No-pumping theorem for non-Arrhenius rates

The no-pumping theorem refers to a Markov system that holds the detailed balance, but is subject to a time-periodic external field. It states that the time-averaged probability currents nullify in the steady periodic (Floquet) state, provided that the Markov system holds the Arrhenius transition rates. This makes an analogy between features of steady periodic and equilibrium states, because in the latter situation all probability currents vanish explicitly. However, the assumption on the Arrhenius rates is fairly specific, and it need not be met in applications. Here a new mechanism is identified for the no-pumping theorem, which holds for symmetric time-periodic external fields and the so called destination rates. These rates are the ones that lead to the locally equilibrium form of the master equation, where dissipative effects are proportional to the difference between the actual probability and the equilibrium (Gibbsian) one. The mechanism also leads to an approximate no-pumping theorem for the Fokker-Planck rates that relate to the discrete-space Fokker-Planck equation.     

Mobility in a periodic potential: A simple derivation

A simple derivation is given for the mobility of Brownian particles in a periodic potential in the overdamped regime. The method makes use of the fact that the steady state current density, in response to a uniform external force, is uniform in space and can be expressed as a product of the particle density and mean velocity field. To lowest order in the external force, the particle density is given by the equilibrium density in the absence of the external force.     

Thermostating by Deterministic Scattering: The Periodic Lorentz Gas

We present a novel mechanism for thermalizing a system of particles in equilibrium and nonequilibrium situations, based on specifically modeling energy transfer at the boundaries via a microscopic collision process. We apply our method to the periodic Lorentz gas, where a point particle moves diffusively through an ensemble of hard disks arranged on a triangular lattice. First, collision rules are defined for this system in thermal equilibrium. They determine the velocity of the moving particle such that the system is deterministic, time-reversible, and microcanonical. These collision rules can systematically be adapted to the case where one associates arbitrarily many degrees of freedom to the disk, which here acts as a boundary. Subsequently, the system is investigated in nonequilibrium situations by applying an external field. We show that in the limit where the disk is endowed by infinitely many degrees of freedom it acts as a thermal reservoir yielding a well-defined nonequilibrium steady state. The characteristic properties of this state, as obtained from computer simulations, are finally compared to those of the so-called Gaussian thermostated driven Lorentz gas.     

Controlling chaos to unstable periodic orbits and equilibrium state solutions for the coupled dynamos system

In the case where the knowledge of goal states is not known, the controllers are constructed to stabilize unstable steady states for a coupled dynamos system. A delayed feedback control technique is used to suppress chaos to unstable focuses and unstable periodic orbits. To overcome the topological limitation that the saddle-type steady state cannot be stabilized, an adaptive control based on LaSalle's invariance principle is used to control chaos to unstable equilibrium (i.e. saddle point, focus, node, etc.). The control technique does not require any computer analysis of the system dynamics, and it operates without needing to know any explicit knowledge of the desired steady-state position.     

The steady state response of structures to periodic excitation

In order to calculate the steady state response of structures to non-harmonic periodic excitation, it is customary to expand the generalized forces of the periodic load in Fourier series. In this paper it is shown that this procedure is unnecessary. When either real modal analysis or complex modal analysis is used to calculate the response, a closed form solution, in the form of an integral of the steady state response of each normal co-ordinate (in real modal analysis), or those of the generalized co-ordinates (in complex modal analysis), can be obtained directly in accordance with the periodicity conditions of the steady state response.     

Fokker-Planck equation approach to fluctuations about nonequilibrium steady states

We examine the properties of steady states in systems which interact at the boundary with a nonequilibrium environment. The examination is based on a nonlinear Fokker-Planck equation, the structure of which is determined by the fact that it also governs the time evolution of the equilibrium fluctuations of the system. The nonlinearities in the Fokker-Planck equation may have two origins: thermodynamic nonlinearities which arise if the thermodynamic potential is not a bilinear function of the state variables, and nonlinear mode coupling which arises if the transport coefficients depend on the state. While these nonlinearities have only a small effect on the equilibrium fluctuations of a system away from critical points, they are shown to be important for the determination of fluctuations about nonequilibrium steady states. In particular the state dependence of the transport coefficients may lead to deviations from local equilibrium and to a breakdown of detail balance. An explicit formula for the time correlations of fluctuations about the nonequilibrium steady state is obtained. The formula leads to long-range correlations in fluids in the presence of a temperature gradient. The result is compared with earlier approaches to the same problem. Finally, we study the linear response to external forces and obtain a generalization of the fluctuation-dissipation formula relating the response functions with the nonequilibrium correlation functions.     

