arXiv subjects
Publications and source records attributed to Kenichiro Aoki.
The relation between the expectation values computed in the random walk theory, and the heat kernel method for the diffusion equation is explained concretely. The random walk is also realized by simulations and their statistical uncertainties are analyzed.
We observe thermal motions of ions, and molecules in water through light extinction, at the individual particle level. The motions appear as time dependent intensity variations, characterized through their averaged spectra. Theoretical spectrum derived from random motions of one particle describes these measured spectra. This theory is used to extract diffusion constants of liquid mixtures and solutions, that correspond to binary diffusion, and thermal diffusion, which are consistent with previous macroscopic measurements. We also estimate the sizes of the particles.
Water, in its three phases, is ubiquitous, and the surface properties of ice is important to clarifying the process of melting, as well as to various other fields, including geophysics. As such, the subject has been studied both theoretically and experimentally, for over a hundred years, while being an active field of research today. It has been established that surface melting, or premelting, exists below the melting point, and a `liquid-like layer' (LLL) exists on the surface of ice. Here, we use the surface thermal fluctuation spectra to study the properties of LLL, including its thickness, for pure ice, and for ice with impurities. We find that the properties of LLL are consistent with those of bulk liquid water, and for layers thicker than 10\,nm, their properties are experimentally indistinguishable from those of liquid water. Measured thicknesses are found to be much smaller than the previous experimental measurements close to the bulk melting temperature. We find that the additions of impurities at ppm levels cause LLL to be thicker, as well to be quite inhomogeneous, with properties depending on the dopant. This is revealed by scanning the surface at $μ$m level resolution, and can contribute to the slipperiness of ice in natural settings.
The symmetries of the minimal $ϕ^4$ theory on the lattice are systematically analyzed. We find that symmetry can restrict trajectories to subspaces, while their motions are still chaotic. The chaotic dynamics of autonomous Hamiltonian systems are discussed, in relation to the thermodynamic laws. Possibilities of configurations with non-equal ideal gas temperatures in the steady state, in Hamiltonian systems, are investigated, and examples of small systems in which the ideal gas temperatures are different within the system are found. The pairing of local (finite-time) Lyapunov exponents are analyzed, and their dependence on various factors, such as the energy of the system, the characteristics of the initial conditions are studied, and discussed. We find that for the $ϕ^4$ theory, higher energies lead to faster pairing times. We also find that symmetries can impede the pairing of local Lyapunov exponents, and the convergence of Lyapunov exponents.
Optical coherence tomography (OCT) uses low-coherence reflectometry to obtain cross-sectional images of inhomogeneous media, such as biological tissue. OCT is particularly useful in the biomedical ea, since the imaging can be performed non-invasively, and in a relatively short time. As such, OCT is not only used in research, but has proven to be effective in clinical practice, with the fields of its application continuing to grow. In OCT, essentially, the reflectance of each cross-section along the depth direction is obtained with high spatial resolution ($\bm {\sim10\,μ}$m), revealing the structure of the sample. We have developed a system that combines OCT with thermal fluctuation measurements, which extracts physical properties, such as the viscoelastic characteristics, of each cross-section in the depth direction. We demonstrate the efficacy of this system using various samples. Furthermore, the noise is reduced to below shot-noise levels using averaged correlations, making it possible to extract spectral information otherwise unobtainable. The measurement system can in principle lead to tissue characterization and pathological diagnosis.
Spectra of thermal fluctuations of a wide range of interfaces, from liquid/air, viscoelastic material/air, liquid/liquid, to liquid/viscoelastic material interfaces, were measured over 100 Hz to 10 MHz frequency range. The obtained spectra were compared with the fluctuation theory of interfaces, and found to be mostly in quite good agreement, when the theory was generalized to apply to thermal fluctuations of liquid/viscoelastic material interfaces. The spectra were measured using a system that combines light reflection, statistical noise reduction through averaged correlations, and confocal microscopy. It requires only a small area of the interface ($\sim1\,μ$m$^2$) , relatively short times for measurements ($\lesssim$few min), and can also be applied to highly viscous materials.
We revisit the equilibrium one-dimensional $ϕ^4$ model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable. At the same time some of the orbits are found to exhibit positive Lyapunov exponents! The periodic orbits confine every particle in a periodic chain to trace out either the same or a mirror-image trajectory in its two-dimensional phase space. These "computationally stable" sets of pairs of single-particle orbits are either symmetric or antisymmetric to the very last computational bit. In such a periodic chain the odd-numbered and even-numbered particles' coordinates and momenta are either identical or differ only in sign. "Positive Lyapunov exponents" can and do result if an infinitesimal perturbation breaking a perfect two-dimensional antisymmetry is introduced so that the motion expands into a four-dimensional phase space. In that extended space a positive exponent results. We formulate a standard initial condition for the investigation of the microcanonical chaotic number dependence of the model. We speculate on the uniqueness of the model's chaotic sea and on the connection of such collections of deterministic and time-reversible states to the Second Law of Thermodynamics.
Through the study of the power spectra of a monochromatic light beam reflected by metallic mirrors, fluctuations in their reflectance is observed. The power spectra were obtained down to a factor $10^{-6}$ below the Standard Quantum Limit, with a dynamic range of $10^5$ in the frequency and power, using methods we developed. The properties of the spectra are investigated and their dependence on the material is analyzed. The physics underlying the phenomenon is also discussed. These fluctuations provide a new window for investigating the inner workings of materials.
