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Microscopic work under stationary boundary calibration

R. Arun Chandru

Published Oct 6, 2026• 0 citations
Publisher

Abstract

Calibrating a confining wall at low energies does not specify the work of changing its microscopic profile. We quantify this distinction using positive finite potentials on a fixed Hilbert space and complete thermal initial states. For any fixed finite order of the wall's energy-dependent boundary response, a nonconstant finite-time drive can preserve that response throughout its motion while producing a strictly positive leading dissipated-work coefficient. The physical duration scales with the inverse wall height. A complete off-label transition law resolves rare outcomes of probability of order $r^3$ at energies of order $r^{-2}$, giving mean dissipation of order $r$ and variance of order $r^{-1}$. For sudden changes in arbitrary smooth bounded cavities, the leading work coefficients factor into thermal boundary pressure and microscopic profile energies. This factorization extends to canonical ideal Bose and Fermi gases at fixed particle number. A smooth, scalar-calibrated finite-wall example certifies positive dissipation and a bounded-energy detection event at nonzero duration, including full thermal tails and finite reporting errors. The preparation costs distinguish observing a rare event from estimating its moments. These results specify the dynamical information left outside stationary boundary calibration without implying a resource-free sensing advantage.

DOI
10.5281/zenodo.23192688

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Calibrating a confining wall at low energies does not specify the work of changing its microscopic profile. We quantify this distinction using positive finite potentials on a fixed Hilbert space and complete thermal initial states. For any fixed finite order of the wall's energy-dependent boundary response, a nonconstant finite-time drive can preserve that response throughout its motion while producing a strictly positive leading dissipated-work coefficient. The physical duration scales with the inverse wall height. A complete off-label transition law resolves rare outcomes of probability of order r3r^3r3 at energies of order r−2r^{-2}r−2, giving mean dissipation of order rrr and variance of order r−1r^{-1}r−1. For sudden changes in arbitrary smooth bounded cavities, the leading work coefficients factor into thermal boundary pressure and microscopic profile energies. This factorization extends to canonical ideal Bose and Fermi gases at fixed particle number. A smooth, scalar-calibrated finite-wall example certifies positive dissipation and a bounded-energy detection event at nonzero duration, including full thermal tails and finite reporting errors. The preparation costs distinguish observing a rare event from estimating its moments. These results specify the dynamical information left outside stationary boundary calibration without implying a resource-free sensing advantage.
A particle is confined by a high but finite potential wall. The wall's height is changed, and the particle's initial state is a complete thermal mixture. Almost all particles retain their ordered energy-level label as the wall becomes harder. That familiar measure of successful control nevertheless misses a small population at increasingly high energies. These rare outcomes dominate the leading dissipated work and make the work variance grow. The paper derives this rare-energy law from the actual continuum Hamiltonian, without discarding the initially occupied high-energy states or replacing the finite wall by a perfectly reflecting boundary. It then proves a limitation on pulse shaping: changing only the wall height with a bounded integrated height action cannot remove the leading energetic cost. This remains true when the waveform and duration change with the wall hardness. The mechanism is an endpoint displacement extending over an entire half-line in the boundary coordinate; the finite-norm correction produced by bounded-action control cannot cancel it. The leading thermodynamic information is also accessible in principle through final-energy measurements. For each fixed protocol, a finite set of appropriately chosen high-energy bands recovers arbitrarily much of the leading dissipative relative entropy. The energy-basis coherence contribution is smaller order. This does not make the measurement inexpensive: the energy range increases and the counts become rare. A separate finite-height continuum certificate supplies one concrete probability floor and work bound, uniformly over the stated action class.