用连续法计算五维对流模型的定常解和周期解

利用连续算法(Continuation algorithm)对五维对流非线性动力系统的定常解和周期解进行了数值计算。在参数平面Ri-Re上计算出实分岔点曲线、极限点曲线、Hopf分岔点曲线,绘出了分岔图。在分岔图上的不同区域,存在性质不同的稳定解如定常吸引子、周期吸引子等。分析了定常解、周期解的分岔过程。计算结果很好地说明大气中由基本态到对流态再到波动态最后到湍流态的物理转换过程。 连续算法对研究非线性动力系统的分岔以及耗散结构是很有效的计算方法。     

Stability of the Boltzmann equation with potential forces on torus

In this paper, we are concerned with the stability of solutions to the Cauchy problem of the Boltzmann equation with potential forces on torus. It is shown that the natural steady state with the symmetry of origin is asymptotically stable in the Sobolev space with exponential rate in time for any initially smooth, periodic, origin symmetric small perturbation, which preserves the same total mass, momentum and mechanical energy. For the non-symmetric steady state, it is also shown that it is stable in L¹-norm for any initial data with the finite total mass, mechanical energy and entropy.     

Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term

We are primarily concerned with the variational problem with long-range interaction. This functional represents the Gibbs free energy of the microphase separation of diblock copolymer melts. The critical points of this variational problem can be regarded as the thermodynamic equilibrium state of the phase separation phenomenon. Experimentally it is well-known in the diblock copolymer problem that the final equilibrium state prefers periodic structures such as lamellar, column, spherical, double-diamond geometries and so on. We are interested in the characterization of the periodic structure of the global minimizer of the functional (corresponding to the strong segregation limit). In this paper we completely determine the principal part of the asymptotic expansion of the period with respect to epsilon (interfacial thickness), namely, we estimate the higher order error term of the period with respect to epsilon in a mathematically rigorous way in one space dimension. Moreover, we decide clearly the dependency of the constant of proportion upon the ratio of the length of two homopolymers and upon the quench depth. In the last section, we study the time evolution of the system. We first study the linear stability of spatially homogeneous steady state and derive the most unstable wavelength, if it is unstable. This is related to spinodal decomposition. Then, we numerically investigate the time evolution equation (the gradient flow of the free energy), and see that the free energy has many local minimizers and the system have some kind of sensitivity about initial data. (c) 1999 American Institute of Physics.     

Two remarks on extremal equilibrium states

First it is shown that each extremal equilibrium state is representable as limit of Gibbs states in finite volumes, and that an analogous statement holds for extremal invariant equilibrium states. Secondly we prove that for negative pair interactions only one equilibrium state exists which minimizes (resp. maximizes) the particle density, but that in general there are more than two extremal invariant equilibrium states with the same particle density. In this context, periodic interactions are studied.     

Simple chemomechanical process for self-generation of rhythms and forms

We show that cross-coupling between the geometrical changes and the chemical reaction-diffusion regimes within a piece of gel embedded in a stationary reactive medium kept far from equilibrium can destabilize the trivial steady state and lead to spatiotemporal dissipative structures. The involved physical processes are the spatial bistability and the swelling properties of the gel. As an illustration of this morphogenetic process, we show that a sphere of gel immersed within such a medium with autocatalytic properties can exhibit periodic radius pulsations, the amplitude of which are controlled by chemistry.     

Statistical mechanical theory of the nonlinear steady state

By making use of perturbation techniques, we develop a theory of the non-linear steady state. We find that the linear term of a mechanical equation such as the Langevin equation is not responsible for the nonlinear terms of its expectation values at the nonequilibrium state arbitrarily far from the thermal equilibrium. The nonlinear steady state is formulated in the two cases where the microscopic conservation law exists and where it does not exist. The expressions for the expectation values of the physical quantities at the steady state are obtained as the functions of other physical quantities which are regarded as the parameters of the steady state. The stability and the instability of the steady state are discussed. A difference in the character of the instability of the steady state from that of the stationary state is discussed. It is noted that the first expansion coefficient should not exhibit an anomaly for instabilities of the steady state. The relation between the mechanical forces appearing in our approach and the corresponding thermal forces is discussed. The variational principle which is valid for the open system is developed.     

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: II. Kinetic potential and variational principles for nonlinear irreversible processes

On the basis of a complete system of fluctuation-dissipation relations, considered in the first part of this series, a variational principle for nonlinear irreversible processes is derived. According to this principle the virtual entropy production functional (analogous to the action in mechanics) has an absolute minimum meaning on the real trajectory of a system. The universal structure of the “kinetic potential” and the “lagrangian” of a system, each contain complete information about fluctuations of macrovariables. The connection of the lagrangian with the markovian kinetic operator of macrovariables is stated. Fundamental properties of dissipative potentials, reflecting microscopic reversibility, are considered. The derived variational principle can be applied to closed systems (the steady state of which is equilibrium) as well as to open ones (when external dynamic forces cause entropy flux through the system and put it into a steady non-equilibrium state). Canonical transformations of macrovariables are considered.     

