First integrals of the axisymmetric shape equation of lipid membranes

The shape equation of lipid membranes is a fourth-order partial differential equation.Under the axisymmetric condi-tion,this equation was transformed into a second-order ordinary differential equation(ODE)by Zheng and Liu(Phys.Rev.E 482856(1993)).Here we try to further reduce this second-order ODE to a first-order ODE.First,we invert the usual process of variational calculus,that is,we construct a Lagrangian for which the ODE is the corresponding Euler-Lagrange equation.Then,we seek symmetries of this Lagrangian according to the Noether theorem.Under a certain restriction on Lie groups of the shape equation,we find that the first integral only exists when the shape equation is identical to the Will-more equation,in which case the symmetry leading to the first integral is scale invariance.We also obtain the mechanical interpretation of the first integral by using the membrane stress tensor.

A single-point measurement scheme for quantum work based on the squeezing state

To investigate the role of initial quantum coherence in work-probability distribution,it is necessary to consider an incomplete or partial measurement,in which the energy cannot be fully discriminated by the detector.In this paper,we use a harmonic oscillator with a coherent or squeezing state to realize this incomplete or partial measurement,and propose a unified framework of quantum work statistics for a closed system with an arbitrary initial state.We find that work is proportional to the change of the real part of the coherent state parameter,i.e.,quantum work can be estimated by the coherent state parameter.The resulting work-probability distribution includes the initial quantum coherence,and can be reduced to the result of the traditional two projective energy measurement scheme(TPM)by squeezing the state of the harmonic oscillator.Our measurement scheme reveals the fundamental connections between measurement error and coherent work.By introducing a‘coherent work-to-noise ratio’,we find the optimal measurement error,which is determined by the energy difference between the superposed energy levels.As an application,we consider a driven two-level system and investigate the effects of driving velocity on work statistics.We find that only when the driving velocity matches the transition frequency of the system can initial quantum coherence play an important role.

Abstract models for heat engines

We retrospect three abstract models for heat engines which include a classic abstract model in textbook of thermal physics,a primary abstract model for finite-time heat engines,and a refined abstract model for finite-time heat engines.The detailed models of heat engines in literature of finite-time thermodynamics may be mapped into the refined abstract model.The future developments based on the refined abstract model are also surveyed.