UEDA GROUP

Department of Physics, The University of Tokyo

2014年度

ご興味ある方の参加、歓迎します。 事前の連絡なく参加頂けますが、 夏学期の論文紹介ゼミについては発表者に連絡頂けましたら発表資料を必要分印刷します。

Schedule of winter semester

(Each seminar starts from 14:45, Tuesday @ #913, Faculty of Science Bldg. 1 (except when specified otherwise))

Oct. 7 Shunsuke Furukawa

Oct. 14 Eriko Kaminishi / Daisuke Takahashi (13:00-, @#1220, Faculty of Science Bldg. 4)

Oct. 21 Tatsuhiko Ikeda / Takeshi Fukuhara (13:00-, @#1220, Faculty of Science Bldg. 4)

Nov. 4 Yui Kuramochi

Nov. 11 Ken Funo

Nov. 18 Tomohiro Shitara

Nov. 25 Yuto Murashita

Dec. 2 Shuhei Yoshida / Kazuya Fujimoto (13:00-, @#1220, Faculty of Science Bldg. 4)

Dec. 16 Yusuke Horinouchi

Jan. 6 Kohaku So (Hongbo Zeng)

Jan. 13 Yuto Ashida

Jan. 20 Sho Higashikawa

Jan. 27 Takeshi Saito / Ryusuke Hamazaki

2014/10/7(Tue.) @#913 14:45-

speaker | Shunsuke Furukawa |

title | Introduction to frustrated magnets: classical versus lattice gas pictures |

abstract | I present an introduction to frustrated magnets, with emphasis on its relation to lattice Bose gases. I first describe the Luttinger-Tisza method for determining the ground states of classical O(N) spin models in the absence of a magnetic field. I then describe a magnon gas picture for quantum spin systems near a saturation magnetic field, and explain that the Bose-Einstein condensation of magnons gives a useful picture for the magnetic order in this regime. The equivalence of the Luttinger-Tisza method with the determination of the magnon dispersion minimum suggests a smooth change of the magnetic order from the zero-field case to the saturation. However, a quantum antiferromagnet on a kagome lattice gives a remarkable counterexample to this scenario. In this system, the magnon dispersion is flat, and the localized nature of magnons leads to a magnetization jump, magnetization plateaus, and a possible magnon supersolid. Similar physics is expected in interacting bosons in an optical kagome lattice with an inverted hopping integral. Finally, I comment on possible phases of bosons loaded into the flat middle band of an optical Lieb lattice. |

2014/10/14(Tue.) @#1220 13:00-

speaker | Eriko Kaminishi |

title | Entanglement prethermalization |

abstract |
The relaxation to a quasi-stationary state before reaching an
equilibrium state is called prethermalization. Prethermalization[1]
and its mechanism has recently attracted great interest associated
with the dynamics of isolated quantum systems such as relaxation and
thermalization processes [2,3]. In order to understand the mechanism
of prethermalization, we study the dynamics of a coherently split
one-dimensional Bose gas with a few particles. As a result, we find a
new type of prethermalization caused by entanglement. By computing
correlation functions between the two Bose gases, we find behavior of
prethermalization even for the small number of particles; the modes
with short wavelength are thermalized but those with long wavelength
are not. In addition to it, we also find that each of the two Bose
gases is well described by the Gibbs state at an effective temperature
if we observe only one of them. In this way, the nonlocal correlation
between two gases plays an important role for observed
prethermalization and we call it ``entanglement prethermalization". [1] J. Berges et al. Phys. Rev. Lett. 93, 142002 (2004) [2] M. Gring et al. Science 337, 1318 - 1322 (2012) [3] T. Langen et al. Nature Physic 9, 640 - 643 (2013) |

speaker | Dr. Daisuke Takahashi (RIKEN) |

title | Bogoliubov theoretical formulation for counting rule of Nambu-Goldstone modes |

