arXiv subjects

Kyosuke Tsumura

Publications and source records attributed to Kyosuke Tsumura.

18 recordsLinked to original sources

arXiv2016

Second-order hydrodynamics for fermionic cold atoms: Detailed analysis of transport coefficients and relaxation times

We give a detailed derivation of the second-order (local) hydrodynamics for Boltzmann equation with an external force by using the renormalization group method. In this method, we solve the Boltzmann equation faithfully to extract the hydrodynamics without recourse to any ansatz. Our method leads to microscopic expressions of not only all the transport coefficients that are of the same form as those in Chapman-Enskog method but also those of the viscous relaxation times $τ_i$ that admit physically natural interpretations. As an example, we apply our microscopic expressions to calculate the transport coefficients and the relaxation times of the cold fermionic atoms in a quantitative way, where the transition probability in the collision term is given explicitly in terms of the $s$-wave scattering length $a_s$. We thereby discuss the quantum statistical effects, temperature dependence, and scattering-length dependence of the first-order transport coefficients and the viscous relaxation times: It is shown that as the temperature is lowered, the transport coefficients and the relaxation times increase rapidly because Pauli principle acts effectively. On the other hand, as $a_s$ is increased, these quantities decrease and become vanishingly small at unitarity because of the strong coupling. The numerical calculation shows that the relation $τ_π=η/P$, which is derived in the relaxation-time approximation and used in most of literature without almost any foundation, turns out to be satisfied quite well, while the similar relation for the relaxation time $τ_J$ of the heat conductivity is satisfied only approximately with a considerable error.

cond-mat.quant-gas↗

arXiv2016

Mesoscopic dynamics of fermionic cold atoms ---Quantitative analysis of transport coefficients and relaxation times---

We give a quantitative analysis of the dynamical properties of fermionic cold atomic gases in normal phase, such as the shear viscosity, heat conductivity, and viscous relaxation times, using the novel microscopic expressions derived by the renormalization group (RG) method, where the Boltzmann equation is faithfully solved to extract the hydrodynamics without recourse to any ansatz. In particular, we examine the quantum statistical effects, temperature dependence, and scattering-length dependence of the transport coefficients and the viscous relaxation times. The numerical calculation shows that the relation $τ_π=η/P$, which is derived in the relaxation-time approximation (RTA) and is used in most of the literature, turns out to be satisfied quite well, while the similar relation for the viscous relaxation time $τ_J$ of the heat conductivity is satisfied only approximately with a considerable error.

cond-mat.quant-gas↗

arXiv2015

Relativistic Causal Hydrodynamics Derived from Boltzmann Equation: a novel reduction theoretical approach

We derive the second-order hydrodynamic equation and the microscopic formulae of the relaxation times as well as the transport coefficients systematically from the relativistic Boltzmann equation. Our derivation is based on a novel development of the renormalization-group method, a powerful reduction theory of dynamical systems, which has been applied successfully to derive the non-relativistic second-order hydrodynamic equation Our theory nicely gives a compact expression of the deviation of the distribution function in terms of the linearized collision operator, which is different from those used as an ansatz in the conventional fourteen-moment method. It is confirmed that the resultant microscopic expressions of the transport coefficients coincide with those derived in the Chapman-Enskog expansion method. Furthermore, we show that the microscopic expressions of the relaxation times have natural and physically plausible forms. We prove that the propagating velocities of the fluctuations of the hydrodynamical variables do not exceed the light velocity, and hence our seconder-order equation ensures the desired causality. It is also confirmed that the equilibrium state is stable for any perturbation described by our equation.

hep-ph↗

arXiv2015

Causal hydrodynamics from kinetic theory by doublet scheme in renormalization-group method

