The integral fluctuation theorem (IFT), ⟨⟨e−Δσ⟩⟩=1\langle\langle e^{-\Delta\sigma}\rangle\rangle=1, is standardly derived under the assumption that the heat bath is initially in the canonical Gibbs ensemble. Heveling, Wang and Gemmer (arXiv:2102.12294; journal version Phys. Rev. E 105, 064112 (2022), with R. Steinigeweg) derived the IFT for isolated quantum system-bath compounds whose bath starts in a microcanonical energy shell or a pure state, provided the coarse-grained bath transition probabilities are 'stiff' (they depend only on the transferred energy) and 'smooth' (single-eigenstate probabilities are close to their energy-bin average), and verified the mechanism on exemplary models. The assumptions themselves, however, are unexplained properties of the dynamics. The open problem is to give this mechanism a microscopic foundation: determine for which quantum many-body model families and parameter regimes stiffness and smoothness provably hold, with quantitative scaling laws and a microscopic mechanism; prove or refute the conjecture that stiffness alone — without the microreversibility-based forward/backward transition-probability relation used in the derivation — implies the IFT; and establish rigorous bounds on the IFT deviation in terms of energy-bin size, temperature, bath size, and stiffness/smoothness violations, including the limit in which bins shrink to single energy eigenvalues. A same-day companion line (Iyoda–Kaneko–Sagawa) proves the IFT for single-eigenstate baths from eigenstate-thermalization-type assumptions in the thermodynamic limit, which resolves that sub-question along a different route and sharpens the surviving core of the stiffness program.

Background

The integral fluctuation theorem (IFT) is an exact identity of nonequilibrium statistical mechanics stating that the exponential average of the stochastic entropy production equals one, ⟨⟨e−Δσ⟩⟩=1\langle\langle e^{-\Delta\sigma}\rangle\rangle=1; by Jensen's inequality it implies the second law ⟨⟨Δσ⟩⟩≥0\langle\langle\Delta\sigma\rangle\rangle \ge 0 while allowing individual second-law-violating fluctuations. In the quantum system-bath setting, the standard derivation assumes the bath is initially in the canonical (Gibbs) ensemble. Heveling, Wang and Gemmer asked what happens when the bath is prepared instead in a microcanonical state (uniform over an energy shell) or a pure quantum state, for example a single energy eigenstate of a nonintegrable many-body system. For an isolated compound with Hamiltonian H=Hsys+Hbath+HintH = H_{\rm sys}+H_{\rm bath}+H_{\rm int} and an initially uncorrelated product state diagonal in the uncoupled eigenbases, they define the entropy production operator σ(t)=−log⁡ρsys(t)+βHbath\sigma(t) = -\log\rho_{\rm sys}(t) + \beta H_{\rm bath} (with β\beta the inverse bath temperature and ρsys(t)\rho_{\rm sys}(t) the reduced system state), whose changes are read out by two projective measurements at times 00 and tt — the two-point measurement scheme, the standard way to define fluctuating work and heat in closed quantum dynamics. Dividing the bath energy scale into bins of size Δ\Delta (large compared to the level spacing, small compared to thermal scales), writing R(jk,AB)R(jk,AB) for the transition probability from the initial system state kk and bath energy bin BB to the final system state jj and bath energy bin AA (with j,kj,k labeling system states and A,BA,B labeling bath energy bins), they call these probabilities stiff if they depend only on the energy difference, R(jk,AB)=R(jk,A−B)R(jk,AB)=R(jk,A-B), and smooth if the transition probability starting from an individual bath energy eigenvalue bb inside bin BB is close to its bin average, i.e. the deviation r(jk,Ab)=R(jk,Ab)−R(jk,AB)r(jk,Ab)=R(jk,Ab)-R(jk,AB) is small (Eq. (24) of the source). Under microreversibility — the property that the Hamiltonian and relevant observables are real in the working basis, which yields a fixed ratio between the transition probabilities of the forward and time-reversed processes (Eq. (16) of the source) — together with an exponentially growing bath density of states, stiffness implies a coarse-grained IFT for microcanonical initial baths, and smoothness lifts the coarse-grained IFT to the microscopic IFT, also for pure initial states. The paper checks the IFT and both assumptions numerically for two exemplary systems and finds smoothness improving with bath size. Two neighboring programs address the same physics under different assumptions. The eigenstate thermalization hypothesis (ETH) — the conjecture that individual high-energy eigenstates of nonintegrable many-body systems are indistinguishable from thermal ensembles for local observables — combined with the Lieb-Robinson bound (a linear light-cone bound on the propagation of operators in quantum many-body dynamics) yields the IFT for single-eigenstate baths in short- and long-time regimes (Iyoda–Kaneko–Sagawa). Dynamical typicality — concentration of measure, implying that most pure states sampled from a given ensemble reproduce its observables — yields pure-state second laws (positivity of entropy production). The stiffness/smoothness program is complementary: it aims at finite baths and mechanism-level, quantitative criteria, but its own assumptions are precisely what remains unproven.

