proportional to τ, whereas in the Onsager framework the local entropy production rate at open circuit (J_e = 0) is σ = κ(∇T)²/T², independent of τ. The manuscript does not identify what physical mechanism produces this extra τ-dependent entropy, nor why the Onsager framework fails to capture it."},{"response":"This is a fair point and we will revise the manuscript to clarify it. The quantity Q in Eq. (8) is a thermodynamic (virtual) quantity, not an actual heat flow measured at open circuit. It represents the heat that would be rejected to the cold reservoir per electron if the thermoelectric converter operated as a classical heat engine absorbing heat π(T_h) at the hot reservoir and producing work qV. This is analogous to the open-circuit analysis of electrochemical cells (batteries, fuel cells), where one computes thermodynamic quantities (e.g., the Nernst voltage) from the Gibbs free energy change even though no current flows at open circuit. The analysis determines the thermodynamic limit and the entropy balance associated with the conversion process, not an actual measurable heat flux. We will add a paragraph to §3 making this analogy explicit and stating clearly that Q is a thermodynamic bookkeeping quantity within the heat-engine framework, not a directly measurable heat flow at open circuit. We will also note that when current does flow (finite load), the actual heat flows include additional Onsager-type contributions (Joule heating, thermal conduction) that are separate from the thermodynamic Q defined here.","revision_made":"yes","referee_comment":"§3, Eq. (8): The heat rejected to the cold reservoir Q = π(Th) - π(Tc) - qV is defined within the heat-engine analogy, but at open circuit no electron transport occurs, so no Peltier or Thomson heat is physically transported between reservoirs. The author should clarify what physical quantity Q represents at open circuit and how it relates to measurable heat flow."},{"response":"The referee is correct that Eq. (7) alone gives the Seebeck voltage without requiring the two-reservoir framing, and that the Gouy-Stodola decomposition in Eq. (10) is, in a mathematical sense, a rearrangement of established results. The physical content of the decomposition, if it has any, lies in the interpretation: it identifies how much of the thermoelectric voltage can be associated with reversible Carnot conversion and how much is 'lost' due to the τ-dependent entropy generation. Whether this interpretation carries physical significance beyond bookkeeping depends on whether the entropy generation ΔS_i represents a real thermodynamic loss or an artifact of the two-reservoir model. This is precisely the issue raised in the first major comment, and we acknowledge that the manuscript does not currently establish the physical significance of the decomposition independently of that question. In the revision, we will (a) state explicitly that Eq. (10) is a rearrangement of Eq. (7) combined with the entropy balance, (b) clarify that its physical significance is contingent on the interpretation of ΔS_i as discussed in our response to the first comment, and (c) avoid overstating the decomposition as a falsifiable prediction until the relationship to the Onsager framework is more firmly established. We agree that the current manuscript language ('falsifiable, quantitative prediction') overstates what the decomposition demonstrates on its own.","revision_made":"partial","referee_comment":"§3, Eq. (10): The 'Carnot work' reference π(Th)(1 - Tc/Th) assumes all heat is absorbed at Th and rejected at Tc, which does not describe the thermoelectric's operation across a continuous gradient. The author should address whether this decomposition has physical content beyond a bookkeeping identity, given that the Seebeck voltage qV = ∫ ε dT (Eq. 7) is already well-established and does not require the two-reservoir framing."}],"tokens_in":3637,"tokens_out":4185,"duration_ms":330442,"standing_objections":["The referee's first comment identifies a genuine and unresolved tension: the τ-dependent entropy generation ΔS_i in Eq. (9) is not captured by the Onsager framework, which gives σ = κ(∇T)²/T² at open circuit with no τ dependence. We can argue that the two frameworks address different questions (global thermodynamic balance vs. local transport), but we cannot fully explain why, if ΔS_i represents real entropy generation, it does not appear in the Onsager entropy production. It is possible that ΔS_i is an artifact of forcing a continuous-gradient device into a two-reservoir model. We will acknowledge this tension explicitly in the revised manuscript, but we cannot resolve it within the scope of the current paper."]},"desk_editor":{"model":"glm-5.2","letter":"The paper extends Thomson's classical heat-engine argument for thermoelectric conversion from infinitesimal to finite temperature differences. The infinitesimal derivation (Eqs. 1–6) is clean and correctly reproduces Kelvin's relations. The finite-difference extension (Eqs. 7–10) is mathematically straightforward — integrate the Seebeck coefficient, do an entropy balance, and you get a τ-dependent entropy generation term plus a Gouy-Stodola decomposition of the voltage into a Carnot-like term and a lost-work term. That decomposition is a new and genuinely useful way to look at thermoelectric voltage, and the paper deserves credit for it. The derivation is parameter-free and self-contained. What is new: the finite-ΔT extension itself, and the Gouy-Stodola framing. Both are legitimate. The soft spot is the physical interpretation, and it is not minor. The entropy generation in Eq. 9, ΔS_i = ∫(1/Tc − 1/T) τ dT, arises specifically when the author collapses the intermediate-temperature reservoirs — which kept the infinitesimal process reversible at each step — into a single cold reservoir at Tc. The author himself flags this collapse as contravening the premise of classical heat engines, then proceeds. The problem is that in the Onsager framework, open-circuit entropy production is σ = κ(∇T)²/T², which is independent of τ. So the τ-dependent term in Eq. 9 has no counterpart in the standard irreversible thermodynamics treatment. The paper says these losses are 'additional to' Onsager losses but does not identify a physical mechanism that would produce τ-dependent entropy at zero current. At open circuit, no electron transport occurs, so no Peltier or Thomson heat is physically being transported. The 