abstract |
Nambu-Goldstone modes (NGMs) are bosonic gapless excitations universally appearing in spontaneously symmetry-broken systems. The classification of NGMs based on their dispersion relations and their counting rule are one of recent hot topics in both condensed-matter and high-energy physics [1-3]. In this seminar I would like to present our recent work [4], in which we have solved the counting problem of NGMs by the Bogoliubov theory. Our formulation mainly consists of two key concepts: (i) zero-mode solutions of the Bogoliubov equations and (ii) their orthogonal relations based on inner products between Bogoliubov quasiparticle wavefunctions. I also give a talk on recent new results such as counting theory of quasi-NGMs [5] and interpolating formulae for non-integer dispersion relations of NGMs localized near topological defects [6]. [1] H. Watanabe and T. Brauner, Phys. Rev. D 84, 125013 (2011). [2] H. Watanabe and H. Murayama, Phys. Rev. Lett. 108, 251602 (2012). [3] Y. Hidaka, Phys. Rev. Lett. 110, 091601 (2013). [4] DT and M. Nitta, arXiv:1404.7696. [5] M. Nitta and DT, in preparation. [6] DT, M. Kobayashi and M. Nitta, in preparation. |

2014/10/21(Tue.) @#1220 13:00-

speaker | Tatsuhiko Ikeda |

title | Accuracy of the microcanonical ensemble in small isolated quantum systems. |

abstract |
Justifying the microcanonical ensemble (ME) only from quantum mechanics has recently attracted renewed attention due to experimental realizations of isolated quantum systems such as ultracold atomic gases [1]. In spite of the time-reversal symmetry of time evolution, effective stationary states have been shown to occur and the eigenstate thermalization hypothesis (ETH) has been proposed as the underlying mechanism for these states being well-described by the ME in the thermodynamic limit [2]. In contrast to the previous studies on why the ME works in the thermodynamic limit, we study how accurately it does in small systems. By numerically investigating the quantum quenches in the hard-core Bose-Hubbard model, we show that the accuracy scales as $D^{-1}$ with the dimension $D$ of the Hilbert space [3]. We argue that the $D^{-1}$ improvement of accuracy is much faster than the ETH ensures and derived from the fact that we cannot induce correlations between numerous many-body eigenstates by a quantum quench. [1] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011). [2] M. Rigol, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008). [3] T. N. Ikeda and M. Ueda, in preparation. |

speaker | Dr. Takeshi Fukuhara (RIKEN) |

title | Quantum dynamics of magnons in optical lattices |

abstract |
Ultracold atoms in optical lattices have offered unique opportunities to study quantum many-body systems presented by fundamental models such as the Hubbard model or the Heisenberg spin model. Recently, it has become possible to observe and control such systems at the single-atom and single-site level. This technique enables us to access out-of-equilibrium dynamics of such systems, where dissipation and decoherence are small. In this talk I will present the quantum dynamics of mobile spin impurities, or magnons, in a one-dimensional Heisenberg spin chain realized with ultracold bosonic atoms in an optical lattice. Starting from a fully magnetized spin chain, we locally excited magnons by selectively flipping spins, and tracked their subsequent dynamics with high resolution in space and time. In the case of single-magnon case, coherent propagation of the magnon was observed [1]. We also experimentally confirmed that the single-magnon dynamics results in generation and propagation of entanglement in the spin system. Different from the single-magnon dynamics, the interaction between magnons becomes important when exciting two magnons, and it leads to the formation of bound states. These two-magnon bound states were predicted to exist in Heisenberg chains by H. Bethe more than 80 years ago. Investigating the spatial correlations of the two magnons after dynamical evolution, we identified the magnon bound states, and observed their dynamics [2]. [1] T. Fukuhara et al., Nature Phys. 9, 235 (2013). [2] T. Fukuhara et al., Nature 502, 76 (2013). |

2014/11/4(Tue.) @#913 14:45-

speaker | Yui Kuramochi |

title | Relative-entropy conservation laws in quantum measurements |

abstract |
We consider the information flows in quantum measurements concerning a observable on the system. One approach based on the Shannon entropy was discussed by Ban[1] and he found a class of quantum measurements in which mutual information between the measurement outcome and the system's observable is equal to a decrease in the Shannon entropy of the system's observable.
We discuss the information flow of an observable from the view point of discriminating the initial state of the system and quantify the information as the relative entropy. We prove the relative-entropy conservation law under a more general condition than that of Shannon entropy conservation. The general theorem is applied to optical destructive measurements and it is found that the Shannon entropy does not conserve in these measurements while the relative entropy does. We further consider the construction of the relative-entropy conserving observable of the system for a given measurement process. [1] M. Ban, J. Phys. A: Math. Gen. 32, 1643 (1999). |

