Boltzmann entropy for dense fluids not in local equilibrium

Using computer simulations, we investigate the time evolution of the (Boltzmann) entropy of a dense fluid not in local equilibrium. The macrovariables M describing the system are the (empirical) particle density f=[f(x,v)] and the total energy E. We find that S(f(t),E) is a monotone increasing in time even when its kinetic part is decreasing. We argue that for isolated Hamiltonian systems monotonicity of S(M(t))=S(M(X(t))) should hold generally for "typical" (the overwhelming majority of) initial microstates (phase points) X0 belonging to the initial macrostate M0, satisfying M(X0)=M(0). This is a consequence of Liouville's theorem when M(t) evolves according to an autonomous deterministic law.     

存在初级象差下电子和离子光学系统中刘维定理的应用

在电子和离子光学理论(特别是象差理论)中,广泛地应用了矩阵的运算方法。本文讨论了旋转对称电子光学系统与重叠场(环形电场与非均匀磁场的叠加)离子光学系统中存在初级象差下的变换矩阵,证明了这些变换矩阵的行列式数值恒为1.这个普遍结论是刘维定理的具体应用。这对于电子和离子光学象差理论及计算机辅助设计具有一定意义。 关键词：     

A Nonequilibrium Extension of the Clausius Heat Theorem

We generalize the Clausius (in)equality to overdamped mesoscopic and macroscopic diffusions in the presence of nonconservative forces. In contrast to previous frameworks, we use a decomposition scheme for heat which is based on an exact variant of the Minimum Entropy Production Principle as obtained from dynamical fluctuation theory. This new extended heat theorem holds true for arbitrary driving and does not require assumptions of local or close to equilibrium. The argument remains exactly intact for diffusing fields where the fields correspond to macroscopic profiles of interacting particles under hydrodynamic fluctuations. We also show that the change of Shannon entropy is related to the antisymmetric part under a modified time-reversal of the time-integrated entropy flux.     

Fluctuation Theorem for Currents and Schnakenberg Network Theory

A fluctuation theorem is proved for the macroscopic currents of a system in a nonequilibrium steady state, by using Schnakenberg network theory. The theorem can be applied, in particular, in reaction systems where the affinities or thermodynamic forces are defined globally in terms of the cycles of the graph associated with the stochastic process describing the time evolution.     

The internal state of a gas of particles with spin

A Hamiltonian theory of motion for particles with internal structure, based on a dynamical bracket algebra, serves to define a generalized phase space for axially symmetric particles. Hydrodynamics of a gas of particles are explored through Liouville's and Boltzmann's equations and the equilibrium distribution function is obtained. Some modification of the pressure and density are necessary because of the spin and for a cold Fermi gas in thermal equilibrium they are shown explicitly.This work is based on a Ph.D. thesis of Y. Feldman submitted to Tel Aviv University in 1979.     

Condensed matter effects in the solar neutrino problem

In addition to our earlier corrections to fusion cross sections, we proposed that previously overlooked condensed matter effects (CME) can help to account for the missing solar neutrino flux. There are three important CME. One is due to a reduction in collision frequency due to an exchange of kinetic and potential energies in collision processes. Another is an excluded volume effect. The third is a shadowing effect due to the presence of spectator species which do not participate in fusion. These CME become appreciable in the high densities encountered in stellar media where they significantly affect fusion rates, since the solar core plasma cannot accurately be described as a collisionless ideal gas. Contrary to Bahcall and Gould (1993), we do not violate Liouville's theorem, the Maxwellian distribution, nor thermodynamics in our proposed solution to the solar neutrino problem.     

Integrability and pseudointegrability in billiards illustrated by the harmonic wedge

We discuss Liouville's theorem for nonsmooth integrable systems of the billiard type and give a scheme of calculation of angle-action variables for the flow. We also deal with the problem of pseudointegrability. We discuss the relationship between the continuous-time (flow) and the discrete-time (map) approaches. We treat all these aspects through a specific billiard—a wedge embedded in a two-dimensional isotropic harmonic potential. Varying the parameters provides two integrable and two pseudointegrable cases. It turns out that the dynamics of one of the latter is indeed integrable in a certain sense. We also address the problem of applying perturbation theory.     

Berry phase correction to electron density of states in solids

Liouville's theorem on the conservation of phase-space volume is violated by Berry phase in the semiclassical dynamics of Bloch electrons. This leads to a modification of the phase-space density of states, whose significance is discussed in a number of examples: field modification of the Fermi-sea volume, connection to the anomalous Hall effect, and a general formula for orbital magnetization. The effective quantum mechanics of Bloch electrons is also sketched, where the modified density of states plays an essential role.     

保守系统的混沌控制

保守系统的混沌控制是一个重要而富有挑战性的研究课题。由于Liouville定理的限制和初始条件的特殊作用，使得适用于耗散系统的混沌控制方法不能直接用于保守系统。本文通过对耗散系统和保守系统混沌运动的特征进行分析和比较，阐述了保守系统混沌运动的规律，总结了近期研究过程中一些典型的基本理论和方法，综述了近年来保守系统混沌控制的相关进展和我们在保守系统的混沌控制方面所做的工作，并对保守系统混沌控制的应用和发展方向进行了展望。     

Fluctuations in the diffusion of particles in a non-uniform medium

We study the thermal fluctuations of density and flow velocity of particles diffusing in a non-uniform medium with memory character. The fluctuation spectra are derived from the macroscopic equations with the aid of the generalized Nyquist theorem. A comparison is made with other methods.     

