Nonhydrodynamic initial conditions are not soon forgotten.

Solutions to hydrodynamic equations, which are used for a vast variety of physical problems, are assumed to be specified by boundary conditions and initial conditions on the hydrodynamic variables only. Initial values of other variables are assumed to be irrelevant for a hydrodynamic description. We show that this assumption is not correct because of the existence of long-time-tail effects that are ubiquitous in systems governed by hydrodynamic equations. We illustrate this breakdown of a hydrodynamic description by means of the simple example of diffusion in a disordered electron system.

Rigidity and Superfast Signal Propagation in Fluids and Solids in Non-Equilibrium Steady States.

In the 1980s, it was theoretically predicted that correlations of various observables in a fluid in a non-equilibrium steady state (NESS) are extraordinarily long-ranged, extending, in a well-defined sense, over the size of the system. This is to be contrasted with correlations in an equilibrium fluid, whose range is typically just a few particle diameters. These NESS correlations were later confirmed by numerous experimental studies. Unlike long-ranged correlations at critical points, these correlations are generic in the sense that they exist for any temperature as long as the system is in a NESS. In equilibrium systems, generic long-ranged correlations are caused by spontaneously broken continuous symmetries and are associated with a generalized rigidity, which in turn leads to a new propagating excitation or mode. For example, in a solid, spatial rigidity leads to transverse sound waves, while, in a superfluid, phase rigidity leads to temperature waves known as second sound at finite temperatures and phonons at zero temperature. More generally, long-ranged spatial correlations imply rigidity irrespective of their physical origin. This implies that a fluid in a NESS should also display a type of rigidity and related anomalous transport behavior. Here we show that this is indeed the case. For the particular case of a simple fluid in a constant temperature gradient, the anomalous transport behavior takes the form of a super-diffusive spread of a constant-pressure temperature perturbation. We also discuss the case of an elastic solid, where we predict a spread that is faster than ballistic.

Work, work fluctuations, and the work distribution in a thermal nonequilibrium steady state.

Long-ranged correlations generically exist in nonequilibrium fluid systems. In the case of a nonequilibrium steady state caused by a temperature gradient, the correlations are especially long-ranged and strong. The anomalous light scattering predicted to exist in these systems is well-confirmed by numerous experiments. Recently, the Casimir force or pressure due to these fluctuations or correlations has been discussed in great detail. In this paper, the notion of a Casimir work is introduced, and an alternative way to measure the nonequilibrium Casimir force is suggested. In particular, the nonequilibrium Casimir force is related to nonequilibrium heat, and not, as in equilibrium, to a volume derivative of an average energy. The nonequilibrium work fluctuations are determined and shown to be very anomalous compared to equilibrium work fluctuations. The nonequilibrium work distribution is also computed, and it is contrasted with work distributions in systems with short-range correlations. Again, there is a striking difference in the two cases. Formal theories of work and work distributions in nonequilibrium steady states are not explicit enough to illustrate any of these interesting features.

Nonequilibrium is different.

Nonequilibrium and equilibrium fluid systems differ due to the existence of long-range correlations in nonequilibrium that are not present in equilibrium, except at critical points. Here we examine fluctuations of the temperature, of the pressure tensor, and of the heat current in a fluid maintained in a nonequilibrium stationary state (NESS) with a fixed temperature gradient, a system in which the nonequilibrium correlations are especially long-ranged. For this particular NESS, our results show that (i) the mean-squared fluctuations in nonequilibrium differ markedly in their system-size scaling compared to their equilibrium counterparts, and (ii) there are large, nonlocal correlations of the normal stress in this NESS. These terms provide important corrections to the fluctuating normal stress in linearized Landau-Lifshitz fluctuating hydrodynamics.

Wave packet autocorrelation functions for quantum hard-disk and hard-sphere billiards in the high-energy, diffraction regime.