2014/11/11(Tue.) @#913 14:45-

speaker | Ken Funo |

title | Quantum fluctuation theorems under feedback control |

abstract | Nonequilibrium equalities such as Fluctuation theorems and Jarzynski equalities have attracted considerable interest, since they give equalities on physical quantities in nonequilibrium processes, in contrast with the conventional second law, which gives inequalities. We consider fluctuation theorems in the quantum system and try to extend them to situations involving feedback control by utilizing correlations between subsystems. In the case of measurement-based feedback control, fluctuation theorems for both the system and memory are modified by including information contents characterizing the acquired information due to the measurement. We also take into account the absolute irreversible processes to quantify the efficiency of the feedback control and the effect of the irreversibility of the memory during the measurement process. Finally, we will shortly talk about the case when we have initially entanglement states where we can utilize them to perform coherent feedback control. |

2014/11/18(Tue.) @#913 14:45-

speaker | Tomohiro Shitara |

title | Information-disturbance relation in quantum measurement based on estimation theory |

abstract | It has been widely known that the more information we extract by quantum measurement, the more disturbed the system is. Many studies have been done to quantitatively formulate relations between information and disturbance. Most of the studies are based on quantum information theory, and formulate information and disturbance by Shannon-entropic quantities. We formulate information and disturbance in a general setting based on the Fisher information, which is a crucial quantity in quantum state estimation, and derive inequalities between them. Information is defined as the classical Fisher information obtained from the measurement, which gives the inverse of the accuracy of estimation. Disturbance is defined as the loss of the quantum Fisher information. Since there are (infinitely) many types of quantum Fisher information, the definition of disturbance is not unique. We reveal that for pure measurement, RLD (right logarithmic derivative) Fisher information gives the minimum disturbance, hence the tightest inequality. |

2014/11/25(Tue.) @#913 14:45-

speaker | Yuto Murashita |

title | Absolute irreversibility in nonequilibrium equalities and the Gibbs paradox |

abstract |
Nonequilibrium equalities have attracted considerable attention in the context of statistical mechanics and information thermodynamics. Although nonequilibrium equalities apply to rather general nonequilibrium situations, integral nonequilibrium equalities break down in such situations as free expansion. We propose a new concept of absolute irreversibility that encompasses those situations to which conventional integral nonequilibrium equalities cannot apply. In mathematical terms, absolute irreversibility is identified as the singularity of probability measure. Using Lebesgue窶冱 decomposition theorem in measure theory, we obtain generalized nonequilibrium equalities applicable to absolutely irreversible processes [1].
The Gibbs paradox deals with the difference of the entropy production between the mixing of different kinds of gas and the mixing of identical gas. Standard resolutions of the Gibbs paradox are based on the quantum indistinguishability of identical particles or the extensivity of the thermodynamic entropy. These resolutions break down in mesoscopic classical systems. We propose a new resolution applicable to the mesoscopic classical regime based on our nonequilibrium equalities, because the entropy production of the gas mixing can be precisely quantified by the absolute irreversible probability. [1] Yuto Murashita, Ken Funo, and Masahito Ueda, Phys. Rev. E 90, 042110 (2014). |

2014/12/2(Tue.) @#1220 13:00-

speaker | Shuhei Yoshida |

title | Universal Relations in Strongly Interacting P-Wave Fermi Gases |

abstract |
When an effective interaction among particles is strong, there are few well-established theoretical methods which are capable of coping with it. They are especially important after the realization of a strong coupling via the Feshbach resonance in cold-atomic systems which offer high controllability and accurate measurements of various physical quantities. It was pointed out, however, that there are several universal relations of physical quantities which hold in strongly-interacting systems regardless of their macroscopic phase, the number of particles, and so on. They have been intensively studied in the field of the Bardeen-Cooper-Schrieffer-to-Bose-Einstein-Condensation crossover, both theoretically and experimentally. One other interesting system than the BCS-BEC crossover is a gas of spinless fermions. By selectively trapping atoms in a single hyperfine state, one can naturaly realize such a system, in which an s-wave interaction is prohibited by the Pauli's exclusion principle and a p-wave interaction dominates. A strong p-wave interaction is also realized by the Feshbach resonance. In this seminar, I will discuss universal relations in gases of spinless fermions in the presence of a strong p-wave interaction. |

