Publications (86)233.92 Total impact

Article: Typicality of thermal equilibrium and thermalization in isolated macroscopic quantum systems
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ABSTRACT: Based on the view that thermal equilibrium should be characterized through macroscopic observations, we develop a general theory about typicality of thermal equilibrium and the approach to thermal equilibrium in macroscopic quantum systems. We first formulate the notion that a pure state in an isolated quantum system represents thermal equilibrium. Then by assuming, or proving in certain classes of nontrivial models (including that of two bodies in thermal contact), largedeviation type bounds (which we call thermodynamic bounds) for the microcanonical ensemble, we prove that to represent thermal equilibrium is a typical property for pure states in the microcanonical energy shell. We also establish the approach to thermal equilibrium under two different assumptions; one is that the initial state has a moderate energy distribution, and the other is the energy eigenstate thermalization hypothesis. We also discuss three easily solvable models in which these assumptions can be verified.  [Show abstract] [Hide abstract]
ABSTRACT: We focus on the issue of proper definition of entanglement entropy in lattice gauge theories, and examine a naive definition where gauge invariant states are viewed as elements of an extended Hilbert space which contains gauge noninvariant states as well. Working in the extended Hilbert space, we can define entanglement entropy associated with an arbitrary subset of links, not only for abelian but also for nonabelian theories. We then derive the associated replica formula. We also discuss the issue of gauge invariance of the entanglement entropy. In the Z N gauge theories in arbitrary space dimensions, we show that all the standard properties of the entanglement entropy, e.g. the strong subadditivity, hold in our definition. We study the entanglement entropy for special states, including the topological states for the Z N gauge theories in arbitrary dimensions. We discuss relations of our definition to other proposals.  [Show abstract] [Hide abstract]
ABSTRACT: We study thermodynamic operations which bring a nonequilibrium steady state (NESS) to another NESS in physical systems under nonequilibrium conditions. We model the system by a suitable Markov jump process, and treat thermodynamic operations as protocols according to which the external agent varies parameters of the Markov process. Then we prove, among other relations, a NESS version of the Jarzynski equality and the extended Clausius relation. The latter can be a starting point of thermodynamics for NESS. We also find that the corresponding nonequilibrium entropy has a microscopic representation in terms of symmetrized Shannon entropy in systems where the microscopic description of states involves "momenta". All the results in the present paper are mathematically rigorous.  [Show abstract] [Hide abstract]
ABSTRACT: We study thermodynamic operations which bring a nonequilibrium steady state (NESS) to another NESS in physical systems under nonequilibrium conditions. We model the system by a suitable Markov jump process, and treat thermodynamic operations as protocols according to which the external agent varies parameters of the Markov process. Then we prove, among other relations, a NESS version of the Jarzynski equality and the extended Clausius relation. The latter can be a starting point of thermodynamics for NESS. We also find that the corresponding nonequilibrium entropy has a microscopic representation in terms of symmetrized Shannon entropy in systems where the microscopic description of states involves "momenta". All the results in the present paper are mathematically rigorous. 
Article: The approach to equilibrium in a macroscopic quantum system for a typical nonequilibrium subspace
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ABSTRACT: We study the problem of the approach to equilibrium in a macroscopic quantum system in an abstract setting. We prove that, for a typical choice of "nonequilibrium subspace", any initial state (from the energy shell) thermalizes, and in fact does so very quickly, on the order of the Boltzmann time $\tau_\mathrm{B}:=h/(k_\mathrm{B}T)$. This apparently unrealistic, but mathematically rigorous, conclusion has the important physical implication that the moderately slow decay observed in reality is not typical in the present setting. The fact that macroscopic systems approach thermal equilibrium may seem puzzling, for example, because it may seem to conflict with the timereversibility of the microscopic dynamics. According the present result, what needs to be explained is, not that macroscopic systems approach equilibrium, but that they do so slowly. Mathematically our result is based on an interesting property of the maximum eigenvalue of the Hadamard product of a positive semidefinite matrix and a random projection matrix. The recent exact formula by Collins for the integral with respect to the Haar measure of the unitary group plays an essential role in our proof. 
Article: Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace
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ABSTRACT: The fact that macroscopic systems approach thermal equilibrium may seem puzzling, for example, because it may seem to conflict with the timereversibility of the microscopic dynamics. We here prove that in a macroscopic quantum system for a typical choice of "nonequilibrium subspace", any initial state indeed thermalizes, and in fact does so very quickly, on the order of the Boltzmann time $\tau_\mathrm{B}:=h/(k_\mathrm{B}T)$. Therefore what needs to be explained is, not that macroscopic systems approach equilibrium, but that they do so slowly.  [Show abstract] [Hide abstract]