Periodic orbits for the classical $ϕ^4$ theory on the one dimensional lattice are systematically constructed by extending the normal modes of the harmonic theory, for periodic, fixed and free boundary conditions. Through the process, we clarify, in general, which normal modes of the linear theory can or can not be extended to the full non-linear theory. We then analyze the stability of these orbits, clarifying the link between the stability, parametric resonance and the Lyapunov spectra for these orbits.The construction of the periodic orbits and the stability analysis is applicable to theories governed by Hamiltonians with quadratic inter-site potentials and a general on-site potential. We also apply the analysis to theories with on-site potentials that have qualitatively different behavior from the $ϕ^4$ theory, with some concrete examples.
Using resonant light absorption, random walk motions of rubidium atoms in nitrogen buffer gas are observed directly. The transmitted light intensity through atomic vapor is measured and its spectrum is obtained, down to orders of magnitude below the shot noise level to detect fluctuations caused by atomic motions. To understand the measured spectra, the spectrum for atoms performing random walks in a gaussian light beam is computed and its analytical form is obtained. The spectrum has $1/f^2$ ($f$: frequency) behavior at higher frequencies, crossing over to a different, but well defined behavior at lower frequencies. The properties of this theoretical spectrum agree excellently with the measured spectrum. This understanding also enables us to obtain the diffusion constant, the photon cross section of atoms in buffer gas and the atomic number density, from a single spectral measurement. We further discuss other possible applications of our experimental method and analysis.
We develop a system for measurements of power spectra of transmitted light intensity fluctuations, in which the extraneous noise, including shot noise, is reduced. In essence, we just apply light, measure the power of the transmitted light and derive its power spectrum. We use this to observe the spontaneous noise spectra of photon atom interactions. Applying light with frequency modulation, we can also observe the spontaneous noise reflecting the coherence between the hyperfine levels in the excited state. There are two in novel components in the measurement system, the noise reduction scheme and the stabilization of the laser system. The noise reduction mechanism can be used to reduce the shot noise contribution to arbitrarily low levels through averaging, in principle. This is combined with differential detection to keep unwanted noise at low levels. The laser system is stabilized to obtain spectral width below 1\,kHz without high frequency ($\gtrsim10\,$MHz) noise. These methods are described systematically and the performance of the asurement system is examined through experimental results.
Fluctuations in light absorption by atoms are observed by applying laser light on rubidium atoms and measuring the transmitted light intensity fluctuations. These fluctuations are spontaneous noise, which are generic to photon atom interactions. By making use of the sub-shot noise random signal detection technology, we have measured the spectra at sub-shot noise levels to reveal their rich nature, which had previously been unobserved. The effects of atoms transiting the laser beam, Rabi flopping in the optical transitions and Larmor precession of the magnetic moment are observed in the spectra. The properties of the fluctuations reflect not only the quantum behavior of atoms, but also that of light.
A Michelson interferometer with noise reduction to sub-shot noise levels is proposed and realized. Multiple measurements of a single signal beam are taken and the quantum property of light plays an essential role in the principle underlying this interferometry. The method makes use of the coherent state of light and requires only a simple modification to the standard Michelson interferometer. The surface fluctuation spectra of liquids are measured using this method down to a few orders of magnitude below the shot noise level. The spectrum derived from hydrodynamical considerations agrees well with the observed results for water. However, for oil, slight deviations are seen at high frequencies ($\gtrsim1\,$MHz), perhaps indicating its more complex underlying physics. The measurement requires a relatively low light power and a short time, so that it has a wide range of applicability.
We study the spectral properties of thermal fluctuations on simple liquid surfaces, sometimes called ripplons. Analytical properties of the spectral function are investigated and are shown to be composed of regions with simple analytic behavior with respect to the frequency or the wave number. The derived expressions are compared to spectral measurements performed orders of magnitude below shot noise levels, which is achieved using a novel noise reduction method. The agreement between the theory of thermal surface fluctuations and the experiment is found to be excellent.
We explain simple laboratory experiments for making quantitative measurements of the Doppler effect from sources with acceleration. We analyze the spectra and clarify the conditions for the Doppler effect to be experimentally measurable, which turn out to be non-trivial when acceleration is involved. The experiments use sources with gravitational acceleration, in free fall and in motion as a pendulum, so that the results can be checked against fundamental physics principles. The experiments can be easily set up from ``off the shelf'' components only. The experiments are suitable for a wide range of students, including undergraduates not majoring in science or engineering.
The concise letter points out that the estimates of the global potential of wind power exceeds the amount of kinetic energy in the relevant layer of atmosphere by far more than an order of magnitude. Originally submitted to the Letters section of the Proceedings of the National Academy of Sciences in August 2009.
We measure spectral properties of surface thermal fluctuations of liquids, solids, complex fluids and biological matter using light scattering methods. The random thermal fluctuations are delineated from random noise at sub-shot noise levels. The principle behind this extraction, which is quite general and is not limited to surface measurements, is explained. An optical lever is used to measure the spectrum of fluctuations in the inclinations of surfaces down to $\sim 10^{-17}\rm rad^2/Hz$ at $1\sim10 μ$W optical intensity, corresponding to $\sim 10^{-29} \rm m^2/\rm Hz$ in the vertical displacement, in the frequency range $1{\rm}\rm kHz\sim10 MHz$. The dynamical evolution of the surface properties is also investigated. The measurement requires only a short amount of time and is essentially passive, so that it can be applied to a wide variety of surfaces.
Thermal transport properties of the FPU $β$ model with a quadratic pinning term are investigated for various couplings and temperatures. In particular, the size dependence of the thermal conductivity, $κ\propto L^α$, is studied. $α$ agrees with that of the FPU $β$ model (with no pinning) at high temperatures but decreases at low temperatures. This crossover behavior occurs at a temperature depending on the strength of the quadratic pinning.