An Optimization Principle for Deriving Nonequilibrium Statistical Models of Hamiltonian Dynamics

A general method for deriving closed reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. Given a vector of resolved variables, selected to describe the macroscopic state of the system, a family of quasi-equilibrium probability densities on phase space corresponding to the resolved variables is employed as a statistical model, and the evolution of the mean resolved vector is estimated by optimizing over paths of these densities. Specifically, a cost function is constructed to quantify the lack-of-fit to the microscopic dynamics of any feasible path of densities from the statistical model; it is an ensemble-averaged, weighted, squared-norm of the residual that results from submitting the path of densities to the Liouville equation. The path that minimizes the time integral of the cost function determines the best-fit evolution of the mean resolved vector. The closed reduced equations satisfied by the optimal path are derived by Hamilton-Jacobi theory. When expressed in terms of the macroscopic variables, these equations have the generic structure of governing equations for nonequilibrium thermodynamics. In particular, the value function for the optimization principle coincides with the dissipation potential that defines the relation between thermodynamic forces and fluxes. The adjustable closure parameters in the best-fit reduced equations depend explicitly on the arbitrary weights that enter into the lack-of-fit cost function. Two particular model reductions are outlined to illustrate the general method. In each example the set of weights in the optimization principle contracts into a single effective closure parameter.     

Basic concepts of synergetics

The paper presents a brief outline of microscopic as well as of macroscopic synergetics. In microscopic synergetics we start from evolution equations for microscopic variables or densities in which fluctuating forces and control parameters are included. When control parameters are changed, the systems are studied close to instability points. The concepts of order parameters, enslaving, critical fluctuations, and critical slowing down are presented. In macroscopic synergetics unbiased estimates on distribution functions and underlying processes are made based on observed moments or correlation functions. In such a case, a Fokker-Planck equation or a corresponding Langevin equation may be derived.     

Phase synchronization between collective rhythms of globally coupled oscillator groups: noiseless nonidentical case

We theoretically study the synchronization between collective oscillations exhibited by two weakly interacting groups of nonidentical phase oscillators with internal and external global sinusoidal couplings of the groups. Coupled amplitude equations describing the collective oscillations of the oscillator groups are obtained by using the Ott-Antonsen ansatz, and then coupled phase equations for the collective oscillations are derived by phase reduction of the amplitude equations. The collective phase coupling function, which determines the dynamics of macroscopic phase differences between the groups, is calculated analytically. We demonstrate that the groups can exhibit effective antiphase collective synchronization even if the microscopic external coupling between individual oscillator pairs belonging to different groups is in-phase, and similarly effective in-phase collective synchronization in spite of microscopic antiphase external coupling between the groups.     

Phase synchronization between collective rhythms of globally coupled oscillator groups: noisy identical case

We theoretically investigate the collective phase synchronization between interacting groups of globally coupled noisy identical phase oscillators exhibiting macroscopic rhythms. Using the phase reduction method, we derive coupled collective phase equations describing the macroscopic rhythms of the groups from microscopic Langevin phase equations of the individual oscillators via nonlinear Fokker-Planck equations. For sinusoidal microscopic coupling, we determine the type of the collective phase coupling function, i.e., whether the groups exhibit in-phase or antiphase synchronization. We show that the macroscopic rhythms can exhibit effective antiphase synchronization even if the microscopic phase coupling between the groups is in-phase, and vice versa. Moreover, near the onset of collective oscillations, we analytically obtain the collective phase coupling function using center-manifold and phase reductions of the nonlinear Fokker-Planck equations.     