speaker | Kazuya Fujimoto (Dept. of Mathematics and Physics, Osaka City University) |

title | Spin correlation in turbulence of spin-1 ferromagnetic spinor Bose-Einstein condensate |

abstract |
Turbulence is strong non-equilibrium phenomenon exhibiting unpredictable and chaotic behaviors, being one of the unresolved and important problems in modern physics. Recently, turbulence in atomic Bose-Einstein condensates(BECs) has attracted considerable attention because this system has the high controllability, the excellent observation by optical techniques, and the realization of novel quantum fluids such as multi-component BECs, which can offer new stage for turbulence study in quantum fluids.
In this seminar, I would like to present our recent theoretical and numerical works for the turbulence in spin-1 ferromagnetic spinor BEC [1,2]. In this turbulence, the spin field as well as the velocity one is much disturbed, so that we call it spin turbulence. We focus on the spin correlation function in the spin turbulence, finding the -7/3 power law in this correlation by using the spin-1 spinor Gross-Pitaevskii equation, which means the Kolmogorov turbulence appears in the spin degrees of freedom. Also, we consider that it is possible to observe the -7/3 power law, which leads to the first confirmation of the Kolmogorov turbulence in the atomic BECs.
[1] K. Fujimoto and M. Tsubota, Phys. Rev. A, 85, 033642 (2012). [2] K. Fujimoto and M. Tsubota, Phys. Rev. A, 85, 053641 (2012). |

2014/12/16(Tue.) @#913 14:45-

speaker | Yusuke Horinouchi |

title | Onset of a limit cycle and universal three-body parameter in Efimov physics |

abstract | Efimov effect is the only experimentaly realized universal phenomenon that exibits a renormalization-group limit cycle, reflecting its discrete scale invariance: When three particles interact resonantly, they form an infinite geometric series of three-body bound states when the interaction is too weak to support a two-body bound state. The Efimov effect is considered to be universal in the sence that the system is parametrized by a single parameter, i.e. the three-body parameter that sets the scattering threshold of the lowest-lying Efimov trimer. Recent experiments in ultracold atom experiments have unexpectedly revealed that the three-body parameter itself is universal when measured in units of an effective range. This universality has been numerically vindicated [J. Wang et al., PRL 108, 263001 (2012)] and its physical origin has been clarified [P. Naidon et al., PRL 112, 105301 (2014)]. We here address an as yet untapped problem: Does this experimentally found universality bring about a new universality in the renormalization-group limit cycle? We answer this question in the affirmative by using the Functional Renormalization Group, nonperturbative character of which plays a decisive role in revealing this new universality. A close connection between a topological property of the limit cycle and universal four-body physics is also suggested. |

2015/1/6(Tue.) @# 14:45-

speaker | Kohaku So (Hongbo Zeng) |

title | Mathematical Physics of Cold Atoms: the Gross-Pitaevskii theory |

abstract | Gross-Pitaevskii (GP) theory is a widely used effective theory to describe both static and dynamical behaviours of Bose-Einstein condensate. Although GP theory is usually heruistically derived from effective interaction model and mean-field treatment, in view of the significant role played by the theory in modern cold-atom physics, it is interesting to ask whether, and when, it can be rigorously justified from the microscopic many-body Hamiltonian. In this semianr, I will review a mathematical phsyics study concerning this problem [Lieb et al, PRA 61, 043602 (2000)]. It was shown that exact quantum mechancial ground energy and density profile and those of GP theory asymptotically coincide in the large particle number and dilute limit. Also, I will touch on problems and accomplishments concerning the proof of the existence of BEC. |

2015/1/13(Tue.) @# 14:45-

speaker | Yuto Ashida |

title | Diffraction-limit-free nondestructive measurement of ultracold atoms in an optical lattice |

abstract | Diffraction limit has long imposed severe restrictions on optical experiments. Recent achievements of single-site-resolved detections are not exceptions: the diffraction limit requires a large number of photodetections which, in turn, forces us to use a near-resonant probe light causing destructions of atomic states. As a result, all the experiments of single-site-resolved detections performed to date are destructive measurements. In this talk, I will introduce a method of surpassing the diffraction limit and achieving a nondestructive measurement of atoms in an optical lattice. I will also discuss an application to other imaging systems such as fluorecent molecules in life science. |

