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The Fokker-Planck equation is one of the fundamental equations in nonequilibrium statistical mechanics, and this equation is known to be derived from the Wasserstein gradient flow equation with a free energy. This gradient flow equation describes relaxation processes and is formulated on a Riemannian manifold. Meanwhile contact Hamiltonian systems are also known to describe relaxation processes. Hence a relation between these two equations is expected to be clarified, which gives a solid foundation in geometric statistical mechanics. In this paper a class of contact Hamiltonian systems is derived from a class of the Fokker-Planck equations on Riemannian manifolds. In the course of the derivation, the Fokker-Planck equation is shown to be written as a diffusion equation with a weighted Laplacian without any approximation, which enables to employ a theory of eigenvalue problems., Comment: 39 pages

We draw connections between contact topology and Maxwell fields in vacuo on 3-dimensional closed Riemannian submanifolds in 4-dimensional Lorentzian manifolds. This is accomplished by showing that contact topological methods can be applied to reveal topological features of a class of solutions to Maxwell's equations. This class of Maxwell fields is such that electric fields are parallel to magnetic fields. In addition these electromagnetic fields are composed of the so-called Beltrami fields. We employ several theorems resolving the Weinstein conjecture on closed manifolds with contact structures and stable Hamiltonian structures, where this conjecture refers to the existence of periodic orbits of the Reeb vector fields. Here a contact form is a special case of a stable Hamiltonian structure. After showing how to relate Reeb vector fields with electromagnetic 1-forms, we apply a theorem regarding contact manifolds and an improved theorem regarding stable Hamiltonian structures. Then a closed field line is shown to exist, where field lines are generated by Maxwell fields. In addition, electromagnetic energies are shown to be conserved along the Reeb vector fields.

We study a nonequilibrium mean field Ising model in the low temperature phase regime, where metastable equilibrium states develop a cuspidal (spinodal) singularity. We focus on celebrated Glauber dynamics, and design a contact Hamiltonian flow which captures some of its rough features in this regime. We prove, however, that there is an inevitable discrepancy between the scaling laws for the relaxation time in the Glauber and the contact Hamiltonian dynamical systems., Comment: 2 figures

Thermodynamics provides a unified perspective of thermodynamic properties of various substances. To formulate thermodynamics in the language of sophisticated mathematics, thermodynamics is described by a variety of differential geometries, including contact and symplectic geometries. Meanwhile affine geometry is a branch of differential geometry and is compatible with information geometry, where information geometry is known to be compatible with thermodynamics. By combining above, it is expected that thermodynamics is compatible with affine geometry, and is expected that several affine geometric tools can be introduced in the analysis of thermodynamic systems. In this paper affine geometric descriptions of equilibrium and nonequilibrium thermodynamics are proposed. For equilibrium systems, it is shown that several thermodynamic quantities can be identified with geometric objects in affine geometry, and that several geometric objects can be introduced in thermodynamics. Examples of these include: specific heat is identified with the affine fundamental form, a flat connection is introduced in thermodynamic phase space. For nonequilibrium systems, two classes of relaxation processes are shown to be described in the language of an extension of affine geometry. Finally this affine geometric description of thermodynamics for equilibrium and nonequilibrium systems is compared with a contact geometric description.

In this paper, a dynamical process in a statistical thermodynamic system of spins exhibiting a phase transition is described on a contact manifold, where such a dynamical process is a process that a metastable equilibrium state evolves into the most stable symmetry broken equilibrium state. Metastable and the most stable equilibrium states in the symmetry broken phase or ordered phase are assumed to be described as pruned projections of Legendre submanifolds of contact manifolds, where these pruned projections of the submanifolds express hysteresis and pseudo-free energy curves. Singularities associated with phase transitions are naturally arose in this framework as has been suggested by Legendre singularity theory. Then a particular contact Hamiltonian vector field is proposed so that a pruned segment of the projected Legendre submanifold is a stable fixed point set in a region of a contact manifold, and that another pruned segment is a unstable fixed point set. This contact Hamiltonian vector field is identified with a dynamical process departing from a metastable equilibrium state to the most stable equilibrium one. To show the statements above explicitly an Ising type spin model with long-range interactions, called the Husimi-Temperley model, is focused, where this model exhibits a phase transition., Comment: A minor error on figures in arXiv version 2 has been fixed