We develop a general framework in the renormalization-group (RG) method for extracting a mesoscopic dynamics from an evolution equation by incorporating some excited (fast) modes as additional components to the invariant manifold spanned by zero modes. We call this framework the doublet scheme. The validity of the doublet scheme is first tested and demonstrated by taking the Lorenz model as a simple three-dimensional dynamical system; it is shown that the two-dimensional reduced dynamics on the attractive manifold composed of the would-be zero and a fast modes are successfully obtained in a natural way. We then apply the doublet scheme to construct causal hydrodynamics as a mesoscopic dynamics of kinetic theory, i.e., the Boltzmann equation, in a systematic manner with no ad-hoc assumption. It is found that our equation has the same form as thirteen-moment causal hydrodynamic equation, but the microscopic formulae of the transport coefficients and relaxation times are different. In fact, in contrast to the Grad equation, our equation leads to the same expressions for the transport coefficients as given by the Chapman-Enskog expansion method and suggests novel formulae of the relaxation times expressed in terms of relaxation functions which allow a natural physical interpretation of the relaxation times. Furthermore, our theory nicely gives the explicit forms of the distribution function and the thirteen hydrodynamic variables in terms of the linearized collision operator, which in turn clearly suggest the proper ansatz forms of them to be adopted in the method of moments.

physics.flu-dyn↗

arXiv2015

Derivation of second-order relativistic hydrodynamics for reactive multi-component systems

We derive the second-order hydrodynamic equation for reactive multi-component systems from the relativistic Boltzmann equation. In the reactive system, particles can change their species under the restriction of the imposed conservation laws during the collision process. Our derivation is based on the renormalization group (RG) method, in which the Boltzmann equation is solved in an organized perturbation method as faithfully as possible and possible secular terms are resummed away by a suitable setting of the initial value of the distribution function. The microscopic formulae of the relaxation times and the lengths are explicitly given as well as those of the transport coefficients for the reactive multi-component system. The resultant hydrodynamic equation with these formulae has nice properties that it satisfies the positivity of the entropy production rate and the Onsager's reciprocal theorem, which ensure the validity of our derivation.

hep-ph↗

arXiv2015

Rare transition event with self-consistent theory of large-amplitude collective motion

A numerical simulation method, based on Dang et al.'s self-consistent theory of large-amplitude collective motion, for rare transition events is presented. The method provides a one-dimensional pathway without knowledge of the final configuration, which includes a dynamical effect caused by not only a potential but also kinetic term. Although it is difficult to apply the molecular dynamics simulation to a narrow-gate potential, the method presented is applicable to the case. A toy model with a high-energy barrier and/or the narrow gate shows that while the Dang et al. treatment is unstable for a changing of model parameters, our method stable for it.

cond-mat.other↗

arXiv2012

Derivation of relativistic hydrodynamic equations consistent with relativistic Boltzmann equation by renormalization-group method

We review our work on the application of the renormalization-group method to obtain first- and second-order relativistic hydrodynamics of the relativistic Boltzmann equation (RBE) as a dynamical system, with some corrections and new unpublished results. For the first-order equation, we explicitly obtain the distribution function in the asymptotic regime as the invariant manifold of the dynamical system, which turns out to be nothing but the matching condition defining the energy frame, i.e., the Landau-Lifshitz one. It is argued that the frame on which the flow of the relativistic hydrodynamic equation is defined must be the energy frame, if the dynamics should be consistent with the underlying RBE. A sketch is also given for derivation of the second-order hydrodynamic equation, i.e., extended thermodynamics, which is accomplished by extending the invariant manifold so that it is spanned by excited modes as well as the zero modes (hydrodynamic modes) of the linearized collision operator. On the basis of thus constructed resummed distribution function, we propose a novel ansatz for the functional form to be used in Grad moment method; it is shown that our theory gives the same expressions for the transport coefficients as those given in the Chapman-Enskog theory as well as the novel expressions for the relaxation times and lengths allowing natural interpretation.

nucl-th↗

arXiv2012

Uniqueness of Landau-Lifshitz Energy Frame in Relativistic Dissipative Hydrodynamics

We show that the relativistic dissipative hydrodynamic equation derived from the relativistic Boltzmann equation by the renormalization-group method uniquely leads to the one in the energy frame proposed by Landau and Lifshitz, provided that the macroscopic-frame vector, which defines the local rest frame of the fluid velocity, is independent of the momenta of constituent particles, as it should. We argue that the relativistic hydrodynamic equations for viscous fluids must be defined on the energy frame if it is consistent with the underlying relativistic kinetic equation.

