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Geometric Approach to Nonequilibrium Statistical Mechanics

智洋 設樂

Published Mar 22, 2018• 0 citations
Publisher

Abstract

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.

DOI
10.15083/00077897

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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.
The quantum Fisher information represents a continuous family of metrics on the space of quantum states and places the fundamental limit on the accuracy of quantum state estimation. We show that the entire family of quantum Fisher information can be determined from linear-response theory through generalized covariances. We derive the generalized fluctuation-dissipation theorem that relates linear-response functions to generalized covariances and hence allows us to determine the quantum Fisher information from linear-response functions, which are experimentally measurable quantities. As an application, we examine the skew information, which is a quantum Fisher information, of a harmonic oscillator in thermal equilibrium, and show that the equality of the skew-information-based uncertainty relation holds.
We develop an information-geometric framework for describing quantum state-update processes associated with measurement and statistical distinguishability. The approach is based on the quantum relative entropy and the quantum Fisher information metric, which together induce a natural Riemannian geometry on the manifold of quantum states. Using the second-order expansion of relative entropy, we show how the Fisher metric governs the local structure of distinguishability between nearby states and defines a corresponding thermodynamic length. This geometric structure provides an effective description of finite quantum state transitions in terms of fluctuation geometry and information-space distance. The formalism is applied to thermal two-level systems and harmonic oscillator states, illustrating how the Fisher metric encodes susceptibilities, fluctuations, and geometric transition costs. We also discuss the relation between thermodynamic length, dissipation bounds, and optimal paths in state space. Within this framework, wave function collapse is interpreted not as a microscopic dynamical mechanism, but as an effective state-update process that admits a geometric characterization in the manifold of density operators. The resulting perspective unifies concepts from quantum information theory, thermodynamics, and differential geometry within a common operational framework based on statistical distinguishability. Possible connections with quantum speed limits, entanglement geometry, and holographic relations between relative entropy and gravitational dynamics are briefly discussed.
Recent advancements have revealed new links between information geometry and classical stochastic thermodynamics, particularly through the Fisher information (FI) with respect to time. Recognizing the nonuniqueness of the quantum Fisher metric in Hilbert space, we exploit the fact that any quantum Fisher information (QFI) can be decomposed into a metric-independent incoherent part and a metric-dependent coherent contribution. We demonstrate that the incoherent component of any QFI can be directly linked to entropic acceleration, and for GKSL dynamics with local detailed balance, to the rate of change of generalized thermodynamic forces and entropic flow, paralleling the classical results. Furthermore, we tighten a classical uncertainty relation between the geometric uncertainty of a path in state space and the time-averaged rate of information change and demonstrate that it also holds for quantum systems. We generalize a classical geometric bound on the entropy rate for far-from-equilibrium processes by incorporating a nonnegative quantum contribution that arises from the geometric action due to coherent dynamics. Finally, we apply an information-geometric analysis to the recently proposed quantum-thermodynamic Mpemba effect, demonstrating this framework's ability to capture thermodynamic phenomena.
Recent developments of the analysis of the second law of thermodynamics by using the information theoretic quantities has led to the discovery of the so-called nonequilibrium free energy and has allowed us to formulate thermodynamic relations for arbitrary nonequilibrium initial and final states.By using the nonequilibrium free energies, we can quantify thermodynamic costs of information encoding and erasure processes as well as to quantify the extractable work from information heat engines in a unified way.In those general situations, reducing dissipation allows us to increase the extractable work from the system (or to reduce the work cost), and reducing work fluctuation allows us to narrow the work probability distribution and to extract a fluctuation-free work from the system.Suppressing both work fluctuation and dissipation as much as possible is vital to control mesoscopic and nano systems that work at the level of thermal fluctuations.Previous studies have searched for a vanishing work fluctuation protocol by using techniques of quantum information theory, known as the single-shot statistical mechanics (and sometimes called the deterministic work extraction protocol) and the vanishing dissipation protocol, known as the thermodynamically reversible protocol by using the second law of thermodynamics.In this thesis, we address issues concerning a method to control nonequilibrium systems by using fluctuation theorems and trade-off relation between work fluctuation and fluctuation in dissipation.We derive quantum fluctuation theorems under measurement and feedback control, which allow us to study how to suppress dissipation during a general quantum measurement and a feedback control.We then consider a method to suppress work fluctuation and fluctuation in dissipation simultaneously as much as possible by showing the trade-off relation between work and dissipation in terms of their standard deviations and derive an explicit protocol that achieves the lower bound of the trade-off relation.We show that the explicit protocol reproduces the deterministic work extraction protocol in the limit of vanishing work fluctuation and the thermodynamically reversible protocol in the limit of vanishing fluctuation in dissipation.
The nonequilibrium thermodynamics of interacting quantum many-body systems is investigated within the framework of thermal time-dependent density functional theory using a generalized linear-response formulation for the full quantum work statistics. A first-principles route is established to reconstruct the relaxation function that underlies linear-response theory, thereby moving beyond phenomenological descriptions and enabling a consistent evaluation of all moments of the dissipated-work distribution in interacting systems. The predictive power of the approach is demonstrated for the Hubbard model subject to a staggered external potential, where the evolution of the relaxation dynamics during the Mott-to-band-insulator crossover reveals how distinct many-body phases shape the out-of-equilibrium thermodynamic response. These results provide a microscopic and transferable framework for quantum thermodynamics in correlated systems, bridging thermal density functional theory and nonequilibrium work statistics.
The quantum geometric tensor and quantum Fisher information have recently been shown to provide a unified geometric description of the linear response of many-body systems. However, a similar geometric description of higher-order perturbative phenomena including nonlinear response in generic quantum systems is lacking. In this work, we develop a general framework for the time-dependent quantum geometry of many-body systems by treating external perturbing fields as coordinates on the space of density matrices. We use the Bures distance between the initial and time-evolved density matrix to define geometric quantities through a perturbative expansion. To lowest order, we derive a time-dependent generalization of the Bures metric related to the spectral density of linear response functions, unifying previous results for the quantum metric in various limits and providing a geometric interpretation of Fermi’s golden rule. At next order in the expansion, we define a time-dependent Bures-Levi-Civita connection for general many-body systems. We show that the connection is the sum of one contribution that is related to a second-order nonlinear response function, and a second contribution that captures the higher geometric structure of first-order perturbation theory. We show that in the quasistatic, zero-temperature limit for noninteracting fermions, this Bures connection reduces to the known expression for band-theoretic Christoffel symbols. Our work provides a systematic framework to explore many-body quantum geometry beyond the quantum metric and highlights how higher-order correlation functions can probe this geometry.