Basic concepts and recent progresses of the tensor network formulation are reviewed. The key feature of the formulation is that it keeps the spatial structure of the entanglement within limited degrees of freedom. A relation between the tensor network formulation and the real-space renormalization group formulation is also explained.
Optical tomography is one of medical imaging modalities such as X-ray CT, ultrasound, and MRI. Near-infrared light is used because light in this range of wavelength is absorbed relatively weakly in biological tissue. Such light, however, is strongly scattered in biological tissue and the specific intensity of light obeys the radiative transport equation, which is a linear Boltzmann equation. Optical tomography is formulated as an inverse problem of determining coefficients of the radiative transport equation from boundary measurements. Since it is difficult to solve the radiative transport equation analytically and even numerically, the radiative transport equation is usually approximated by the diffusion equation. As one can easily imagine, the use of this diffusion approximation results in low resolution of reconstructed images. Recent progress in this field makes it possible to obtain tomographic images using the radiative transport equation without relying on the diffusion approximation.
Dimensionality plays a key role in the physics of solids and liquids. Fluctuation shows up differently in different spatial dimensions, as typically observed in phase transitions. In this paper, we introduce our recent study on the dimensionality dependence of glassy fluctuation. We clarify that enhancement of fluctuation in a 2D glass takes place because of a mechanism similar to Mermin-Wagner theorem for a 2D crystal. It is justified by direct calculation of the Debye-asymptote of the vibrational density of states of acoustic vibrations, which also accounts for enhanced fluctuation recently observed in 2D glassy systems. It leads to system-size dependent behavior of the estimated relaxation time and dynamic correlation length in the 2D system in terms of the density-based correlation functions. However, such size dependence is eliminated by introduction of an alternative correlator that characterizes relative rearrangement motion corresponding to the inherent structural relaxation.
Partition of energy in chain-type systems where masses are connected by springs or rigid links is investigated. When the chain is composed of masses connected with rigid links, we found that the values of average kinetic energy of the masses are not the same but those which are near to the ends of the chain have larger values than those in the middle of the chain. We clarified that this inhomogeneity results from the generalized equipartition of energy. When the chain is composed of masses connected with springs, average kinetic energies of the masses are the same in the thermal equilibrium. However, inhomogeneity of kinetic energy is observed as transient, then relaxes to homogeneous thermal equilibrium. The relaxation time τ to the equilibrium takes long when the spring constant k is large. We found that τ∝ exp (c√k), which is consistent with Boltzmann-Jeans law. The Boltzmann-Jeans law also holds when the chain is in solvent.
Nuclear emulsion is a powerful tracking device that can record the three-dimensional trajectory of charged particles within 1 μm spatial resolution. We are promoting GRAINE project which is 10 MeV–100 GeV cosmic γ-ray observations with a precise (0.08° at 1–2 GeV) and polarization-sensitive large-aperture-area (～10 m2) emulsion telescope by repeating long duration balloon flights. We are developing multi-stage shifter which allows us to give a timing information to emulsion tracks with ～seconds or below. The multi-stage shifter opened feasibilities of precise cosmic γ-ray observations, GRAINE, as well as precise measurements of ν –N interactions, J-PARC T60. ～Millisecond time resolution in a balloon-borne experiment, ～second time resolution for 126.7 days in an accelerator ν experiment and ～106 time-resolved numbers are being achieved. New model of multi-stage shifter is also being developed for future experiments.