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arXiv:2403.16383

Nonequilibrium Bounds for Canonical Nonlinearity Under Single-Shot Work

Koretaka Yuge, Sakamoto, Yutaro

Published Mar 25, 2024• 0 citations
arXivHTMLPDF

Abstract

For classical discrete systems under constant composition (specifically substitutional alloys), canonical average acts as a map from a set of many-body interatomic interactions to a set of configuration in thermodynamic equilibrium, which is generally nonlinear. In terms of the configurational geometry (i.e., information about configurational density of states), the nonlinearity has been measured as special vector on configuration space, which is extended to Kullback-Leibler (KL) divergence on statistical manifold. Although they successfully provide new insight into how the geometry of lattice characterizes the nonlinearity, their application is essentially restricted to thermodynamic equilibrium. Based on the resource theory (especially, thermo-majorization), we here extend the applicability of the nonlinearity to nonequilibrium states obtained through single-shot work on Gibbs state. We reveal that the extended nonlinearity for nonequilibrium state is bounded from upper and lower by the information about one of the optimal Renyi divergences for equilibrium states in between practical and linear systems, and temperature and work.

DOI
10.48550/arxiv.2403.16383

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For classical discrete systems under constant composition (specifically substitutional alloys), canonical average acts as a map from a set of many-body interatomic interactions to a set of configuration in thermodynamic equilibrium, which is generally nonlinear. In terms of the configurational geometry (i.e., information about configurational density of states), the nonlinearity has been measured as special vector on configuration space, which is extended to Kullback-Leibler (KL) divergence on statistical manifold. Although they successfully provide new insight into how the geometry of lattice characterizes the nonlinearity, their application is essentially restricted to thermodynamic equilibrium. Based on the resource theory (especially, thermo-majorization), we here extend the applicability of the nonlinearity to nonequilibrium states obtained through single-shot work on Gibbs state. We reveal that the extended nonlinearity for nonequilibrium state is bounded from upper and lower by the information about one of the optimal Renyi divergences for equilibrium states in between practical and linear systems, and temperature and work.
The typicality of the canonical state shows that majority of the states are indistinguishable from equilibrium, and thus the nonequilibrium states are exceptionally rare in the extremely high-dimensional Hilbert space. On the contrary, we can easily apply an external force acting on the system, and then the actual density matrix quantitatively deviates from the canonical state specified by the system Hamiltonian at each instance. To express how the external forcing amounts to the deviation from equilibrium, we give a universal thermodynamic expression of the lower bound of Hilbert-Schmidt distance between the actual nonequilibrium and corresponding canonical states. The lower bound is expressed only by the amount of forcing and its consequent entropy production rate.
In this thesis, we study nonequilibrium statistical mechanics from a geometric perspective.In particular, we (i) examine the relation between quantum information geometry and linear response theory, and (ii) analyze nonequilibrium processes in thermodynamic control and derive a geometric expression for work.The first original study in this thesis is to understand quantum information geometry based on linear response theory.We show that the quantum Fisher information can be determined by measuring the linear response functions.For that purpose, we generalize the fluctuationdissipation theorem, and establish the quantitative relation between linear response functions and the generalized covariance, which contains the same amount of information on the quantum state.Based on the generalized fluctuation-dissipation theorem, we can determine the generalized covariance and the quantum Fisher information by measuring linear response functions, such as the dynamical susceptibilities and the complex admittances for all frequencies.We demonstrate that our result is applicable to experimental determination of the skew information, and a validation of skew information-based uncertainty relations.The second original study is on the analysis of work in thermodynamic control.We extend the thermodynamic metric-based expression for work into two directions.One is to obtain a systematic expansion of the average work from a phenomenological argument, and the other is to obtain an expansion of the work distribution for overdamped Langevin systems.First, we derive a systematic expansion of the work in terms of a small parameter ϵ that characterizes how slowly we control