ABSTRACT: We prove two theorems concerning the time evolution in general isolated quantum systems. The theorems are relevant to the issue of the time scale in the approach to equilibrium. The first theorem shows that there can be pathological situations in which the relaxation takes an extraordinarily long time, while the second theorem shows that one can always choose an equilibrium subspace the relaxation to which requires only a short time for any initial state.  [Show abstract] [Hide abstract]
ABSTRACT: In a system of interacting f=1 bosons (in the subspace where the total spin in the z direction is vanishing), we prove inequalities for the ground state expectation value of the density of spin0 bosons. The inequalities imply that the ground state possesses “polar” or “antiferromagnetic” order when the quadratic Zeeman term q is large enough. In the low density limit, the inequalities establish the existence of a sharp transition at q=0 when q is varied.  [Show abstract] [Hide abstract]
ABSTRACT: We prove basic theorems about the ground states of the S=1 BoseHubbard model. The results are quite universal and depend only on the coefficient U_{2} of the spindependent interaction. We show that the ground state exhibits saturated ferromagnetism if U_{2}<0, is spinsinglet if U_{2}>0, and exhibits "SU(3)ferromagnetism" if U_{2}=0, and completely determine the degeneracy in each region.  [Show abstract] [Hide abstract]
ABSTRACT: A version of the second law of thermodynamics states that one cannot lower the energy of an isolated system by a cyclic operation. We prove this law without introducing statistical ensembles and by resorting only to quantum mechanics. We choose the initial state as a pure quantum state whose energy is almost E_0 but not too sharply concentrated at energy eigenvalues. Then after an arbitrary unitary time evolution which follows a typical "waiting time", the probability of observing the energy lower than E_0 is proved to be negligibly small.  [Show abstract] [Hide abstract]
ABSTRACT: Recently, in their attempt to construct steady state thermodynamics (SST), Komatsu, Nakagwa, Sasa, and Tasaki found an extension of the Clausius relation to nonequilibrium steady states in classical stochastic processes. Here we derive a quantum mechanical version of the extended Clausius relation. We consider a small system of interest attached to large systems which play the role of heat baths. By only using the genuine quantum dynamics, we realize a heat conducting nonequilibrium steady state in the small system. We study the response of the steady state when the parameters of the system are changed abruptly, and show that the extended Clausius relation, in which "heat" is replaced by the "excess heat", is valid when the temperature difference is small. Moreover we show that the entropy that appears in the relation is similar to von Neumann entropy but has an extra symmetrization with respect to timereversal. We believe that the present work opens a new possibility in the study of nonequilibrium phenomena in quantum systems, and also confirms the robustness of the approach by Komtatsu et al.  [Show abstract] [Hide abstract]
ABSTRACT: We consider a (small) quantum mechanical system which is operated by an external agent, who changes the Hamiltonian of the system according to a fixed scenario. In particular we assume that the agent (who may be called a demon) performs measurement followed by feedback, i.e., it makes a measurement of the system and changes the protocol according to the outcome. We extend to this setting the generalized Jarzynski relations, recently derived by Sagawa and Ueda for classical systems with feedback. One of the two relations by Sagawa and Ueda is derived here in errorfree quantum processes, while the other is derived only when the measurement process involves classical errors. The first relation leads to a second law which takes into account the efficiency of the feedback. 
Article: Entropy and Nonlinear Nonequilibrium Thermodynamic Relation for Heat Conducting Steady States
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ABSTRACT: Among various possible routes to extend entropy and thermodynamics to nonequilibrium steady states (NESS), we take the one which is guided by operational thermodynamics and the Clausius relation. In our previous study, we derived the extended Clausius relation for NESS, where the heat in the original relation is replaced by its "renormalized" counterpart called the excess heat, and the GibbsShannon expression for the entropy by a new symmetrized GibbsShannonlike expression. Here we concentrate on Markov processes describing heat conducting systems, and develop a new method for deriving thermodynamic relations. We first present a new simpler derivation of the extended Clausius relation, and clarify its close relation with the linear response theory. We then derive a new improved extended Clausius relation with a "nonlinear nonequilibrium" contribution which is written as a correlation between work and heat. We argue that the "nonlinear nonequilibrium" contribution is unavoidable, and is determined uniquely once we accept the (very natural) definition of the excess heat. Moreover it turns out that to operationally determine the difference in the nonequilibrium entropy to the second order in the temperature difference, one may only use the previous Clausius relation without a nonlinear term or must use the new relation, depending on the operation (i.e., the path in the parameter space). This peculiar "twist" may be a clue to a better understanding of thermodynamics and statistical mechanics of NESS.  [Show abstract] [Hide abstract]