'lost work' Tc·ΔS_i looks like the difference between an inapplicable Carnot reference and the actual voltage, not a measure of physical dissipation. The stress-test concern lands here: without an independent mechanism, Eq. 9 reads as a bookkeeping artifact of the two-reservoir model rather than a physical irreversibility. That said, the Gouy-Stodola decomposition itself is mathematically correct given the assumptions, and the infinitesimal derivation is sound. The paper is a short conceptual note aimed at readers interested in the thermodynamic foundations of thermoelectricity. It deserves a serious referee who can push on the two-reservoir assumption and ask whether the result should be reframed as a statement about the heat-engine analogy rather than a claim about physical irreversibility. I would accept this for peer review.","headline":"Extending Thomson's thermoelectric argument to finite ΔT yields a Gouy-Stodola decomposition of thermoelectric voltage, but the claimed intrinsic irreversibility at open circuit likely rests on a modeling artifact","tokens_in":4660,"tokens_out":1368,"would_cite":false,"duration_ms":79425,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["84.60.Rb","05.70.-a"],"model":"glm-5.2","headline":"Thermoelectric voltage is irreversible, even with no current","keywords":[],"falsifier":"Show that the entropy generation term in Equation (9) vanishes or becomes negligible when the intermediate-temperature reservoirs are properly retained in the finite-temperature-difference analysis, rather than collapsed into a single cold reservoir.","tokens_in":3963,"feed_emoji":"🔋","tokens_out":745,"duration_ms":111625,"temperature":0.7,"pith_summary":"The paper argues that thermoelectric voltage generation is inherently irreversible whenever the Thomson coefficient is nonzero, even at open circuit where no current flows. The author extends Thomson's classical heat-engine analysis of a thermoelectric converter from an infinitesimal temperature difference to a finite one. At infinitesimal temperature differences, the process appears reversible: entropy carried by electrons balances entropy dumped into the cold reservoir. But when stretched to a finite temperature gap between a hot reservoir Th and a cold reservoir Tc, the entropy balance no longer cancels. The residual entropy generation is proportional to the Thomson coefficient integrated over the temperature range. The resulting voltage decomposes into a reversible Carnot-work term minus an irreversible lost-work term, matching the Gouy-Stodola equation from classical heat-engine theory.","feed_headline":"Open-circuit thermoelectric voltage is inherently irreversible","feed_subtitle":"Extending Thomson's heat-engine argument to finite temperature gaps shows entropy generation whenever the Thomson coefficient is nonzero, c","key_machinery":"The Thomson coefficient tau, defined as the rate of change of Peltier heat with temperature. The Gouy-Stodola equation, which expresses heat-engine output as reversible work minus lost work due to entropy generation. Kelvin's two relations connecting the Seebeck coefficient, Peltier heat, and Thomson coefficient.","core_discovery":"The central result is Equation (10): qV = [pi(Th) - pi(Tc)] - Tc * delta_S_i, where the first bracket is the reversible Carnot work and the second term is lost work from irreversible entropy generation. The entropy generation delta_S_i equals the integral of (tau/T) dT from Tc to Th, where tau is the Thomson coefficient. This term vanishes only when tau is zero everywhere. The paper thus identifies the Thomson coefficient as the physical origin of irreversibility in thermoelectric voltage generation at open circuit, distinct from Joule heating or thermal conduction losses.","pith_inferences":[],"forward_implications":["If correct, the result adds a fundamental loss channel to thermoelectric generators that cannot be eliminated by reducing electrical resistance or thermal leakage, setting a ceiling below Carnot efficiency even at open circuit.","The irreversibility is tied to the material's Thomson coefficient, so material design aimed at minimizing tau over the operating temperature range could reduce this loss.","The decomposition of voltage into reversible and irreversible parts via Equation (10) provides a diagnostic tool: measuring tau(T) across the temperature range quantifies the lost work directly.","The result clarifies that Joule heating and thermal conduction are separate, additional losses layered on top of the Thomson-coefficient irreversibility identified here."],"fun_headline_variants":["Thomson coefficient drives entropy generation in thermoelectric voltage","Open-circuit thermoelectric voltage splits into reversible and irreversible parts","Thermoelectric voltage matches classical heat-engine equation with Thomson losses","Thomson coefficient identified as source of irreversibility in thermoelectric generators"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The derivation extends Thomson's infinitesimal-temperature-difference argument to a finite gap by collapsing the chain of intermediate-temperature reservoirs into a single cold reservoir at Tc. The author acknowledges this collapse contradicts the classical heat-engine premise of heat exchange with exactly two reservoirs. If the entropy generation is an artifact of this reservoir collapse rather than a physical property of the thermoelectric material, the central irreversibid","fun_headline_variants_meta":{"raw":{"variants":["Thomson coefficient drives entropy generation in thermoelectric voltage","Open-circuit thermoelectric voltage splits into reversible and irreversible parts","Thermoelectric voltage matches classical heat-engine equation with Thomson losses","Thomson coefficient identified as source of irreversibility in thermoelectric generators"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":466,"prompt_tokens":390,"completion_tokens":76,"prompt_tokens_details":null},"tokens_in":390,"tokens_out":76,"duration_ms":49302,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T11:26:39.240276+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Show that the entropy generation term in Equation (9) vanishes or becomes negligible when the intermediate-temperature reservoirs are properly retained in the finite-temperature-difference analysis, rather than collapsed into a single cold reservoir.","supporting_citations":[],"review_version":1}