Nonlinear transport and dynamics of fluctuations

The time evolution of the macroscopic variables of a system initially in a state far from thermal equilibrium is studied from a statistical mechanical point of view. Exact nonlinear transport equations for the mean values and linear nonstationary Langevin equations for the fluctuations around the mean path are derived. Connections between the dynamics of fluctuations and the transport equations are discussed. The Langevin random forces depend on the macroscopic state and they are related to the transport kernels by a fluctuation-dissipation formula.     

不可压湍流的控制方程

使用侧偏统计平均方法理性推导了不可压湍流的封闭方程组．侧偏统计平均提供了湍流脉动量的一阶统计信息，在引入加权漂移速度对称性及正交各向异性后，使用动量传输链概念模化封闭了整个方程组．方程组不含任何经验系数，不使用壁面函数，保留了ＮＳ方程的均化的非线性特性．其级数形式的能量方程与非线性现象多尺度层次现象相对应，具备了描述湍流统计平均流动及拟序结构流动的双重功能．用平面射流，圆射流，层流－湍流转捩及后向台阶流等算例初步验证了方程对真实湍流的适定性．     

Microdynamics and equations of motion for macrovariables

By means of a new time-dependent projection operator an exact generalized Langevin equation for the macrovariables of a system is derived. This equation is in general nonlinear and also valid far from equilibrium. The projection operator picks up the macroscopic part of an observable which is defined in such a way that it's mean value depends only on the macroscopic state given by the mean values of the considered macrovariables. The exact equation can be separated into an evolution equation for the mean values and an equation for the fluctuations. The second equation contains a nonlinear random force and a term which shows up to be the linearization of the mean value equations around the mean path. The connection with previous works is discussed.     

Evolution of wave turbulence under "gusty" forcing

We consider nonlinear evolution of a random wave field under gusty forcing, fluctuating around a constant mean. Here the classical wave turbulence theory that assumes a proximity to stationarity is not applicable. We show by direct numerical simulation that the self-similarity of wave field evolution survives under fluctuating forcing. The wave field statistical characteristics averaged over fluctuations of forcing evolve as if there were a certain constant "effective wind." The results justify the use of the kinetic equations with forcing averaged over gusts as a good first approximation.     

Strategies for fluctuation renormalization in nonlinear transport theory

We are concerned here with the problems encountered in the derivation of nonlinear transport equations from a correspondingly nonlinear Langevin equation. A dynamical coupling between the time-dependent averages and the fluctuations must be accounted for by a procedure which leads to a renormalization of the nonlinear transport equation. Generalizing the familiar phenomenological approach to Brownian motion to nonlinear dynamics, we illustrate how the problem arises and show how the fluctuation renormalization can be obtained exactly by a formal procedure or approximately by more tractable methods.     

On the theory of fluctuations around non-equilibrium steady states: A generalized time-convolutionless projector formalism

A new method of finding nonlinear Langevin type equations of motion for relevant macrovariables and the corresponding master equation for systems far from thermal equilibrium is presented by generalizing the time-convolutionless formalism proposed previously for equilibrium hamiltoian systems by Tokuyama and Mori. The Langevin type equation consists of a fluctuating force, and the nonlinear drift coefficients which are always identical to those of the master equation. A simple formula which relates the drift coefficients to the time correlation of the fluctuating forces is derived. This is a generalization of the fluctuation-dissipation theorem of the second kind in equilibrium systems and is valid not only for transport phenomena due to internal fluctuations but also for transport phenomena due to externally-driven fluctuations. A new cumulant expansion of the master equation is also obtained. The conditions under which a Langevin and a Fokker-Planck equation of a generalized type for non-equilibrium open systems can be derived are clarified.The theory is illustrated by studying hydrodynamic fluctuations near the Rayleigh-Bénard instability. The effects of two kinds of fluctuations, internal fluctuations of irrelevant macrovariables and external (thermal) noises, on the convective instability are investigated. A stochastic Ginzburg-Landau type equation for the order parameter and the corresponding nonlinear Fokker-Planck equation are derived.     

The rich phenomenology of Brownian particles in nonlinear potential landscapes

Non-interacting Brownian particles obey Langevin equations fulfilling a fluctuation–dissipation relation between friction and thermal noise. Under a linear potential (constant force) Einstein found a relation between diffusion and transport through mobility. In nonlinear potentials this prediction is only satisfied within the limits of very small and large constant external forces. Moreover, other more interesting behaviors do appear, such as: dispersionless transport, sorting, giant diffusion, subdiffusion, superdiffusion, subtransport, etc. All these phenomena depend on the characteristics of the nonlinear potential landscape: periodic or random, the symmetries and boundary conditions. Moreover, the presence of transport is the keystone of most of this phenomenology. In this review, we present numerical simulations illustrating these facts and theoretical analysis when possible.     

