非平衡系统的理论研究由于缺乏一个通用的框架而变得复杂。在这项工作中，我们首先证明了Ao最近引入的一个变换（J. Phys. A { \bf 37}，L25（2004））与Graham（Z. Physik B { \bf 26}，397（1977））和Eyink { \it等人} (J. Stat. Phys. { \bf 83}，385（1996）），这也可以被看作是Helmholtz定理在向量微积分中的推广应用。然后，我们表明，系统所描述的普通随机微分方程与白噪声可以映射到恒温哈密顿系统。耗散系统的稳态对应于相应哈密顿系统的平衡态。这些结果为非平衡动力学，特别是非平衡稳态的相应研究提供了坚实的理论基础。映射允许应用的哈密顿系统的耗散非哈密顿系统，热力学平衡态的非平衡稳态的既定技术和结果。我们讨论了目前的工作的几个影响。
英文摘要：
Theoretical studies of nonequilibrium systems are complicated by the lack of a general framework. In this work we first show that a transformation introduced by Ao recently (J. Phys. A {\bf 37}, L25 (2004)) is related to previous works of Graham (Z. Physik B {\bf 26}, 397 (1977)) and Eyink {\it et al.} (J. Stat. Phys. {\bf 83}, 385 (1996)), which can also be viewed as the generalized application of the Helmholtz theorem in vector calculus. We then show that systems described by ordinary stochastic differential equations with white noise can be mapped to thermostated Hamiltonian systems. A steady-state of a dissipative system corresponds to the equilibrium state of the corresponding Hamiltonian system. These results provides a solid theoretical ground for corresponding studies on nonequilibrium dynamics, especially on nonequilibrium steady state. The mapping permits the application of established techniques and results for Hamiltonian systems to dissipative non-Hamiltonian systems, those for thermodynamic equilibrium states to nonequilibrium steady states. We discuss several implications of the present work.
Entanglement Hamiltonians in dissipative free fermions and the time-dependent GGE
SISSA（的里雅斯特高等研究学院）；INFN Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Dipartimento di Fisica dell’Università di Pisa（比萨大学物理系）；INFN Sezione di Pisa（意大利国家核物理研究所比萨分部）
We discuss spontaneous breaking of internal symmetry and its Nambu-Goldstone (NG) modes in dissipative systems. We find that there exist two types of NG modes in dissipative systems corresponding to type-A and type-B NG modes in Hamiltonian systems. To demonstrate the symmetry breaking, we consider a $O(N)$ scalar model obeying a Fokker-Planck equation. We show that the type-A NG modes in the dissipative system are diffusive modes, while they are propagating modes in Hamiltonian systems. We point out that this difference is caused by the existence of two types of Noether charges, $Q_R^α$ and $…
We study the effects of dissipative boundaries in many-body systems at continuous quantum transitions, when the parameters of the Hamiltonian driving the unitary dynamics are close to their critical values. As paradigmatic models, we consider fermionic wires subject to dissipative interactions at the boundaries, associated with pumping or loss of particles. They are induced by couplings with a Markovian baths, so that the evolution of the system density matrix can be described by a Lindblad master equation. We study the quantum evolution arising from variations of the Hamiltonian and dissipati…
We investigate the effects of dissipation on the quantum dynamics of many-body systems at quantum transitions, especially considering those of the first order. This issue is studied within the paradigmatic one-dimensional quantum Ising model. We analyze the out-of-equilibrium dynamics arising from quenches of the Hamiltonian parameters and dissipative mechanisms modeled by a Lindblad master equation, with either local or global spin operators acting as dissipative operators. Analogously to what happens at continuous quantum transitions, we observe a regime where the system develops a nontrivia…
Non-Hermitian Hamiltonians provide a simple picture for inspecting dissipative systems with natural or induced gain and loss. We investigate the Floquet dynamical phase transition in the dissipative periodically time driven XY and extended XY models, where the imaginary terms represent the physical gain and loss during the interacting processes with the environment. The time-independent effective Floquet non-Hermitian Hamiltonians disclose three regions by analyzing the non-Hermitian gap: pure real gap (real eigenvalues), pure imaginary gap, and complex gap. We show that each region of the sys…
Quantum information can be encoded in the set of steady-states (SSS) of a driven-dissipative system. Non steady-states are separated by a large dissipative gap that adiabatically decouples them way while the dynamics inside the SSS is governed by an effective, dissipation-projected, Hamiltonian. The latter results from a highly non-trivial interplay between a weak driving with the fast relaxation process that continuously projects the system back to the SSS. This amounts to a novel type of environment-induced quantum Zeno effect. We prove that the dissipation-projected dynamics is of geometric…
Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in this line of work is how to discretize the system in such a way that its stability and rates of convergence are preserved. In this paper we propose a geometric framework in which such discretizations can be realized systematically, enabling the derivation of "rate-matching" algorithms without the need for a discrete convergence analysis. More specifically, we…
A periodically driven quantum system, when coupled to a heat bath, relaxes to a non-equilibrium asymptotic state. In the general situation, the retrieval of this asymptotic state presents a rather non-trivial task. It was recently shown that in the limit of an infinitesimal coupling, using so-called rotating wave approximation (RWA), and under strict conditions imposed on the time-dependent system Hamiltonian, the asymptotic state can attain the Gibbs form. A Floquet-Gibbs state is characterized by a density matrix which is diagonal in the Floquet basis of the system Hamiltonian with the diago…
Beyond the lemon: Spectra of random Lindblad operators with a Hamiltonian component
Institute of Theoretical Physics, Jagiellonian University（雅盖隆大学理论物理研究所）；Doctoral School of Exact and Natural Sciences, Jagiellonian University（雅盖隆大学精确与自然科学博士学院）；Department of Computer Science, Oslo Metropolitan University（奥斯陆城市大学计算机科学系）；Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）
If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to …