ABSTRACT: We treat the problem of the approach to thermal equilibrium by only resorting to quantum dynamics of an isolated macroscopic system. Inspired by the two important works in 2009 and in 1929, we have noted that a condition we call "thermodynamic normality" for a macroscopic observable guarantees the approach to equilibrium (in the sense that a measurement of the observable at time $t$ almost certainly yields a result close to the corresponding microcanonical average for a sufficiently long and typical $t$). A crucial point is that we make no assumptions on the initial state of the system, except that its energy is distributed close to a certain macroscopic value. We also present three (rather artificial) models in which the thermodynamic normality can be established, thus providing concrete examples in which the approach to equilibrium is rigorously justified. Note that this kind of results which hold for ANY initial state are never possible in classical systems. We are thus dealing with a mechanism which is peculiar to quantum systems. The present note is written in a selfcontained (and hopefully readable) manner. It only requires basic knowledge in quantum physics and equilibrium statistical mechanics. Comment: 27 pages, 2 figures. The version 3 is a major revision (even the title has been changed). A new example has been added. We now call the main notion "thermodynamic normality", and some discussions have been added. Minor changes in v.4.  [Show abstract] [Hide abstract]
ABSTRACT: We describe our recent attempts toward statistical mechanics and thermodynamics for nonequilibrium steady states (NESS) realized, e.g., in a heat conducting system. Our first result is a simple expression of the probability distribution (of microscopic states) of a NESS. Our second result is a natural extension of the thermodynamic Clausius relation and a definition of an accompanying entropy in NESS. This entropy coincides with the normalization constant appearing in the above mentioned microscopic expression of NESS, and has an expression similar to the Shannon entropy (with a further symmetrization). The NESS entropy proposed here is a clearly defined measurable quantity even in a system with a large degrees of freedom. We numerically measure the NESS entropy in hardsphere fluid systems with a heat current, by observing energy exchange between the system and the heat baths when the temperatures of the baths are changed according to specified protocols.  [Show abstract] [Hide abstract]
ABSTRACT: We introduce and study two classes of Hubbard models with magnetic flux or with spinorbit coupling, which have a flat lowest band separated from other bands by a nonzero gap. We study the Chern number of the flat bands, and find that it is zero for the first class but can be nontrivial in the second. We also prove that the introduction of onsite Coulomb repulsion leads to ferromagnetism in both the classes.  [Show abstract] [Hide abstract]
ABSTRACT: It is believed that strong ferromagnetic orders in some solids are generated by subtle interplay between quantum manybody effects and spinindependent Coulomb interactions between electrons. Here we describe our rigorous and constructive approach to ferromagnetism in the Hubbard model, which is a standard idealized model for strongly interacting electrons in a solid.  [Show abstract] [Hide abstract]
ABSTRACT: Starting from microscopic mechanics, we derive thermodynamic relations for heat conducting nonequilibrium steady states. The extended Clausius relation enables one to experimentally determine nonequilibrium entropy to the second order in the heat current. The associated Shannonlike microscopic expression of the entropy is suggestive. When the heat current is fixed, the extended Gibbs relation provides a unified treatment of thermodynamic forces in the linear nonequilibrium regime.  [Show abstract] [Hide abstract]
ABSTRACT: Recently a novel concise representation of the probability distribution of heat conducting nonequilibrium steady states was derived. The representation is valid to the second order in the ``degree of nonequilibrium'', and has a very suggestive form where the effective Hamiltonian is determined by the excess entropy production. Here we extend the representation to a wide class of nonequilibrium steady states realized in classical mechanical systems where baths (reservoirs) are also defined in terms of deterministic mechanics. The present extension covers such nonequilibrium steady states with a heat conduction, with particle flow (maintained either by external field or by particle reservoirs), and under an oscillating external field. We also simplify the derivation and discuss the corresponding representation to the full order. Comment: 27 pages, 3 figures  [Show abstract] [Hide abstract]
ABSTRACT: Starting from a classical mechanics of a ``colloid particle'' and $N$ ``water molecules'', we study effective stochastic dynamics of the particle which jumps between deep potential wells. We prove that the effective transition probability satisfies (local) detailed balance condition. This enables us to rigorously determine precise form of the transition probability when barrier potentials have certain regularity and symmetry.
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4k  Citations  
233.92  Total Impact Points  
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Institutions

19892015

Gakushuin University
 Department of Physics
Edo, Tōkyō, Japan


19871988

Princeton University
 Department of Physics
Princeton, New Jersey, United States


19851987

The University of Tokyo
 Department of Physics
Tōkyō, Japan