Transport Theory and Collective Modes. I. The Case of Moderately Dense Gases

The complex spectral representation of the Liouville operator introduced by Prigogine and others is applied to moderately dense gases interacting through hard-core potentials in arbitrary d-dimensional spaces. Kinetic equations near equilibrium are constructed in each subspace as introduced in the spectral decomposition for collective, renormalized reduced distribution functions. Our renormalization is a nonequilibrium effect, as the renormalization effect disappears at equilibrium. It is remarkable that our renormalized functions strictly obey well-defined Markovian kinetic equations for all d, even though the ordinary distribution functions obey nonMarkovian equations with memory effects. One can now define transport coefficients associated to the collective modes for all dimensional systems including d = 2. Our formulation hence provides a microscopic meaning of the macroscopic transport theory. Moreover, this gives an answer to the long-standing question whether or not transport equations exist in two-dimensional systems. The non-Markovian effects for the ordinary distribution function, such as the long-time tails for arbitrary n-mode coupling, are estimated by superposition of the Markovian evolutions of the dressed distribution functions.     

Regression law and the renormalization of the transport coefficients

Using Mazur's lemma we show that the coarse-grained variables used in nonequilibrium statistical mechanics are the Onsager's regression variables. With this result we find a regression law for the fluctuations which is both non-Markovian and nonlinear. Considering the Markovian approximation and generalizing Onsager's ideas leading to the symmetry of the transport matrix, we formulate Mori and Fujisaka's method for the renormalization of transport coefficients due to nonlinear interactions.     

On the theory of nonlinear response in a nonideal plasma

A theory of nonlinear response is developed for studying nonlinear phenomena and nonlinear transport processes in nonideal Coulomb systems. Temporal plasma echo and transformation of waves in a nonideal Coulomb system are studied on the basis of the theory of nonlinear response to mechanical perturbations. General constraints imposed on nonlinear response functions are considered, and the model for determining quadratic response functions is formulated. The conditions for the emergence of temporal plasma echo and wave transformation are determined. It is shown that these nonlinear effects in a nonideal plasma can be initiated by ultrashort field pulses. A theory of transport is developed for determining the Burnett transport properties of a nonideal multielement plasma. A procedure is proposed for comparing the phenomenological conservation equations for a charged continuous medium and equations of motion for the operators of corresponding dynamic variables. The Mori algorithm is used for deriving the equations of motion for operators of dynamic variables in the form of generalized Langevin equations. The linearized Burnett approximation is considered in detail. The properties of the matrices of coefficients of higher-order derivatives in the system of conservation equations in the linearized Burnett approximation, which are important for hydrodynamic applications, are discussed. Various versions of the theory of nonlinear response are compared.     

Phase measurement of waves that obey nonlinear equations

We consider the problem of phase retrieval for classical and quantum wave fields that obey a wide class of nonlinear wave equations. This problem is tackled by means of a suitable generalization of existing methods based on the linear transport-of-intensity equation. The extension of these ideas to systems of coupled nonlinear wave equations is also given.     

Nonclassical light generation in quasi-phase-matched parametric self-frequency conversion

We present a quantum theory of the parametric self-conversion of the laser radiation frequency in active nonlinear crystals with a regular domain structure. Such crystals feature simultaneous lasing and quasi-phase-matched parametric conversion of the laser radiation frequency. These processes are described using the Heisenberg-Langevin equations in two regimes of the subharmonic generation: super-and subthreshold. The spectral properties of the quadrature components of the laser frequency and its subharmonic and the photon statistics have been studied as dependent on the pump power, crystal length, and reflectance of the laser cavity output mirror. Using the obtained analytical expressions, these characteristics are calculated for a active nonlinear Nd:Mg:LiNbO₃ crystal with a regular domain structure. In the subthreshold regime, the maximum decrease in the spectral density of fluctuations in the subharmonic quadrature component relative to the standard quantum limit may reach 90%; in the above-threshold regime, these fluctuations are virtually not suppressed. A decrease in the spectral density of fluctuations of the laser frequency quadrature does not exceed 10%. In the subthreshold excitation regime, the subharmonic photons obey a super-Poisson statistics; in the above-threshold regime, the photon statistics is Poisson-like.     