Deviations from minimum entropy production at steady states of reacting chemical systems arbitrarily close to equilibrium

Our analysis of reacting systems displaced from equilibrium by a matter flux across the boundaries has shown that the state of minimum entropy production differs from the steady state, even in the near equilibrium regime. When the displacement δ from equilibrium is small, the derivative of the dissipation at the steady state and the dissipation itself are both of order δ². The state of least dissipation is displaced from the steady state by terms of order δ² in the species concentrations. The theorem of minimum entropy production may be derived by first truncating the series expansions for the reaction rates, affinities and dissipation assuming that δ is small, and then differentiating to locate the minimum of the dissipation within the resulting idealized model. For chemical systems with an arbitrarily small but macroscopic displacement from equilibrium, this truncation procedure establishes that the dissipation is comparatively small in a neighborhood of the steady state; but it causes large relative errors in the values of the concentration derivatives and time derivatives of the dissipation within that neighborhood, because the operations of series truncation and differentiation do not commute near the steady state, when δ is small but nonzero. Near the steady state, the concentration derivative of the term of order δ³ in the dissipation is comparable to or larger than the derivative of the δ² term.     

RF impedance of intrinsic Josephson junction in flux-flow state with a periodic pinning potential and its optimum condition for RF radiation

We reported dynamics of Josephson vortices interacting with electromagnetic waves in strongly coupled long Josephson junctions stack, such as an intrinsic Josephson junction (IJJ), by numerical simulations based on coupled sine-Gordon equations considering a periodic pinning potential of sinusoidal form. The numerical simulation results for the influence of the electromagnetic waves on flux-flow properties show that the periodic pinning potential induces an in-phase motion of Josephson vortices over the junction stacks, which achieve high performances of IJJ flux-flow oscillator. In order to prove it from another viewpoint, we calculate RF impedance of long Josephson junction stacks in flux-flow state. A remarkable negative real part region of RF impedance appears at 1st harmonic step, it means that the long Josephson junction stacks in flux-flow state acts as an oscillator at the negative real part region. In this study, we evaluate the optimum condition for RF radiation with the periodic pinning potential.     

Rotation in an asymmetric multidimensional periodic potential due to colored noise

We analyze the motion of an overdamped classical particle in a multidimensional periodic potential, driven by an external noise. We demonstrate that in the steady state the presence of temporal correlations in the noise and spatial asymmetry within a period of the potential could lead to particle rotation. The rotation is a direct consequence of a change in the sign of the noise-induced drift motion in each dimension. By choosing different potentials, we can generate a variety of flow patterns from laminar drifts to rotations.     

积与积减原理

文章分析了重力势能、引力势能、电荷势能、化学势能、热量势能、质量积、动量积等多种势能,发现它们均可表达为一种守恒广延量和对应的强度量的乘积,因此可将其统一定义为"积".基于积这一概念,文章得到了孤立系统内守恒广延量传递过程的积减原理,即孤立系统内进行的守恒广延量传递过程中系统的积总是减小的.进一步,文章还基于积的概念发展了孤立系统和封闭系统的势平衡判据,发现孤立系统达到势平衡状态时,系统的积达到最小值(最小积原理);当封闭系统达到势平衡状态时,系统的准自由积达到最小值(最小准自由积原理).上述结论应用于传热学中即可得到热量传递过程的(火积)减原理及相应的热平衡判据.与热力学中的核心概念熵相对应,由于物理量(火积)可以描述传热过程的不可逆性,作为传热过程的优化准则,度量系统的无序度,并给出系统的热平衡判据,因此(火积)是传热学中的核心概念. 关键词： 势能 积 积减原理 平衡判据     

简并光学参量振荡器混沌反控制

提出一种实现简并光学参量振荡器混沌反控制的方法，用正弦信号调制简并光学参量振荡器的基模衰减率，使简并光学参量振荡器从定态输出转化为混沌态.数值模拟结果表明，选择不同的调制幅度和调制角频率，只要满足系统的最大李雅谱诺夫指数大于零，即可实现不同的混沌轨道重构.通过比较最大李雅谱诺夫指数λ_max随调制幅度和调制角频率变化曲线， 指出系统从周期态调制到混沌态比从无亚谐波输出的定态调制到混沌态更容易，有更宽的调制幅度和调制角频率选择范围. 关键词： 简并光学参量振荡器 混沌反控制 调制     

Electron-hole droplets size at equilibrium

A finite lifetime τ implies that the liquid phase can only form droplets of finite size. Introducing this finite τ in the usual nucleation theory, one can obtain analytical expressions for the electron-hole droplets size, number and exciton density at steady state, in the limit of large τ. This equilibrium size results, in fact, from a Maxwell construction on an appropriate chemical potential.