2014/1/20(Tue.) @# 14:45-

speaker | Sho Higashikawa |

title | Characterization of spontaneous symmetry breaking and application to SU(N) symmetric system |

abstract | Recent realization of an SU(N) symmetric system by multi-component ultracold atomic gases have opened up a great avenue for exploring exotic phases unique to high symmetry groups. However, theoretical analysis becomes more and more difficult as N grows due to the complexity of the Hamiltonian and nontrivial symmetry breaking. We show that Lie algebra and its representation provide a simple and powerful method to deal with this complexity and classify various symmetry broken phases for different N in a unified manner. Also, we identify unbroken symmetry and the number of Nambu-Goldstone bosons with both linear and parabolic dispersion for these phases. |

2015/1/27(Tue.) @# 14:45-

speaker | Takeshi Saito |

title | Interacting particles in synthetic gauge fields and dimensions: Emergence of devil's staircase |

abstract |
A synthetic dimension is an extended 'dimension' provided by the internal atomic degrees of freedom. Raman transitions between the internal states allow us to introduce hoppings with additional phase factors along the synthetic dimension.
With this idea, it has been proposed that two-dimensional (2D) quantum gases with synthetic gauge fields can be realized from one-dimensional (1D) multi-component gases and that some interesting features such as the emergence of chiral edge states can be observed in the non-interacting case [1].
Here we study the interaction effects in such systems. The interaction is non-decaying with the distance along the synthetic dimension, sharply contrasting with usual 2D systems.
Within the lowest-Landau-level approximation, we derive an effective 1D lattice model for these systems, in which hopping amplitudes and interactions decay like a Gaussian with the distance along the physical dimension.
We discuss that the emergent long-range interactions here stabilize a cascade of crystal ground states known as 'devil's staircase'. [1] A. Celi et al., Phys. Rev. Lett. 112, 043001 (2014). |

speaker | Ryusuke Hamazaki |

title | Generalized Gibbs ensemble in nonintegrable systems with abelian discrete symmetries |

abstract |
Recent progress of experimental techniques has enabled us to study the dynamics of isolated quantum systems. One of the particularly interesting questions is how an isolated system approaches the ensemble predicted by the standard statistical mechanics. This “thermalization” is in fact absent in integrable systems, where infinitely many conserved quantities restrict time evolution. It is predicted that these systems relax to what is called generalized Gibbs ensemble(GGE), which maximizes Shannon entropy under the constraint of conserved quantities. On the other hand, with a lot of numerical evidence, generic nonintegrable systems are believed to thermalize by the mechanism of eigenstate thermalization hypothesis(ETH). However, since discrete symmetries automatically entail a finite number of conserved quantities, nonintegrable systems with such symmetries may relax to non-thermal states, as is the case with integrable ones. We therefore numerically investigate the relaxation of a nonintegrable hard-core Bose-Hubbard system, which has some parity symmetries. We firstly show the stationary state is indeed described better by a GGE-like distribution, which contains the information of parity, than by the canonical ensemble. We next argue the above result is attributed to the scenario of generalized ETH for each symmetry sector, instead of usual ETH. |

Schedule of summer semester (start from 13:00 @ #913)

July 9 Haruki Watanabe (University of California, Berkeley)

Abstract

April 17 (Thu) Tatsuhiko N. Ikeda

"The finite group velocity of quantum spin systems"

E. Lieb and D. W. Robinson, Comm. Math. Phys. 28, 3, (1972)

"Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order"

S. Bravyi, M. B. Hastings, and F. Verstraete, Phys. Rev. Lett. 97, 050401 (2006)

April 24 (Thu) Yui Kuramochi

"Manipulation of photons in a cavity by dispersive atom-field coupling: Quantum-nondemolition measurements and generation of 窶倪\牢chrodinger cat窶吮\・states"

M. Brune, S. Haroche, J. M. Raimond, L. Davidovich, and N. Zagury, Phys. Rev. A 45, 5193 (1992)

"Quantum jumps of light recording the birth and death of a photon in a cavity"

S. Gleyzes et al. Nature 446, 297-300 (2007)

May 1 (Thu) Ken Funo

""Irreversibility" of the flux of the renormalization group in a 2D field theory"