Problem Statement

Give the stiffness/smoothness mechanism of the integral fluctuation theorem for non-canonical quantum baths a microscopic foundation. In the setup of Heveling, Wang and Gemmer — an isolated quantum system-bath compound H=Hsys+Hbath+HintH = H_{\rm sys}+H_{\rm bath}+H_{\rm int} with an initially uncorrelated diagonal product state, entropy production operator σ(t)=−log⁡ρsys(t)+βHbath\sigma(t) = -\log\rho_{\rm sys}(t)+\beta H_{\rm bath} read out by two projective measurements, and bath energies binned into intervals of size Δ\Delta — establish the following jointly linked results, whose common objective is to determine whether stiffness and smoothness are the general operative mechanism behind the validity of ⟨⟨e−Δσ⟩⟩=1\langle\langle e^{-\Delta\sigma}\rangle\rangle = 1 for microcanonical and pure-state baths. First (generality and mechanism): determine for which microscopic model families and parameter regimes — integrable versus nonintegrable, as a function of bath size, system-bath coupling strength, temperature, and driving protocol — the stiffness property (coarse-grained transition probabilities obeying R(jk,AB)=R(jk,A−B)R(jk,AB)=R(jk,A-B)) and the smoothness property (single-eigenstate transition probabilities close to their bin average, r(jk,Ab)≈0r(jk,Ab)\approx 0) hold. In particular, resolve this part by either of two independently standing routes: (a) a quantitative scaling law and microscopic mechanism for stiffness, analogous to the known 1/2L1/\sqrt{2^L}-type finite-size scaling of smoothness; or (b) a characterization of classes of model families or parameter regimes in which stiffness or smoothness provably fails. Either route on its own resolves this part: route (a) establishes the affirmative scaling and mechanism, while route (b) bounds the generality claimed here. Second (role of microreversibility): prove or refute the conjecture that the IFT for these non-canonical bath preparations follows from stiffness alone — that is, that it remains valid even when the microreversibility-based relation between forward and backward transition probabilities (Eq. (16) of the source) is violated, as may occur when time-reversal symmetry is broken, for example by magnetic fields. Third (quantitative error control): establish rigorous bounds on the deviation ∣⟨⟨e−Δσ⟩⟩−1∣|\langle\langle e^{-\Delta\sigma}\rangle\rangle - 1| in terms of the bin size Δ\Delta, the inverse temperature β\beta, the bath size, and the measured violations of stiffness and smoothness — including controlled behavior in the limit where bin sizes shrink to single energy eigenvalues, treated within the stiffness/smoothness framework rather than by importing eigenstate-thermalization-type assumptions. These three deliverables jointly support the single objective above: a proof in the second part reorganizes the first and third, a counterexample in the second part bounds the generality claimed in the first, and the third quantifies the regime of validity of both.

References

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