An isolated quantum system evolves reversibly, yet a small subsystem can lose nearly all information about its initial state. Conserved quantities may retain some of this information, depending on how strongly they overlap with local observables. Here we consider a subsystem in an arbitrary initial state coupled to a maximally mixed bath. We show that the worst-input mean-square equilibration error, maximised over window origins, is bounded from above and below by the mean square of a complete local autocorrelation, with the stationary part subtracted. The two mean-square quantities are linearly comparable. A correlation record over a finite interval therefore controls the equilibration error in every interval of the same length, and the result extends to nonnegative observation weights with nonnegative Fourier transform. The result allows arbitrary degeneracies of energies and gaps. We apply it to two static Hamiltonian families. For fixed spin subsystems in the one-particle sector of a bounded-range periodic clock, fixed-accuracy thermalisation is guaranteed on windows that grow only slightly faster than linearly with system size, despite exponentially small positive energies. A random commuting model with logarithmic interaction range retains an extensive set of conserved observables. Its expected mean-square error is bounded by a finite-size term and an inverse-time term. The inverse-time dependence is unavoidable because of the initial transient. These examples show how local relaxation can coexist with exact conservation laws and a finely spaced spectrum.
Coherent boundary interference can purify the retained part of an increasingly mixed Gibbs state, while its work statistics probe inequivalent spectral sectors. We establish this connection for a confined particle subject to a coherently controlled potential wall. A common boundary source governs the conditional state, while its positive spectral measure determines the leading population--coherence allocation of dissipated work and the Ramsey response. Resonant interference produces an asymptotically pure retained state with an explicitly paid success probability. For a sharp wall, a discrete near-elastic sector contributes a logarithm to absolute work that is absent from the quadratic-weight continuum limit. Its contribution makes the ratio of absolute to network diverge on specified thermal paths. Smooth-wall thermodynamic limits, a finite smooth-device certificate with timing and readout errors, and a finite-count reconstruction scheme connect these results to operational resources.
A useful thermodynamic model predicts what a device will do without tracking its full microscopic state. For a finite quantum working medium, the difficult question is which quantities should actually be measured. We develop a task-based method that keeps exact energy accounting separate from calibrated macroscopic prediction. Operator geometry identifies what a measured record determines and bounds what remains uncertain. We then compare direct measurements of dynamical observables with earlier measurements of the same coarse record, selecting the representation together with its measurement cost. A Bose--Hubbard example yields a compact response table tested over a continuous operating region and on unseen microscopic trajectories. A driven Ising chain shows why history is not always the cheaper choice. The result is a way to turn detailed quantum dynamics into a small, reusable, and explicitly qualified thermodynamic description.
A detector records an event through a succession of physical changes, from its initial response to the recovery of its working state. We construct a complete finite cycle on the local physical--observation phase space of PODA. An incoming carrier traverses fixed interaction regions, with its arrival, speed, and energy transfer governed by an autonomous Hamiltonian bounded below. Six canonical gates write a smooth event profile, amplify the pointer separation, export the record, reverse the working displacement, and exchange the remaining pointer state with a prepared blank. Their exact reference map identifies the selector and mechanical recoil left by record export. The retained core returns on the event-cell plateaus; narrow transition strips contribute a bounded probability of failure and a recoil that depends on the resolution. With a positive free Hamiltonian present throughout the passage, energy conservation and a finite-time perturbation estimate establish faithful writing, finite retention, and return within prescribed tolerances. An explicit binary realization admits nonzero preparation errors, and independent trajectory integrations check the equations and the endpoint map. These results complete the single cycle for the specified local model and incoming family. The calculation follows the archive, carrier, and waste through the output section, making explicit the resources consumed by reset. Preparation of fresh blanks, replenishment of those resources, and storage beyond the proven holding interval remain physical requirements of continued operation.
We derive a universal bound on the efficiency with which "dissipated" work can generate distinguishable changes in a quantum many-body state at a finite temperature, as quantified by the quantum Fisher information. The bound follows solely from the analytic structure of equilibrium many-body correlators and is independent of all microscopic details. It takes a frequency-resolved form with a characteristic crossover at the Planckian scale, ω⋆∼kBT/ℏω_\star\sim k_B T/\hbarω⋆​∼kB​T/ℏ. We find that Planckian scatterers sit at the edge of optimality, displaying maximal relaxation rate before information-dissipation efficiency collapses. This suggests strange metals are not just fast dissipators, but the fastest that remain efficient in generating distinguishability. The bounded quantity can be evaluated directly from optical conductivity measurements in strongly correlated electronic systems, offering a unique window into how dissipation generates distinguishable changes.