REVIEW
Thermodynamic cost of locally maintaining a steady state
Statistical Mechanics · Submitted 28 Oct 2026 · 120 references
Pith's verdict: The Gaussian counterexample is clean and worth knowing; the headline sigma-hat >= c xi^2 bound is internally inconsistent as printed and needs corrected formulas before the paper can appear.
T0 · AI review
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
This paper claims that holding a steady state with long-range conditional correlations using only local, finite-range couplings to equilibrium heat baths forces a minimum entropy production that grows as the square of the Markov length…
The reading
What carries the argument
The load-bearing object is the precision matrix $K = C^{-1}$, the inverse steady-state covariance, because for a Gaussian state the CMI is read from the inverse: the Markov length $\xi$ is set by the pole of $K(k)$ closest to the real axis, while poles of $C$ set ordinary correlations. The counterexample uses $B(k)=1$ and $D(k)=4\sin^2(k/2)$, giving a $K$ with a real-axis pole at $k=0$ and infinite Markov length despite one-site correlations. Entropy accounting runs through the multibath identity $\sigma_k = \lambda(T_{\mathrm{eff}}(k)\mathbb{E}_k[1/T]-1)$, where a range-$r$ coupling makes $D(k)$ a trigonometric polynomial of degree $r$; Bernstein's inequality caps its curvature at $r^2\lambda T_{\mathrm{hottest}}$, yielding $\xi \lesssim r\sqrt{T_{\mathrm{hottest}}/T_{\mathrm{coldest}}}$, and a majorant optimization over order-$r$ polynomials $g(k)\ge 1/D(k)$ gives the closed-form bound $L_r(m) = 2\sinh((r+2)\kappa)/\sinh\kappa - (r+2) \sim \xi^2(r+1)(r+2)(r+3)/2$. For jump processes the decisive mechanism is string order: the appendix proves that persistent string order in a steady state of a local jump process forces the string-flipping activity to vanish exactly, i.e. an exact conservation law, which contradicts uniqueness of the steady state.
Extended reading notes
Core claim
The central discovery is a sharp separation between continuous and discrete state spaces. For Gaussian (Ornstein–Uhlenbeck) field theories the paper exhibits a strictly local process, $\dot\phi = -\phi + \eta$ with conserved nearest-neighbor noise, whose steady state has a precision matrix $K(k) = 1/(4\sin^2(k/2))$ with a real-axis pole at $k=0$: ordinary correlations vanish beyond one site but the conditional mutual information decays only as a power law, so the Markov length is infinite, and detailed balance holds (the circulation $Q=0$). The construction generalizes to any rational $K(k)=\nu(k)/\theta(k)$, producing a ladder of hyperuniform steady states with $S(k)\sim k^{2n}$. Yet for any complete implementation by a local quadratic Hamiltonian with uniformly bounded stiffness and finite-range linear couplings to Markovian equilibrium baths, the paper proves entropy production obeys $\hat\sigma \ge c\xi^2$ per site per relaxation time for large $\xi$ (Eq. 15), with an explicit nearest-neighbor two-bath construction attaining it; the algebraic-CMI endpoint requires infinite entropy production. In contrast, a full-support local detailed-balance jump process with irreducible frozen blocks must stabilize a local Hamiltonian, so long-range CMI forces detailed-balance violation; a shot-noise floor blocks hyperuniformity, and persistent string order in a local jump process is shown to force an exact conservation law.
Load-bearing premise
The entropy lower bound holds only for implementations whose baths are finite-range Markovian equilibrium reservoirs with finite correlation length and whose Hamiltonian is a local quadratic form with uniformly bounded stiffness; a gapless or infinite-correlation-length bath could imprint a long Markov length on the system by marginalization at zero entropy production.
Editorial extensions
If this is right
- Any local quadratic implementation of a steady state whose CMI decays with Markov length $\xi$ must produce at least $c\xi^2$ entropy per site per relaxation time, so the algebraic-CMI endpoint ($\xi\to\infty$) is thermodynamically unattainable in this implementation class.
- The achievable Markov length for range-$r$ couplings is bounded by $\xi \lesssim r\sqrt{T_{\mathrm{hottest}}/T_{\mathrm{coldest}}}$, and the explicit nearest-neighbor two-bath construction (17) saturates the entropy bound at $r=1$.
- A full-support local detailed-balance jump process with irreducible frozen blocks must stabilize a local Hamiltonian, so any discrete steady state with long-range CMI must violate detailed balance.
- Hyperuniformity is blocked in discrete systems by a shot-noise floor, $\mathrm{var}(Q)/N \ge r_2^2/(4C) > 0$, whenever local charge-changing activity is nonzero, so discrete analogues of the Gaussian counterexample cannot be reached rapidly.
- Quantum spin systems evade the discrete obstruction — the adapted model of a recent reference has a unique rapidly mixing steady state with long-range CMI — but its thermal implementation also pays infinite entropy production at the endpoint and only logarithmically in $\xi$ when thermally regularized.
Reading between the lines
- A general principle suggested by the paper: locality of the generator together with equilibrium (detailed balance) constrains the conditional-correlation structure of the steady state, and the Gaussian counterexample shows that continuous state spaces soften this constraint at the price of a fluctuation-dissipation violation that any local implementation must carry.
