[Show abstract][Hide abstract] 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), large-deviation 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 contain gauge
non-invariant 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 non-abelian 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.
Journal of High Energy Physics 02/2015; 2015(6). DOI:10.1007/JHEP06(2015)187 · 6.11 Impact Factor
[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.
[Show abstract][Hide abstract] 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
time-reversibility 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 semi-definite 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.
[Show abstract][Hide abstract] ABSTRACT: The fact that macroscopic systems approach thermal equilibrium may seem
puzzling, for example, because it may seem to conflict with the
time-reversibility 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.
New Journal of Physics 02/2014; 17(4). DOI:10.1088/1367-2630/17/4/045002 · 3.56 Impact Factor
[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 spin-0 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 Bose-Hubbard model. The results are quite universal and depend only on the coefficient U_{2} of the spin-dependent interaction. We show that the ground state exhibits saturated ferromagnetism if U_{2}<0, is spin-singlet 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
time-reversal. 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 error-free 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.
[Show abstract][Hide abstract] 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 Gibbs-Shannon expression for the entropy by a new symmetrized
Gibbs-Shannon-like 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.
Journal of Statistical Physics 09/2010; DOI:10.1007/s10955-010-0095-5 · 1.20 Impact Factor
[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 self-contained (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 spin-orbit 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 on-site 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 many-body
effects and spin-independent 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 Shannon-like 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.