过渡金属化合物OsB2与OsO2低压缩性的第一性原理计算研究

利用基于密度泛函理论平面波赝势法的第一性原理计算,研究了过渡金属化合物OsB2和OsO2的金红石相、黄铁矿相与萤石相三种结构在高压下的状态方程和结构特性以及OsO2可能的高压相变.理论计算结果支持OsB2与OsO2的萤石相是潜在超低可压缩性的硬性材料.同时,也分析了它们的电子结构,力求理解大体变模量和高硬度的微观机制.结果表明,可以利用过渡金属高的价电子浓度,掺入硼、氧、碳、氮等轻的元素形成强的方向键,这可能提供了一种合成超硬材料的新途径.     

Generalized quantum hydrodynamics of a trapped dilute Bose gas

Quantal kinetic equations for particle and current densities of condensate and non-condensate in a confined Bose-condensed fluid are set up by expansion of the one-body density matrix about its diagonal. A microscopic Landau equation for superfluid flow in the inhomogeneous system is derived. Current-density functional theory in the local (long-wavelength) approximation is then used to propose a unified treatment of various damping mechanisms.     

Spontaneous Breaking of Translational Invariance and Spatial Condensation in Stationary States on a Ring. II. The Charged System and the Two-Component Burgers Equations

We further study the stochastic model discussed in ref. 2 in which positive and negative particles diffuse in an asymmetric, CP invariant way on a ring. The positive particles hop clockwise, the negative counter-clockwise and oppositely-charged adjacent particles may swap positions. We extend the analysis of this model to the case when the densities of the charged particles are not the same. The mean-field equations describing the model are coupled nonlinear differential equations that we call the two-component Burgers equations. We find roundabout weak solutions of these equations. These solutions are used to describe the properties of the stationary states of the stochastic model. The values of the currents and of various two-point correlation functions obtained from Monte-Carlo simulations are compared with the mean-field results. Like in the case of equal densities, one finds a pure phase, a mixed phase and a disordered phase.     

Time-dependent statistics of binary linear lattices

The time-dependent statistics of binary linear lattices is investigated on the basis of a master equation at the microscopic level. It is assumed that the kinetics may be formulated as transformations of specified sequences of clusters ofA units andB units into other specified sequences. On the basis of aStosszahlansatz, a master equation at the macroscopic level is derived. In the limit of a large system, the densities of clusters of all types satisfy rate equations similar to the equations of chemical kinetics. AnH-theorem is proven and the nonequilibrium thermodynamics of the system is studied. The theory has application to the kinetics of the helix-coil phase transition in biopolymers.     

Constraints on neutron star radii based on chiral effective field theory interactions

We show that microscopic calculations based on chiral effective field theory interactions constrain the properties of neutron-rich matter below nuclear densities to a much higher degree than is reflected in commonly used equations of state. Combined with observed neutron star masses, our results lead to a radius R=9.7-13.9 km for a 1.4M⊙ star, where the theoretical range is due, in about equal amounts, to uncertainties in many-body forces and to the extrapolation to high densities.     

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.     

Derivation of the nonlinear hydrodynamic equations using multi-mode techniques

A general mode-mode coupling theory is developed for the microscopic mass, energy and momentum densities of a simple classical fluid. A projection operator method is employed to derive a generalized Langevin equation that contains nonlinearities of all orders with both convective and dissipative terms. A general nonequilibrium ensemble average, which contains local equilibrium as a special case, is employed to derive nonlinear transport equations that are nonlocal in both space and time.The nonlinear Euler and Navier-Stokes equations are recovered using a factorization procedure based on an inverse system size approximation. We show that in the context of mode-mode coupling theory, nonlinearities of all orders must be retained to derive the full nonlinear transport equations. We also slow that the space and time dependent nonequilibrium pressure and transport coefficients are functions of the nonequilibrium mass and internal energy densities. The thermodynamic closure relationships follow as a natural consequence of mode-mode coupling theory. For a system linearly displaced from equilibrium we demonstrate the role of the corrections to our factorization approximation in renormalizing the transport coefficients.     

