Spiral symmetry and general Bloch's theorem

In this paper, spiral symmetry in cylindrical coordinate and general Bloch's theorem induced from it are discussed. This general Bloch's theorem is useful for considering the properties related to single-walled carbon nanotubes.Comment: 4 page

Compatibility between shape equation and boundary conditions of lipid membranes with free edges

Only some special open surfaces satisfying the shape equation of lipid membranes can be compatible with the boundary conditions. As a result of this compatibility, the first integral of the shape equation should vanish for axisymmetric lipid membranes, from which two theorems of non-existence are verified: (i) There is no axisymmetric open membrane being a part of torus satisfying the shape equation; (ii) There is no axisymmetric open membrane being a part of a biconcave discodal surface satisfying the shape equation. Additionally, the shape equation is reduced to a second-order differential equation while the boundary conditions are reduced to two equations due to this compatibility. Numerical solutions to the reduced shape equation and boundary conditions agree well with the experimental data [A. Saitoh \emph{et al.}, Proc. Natl. Acad. Sci. USA \textbf{95}, 1026 (1998)].Comment: 6 journal pages, 4 figure

Structures, Symmetries, Mechanics and Motors of carbon nanotubes

The structures and symmetries of single-walled carbon nanotubes (SWNTs) are introduced in detail. The physical properties of SWNTs induced by their symmetries can be described by tensors in mathematical point of view. It is found that there are 2, 4, and 5 different parameters in the second, third, and fourth rank tensors representing electronic conductivity (or static polarizability), the second order nonlinear polarizability, and elastic constants of SWNTs, respectively. The values of elastic constants obtained from tight-binding method imply that SWNTs might be very weakly anisotropic in mechanical properties. The further study on the mechanical properties shows that the elastic shell theory in the macroscopic scale can be applied to carbon nanotubes (CNTs) in the mesoscopic scale, as a result, SWNTs can be regarded as an isotropic material with Poisson ratio, effective thickness, and Young's modulus being $\nu=0.34$, $h=0.75$\AA, $Y=4.70$TPa, respectively, while the Young's moduli of multi-walled carbon nanotubes (MWNTs) are apparent functions of the number of layers, $N$, varying from 4.70TPa to 1.04TPa for N=1 to $\infty$. Based on the chirality of CNTs, it is predicted that a new kind of molecular motor driven by alternating voltage can be constructed from double walled carbon nanotubes (DWNTs).Comment: 18 pages+5 figure; will appear as a Chapter in "Nanotubes: New Research" (Nova Science Publishers, 2005

Lipid membranes with free edges

Lipid membrane with freely exposed edge is regarded as smooth surface with curved boundary. Exterior differential forms are introduced to describe the surface and the boundary curve. The total free energy is defined as the sum of Helfrich's free energy and the surface and line tension energy. The equilibrium equation and boundary conditions of the membrane are derived by taking the variation of the total free energy. These equations can also be applied to the membrane with several freely exposed edges. Analytical and numerical solutions to these equations are obtained under the axisymmetric condition. The numerical results can be used to explain recent experimental results obtained by Saitoh \emph{et al}. [Proc. Natl. Acad. Sci. \textbf{95}, 1026 (1998)].Comment: 15 pages, 6 figure

Recent theoretical advances in elasticity of membranes following Helfrich's spontaneous curvature model

Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solutions to these equations and their corresponding configurations are demonstrated.Comment: 10 pages, 8 figure

Comment on "Highly Extended Image States around Nanotubes"

A Comment on the Letter by Granger et.al., Phys. Rev. Lett. 89, 135506 (2002).Comment: 2 page

Double-walled carbon nanotubes as hundred gigahertz oscillators

Based on the van der Waals interaction, the periodically nonlinear potential of a singe-walled carbon nanotube (SWNT) with finite length in an infinite length SWNT is analytically obtained. It is found that the inner SWNT can oscillate in the outer SWNT with frequency beyond ten Gigahertz, even up to a hundred Gigahertz.Comment: 9 pages, 8 figures, to PR