- The predicted mode-dependent effective temperature $T_{\mathrm{eff}}(k)\sim k^2$ for local implementations is an experimentally testable signature: passive correlation data cannot distinguish the equilibrium and nonequilibrium readings of the same process, but response measurements can, which matters for how hyperuniform classical field systems should be probed.
- The shot-noise floor (19) is derived for linear dynamics; a natural extension is to test whether interacting discrete systems with conserved charge obey the same $\mathrm{var}(Q)$ bound, which would generalize the hyperuniformity obstruction beyond the Gaussian class.
- If the conjectured impossibility for rapid-mixing jump processes holds, it places a thermodynamic-like limit on the information-processing capacity of classical discrete stochastic systems, in contrast to continuous and quantum ones — suggesting a resource-theoretic reading in which long-range conditional mutual information is a protected resource.
Original abstract
The Gibbs states of local Hamiltonians have short-range conditional correlations. To maintain a steady state with long-range conditional correlations, one might expect that a stochastic process must ``pay'' either in nonlocal moves or entropy production. Surprisingly, we find strictly local Gaussian processes that obey detailed balance and have unique steady states with arbitrarily long-range conditional correlations. We show that such processes must still produce entropy, if implemented using a local quadratic Hamiltonian coupled locally to equilibrium, rapidly relaxing baths. We lower-bound this entropy production in terms of the steady state's conditional mutual information (CMI); if the CMI decays exponentially with distance, the associated length-scale is called the Markov length $\xi$. The entropy production per site per relaxation time obeys $\hat\sigma \ge c \xi^2$ for large $\xi$, and with $c$ an implementation-dependent constant. We also bound the tradeoff between coupling range and entropy production. We contrast these Gaussian processes with jump processes on discrete state spaces, for which, under certain irreducibility assumptions, steady states with long-range CMI are only possible if detailed balance is violated. We conjecture (and present evidence) that rapidly mixing local jump processes cannot stabilize steady states with long-range CMI.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The Gibbs states of local Hamiltonians have short-range conditional correlations. To maintain a steady state with long-range conditional correlations, one might expect that a stochastic process must ``pay'' either in nonlocal moves or entropy production. Surprisingly, we find strictly local Gaussian processes that obey detailed balance and have unique steady states with arbitrarily long-range conditional correlations. We show that such processes must still produce entropy, if implemented using a local quadratic Hamiltonian coupled locally to equilibrium, rapidly relaxing baths. We lower-bound this entropy production in terms of the steady state's conditional mutual information (CMI); if the CMI decays exponentially with distance, the associated length-scale is called the Markov length $\xi$. The entropy production per site per relaxation time obeys $\hat\sigma \ge c \xi^2$ for large $\xi$, and with $c$ an implementation-dependent constant. We also bound the tradeoff between coupling range and entropy production. We contrast these Gaussian processes with jump processes on discrete state spaces, for which, under certain irreducibility assumptions, steady states with long-range CMI are only possible if detailed balance is violated. We conjecture (and present evidence) that rapidly mixing local jump processes cannot stabilize steady states with long-range CMI.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Gibbs states of local Hamiltonians have short-range conditional correlations. | background 1.00 | 0.09 | 0.02 | asserted 1.00 |
| s1 | To maintain a steady state with long-range conditional correlations, one might expect that a stochastic process must ``pay'' either in nonlocal moves or entropy production. | background 0.95 | 0.05 | 0.05 | speculative 0.95 |
| s2 | Surprisingly, we find strictly local Gaussian processes that obey detailed balance and have unique steady states with arbitrarily long-range conditional correlations. | result 0.96 | 0.95 | 0.55 | asserted 1.00 |
| s3 | We show that such processes must still produce entropy, if implemented using a local quadratic Hamiltonian coupled locally to equilibrium, rapidly relaxing baths. | result 0.95 | 0.92 | 0.22 | asserted 1.00 |
| s4 | We lower-bound this entropy production in terms of the steady state's conditional mutual information (CMI); if the CMI decays exponentially with distance, the associated length-scale is called the Markov length $\xi$. | result 0.68 | 0.89 | 0.21 | asserted 1.00 |
| s5 | The entropy production per site per relaxation time obeys $\hat\sigma \ge c \xi^2$ for large $\xi$, and with $c$ an implementation-dependent constant. | result 1.00 | 0.94 | 0.40 | asserted 1.00 |
| s6 | We also bound the tradeoff between coupling range and entropy production. | result 0.67 | 0.91 | 0.05 | asserted 0.98 |
| s7 | We contrast these Gaussian processes with jump processes on discrete state spaces, for which, under certain irreducibility assumptions, steady states with long-range CMI are only possible if detailed balance is violated. | result 0.79 | 0.70 | 0.06 | asserted 0.85 |
| s8 | We conjecture (and present evidence) that rapidly mixing local jump processes cannot stabilize steady states with long-range CMI. | future work 0.56 | 0.69 | 0.15 | speculative 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:2610.38925 by the authors.
References
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.