In this paper, explicit stable integrators based on symplectic and contact geometries are proposed for a non-autonomous ordinarily differential equation (ODE) found in improving convergence rate of Nesterov's accelerated gradient method. Symplectic geometry is known to be suitable for describing Hamiltonian mechanics, and contact geometry is known as an odd-dimensional counterpart of symplectic geometry. Moreover, a procedure, called symplectization, is a known way to construct a symplectic manifold from a contact manifold, yielding Hamiltonian systems from contact ones. It is found in this paper that a previously investigated non-autonomous ODE can be written as a contact Hamiltonian system. Then, by symplectization of a non-autonomous contact Hamiltonian vector field expressing the non-autonomous ODE, novel symplectic integrators are derived. Because the proposed symplectic integrators preserve hidden symplectic and contact structures in the ODE, they should be more stable than the Runge-Kutta method. Numerical experiments demonstrate that, as expected, the second-order symplectic integrator is stable and high convergence rates are achieved.

In this paper, continuous-time master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry. In this formulation, chains in algebraic topology are used, and master equations are described on graphs that consist of vertexes representing states and of directed edges representing transition matrices. It is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations, where the Laplacians are defined as self-adjoint operators with respect to introduced inner products. An isospectral property of these Laplacians is shown for non-zero eigenvalues, and its applications are given. The convergence to the equilibrium state is shown by analyzing this class of diffusion equations. In addition, a systematic way to derive closed dynamical systems for expectation values is given. For the case that the detailed balance conditions are not imposed, master equations are expressed as a form of a continuity equation., Comment: 30 pages

Based on information and para-contact metric geometries, in this paper a class of dynamical systems is formulated for describing time-development of expectation variables. Here such systems for expectation variables are exactly derived from continuous-time master equations describing nonequilibrium processes., Comment: GSI 2019 conference proceeding to be published in Springer's Lecture Note in Computer Science (LNCS) series

In this paper a class of dynamical systems describing deterministic neural network models are formulated from a viewpoint of differential geometry. This class includes the Hopfield model and gradient systems, and is such that the so-called activation functions induce information and Hessian geometries. In this formulation, it is shown that the phase space compressibility of a dynamical system belonging to this class is written in terms of the Laplace operator defined on Hessian manifolds, where phase space compressibility is associated with a volume-form of a manifold, and expresses how such a volume-form is compressed along the vector field of a dynamical system. Since the sigmoid function, as an activation function, plays a role in the study of neural network models, such compressibility is explicitly calculated for this case. Throughout this paper, the so-called dual coordinates known in information geometry are explicitly used.

On Feb. 5 2016 (UTC), an earthquake with moment magnitude 6.4 occurred in southern Taiwan, known as the 2016 (Southern) Taiwan earthquake. In this study, evidences of seismic earthquake precursors for this earthquake event are investigated. Results show that ionospheric anomalies in Total Electric Content (TEC) can be observed before the earthquake. These anomalies were obtained by processing TEC data, where such TEC data are calculated from phase delays of signals observed at densely arranged ground-based stations in Taiwan for Global Navigation Satellite Systems. This shows that such anomalies were detected within 1 hour before the event.

