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Thermodynamic Uncertainty Relations for Coherent Transport

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read For coherent fermionic conductors, the entropy-production rate is bounded from below by a measured current and its fluctuations, through the inequality Q_qu ≥ 1, which holds at arbitrary bias in any multi-terminal geometry.

desk verdict A clean, self-contained derivation of a universal TUR for coherent fermionic transport; the main bound holds, with minor caveats about the broken-TRS constant and the probe-terminal claims. read the letter →

arxiv 2502.07917 v2 pith:BSA6XJAK submitted 2025-02-11 cond-mat.stat-mech cond-mat.mes-hallquant-ph

Kay Brandner , Keiji Saito This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.mes-hallquant-ph MSC 82C0582C70
keywords thermodynamicuncertaintyrelationcoherenttransportentropyproductioncurrentfluctuationsLandauer-Büttikerformalismshotnoisethermoelectricefficiencyquantumdotchain
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 establishes a thermodynamic uncertainty relation for coherent (ballistic, non-interacting) fermionic transport through a multi-terminal sample. It proves that at any terminal α, the combination Q_qu = (S_α/J_α) sinh[σ/(2 k_B J_α)] is always at least 1, where J_α is the mean particle current entering from that terminal, S_α is its full fluctuation (thermal plus shot noise), and σ is the total entropy-production rate. The bound holds for arbitrary chemical and thermal biases, any number of terminals, and any scattering potential, as long as the microdynamics are symmetric under time reversal. It is tight: a two-terminal conductor with a boxcar-shaped transmission function saturates it. This gives a practical way to lower-bound total dissipation from a single measured current and its noise, without measuring heat currents.

What carries the argument

The load-bearing object is the inequality Q_qu = (S_α/J_α) $\sinh$[σ/(2 k_B J_α)] ≥ 1, derived through the scattering formalism. The proof rewrites the entropy production σ in terms of transmission functions and Fermi functions, introduces the convex function Φ[x] = x·arsinh(x), applies a tangent-line (Jensen) bound at an optimally chosen point X = J_α/(2N_α), and then uses the relation S_α ≥ 2N_α ≥ S_α^th to eliminate the auxiliary quantity N_α. For broken time-reversal symmetry, a similar argument with a concavity bound on the binary entropy function yields the same form with the numerical factor ψ_0 ≃ 0.85246 inserted into the argument of the hyperbolic sine.

What would settle it

Construct a multi-terminal fermionic conductor with time-reversal-symmetric scattering but with finite interactions or inelastic collisions inside the sample, then measure J_α, S_α, and the true total entropy production, and check whether the quantity (S_α/J_α) sinh[σ/(2 k_B J_α)] drops below 1 wherever those internal processes dissipate energy.

Watch

Extended reading notes

Core claim

The central discovery is that, for fermionic coherent conductors, the classical thermodynamic uncertainty relation Q_cl = σS/(2 k_B $J^{2}$) ≥ 1 is replaced by Q_qu = (S_α/J_α) $\sinh$[σ/(2 k_B J_α)] ≥ 1. Unlike its classical counterpart, this inequality is never violated by energy filtering and Pauli blocking; it holds arbitrarily far from equilibrium and for any multi-terminal geometry. The proof uses only the Landauer–Büttiker scattering formalism, the reservoir-only entropy balance σ = −∑_α Q_α/T_α, and convexity of x·arsinh(x). In linear response, the hyperbolic sine linearizes and the classical bound is recovered; with a numerical prefactor ψ_0 ≃ 0.85246, the bound extends to samples with broken time-reversal symmetry.

Load-bearing premise

The proof assumes that all entropy is produced in the reservoirs, so no inelastic scattering or interactions inside the sample generate additional dissipation; if they do, the bound constrains only the reservoir part and may miss the physically relevant total entropy production.

Editorial extensions

If this is right

  • A two-terminal thermoelectric heat engine obeys the trade-off relation Q_qu^HE ≥ 1, implying that ideal Carnot efficiency is attainable only with vanishing power output or diverging power fluctuations.
  • A two-terminal thermoelectric refrigerator satisfies a similar trade-off, and its efficiency can be estimated from current mean and fluctuations alone, bypassing difficult heat-current measurements.
  • A chain of quantum dots with specially tuned hopping amplitudes and reservoir couplings produces a nearly boxcar transmission function, essentially saturating the quantum bound while strongly violating the classical one.
  • The bound extends to time-reversal-broken systems, including chiral quantum Hall edge states, at the cost of only a numerical factor ψ_0 in the dissipation estimate.
  • The authors expect (but do not prove) that the bound remains robust against moderate dephasing, internal dissipation, and carrier interactions when these are modeled by probe terminals.
  • In the linear-response limit, the bound reduces exactly to the classical thermodynamic uncertainty relation, showing that violations of the classical bound are a far-from-equilibrium effect.

