The Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems with Three or More Degrees of Freedom – II

We develop a method for the construction of a dividing surface using periodic orbits in Hamiltonian systems with three or more degrees-of-freedom that is an alternative to the method presented in [ Katsanikas & Wiggins, 2021 ]. Similar to that method, for an [Formula: see text] degrees-of-freedom Hamiltonian system, we extend a one-dimensional object (the periodic orbit) to a [Formula: see text] dimensional geometrical object in the energy surface of a [Formula: see text] dimensional space that has the desired properties for a dividing surface. The advantage of this new method is that it avoids the computation of the normally hyperbolic invariant manifold (NHIM) (as the first method did) and it is easier to numerically implement than the first method of constructing periodic orbit dividing surfaces. Moreover, this method has less strict required conditions than the first method for constructing periodic orbit dividing surfaces. We apply the new method to a benchmark example of a Hamiltonian system with three degrees-of-freedom for which we are able to investigate the structure of the dividing surface in detail. We also compare the periodic orbit dividing surfaces constructed in this way with the dividing surfaces that are constructed starting with a NHIM. We show that these periodic orbit dividing surfaces are subsets of the dividing surfaces that are constructed from the NHIM.

The Generalization of the Periodic Orbit Dividing Surface in Hamiltonian Systems with Three or More Degrees of Freedom – I

We present a method that generalizes the periodic orbit dividing surface construction for Hamiltonian systems with three or more degrees of freedom. We construct a torus using as a basis a periodic orbit and we extend this to a ([Formula: see text])-dimensional object in the ([Formula: see text])-dimensional energy surface. We present our methods using benchmark examples for two and three degrees of freedom Hamiltonian systems to illustrate the corresponding algorithm for this construction. Towards this end, we use the normal form quadratic Hamiltonian system with two and three degrees of freedom. We found that the periodic orbit dividing surface can provide us the same dynamical information as the dividing surface constructed using normally hyperbolic invariant manifolds. This is significant because, in general, computations of normally hyperbolic invariant manifolds are very difficult in Hamiltonian systems with three or more degrees of freedom. However, our method avoids this computation and the only information that we need is the location of one periodic orbit.

Molecular Dynamics Simulation of Tolman Length and Interfacial Tension of Symmetric Binary Lennard–Jones Liquid

The Tolman length and interfacial tension of partially miscible symmetric binary Lennard–Jones (LJ) fluids (A, B) was revealed by performing a large-scale molecular dynamics (MD) simulation with a sufficient interfacial area and cutting distance. A unique phenomenon was observed in symmetric binary LJ fluids, where two surfaces of tension existed on both sides of an equimolar dividing surface. The range of interaction εAB between the different liquids and the temperature in which the two LJ fluids partially mixed was clarified, and the Tolman length exceeded 3 σ when εAB was strong at higher temperatures. The results show that as the temperature or εAB increases, the Tolman length increases and the interfacial tension decreases. This very long Tolman length indicates that one should be very careful when applying the concept of the liquid–liquid interface in the usual continuum approximation to nanoscale droplets and capillary phase separation in nanopores.

Study of the Start Point of Optimization Trajectories for Complex Strategies of the Circuit Design Process

The different design trajectories have been analyzed in the design space on the basis of the new system design methodology. Optimal position of the design algorithm start point was analyzed to minimize the CPU time. The initial point selection has been done on the basis of the before discovered acceleration effect of the system design process. The geometrical dividing surface was defined and analyzed to obtain the optimal position of the algorithm start point. The numerical results of the design of passive and active nonlinear electronic circuits confirm the possibility of the optimal selection of the starting point of the design algorithm.

The Role of Time-Dependent Phase Space Structures in Reaction Dynamics and the No-Recrossing Property of Dividing Surfaces

We analyze benchmark models for reaction dynamics associated with a time-dependent saddle point. Our model allows us to incorporate time dependence of a general form, subject to an exponential growth restriction. Under these conditions, we analytically compute the time-dependent normally hyperbolic invariant manifold; its time-dependent stable and unstable manifolds; and a time-dependent dividing surface that has the no-recrossing property. Consideration of the time dependence of these phase space structures is necessary in order to precisely capture reacting and nonreacting trajectories. Moreover, we show that a time-dependent dividing surface is necessary in order to eliminate recrossing in the time-dependent setting. In other words, if the dividing surface is not time-dependent, recrossing may occur.

Visualizing the double-gyroid twin

Periodic gyroid network materials have many interesting properties (band gaps, topologically protected modes, superior charge and mass transport, and outstanding mechanical properties) due to the space-group symmetries and their multichannel triply continuous morphology. The three-dimensional structure of a twin boundary in a self-assembled polystyrene-b-polydimethylsiloxane (PS-PDMS) double-gyroid (DG) forming diblock copolymer is directly visualized using dual-beam scanning microscopy. The reconstruction clearly shows that the intermaterial dividing surface (IMDS) is smooth and continuous across the boundary plane as the pairs of chiral PDMS networks suddenly change their handedness. The boundary plane therefore acts as a topological mirror. The morphology of the normally chiral nodes and strut loops within the networks is altered in the twin-boundary plane with the formation of three new types of achiral nodes and the appearance of two new classes of achiral loops. The boundary region shares a very similar surface/volume ratio and distribution of the mean and Gaussian curvatures of the IMDS as the adjacent ordered DG grain regions, suggesting the twin is a low-energy boundary.

Strong Coupling and Nonextensive Thermodynamics

We propose a Hamiltonian-based approach to the nonextensive thermodynamics of small systems, where small is a relative term comparing the size of the system to the size of the effective interaction region around it. We show that the effective Hamiltonian approach gives easy accessibility to the thermodynamic properties of systems strongly coupled to their surroundings. The theory does not rely on the classical concept of dividing surface to characterize the system’s interaction with the environment. Instead, it defines an effective interaction region over which a system exchanges extensive quantities with its surroundings, easily producing laws recently shown to be valid at the nanoscale.

Adsorption of surfactants at liquid interfaces: thermodynamics

The thickness and hence material content of a surface is generally unknown, and there are two common definitions of a surface/interface. In one the surface is treated as a phase distinct from the surrounding bulk phases, and in the other, due to Gibbs, the Gibbs dividing surface is supposed to be a plane, parallel to the physical interface. The former model gives rise to the surface concentrationΓ‎s of a surfactant, and the Gibbs model introduces the surface excess concentration, Γ‎σ‎. Some thermodynamic quantities for surfaces (e.g. surface chemical potential and Gibbs free energy for surfaces) are defined. Adsorption lowers interfacial tension by an amount termed the surface pressure, and the Gibbs adsorption equation allows the calculation of Γ‎s or Γ‎σ‎ for a surfactant from the variation of interfacial tension of a liquid/fluid interface with surfactant concentration in bulk solution.