the system.The leading-order contribution is given by the thermodynamic metric expression.The next leading-order contributions to the thermodynamic metric contribution can be detected by comparing the excess work in a forward control and a backward control, and are predicted to scale as 1/T 2 as a function of the total control time T .Since the expansion is derived without assuming any specific microscopic dynamics, it is valid as long as the perturbation series expansion is valid.Finally, we examine the work distribution in overdamped Langevin systems.We derive the time evolution equation for the moment generating function of the work, and solve it from the lower-order contributions in ϵ.The O(ϵ) contribution to the generating function reproduces two known facts: the work distribution is Gaussian, and the average work is given by the thermodynamic metric.When we take up to O(ϵ 2 ) contributions into account, the work distribution exhibits nonzero skewness, which means that the fluctuation-dissipation relation is violated with scaling 1/T 2 .Furthermore, from the analytic calculation with numerical supports, we conjecture that the nth cumulant of the work scales as 1/T n-1 for n ≥ 1.
When reformulated as a resource theory, thermodynamics can analyze system behaviors in the single-shot regime. In this, the work required to implement state transitions is bounded by $\ensuremath{\alpha}$-R\'enyi divergences and so differs in identifying efficient operations compared to stochastic thermodynamics. Thus, a detailed understanding of the difference between stochastic and resource-theoretic thermodynamics is needed. To this end, we explore reversibility in the single-shot regime, generalizing the two-level work reservoirs used there to multilevel work reservoirs. This achieves reversibility in any transition in the single-shot regime. Building on this, we systematically develop multilevel work reservoirs in the nondissipation regime with and without catalysts. The resource-theoretic results show that two-level work reservoirs undershoot Landauer's bound, misleadingly implying energy dissipation during computation. In contrast, we demonstrate that multilevel work reservoirs achieve Landauer's bound while producing arbitrarily low entropy.
Exact equations of motion for the space- and time-dependent thermodynamic coordinates of a many-body system are derived directly from the Liouville equation. This is done by definining a generalized canonical density operator depending only upon present values of the thermodynamic coordinates. This operator is used no matter how far the system is from equilibrium. An explicit expression for the entropy of a system possibly not in equilibrium is given in terms of this operator. The equation of motion for the operator is derived, and the coupled, nonlinear, integrodifferential equations of motion for the thermodynamic coordinates follow immediately.
The existence of thermodynamic limits of non-equilibrium reduced density matrix and distribution function is proved on the following assumptions on initial states and temporal evolution of the system: The initial state is described by a canonical distribution, an effective Hamiltonian of which is the sum of local Hamiltonians and some averages of “local” operators are bounded. This proof yields an extension of Kubo's Ansatz on the extensive property of a macrovariable to quantal systems and to non-Markoffian macrovariables in stochastic models. A generating function (generalized thermodynamic potential) formalism is used effectively for discussing the above non-equilibrium problems. Non-linear response and fluctuation are also discussed in the above-mentioned scheme.
Abstract For quantum spin systems in any spatial dimension with a local, translation-invariant Hamiltonian, we prove that asymptotic state convertibility from a quantum state to another one by a thermodynamically feasible class of quantum dynamics, called thermal operations, is completely characterized by the Kullback–Leibler (KL) divergence rate, if the state is translation-invariant and spatially ergodic. Our proof consists of two parts and is phrased in terms of a branch of the quantum information theory called the resource theory. First, we prove that any states, for which the min and max Rényi divergences collapse approximately to a single value, can be approximately reversibly converted into one another by thermal operations with the aid of a small source of quantum coherence. Second, we prove that these divergences collapse asymptotically to the KL divergence rate for any translation-invariant ergodic state. We show this via a generalization of the quantum Stein’s lemma for quantum hypothesis testing beyond independent and identically distributed situations. Our result implies that the KL divergence rate serves as a thermodynamic potential that provides a complete characterization of thermodynamic convertibility of ergodic states of quantum many-body systems in the thermodynamic limit, including out-of-equilibrium and fully quantum situations.