In applied mathematical sciences and theoretical physics, a geometric formulation of each theory has given various benefits. Examples include the idea of geometric phase in the geometric quantum mechanics and information geometry being a geometrization of mathematical statistics and its applications to mathematical engineering. Although it is proposed that equilibrium thermodynamics can be described in Legendre submabifolds of a contact manifold, an odd-dimensional counterpart of a symplectic manifold, nonequilibrium ones have not been intensively studied yet. Here nonequilibrium thermodynamics and statistical mechanics are expected to be important in near future since their applications are found in nanotechnology. When dealing with nonequilibrium thermodynamics, one needs to introduce time, and a vector field in general. Differential geometry provides a set of sophisticated and well-developed tools for analyzing vector fields on manifolds, and it can be expected that such tools reveals several mathematical meanings and some generalizations.
I have shown that integral curves of a class of contact Hamiltonian vector fields on a contact manifold connect points of Legendre submanifolds and those outside Legendre submanifolds. This can physically be interpreted as a class of relaxation processes by giving the standard physical interpretations used in equilibrium cases and their natural extensions. In addition I have shown as a theorem that a real number, a higher dimensional contact manifold, and a convex function on it induce a lower dimensional statistical manifold, invented in information geometry. This theorem is expected to be a fundamental one when relating statements in information geometry and those in contact geometry. These two research fields have independently been studied so far.
Electromagnetism is one of the most fundamental subjects in sciences and its basics are directly related to engineering applications. Although there is a long history of studies of electromagnetism, the correct form of the stress-energy-momentum tensor for media has not been known, even Abraham and Minkowski studied this problem more than 100 years ago. The constitutive relations that connect DH and EB for meta-materials which are expected to be functional devices are not simple, and theoretical predictions for energy and momentum in such a medium depend on the chosen tensor.
We proposed a way to systematically calculate torque and/or force from a given stress-energy-momentum tensor. Using this technique the time-average of torque occuring in rotating media has been calculated when a monochromatic plane wave is injected. Then it has been shown that the theoretical predictions distinguish the tensor we choose. In future by observing the torque experimentally and comparing it with our theoretical predictions, we hope that the correct stress-energy-momentum tensor for media is determined.
Accelerators are used not only for experiments in elementary particle physics, but also for intensive light sources that are for material sciences. To understand physics of an accelator and to design it, it is essential to determine electromagnetic fields in accelerators for given moving charge sources and given boundary conditions. However, it is difficult to determine electromagnetic fields for complex boundary conditions even we use computers. On the other hand, the Maxwell equations, which describe electromagnetic field for given sources, can be reformulated in the language of differential geometry elegantly. This reformulated equations give us a systematic way to deal with such complex boundary conditions for fields.
Using this mathematical language, we attempt to describe electromagnetic fields perturbatively for geometrically curved pipes with given sources based on a simple waveguide problem.
We develop singular perturbation methods, and show applications of them to some weakly nonlinear systems. When expanding a solution with respect to the small parameter associated with a given problem, we sometimes cannot describe the long-time behavior of the system. In such a case, to understand the long-time or global behavior of the system, we need to have ** a better ** perturbation method. The original renormalization method was proposed by Illinois group and is a candidate for such a method.
We propose another renormalization method, and show that this method can be applied to not only partial differential equations, discrete equations that we cannot employ differential calculus, but also delay differential equations in which we encounter some problems due to high dimensional phase space.
There are a variety of classical Hamiltonian systems with many degrees of freedom whose interaction lengths are long, such as self-gravity, vortex lines, and plasma systems. Since these systems do not have additivity, even in the thermodynamic limit in equilibrium state, we could observe a difference between the expectation value of a macroscopic variable calculated by microcanonical statistics and one with the use of canonical statistics. We study such Hamiltonian systems from the point of view of dynamical systems theory and/or statistical mechanics.
We study the expectation value of a macroscopic variable defined in the alpha XY model whose interaction length can be controlled, and show that a finite size effect non-trivially depends on the interaction length, and the effect can be explained using canonical statistical mechanics.
As well as the case of systems with long-range interactions, we are interested in classical Hamiltonian systems with short-range interactions. There are a variety of applications of mathematical and physical studies of systems with short-range interactions to other scientific fields, such as solid systems and connected macroscopic structures. The standard models to study them are the Fermi-Pasta-Ulam model and the Discrete Nonlinear Schrodinger equation (DNLS), and there have been many fundamental studies for them.
Although it was suggested that the largest Lyapunov exponent, which is used to measure the strength of chaos, can be predicted using only one time-periodic orbit in such a lattice model, we did not know the dependence of the choice of the periodic orbit. We show that the predicted largest Lyapunov exponent generally depends on the periodic orbit which we employ, and the predicted expectation value of the macroscopic variable does not depend on the periodic orbit, at least in a high energy regime.
When a common external signal, such as noise, is put to disconnected dynamical systems, we often observe that output signals are synchronized. This phenomenon is not only interesting in the view of nonlinear science, but important in engineering, such as optical systems consisting of semiconductor lasers, because this gives a synchronized state between laser users at remote locations and this scheme could be a next generation communication system.
We show that some noises can induce this kind of synchronization in disconnected semiconductor lasers in numerical simulations and in experiment. In addition to this, we study when this kind of induced synchronization is enhanced for some dynamical systems by defining the strength of synchronization.
In nearly integrable classical Hamiltonian systems whose degrees of freedom is more than three, we observe Arnold diffusion, which is global motion in phase space appeared even in the case that the perturbation strength is very small. Note that, as for systems with two degrees of freedom, motion is restricted due to the survival tori in phase space. The study of Arnold diffusion is important in the sense that this phenomenon can be related to Ergodicity and its time-scale(s).