超音速湍流脉动流场气动光学效应分析

对高超音速飞行器在大气中飞行时所产生的湍流脉动气动光学效应进行了理论分析与计算.根据CFD计算流场时所使用的湍流模型及其控制方程,推导出流场的折射率脉动方差控制方程.用统计方法,求出该脉动流场的系综平均光学传递函数及相位均方差,从不同角度描述了湍流脉动的气动光学效应.计算结果表明,气动光学传输函数的幅度响应函数具有低通特征,使所获得的图像发生像模糊,而相位响应函数则导致红外成像相位非线性偏移.此外,在相同飞行高度下,马赫数越高,图像模糊越严重.电弧风洞实验结果验证了本文理论分析的正确性.     

Diffusive Fluctuations for One-Dimensional Totally Asymmetric Interacting Random Dynamics

We study central limit theorems for a totally asymmetric, one-dimensional interacting random system. The models we work with are the Aldous–Diaconis–Hammersley process and the related stick model. The A-D-H process represents a particle configuration on the line, or a 1-dimensional interface on the plane which moves in one fixed direction through random local jumps. The stick model is the process of local slopes of the A-D-H process, and has a conserved quantity. The results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under hydrodynamic scaling. We look at diffusive fluctuations, by which we mean fluctuations on the scale of the classical central limit theorem. In the scaling limit these fluctuations obey deterministic equations with random initial conditions given by the initial fluctuations. Of particular interest is the effect of macroscopic shocks, which play a dominant role because dynamical noise is suppressed on the scale we are working. Received: 4 October 2001 / Accepted: 12 March 2002     

Lagrangian averages,averaged Lagrangians,and the mean effects of fluctuations in fluid dynamics

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler-Poincare (EP) variational framework of fluid dynamics, for an averaged Lagrangian. This is the Lagrangian averaged Euler-Poincare (LAEP) theorem. Next, we derive a set of approximate small amplitude GLM equations (glm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the glm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The glm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction. Next, the new glm EP motion equations for incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or glm) fluid theory with a Taylor hypothesis closure. Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha equations. Thus, by using the LAEP theorem, we bridge between the GLM equations and the Euler-alpha closure equations, through the small-amplitude glm approximation in the EP variational framework. We conclude by highlighting a new application of the GLM, glm, and alpha-model results for Lagrangian averaged ideal magnetohydrodynamics. (c) 2002 American Institute of Physics.     

Has Bell's inequality a general meaning for hidden-variable theories?

We analyze the proof given by J. S. Bell of an inequality between mean values of measurement results which, according to him, would be characteristic of any local hidden-parameter theory. It is shown that Bell's proof is based upon a hypothesis already contained in von Neumann's famous theorem: It consists in the admission that hidden values of parameters must obey the same statistical laws as observed values. This hypothesis contradicts in advance well-known and certainly correct statistical relations in measurement results: One must therefore reject the type of theory considered by Bell, and his inequality has no general meaning.     

超音速湍流脉动流场气动光学效应分析

对高超音速飞行器在大气中飞行时所产生的湍流脉动气动光学效应进行了理论分析与计算.根据CFD计算流场时所使用的湍流模型及其控制方程,推导出流场的折射率脉动方差控制方程.用统计方法,求出该脉动流场的系综平均光学传递函数及相位均方差,从不同角度描述了湍流脉动的气动光学效应.计算结果表明,气动光学传输函数的幅度响应函数具有低通特征,使所获得的图像发生像模糊,而相位响应函数则导致红外成像相位非线性偏移.此外,在相同飞行高度下,马赫数越高,图像模糊越严重.电弧风洞实验结果验证了本文理论分析的正确性.     

Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler–Poincaré (EP) variational framework of fluid dynamics, for an averaged Lagrangian. We then state the EP Averaging Result—that GLM equations arise from GLM Hamilton’s principles in the EP framework. Next, we derive a new set of approximate small-amplitude GLM equations (gℓm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the gℓm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The gℓm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction (WMFI). The gℓm EP motion equations for compressible and incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or gℓm) fluid theory with a Taylor hypothesis closure (THC). Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha closure equations. We use the EP Averaging Result to bridge between the GLM equations and the Euler-alpha closure equations. Hence, combining the small-amplitude approximation with THC yields in new turbulence closure equations for compressible fluids in the EP variational framework.     

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem

The paper is devoted to the theory of thermal fluctuations in nonlinear macroscopic systems and to the derivation of variational principles of nonlinear nonequilibrium thermodynamics. In the first part of the paper rigorous universal fluctuation-dissipation relations for nonlinear classical and quantum systems, subjected to dynamic as well as thermodynamic perturbations, are derived and analyzed. General expressions for dissipative fluxes and nonlinear transfer coefficients with the help of fluctuation cumulants are found. The canonical structure of nonlinear evolution equations of macrovariables is derived and the rule of introducing langevinian random forces into these equations, in accordance with fluctuation-dissipation relations. A Markovian theory of fluctuations in a stationary nonequilibrium state is constructed.