Extend Special Relativity to the Superluminal Case

First, we extend the special relativity into the superluminal case and put forward a superluminal theory of kinematics, in which we show that the temporal coordinate need exchanging with one of the spatial coordinates in a superluminal inertial frame, and that the coordinate transformations from any superluminal inertial frame to the rest frame (here rest just says in a relative sense) are the same as the Lorentz transformations from some normal inertial frame to the rest frame. Consequently, the causality can not be violated. Secondly, we investigate the superluminal theory of dynamics and find that the total energy of any object moving at a speed of $v$ (faster than the speed of light in vacuum $c$) is equal to the total energy of that object moving at a speed of $u (u<c)$ provided that the product of two speeds satisfy $uv=c^{2}$. Lastly, we conjecture that this superluminal theory can give a novel interpretation to the essence of matter waves put forward by de Broglie.Comment: 3 papges, 2 figure

Constitutive Relation for Nonlinear Response and Universality of Efficiency at Maximum Power for Tight-Coupling Heat Engines

We present a unified perspective on nonequilibrium heat engines by generalizing nonlinear irreversible thermodynamics. For tight-coupling heat engines, a generic constitutive relation of nonlinear response accurate up to the quadratic order is derived from the symmetry argument and the stall condition. By applying this generic nonlinear constitutive relation to finite-time thermodynamics, we obtain the necessary and sufficient condition for the universality of efficiency at maximum power, which states that a tight-coupling heat engine takes the universal efficiency at maximum power up to the quadratic order if and only if either the engine symmetrically interacts with two heat reservoirs or the elementary thermal energy flowing through the engine matches the characteristic energy of the engine. As a result, we solve the following paradox: On the one hand, the universal quadratic term in the efficiency at maximum power for tight-coupling heat engines proved as a consequence of symmetry [M. Esposito, K. Lindenberg, and C. Van den Broeck, Phys. Rev. Lett. 102, 130602 (2009); S. Q. Sheng and Z. C. Tu, Phys. Rev. E 89, 012129 (2014)]; On the other hand, two typical heat engines including the Curzon-Ahlborn endoreversible heat engine [F. L. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975)] and the Feynman ratchet [Z. C. Tu, J. Phys. A 41, 312003 (2008)] recover the universal efficiency at maximum power regardless of any symmetry

Universality of energy conversion efficiency for optimal tight-coupling heat engines and refrigerators

A unified $\chi$-criterion for heat devices (including heat engines and refrigerators) which is defined as the product of the energy conversion efficiency and the heat absorbed per unit time by the working substance [de Tom\'{a}s \emph{et al} 2012 \textit{Phys. Rev. E} \textbf{85} 010104(R)] is optimized for tight-coupling heat engines and refrigerators operating between two heat baths at temperatures $T_c$ and $T_h(>T_c)$. By taking a new convention on the thermodynamic flux related to the heat transfer between two baths, we find that for a refrigerator tightly and symmetrically coupled with two heat baths, the coefficient of performance (i.e., the energy conversion efficiency of refrigerators) at maximum $\chi$ asymptotically approaches to $\sqrt{\varepsilon_C}$ when the relative temperature difference between two heat baths $\varepsilon_C^{-1}\equiv (T_h-T_c)/T_c$ is sufficiently small. Correspondingly, the efficiency at maximum $\chi$ (equivalent to maximum power) for a heat engine tightly and symmetrically coupled with two heat baths is proved to be $\eta_C/2+\eta_C^2/8$ up to the second order term of $\eta_C\equiv (T_h-T_c)/T_h$, which reverts to the universal efficiency at maximum power for tight-coupling heat engines operating between two heat baths at small temperature difference in the presence of left-right symmetry [Esposito \emph{et al} 2009 \textit{Phys. Rev. Lett.} \textbf{102} 130602].Comment: substantial revisio