Nuclear liquid-gas phase transition within the lattice gas model

We study the nuclear liquid-gas phase transition on the basis of a two-component lattice gas model. A Metropolis type of sampling method is used to generate microscopic states in the canonical ensemble. The effective equation of state and fragment mass distributions are evaluated in a wide range of temperatures and densities. A definition of the phase coexistence region appropriate for small systems is proposed. The caloric curve resulting from different types of freeze-out conditions are presented.     

过渡金属化合物OsB2与OsO2低压缩性的第一性原理计算研究

利用基于密度泛函理论平面波赝势法的第一性原理计算, 研究了过渡金属化合物OsB₂和OsO₂的金红石相、黄铁矿相与萤石相三种结构在高压下的状态方程和结构特性以及OsO₂可能的高压相变.理论计算结果支持OsB₂与OsO₂的萤石相是潜在超低可压缩性的硬性材料.同时，也分析了它们的电子结构，力求理解大体变模量和高硬度的微观机制.结果表明，可以利用过渡金属高的价电子浓度，掺入硼、氧、碳、氮等轻的元素形成强的 关键词： 过渡金属化合物 密度泛函理论 低压缩性 高压     

Influence of shell structure and pairing correlations on the nuclear state density

A microscopic calculation of nuclear state densities was performed starting from realistic single-particle levels. Within this concept, a correspondence between microscopic ground state energy corrections and microscopic effects on state densities was deduced. It is shown that the approximations introduced by a simple analytical expression for the nuclear state density are comparable to the uncertainties of microscopically calculated state densities for nuclei in their ground state configuration. Guidelines for the determination of the parameters of this analytical expression were deduced from the microscopic computations.     

Collective motion of active Brownian particles in one dimension

We analyze a model of active Brownian particles with non-linear friction and velocity coupling in one spatial dimension. The model exhibits two modes of motion observed in biological swarms: A disordered phase with vanishing mean velocity and an ordered phase with finite mean velocity. Starting from the microscopic Langevin equations, we derive mean-field equations of the collective dynamics. We identify the fixed points of the mean-field equations corresponding to the two modes and analyze their stability with respect to the model parameters. Finally, we compare our analytical findings with numerical simulations of the microscopic model.     

Evolution equations of the expected phase space ion densities and correlation functions in a D+T plasma

Summary In this paper we formualte a master equation approach describing a D+T thermonuclear plasma in a lumped phase space. From the first moments of this master equation and performing the pass to the continuous limit the evolution equations for the expected phase space ion densities emerge. Also we have obtained the evolution equations of the equal time correlation and covariance functions. Finally we have deduced the hydrodynamic equations that arise from a master equation approach.     

On the theory of austenite-cementite phase equilibria in steels

A microscopic model is proposed, which describes the structure and thermodynamic properties of cementite (an ordered iron carbide with a composition of Fe₃C_{1 ? δ}) and the phase equilibria between cementite and austenite (a disordered solid solution of carbon in face-centered cubic iron). Based on this model, chemical and stress-induced (deformational) interactions of carbon atoms in cementite and austenite structures are quantitatively evaluated. The “lattice” contributions in the equations of phase equilibria, which are related to changes in the crystal structure as a result of the cementite-austenite phase transition, are estimated. The proposed model well describes available data on the thermodynamics of cementite and austenite and provides a basis for the further microscopic investigation of high-temperature phase transformations in steels.     

Relativistic hartree-fock description of high-density nuclear matter

Relativistic Hartree-Fock equations are solved for an infinite system of nucleons and mesons. At high densities there exists a phase transition from a Fermi “sphere” to a Fermi “shell”, characterized by a discontinuity in the heat capacity. The precise value of the critical density is sensitive to the approximations.     

Dynamics of NaNO2 in the paraelectric and in the incommensurate phase

On the basis of a microscopic model with translation-rotation coupling, a set of coupled dynamic equations is derived. The corresponding resonances are studied in the paraelectric and in the incommensurate phase. The theory gives a qualitative explanation of inelastic neutron scattering results.