physics.flu-dyn↗

arXiv2012

New forms of non-relativistic and relativistic hydrodynamic equations as derived by the renormalization-group method - possible functional ansatz in the moment method consistent with Chapman-Enskog theory -

After a brief account of the derivation of the first-order relativistic hydrodynamic equation as a construction of the invariant manifold of relativistic Boltzmann equation, we give a sketch of derivation of the second-order hydrodynamic equation (extended thermodynamics) both in the nonrelativistic and relativistic cases. We show that the resultant equation suggests a novel ansatz for the functional form to be used in Grad moment method, which turns out to give the same expressions for the transport coefficients as those given in the Chapman-Enskog theory as well as the novel expressions for the relaxation times and lenghts allowing natural physical interpretaion.

nucl-th↗

arXiv2011

First-principle Derivation of Stable First-order Generic-frame Relativistic Dissipative Hydrodynamic Equations from Kinetic Theory by Renormalization-group Method

We derive first-order relativistic dissipative hydrodynamic equations (RDHEs) from relativistic Boltzmann equation (RBE) on the basis of the renormalization-group (RG) method. We introduce a macroscopic-frame vector (MFV) to specify the local rest frame (LRF) on which the macroscopic dynamics is described. The five hydrodynamic modes are identified with the same number of the zero modes of the linearized collision operator, i.e., the collision invariants. After defining the inner product in the function space spanned by the distribution function, the higher-order terms, which give rise to the dissipative effects, are constructed so that they are orthogonal to the zero modes in terms of the inner product: Here, any ansatz's, such as the so-called conditions of fit used in the standard methods in an ad-hoc way, are not necessary. We elucidate that the Burnett term dose not affect the RDHEs owing to the very nature of the hydrodynamic modes as the zero modes. Applying the RG equation, we obtain the RDHE in a generic LRF specified by the MFV, as the coarse-grained and covariant equation. Our generic RDHE reduces to RDHEs in various LRFs, including the energy and particle LRFs with a choice of the MFV. We find that our RDHE in the energy LRF coincides with that of Landau and Lifshitz, while the derived RDHE in the particle LRF is slightly different from that of Eckart, owing to the presence of the dissipative internal energy. We prove that the Eckart equation can not be compatible with the underlying RBE. The proof is made on the basis of the observation that the orthogonality condition to the zero modes coincides with the ansatz's posed on the dissipative parts of the energy-momentum tensor and the particle current in the phenomenological RDHEs. We also present an analytic proof that all of our RDHEs have a stable equilibrium state owing to the positive definiteness of the inner product.

hep-ph↗

arXiv2010

Second-order Relativistic Hydrodynamic Equations for Viscous Systems; how does the dissipation affect the internal energy?

We derive the second-order dissipative relativistic hydrodynamic equations in a generic frame with a continuous parameter from the relativistic Boltzmann equation. We present explicitly the relaxation terms in the energy and particle frames. Our results show that the viscosities are frame-independent but the relaxation times are generically frame-dependent. We confirm that the dissipative part of the energy-momentum tensor in the particle frame satisfies $δT^μ_μ= 0$ obtained for the first-order equation before, in contrast to the Eckart choice $u_μδT^{μν} u_ν= 0$ adopted as a matching condition in the literature. We emphasize that the new constraint $δT^μ_μ= 0$ can be compatible with the phenomenological derivation of hydrodynamics based on the second law of thermodynamics.

hep-ph↗

arXiv2009

Critical Opalescence around the QCD Critical Point and Second-order Relativistic Hydrodynamic Equations Compatible with Boltzmann Equation