Macroscopic Fluctuation Theory for Stationary Non-Equilibrium States

We formulate a dynamical fluctuation theory for stationary non-equilibrium states (SNS) which is tested explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Within this theory we derive the following results: the modification of the Onsager–Machlup theory in the SNS; a general Hamilton–Jacobi equation for the macroscopic entropy; a non-equilibrium, nonlinear fluctuation dissipation relation valid for a wide class of systems; an H theorem for the entropy. We discuss in detail two models of stochastic boundary driven lattice gases: the zero range and the simple exclusion processes. In the first model the invariant measure is explicitly known and we verify the predictions of the general theory. For the one dimensional simple exclusion process, as recently shown by Derrida, Lebowitz, and Speer, it is possible to express the macroscopic entropy in terms of the solution of a nonlinear ordinary differential equation; by using the Hamilton–Jacobi equation, we obtain a logically independent derivation of this result.     

The Quantum-Classical Comparison of the Arrival-Time Distribution Through the Probability Current

We consider the arrival time distribution defined through the quantum probability current for a Gaussian wave packet representing free particles in quantum mechanics in order to explore the issue of the classical limit of arrival time. We formulate the classical analogue of the arrival time distribution for an ensemble of free particles represented by a phase space distribution function evolving under the classical Liouville's equation. The classical probability current so constructed matches with the quantum probability current in the limit of minimum uncertainty. Further, it is possible to show in general that smooth transitions from the quantum mechanical probability current and the mean arrival time to their respective classical values are obtained in the limit of large mass of the particles.     

Recurrence in Quantum Mechanics

We first compare the mathematical structure of quantum and classical mechanics when both are formulated in a C^*-algebraic framework. By using finite von Neumann algebras, a quantum mechanical analogue of Liouville's theorem is then proposed. We proceed to study Poincaré recurrence in C^*-algebras by mimicking the measure theoretic setting. The results are interpreted as recurrence in quantum mechanics, similar to Poincaré recurrence in classical mechanics.     

新的Lax可积发展方程族及其无限维双-哈密顿结构

基于一个新的具有三个位势函数(q,r,s)的等谱问题,获得了一族新的含有一个任意函数的Lax可积发展方程.特别地,当位势函数s取不同的函数时,这个方程族约化为若干类型的方程组,进一步利用迹恒等式,给出了这些方程组的双哈密顿结构,并且证明它们是Liouville可积的.此外,给出了守恒密度和对称. 关键词： 等谱问题 哈密顿结构 Lax可积 Liouville可积     

Modified Schrödinger dynamics with attractive densities

The linear Schrödinger equation does not predict that macroscopic bodies should be located at one place only, or that the outcome of a measurement shoud be unique. Quantum mechanics textbooks generally solve the problem by introducing the projection postulate, which forces definite values to emerge during measurements; many other interpretations have also been proposed. Here, in the same spirit as the GRW and CSL theories, we modify the Schrödinger equation in a way that efficiently cancels macroscopic density fluctuations in space. Nevertheless, we do not assume a stochastic dynamics as in GRW or CSL theories. Instead, we propose a deterministic evolution that includes an attraction term towards the averaged density in space of the de Broglie-Bohm position of particles, and show that this is sufficient to ensure macroscopic uniqueness and compatibility with the Born rule. The state vector can then be seen as directly related to physical reality.     

Large numbers hypothesis. III. Kinetic theory,statistical physics,and thermodynamics

Dirac's large numbers hypothesis (LNH) is incorporated into kinetic theory, statistical physics, and thermodynamics using the self-consistent formalism of units covariance. The ingeodesic equation and matter creation introduce modifications of the most fundamental laws of the subject. Liouville's theorem no longer holds, the Boltzmann equation is modified, as is theH-theorem. This affects the second law of thermodynamics in that for canonical LNH neither reversible nor adiabatic processes are possible (as expected). A significant result is that the collision terms have the same form as in standard physics. This means that equilibrium distribution functions are identical to those of standard physics, as required for self-consistency with the precepts of LNH. The net effect of LNH is as though all matter in our Universe were weakly coupled to a large heat bath.NAS-NRC Senior Research Associate 1981–1983.     

Mean square relaxation times for evolution of random fields

We consider the problem of relaxation times for Markov evolution of systems composed of a countable number of locally interacting particles, each one of which has a finite phase space. We give a theorem for comparison of mean square relaxation times of evolutions possessing the same ergodic stationary state. We give a reduction theorem for “attractive” evolutions. The results are applied to a generalization of the Glauber evolution of the one dimensional Ising chain.     

A stochastic model for the dynamics of a classical spin

A stochastic model for the dynamics of a macroscopic or classical spin based on a classical generalized Lagrangian formalism is proposed. The model can be used to describe the evolution of the magnetic moment of superparamagnetic particles. In this sense, it is a generalization of the model proposed by Brown, allowing for fluctuations on the magnitudes of the magnetic moments of the particles. The corresponding covariant Fokker-Planck equation is also obtained.     

Evolution of percolating force chains in compressed granular media

The evolution of effective force chains percolating through a compressed granular system is investigated. We compressed an ensemble of spherical particles monitoring the macroscopic constitutive behavior and the acoustic signals emitted by microscopic rearrangements of particles. We applied the continuous damage model of fiber bundles to describe the evolution of the array of force chains. The model provides a nonlinear constitutive behavior in good quantitative agreement with the experimental results. The amplitude distribution of acoustic signals was found experimentally to follow a power law of an exponent delta=1.15+/-0.05, which is in good agreement with the analytic solution of the model.