In this paper a class of dynamical systems describing expectation variables exactly derived from continuous-time master equations is introduced and studied from the viewpoint of differential geometry, where such master equations consist of a set of appropriately chosen Markov kernels. To geometrize such dynamical systems for expectation variables, information geometry is used for expressing equilibrium states, and contact geometry is used for nonequilibrium states. Here time-developments of the expectation variables are identified with contact Hamiltonian vector fields on a contact manifold. Also, it is shown that the convergence rate of this dynamical system is exponential. Duality emphasized in information geometry is also addressed throughout., Comment: 21 pages

In this paper a class of classical Hamiltonian systems is geometrically formulated. This class is such that a Hamiltonian can be written as the sum of a kinetic energy function and a potential energy function. In addition, these energy functions are assumed strictly convex. For this class of Hamiltonian systems Hessian and information geometric formulation is given. With this formulation, a generalized Toda's dual transform is proposed, where his original transform was used in deriving his integrable lattice system. Then a relation between the generalized Toda's dual transform and the Legendre transform of a class of potential energy functions is shown. As an extension of this formulation, dissipation-less electric circuit models are also discussed in the geometric viewpoint above., Comment: no figure

Maps on a parameter space for expressing distribution functions are exactly derived from the Perron-Frobenius equations for a generalized Boole transform family. Here the generalized Boole transform family is a one-parameter family of maps where it is defined on a subset of the real line and its probability distribution function is the Cauchy distribution with some parameters. With this reduction, some relations between the statistical picture and the orbital one are shown. From the viewpoint of information geometry, the parameter space can be identified with a statistical manifold, and then it is shown that the derived maps can be characterized. Also, with an induced symplectic structure from a statistical structure, symplectic and information geometric aspects of the derived maps are discussed.

This paper studies distributed-parameter systems on Riemannian manifolds with respect to Stokes-Dirac structures in a language of contact geometry with fiber bundles. For the class where energy functionals are quadratic, it is shown that distributed-parameter port-Hamiltonian systems with respect to Stokes-Dirac structures on one, two, and three dimensional Riemannian manifolds are written in terms of contact Hamiltonian vector fields on bundles. Their fiber spaces are contact manifolds and base spaces are Riemannian manifolds. In addition, for a class of distributed-parameter port-Hamiltonian systems, information geometry induced from contact manifolds and convex energy functionals is introduced and briefly discussed., Comment: 33 pages, no figure. arXiv admin note: substantial text overlap with arXiv:1702.05746. text overlap with arXiv:1512.00950

It is shown that Maxwell's equations in media without source can be written as a contact Hamiltonian vector field restricted to a Legendre submanifold, where this submanifold is in a fiber space of a bundle and is generated by either electromagnetic energy functional or co-energy functional. Then, it turns out that Legendre duality for this system gives the induction oriented formulation of Maxwell's equations and field intensity oriented one. Also, information geometry of the Maxwell fields is introduced and discussed., Comment: 26 pages, no fugure

Contact geometry has been applied to various mathematical sciences, and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry. Here, such a dually flat space is related to a Legendre submanifold in a contact manifold. In this paper contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact Hamiltonian vector fields. Based on these descriptions, two ways of lifting vector fields on Legendre submanifolds to contact manifolds are given. For some classes of these lifted vector fields, invariant measures in contact manifolds and stability analysis around Legendre submanifolds are explicitly given. Throughout this paper, Legendre duality is explicitly stated. In addition, to show how to apply these general methodologies to applied mathematical disciplines, electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed., Comment: 38 pages

It has been proposed that equilibrium thermodynamics is described on Legendre submanifolds in contact geometry. It is shown in this paper that Legendre submanifolds embedded in a contact manifold can be expressed as attractors in phase space for a certain class of contact Hamiltonian vector fields. By giving a physical interpretation that points outside the Legendre submanifold can represent nonequilibrium states of thermodynamic variables, in addition to that points of a given Legendre submanifold can represent equilibrium states of the variables, this class of contact Hamiltonian vector fields is physically interpreted as a class of relaxation processes, in which thermodynamic variables achieve an equilibrium state from a nonequilibrium state through a time evolution, a typical nonequilibrium phenomenon. Geometric properties of such vector fields on contact manifolds are characterized after introducing a metric tensor field on a contact manifold. It is also shown that a contact manifold and a strictly convex function induce a lower dimensional dually flat space used in information geometry where a geometrization of equilibrium statistical mechanics is constructed. Legendre duality on contact manifolds is explicitly stated throughout., Comment: 29 pages