Reading between the lines

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

  • Beyond the paper: if the probe-terminal expectation holds, the bound could serve as a general dissipation-inference tool for interacting quantum dot arrays, where heat currents are hard to measure.
  • Beyond the paper: the appearance of the symmetry-dependent prefactor ψ_0 suggests a possible family of uncertainty relations with prefactors tied to the degree of time-reversal breaking, which could be explored numerically for partially symmetric scattering matrices.
  • Beyond the paper: the tightness at boxcar transmission profiles invites searches for finite-parameter transmission shapes that nearly saturate the bound at small N, potentially guiding the design of high-precision, low-dissipation mesoscopic devices.
  • Beyond the paper: the same convexity-based proof strategy might extend to heat-current fluctuations, although the paper notes that an additional parameter would be needed to match physical dimensions.
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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 / 4 minor

Summary. The paper derives a universal thermodynamic uncertainty relation for fermionic coherent transport, Eq. (2): Q_qu = (S_alpha/J_alpha) sinh[sigma/(2 k_B J_alpha)] >= 1, where J_alpha and S_alpha are the mean and zero-frequency fluctuations of the particle current entering terminal alpha and sigma is the total rate of entropy production. The proof uses Landauer-Buttiker expressions for currents and noise, rewrites sigma as an integral of a convex function x arsinh x, applies a tangent bound, and eliminates the channel-overlap factor N_alpha using the AM-GM inequality. The paper also gives a broken-time-reversal version with a numerical constant psi_0, derives power-efficiency-fluctuation trade-offs for heat engines and refrigerators, and illustrates the results with a quantum dot chain whose transmission function approaches a boxcar shape.

Significance. The central result is significant: it provides a parameter-free lower bound on dissipation from a single measured current and its noise, valid arbitrarily far from equilibrium and for arbitrary multi-terminal geometries, and it is tight for narrow boxcar transmissions. The derivation is elegant and self-contained, and the thermoelectric trade-off relations are practically useful. The main limitations are the restriction to non-interacting coherent transport without internal dissipation and the reliance of the broken-TRS extension on a numerically determined constant rather than a fully analytic proof.

major comments (3)
  1. [Section 2 (second remark) and Eq. (9)] The claim that Eq. (2) 'still applies' to systems with incoherent or inelastic scattering and interactions through probe terminals is not derived. The entropy balance sigma = -sum_alpha Q_alpha/T_alpha is exact only when no entropy is produced inside the sample; a probe with zero net particle and heat current can model elastic dephasing but not inelastic dissipation, which requires a nonzero heat flow into the probe and therefore adds a positive contribution to sigma. Since the paper only states 'we expect' robustness, this passage should be explicitly labeled as a conjecture, or a proof should be given that the probe heat contribution cannot make the bound fail.
  2. [Supplemental Material, Eq. (S5) and Eq. (16)] The broken-time-reversal bound Eq. (16) rests on the numerical statement psi[x,y] >= psi_0 = 0.85246 over [0,1]^2. No analytical certificate or interval-arithmetic verification is provided, so the advertised extension to systems with broken time-reversal symmetry is not fully proven. In addition, the step from the psi_0 replacement to Eq. (16) is only sketched ('along the lines described in the main text'), although the variable Y = (f_alpha - f_beta)/(g_alphaalpha + g_betabeta) and the weight differ from the time-reversal-symmetric case; the cancellation condition is Y_0 = J_alpha/S_th^alpha. Please provide a rigorous lower bound on psi and spell out the analog of the tangent argument.
  3. [Eq. (2) and surrounding statement] The derivation requires g^alpha_beta_E = f^alpha_E(1 - f^beta_E) > 0, which holds only for finite reservoir temperatures. At T = 0 the quantities N_alpha and S_alpha can vanish while J_alpha stays nonzero (for example, a perfectly transmitting channel), so Eq. (2) as written is not defined and the proof's tangent argument breaks down. The theorem should be stated for finite reservoir temperatures, with the zero-temperature limit discussed separately.
minor comments (4)
  1. [Eqs. (2) and (16)] The ratios S_alpha/J_alpha are written without absolute values; for J_alpha < 0 the expression is nevertheless positive because the hyperbolic sine has the same sign as J_alpha. To avoid confusion for the reader, it would be clearer to write |J_alpha| in the prefactor.
  2. [Supplemental Material, after Eq. (S5)] The numerical minimization of psi is not documented; please state the method (grid size, precision, or certified interval arithmetic) or provide the code or data used to obtain psi_0.
  3. [Eq. (24)] The notation 'iGamma = t0 = tN' appears to contain a typo; presumably t0 and tN denote boundary hopping amplitudes and the reservoir coupling is Gamma. Please clarify the notation.
  4. [Eq. (2) and limiting cases] The cases J_alpha = 0 and zero temperature are not discussed; the inequality is singular in both cases, and the limiting statements should be made precise if these cases are meant to be included.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (2) is derived from Landauer–Büttiker formulas and convexity, with no fitted parameter or load-bearing self-citation.