The dynamical density fluctuations around QCD critical point (CP) are analyzed using relativistic dissipative fluid dynamics, and we show that the sound mode around the QCD CP is strongly attenuated whereas the thermal fluctuation stands out there. We speculate that if possible suppression or disappearance of a Mach cone, which seems to be created by the partonic jets at RHIC, is observed as the incident energy of the heavyion collisions is decreased, it can be a signal of the existence of the QCD CP. We have presented the Israel-Stewart type fluid dynamic equations that are derived rigorously on the basis of the (dynamical) renormalization group method in the second part of the talk, which we omit here because of a lack of space.

hep-ph↗

arXiv2008

Relaxation Time and Relaxation Function of Quark-Gluon Plasma with Lattice QCD

We propose a method which enables a QCD-based calculation of a relaxation time for a dissipative current in the causal and dissipative hydrodynamic equation derived by Israel and Stewart. We point out that the Israel-Stewart equation is not unique as a causal and dissipative hydrodynamic equation, and the form of the causal and dissipative hydrodynamic equation is determined by the shape of a spectral function reflecting the properties of elementary excitations in the system we consider. Our method utilizes a relaxation function, which can be calculated from QCD using the linear response theory. We show that the relaxation function can be derived from a spectral function for a microscopic representation of the dissipative current. We also show that the Israel-Stewart equation is acceptable only as long as the calculated relaxation function is approximated well by a exponentially damping function, and the relaxation time can be obtained as its damping time constant. Taking a baryon-number dissipative current of a plasma consisting of charm quarks and gluons as a simple example, we present the first calculation of the relaxation function with use of the spectral function derived employing the quenched lattice QCD together with the maximum entropy method. The calculated relaxation function shows a strongly-oscillation damping behaviour due to the charmed vector hadron $J/Ψ$ surviving above the deconfinement phase transition temperature in QCD. This result suggests that the applicability of the Israel-Stewart equation to the baryon-number dissipative current of the charm quark-gluon plasma is quite doubtful. We present an idea for the improvement of the Israel-Stewart equation by deriving the hydrodynamic equation consistent with the strongly-oscillation damping relaxation function.

nucl-th↗

arXiv2008

Stable First-order Particle-frame Relativistic Hydrodynamics for Dissipative Systems

We propose a stable first-order relativistic dissipative hydrodynamic equation in the particle frame (Eckart frame) for the first time. The equation to be proposed was in fact previously derived by the authors and a collaborator from the relativistic Boltzmann equation. We demonstrate that the equilibrium state is stable with respect to the time evolution described by our hydrodynamic equation in the particle frame. Our equation may be a proper starting point for constructing second-order causal relativistic hydrodynamics, to replace Eckart's particle-flow theory.

nucl-th↗

arXiv2007

Tetra-Quark Resonances in Lattice QCD

We study $qq \bar q \bar q$-type four-quark (4Q) systems in SU(3)$_c$ anisotropic quenched lattice QCD, using the $O(a)$-improved Wilson (clover) fermion at $β=5.75$ on $12^3 \times 96$ with renormalized anisotropy $a_s/a_t=4$.For comparison, we first investigate the lowest $q\bar q$ scalar meson from the connected diagram and find its large mass of about 1.32GeV after chiral extrapolation, and thus the lowest $q\bar q$ scalar meson corresponds to $f_0(1370)$.We investigate the lowest 4Q state in the spatially periodic boundary condition, and find that it is just a two-pion scattering state, as is expected. To examine spatially-localized 4Q resonances, we use the Hybrid Boundary Condition (HBC) method, where anti-periodic and periodic boundary conditions are imposed on quarks and antiquarks, respectively. By applying HBC on a finite-volume lattice, the threshold of the two-meson scattering state is raised up, while the mass of a compact 4Q resonance is almost unchanged.In HBC, the lowest 4Q state appears slightly below the two-meson threshold. To clarify the nature of the 4Q system, we apply the Maximum Entropy Method (MEM) for the 4Q correlator and obtain the spectral function of the 4Q system.From the combination analysis of MEM with HBC, we finally conclude that the 4Q system appears as a two-pion scattering state and there is no spatially-localized 4Q resonance in the quark-mass region of $m_s< m_q <2m_s$.