A. B. Zomolodchikov, JETP Lett. 43, 12 (1986)

"A finite entanglement entropy and the c-theorem"

H. Casini and M. Huerta, Phys. Lett. B 600 (2004)

"Renormalization group running of the entanglement entropy of a circle"

H. Casini and M. Huerta, Phys. Rev. D 85, 125016 (2012)

May 8 (Thu) Kohaku So (Hongbo Zeng)

"The stability of matter"

E. H. Lieb, Rev. Mod. Phys. 48, 553 (1976)

May 22 (Thu) Sho Higashikawa

"Resistance Minimum in Dilute Magnetic Alloys"

J. Kondo, Prog. Theor. Phys. 32, 1 (1964)

"Localized Magnetic States in Metals"

P. W. Anderson, Phys. Rev. 124, 41 (1961)

May 28 (Wed) @#201a Tomohiro Shitara

"General state changes in quantum theory"

K. Kraus, Ann. Phys. 64 (1971)

"Positive Functions on C*-Algebras"

W. F. Stinespring, Proc. Amer. Math. Soc. 6 (1955)

June 5 (Thu) Yuto Murashita

"Transport, Collective Motion, and Brownian Motion"

H. Mori, Prog. Theor. Phys. 33, 3 (1964)

June 12 (Thu) Shuhei Yoshida

"Interpretation of Recent Results on He3 below 3 mK: A New Liquid Phase?"

A. J. Leggett, Phys. Rev. Lett. 29, 1227 (1972)

"Evidence for a New Phase of Solid He3"

D. D. Osheroff, R. C. Richardson, and D. M. Lee, Phys. Rev. Lett. 28, 885 (1972)

"New Magnetic Phenomena in Liquid He3 below 3 mK"

D. D. Osheroff, W. J. Gully, R. C. Richardson, and D. M. Lee, Phys. Rev. Lett. 29, 920 (1972)

June 19 (Thu) Yuto Ashida

"Environment-Independent Decoherence Rate in Classically Chaotic Systems"

R. A. Jalabert and H. M. Pastawski, Phys. Rev. Lett. 86, 2490 (2001)

June 26 (Thu) Yusuke Horinouchi

"Diagram Technique for Nonequilibrium Processes"

L. V. Keldysh, JETP 20, 4 (1965)

2014/7/9(Wed) @#206 13:00-

speaker | Haruki Watanabe (University of California, Berkeley) |

title | Nambu-Goldstone bosons in systems without Lorentz invariance |

abstract |
The method of low-energy effective Lagrangians is a very powerful tool to systematically examine the properties of Nambu-Goldstone bosons (NGBs), including interactions among NGBs and with other low-energy degrees of freedom. Based on pioneering works by H. Leutwyler, we have further clarified the general structure of the effective Lagrangian for non-relativstic systems.
After reviewing the effective Lagrangian, we will first discuss the counting rule of NGBs and their dispersion relations [1]. Zero modes (moduli) of topological solutions can be understood on the same footing [2].
The second part of the talk is devoted to the effect of interactions among low-energy degrees of freedom. The peculiar fractional-power dispersion relation of ripplons (E = p^1.5) on a superfluid-superfluid interface can be understood as a result of interactions between NGBs in the bulk and on the surface [3]. If time permits, we will also discuss non-Fermi liquid behaviors (breakdown of Landau窶冱 Fermi liquid description) due to interactions between fermions and a certain class of NGBs [4]. [1] HW and H. Murayama, Phys. Rev. Lett. 108, 251602 (2012). [2] HW and H. Murayama, Phys. Rev. Lett. 112, 191804 (2014). [3] HW and H. Murayama, Phys. Rev. D 89 101701 (R) (2014). [4] HW and A. Vishwanath, arXiv:1404.3728. |

July 10 (Thu) Masataka Aizawa

"The Singularities of Gravitational Collapse and Cosmology"

S. W. Hawking and R. Penrose, Proc. R. Soc. Lond. A 27, 314 (1970)

July 24 (Thu) @#201a Daisuke Miyafuji

"Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems"

R. Kubo, J. Phys. Soc. Jpn. 12 (1957)

"Statistical-Mechanical Theory of Irreversible Processes. II. Response to Thermal Disturbance"

R. Kubo, M. Yokota, and S. Nakajima, J. Phys. Soc. Jpn. 12 (1957)