full rationale

The central inequality (2) is derived directly from the Landauer–Büttiker expressions (3)–(8), the reservoir entropy balance (9), an algebraic rearrangement into the convex function x·arsinh(x), and elementary inequalities. In particular, Eq. (10) is rewritten via the identity relating log-ratios of Fermi functions to arsinh of X, and the choice X = J_alpha/(2 N_alpha) makes the first-order term in the convexity bound vanish by construction. The replacement of 2N_alpha by S_alpha uses only 2N_alpha <= Sth_alpha <= S_alpha and monotonicity of arsinh; nothing is fitted to force the inequality. The secondary bound (16) involves a numerically minimized universal constant psi0, which is a mathematical lower bound for a fixed function, not a parameter fitted to the target result. Self-citations (Refs. 17, 21, 24, 45, 82) appear only as background, context, or illustrations of saturation and classical-bound violations; the proof of Eq. (2) does not rely on these citations as premises. The quantum-dot chain is an illustrative model, and the transmission formula (25) is verified by computer algebra rather than assumed for the main derivation. The statement that probe terminals should make the bound robust against dephasing and dissipation is explicitly labeled an expectation ('We therefore expect'), which is a limitation or open point, not a circular step. Overall, the derivation is self-contained and does not reduce any output to its input by construction.

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

The central bound follows from standard Landauer-Buttiker scattering theory and a convexity argument without fitted parameters. The model parameters (Gamma, t_j, w, N) are chosen for illustration, not fitted to data. The single non-analytic input is the numerical constant psi_0 in the broken-time-reversal extension, which is asserted from numerical minimization.

assumptions (5)
  • domain assumption Landauer-Buttiker scattering formulas (Eqs. 3-8) describe currents and zero-frequency noise of noninteracting coherent fermionic conductors.
    The entire derivation is built on these standard formulas from Refs 42-44; no inelastic scattering or interactions inside the sample are included.
  • domain assumption All entropy production occurs in the reservoirs, sigma = -sum_alpha Q_alpha/T_alpha (Eq. 9), with no dissipation inside the conductor.
    This premise is stated in the text; the probe-terminal argument for dephasing and interactions is an expectation, not a proof.
  • domain assumption Time-reversal symmetry implies reciprocity T^{alpha beta}_E = T^{beta alpha}_E; for broken TRS, unitarity gives sum rules sum_beta T^{alpha beta}_E = sum_beta T^{beta alpha}_E.
    Used in Eq. (10) and in the broken-TRS derivation, from standard scattering theory (Refs 42-44).
  • standard math Phi(x) = x arsinh x is convex on the real line, so a tangent-line bound applies.
    Elementary calculus; the convexity is used in the step from Eq. (12) to Eq. (15).
  • ad hoc to paper The bound psi[x,y] >= psi_0 = 0.85246 holds on [0,1]^2, asserted by numerical minimization in the SM.
    The broken-TRS factor psi_0 is not proven analytically; the SM states 'Numerical minimization shows' without sharing the code or a certificate.

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Cite this review

Pith. "Pith review of Thermodynamic Uncertainty Relations for Coherent Transport." pith.science (2026). https://pith.science/paper/BSA6XJAK

@misc{pith2026250207917,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic Uncertainty Relations for Coherent Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BSA6XJAK}},
  note         = {Machine review of arXiv:2502.07917}
}
read the original abstract

We derive a universal thermodynamic uncertainty relation for Fermionic coherent transport, which bounds the total rate of entropy production in terms of the mean and fluctuations of a single particle current. This bound holds for any multi-terminal geometry and arbitrary chemical and thermal biases, as long as no external magnetic fields are applied. It can further be saturated in two-terminal settings with boxcar-shaped transmission functions and reduces to its classical counterpart in linear response. Upon insertion of a numerical factor, our bound also extends to systems with broken time-reversal symmetry. As an application, we derive trade-off relations between the figures of merit of coherent thermoelectric heat engines and refrigerators, which show that such devices can attain ideal efficiency only at vanishing mean power or diverging power fluctuations. To illustrate our results, we work out a model of a coherent conductor consisting of a chain of quantum dots.

Figures

Figures reproduced from arXiv: 2502.07917 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Chain of quantum dots as a coherent refrigerator. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Reference graph

Works this paper leans on

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