hep-lat↗

arXiv2007

Peristaltic modes of single vortex in the Abelian Higgs model

Using the Abelian Higgs model, we study the radial excitations of single vortex and their propagation modes along the vortex line. We call such beyond-stringy modes {\it peristaltic} modes of single vortex. With the profile of the static vortex, we derive the vortex-induced potential, i.e., single-particle potential for the Higgs and the photon field fluctuations around the static vortex, and investigate the coherently propagating fluctuations which corresponds to the vibration of the vortex. We derive, analyze and numerically solve the field equations of the Higgs and the photon field fluctuations around the static vortex with various Ginzburg-Landau parameter $κ$ and topological charge $n$. Around the BPS value or critical coupling $κ^2$=1/2, there appears a significant correlation between the Higgs and the photon field fluctuations mediated by the static vortex. As a result, for $κ^2$=1/2, we find the characteristic {\it new} discrete pole of the peristaltic mode corresponding to the quasi-bound-state of coherently fluctuating fields and the static vortex. We investigate its excitation energy, correlation energy of coherent fluctuations, spatial distributions, and the resulting magnetic flux behavior in detail. Our investigation covers not only usual Type-II vortices with $|n|$=1 but also Type-I and Type-II vortices with $n\in Z$ for the application to various general systems where the vortex-like objects behave as the essential degrees of freedom.

hep-ph↗

arXiv2005

Application of the Renormalization-group Method to the Reduction of Transport Equations

We first give a comprehensive review of the renormalization group method for global and asymptotic analysis, putting an emphasis on the relevance to the classical theory of envelopes and on the importance of the existence of invariant manifolds of the dynamics under consideration. We clarify that an essential point of the method is to convert the problem from solving differential equations to obtaining suitable initial (or boundary) conditions:The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. The RG method is applied to derive the Navier-Stokes equation from the Boltzmann equation, as an example of the reduction of dynamics. We work out to obtain the transport coefficients in terms of the one-body distribution function.

hep-th↗

arXiv2005

Lattice QCD Evidence for Exotic Tetraquark Resonance

We study the manifestly exotic tetraquark D$_{\rm s0}^{++}(cu\bar s \bar d)$ and the scalar tetraquark $f_0(ud \bar u \bar d)$ in SU(3)$_c$ anisotropic quenched lattice QCD with the $O(a)$-improved Wilson (clover) fermion with various quark masses including the idealized SU(4)$_f$ case. For $f_0(ud\bar u \bar d)$ etc., we only consider connected diagrams at the quenched level, i.e., the tetraquark $f_0(ud\bar u\bar d)$ is identical with D$_{\rm s0}^{++} (cu\bar s \bar d)$ in the idealized SU(4)$_f$ case. First, for comparison, we study the lowest $q\bar q$ scalar meson, and find that it has a large mass of about 1.37GeV after chiral extrapolation, which corresponds to $f_0(1370)$. Second, we investigate the lowest 4Q state in the spatial periodic boundary condition, and find that it is just a scattering state of two pseudoscalar mesons, as is expected. Third, to extract spatially-localized 4Q resonances, we use the Hybrid Boundary Condition (HBC) method, where anti-periodic and periodic boundary conditions are imposed on quarks ($c$,$u$) and antiquarks ($\bar s$,$\bar d$), respectively. By applying the HBC on a finite-volume lattice, the threshold of the two-meson scattering state is raised up, while the mass of a compact 4Q resonance is almost unchanged. In HBC, we find a nontrivial 4Q resonance state about 100 MeV below the two-meson threshold in some quark-mass region. Its chiral behavior largely differs from a two-meson scattering state. The scalar tetraquark $f_0(ud\bar u\bar d)$ is found to have the mass of about 1.1GeV after chiral extrapolation, and seems to correspond to $f_0(980)$. Then, the manifestly exotic tetraquark D$_{\rm s0}^{++}(cu\bar s \bar d)$ would exist around 1GeV in the idealized SU(4)$_f$ chiral limit. Finally, MEM analysis is applied to obtain the spectral function of the 4Q system.

hep-lat↗