We consider the time evolution of a wave packet representing a quantum particle moving in a geometrically open billiard that consists of a number of fixed hard-disk or hard-sphere scatterers. Using the technique of multiple collision expansions we provide a first-principle analytical calculation of the time-dependent autocorrelation function for the wave packet in the high-energy diffraction regime, in which the particle's de Broglie wavelength, while being small compared to the size of the scatterers, is large enough to prevent the formation of geometric shadow over distances of the order of the particle's free flight path. The hard-disk or hard-sphere scattering system must be sufficiently dilute in order for this high-energy diffraction regime to be achievable. Apart from the overall exponential decay, the autocorrelation function exhibits a generally complicated sequence of relatively strong peaks corresponding to partial revivals of the wave packet. Both the exponential decay (or escape) rate and the revival peak structure are predominantly determined by the underlying classical dynamics. A relation between the escape rate, and the Lyapunov exponents and Kolmogorov-Sinai entropy of the counterpart classical system, previously known for hard-disk billiards, is strengthened by generalization to three spatial dimensions. The results of the quantum mechanical calculation of the time-dependent autocorrelation function agree with predictions of the semiclassical periodic orbit theory.

Lyapunov spreading of semiclassical wave packets for the Lorentz gas: theory and applications.

We consider the quantum-mechanical propagator for a particle moving in a d -dimensional Lorentz gas, with fixed, hard-sphere scatterers. To evaluate this propagator in the semiclassical region, and for times less than the Ehrenfest time, we express its effect on an initial Gaussian wave packet in terms of quantities analogous to those used to describe the exponential separation of trajectories in the classical version of this system. This result relates the spread of the wave packet to the rate of separation of classical trajectories, characterized by positive Lyapunov exponents. We consider applications of these results, first to illustrate the behavior of the wave-packet autocorrelation functions for wave packets on periodic orbits. The autocorrelation function can be related to the fidelity, or Loschmidt echo, for the special case that the perturbation is a small change in the mass of the particle. An exact expression for the fidelity, appropriate for this perturbation, leads to an analytical result valid over very long time intervals, inversely proportional to the size of the mass perturbation. For such perturbations, we then calculate the long-time echo for semiclassical wave packets on periodic orbits.

Diffusive-ballistic crossover in 1D quantum walks.

We show that particle transport, as characterized by the equilibrium mean square displacement, in a uniform, quantum multibaker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semiclassical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a particle in the system. These results exhibit a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck's constant. We expect that, for a large class of 1D quantum random walks similar to the quantum multibaker, a sufficient condition for diffusion in the semiclassical limit is classically chaotic dynamics in each cell. The systems described generalize known quantum random walks and may have applications for quantum computation.

Quantum multibaker maps: extreme quantum regime.

We introduce a family of models for quantum mechanical, one-dimensional random walks, called quantum multibaker maps (QMB). These are Weyl quantizations of the classical multibaker models previously considered by Gaspard, Tasaki, and others. Depending on the properties of the phase's parametrizing the quantization, we consider only two classes of the QMB maps: uniform and random. Uniform QMB maps are characterized by phases that are the same in every unit cell of the multibaker chain. Random QMB maps have phases that vary randomly from unit cell to unit cell. The eigenstates in the former case are extended while in the latter they are localized. In the uniform case and for large variant Planck's over 2pi, analytic solutions can be obtained for the time-dependent quantum states for periodic chains and for open chains with absorbing boundary conditions. Steady state solutions and the properties of the relaxation to a steady state for a uniform QMB chain in contact with "particle" reservoirs can also be described analytically. The analytical results are consistent with, and confirmed by, results obtained from numerical methods. We report here results for the deep quantum regime (large variant Planck's over 2pi ) of the uniform QMB, as well as some results for the random QMB. We leave the moderate and small variant Planck's over 2pi results as well as further consideration of the other versions of the QMB for further publications.

Entropy production of diffusion in spatially periodic deterministic systems.

This paper presents an ab initio derivation of the expression given by irreversible thermodynamics for the rate of entropy production for different classes of diffusive processes. The first class is Lorentz gases, where noninteracting particles move on a spatially periodic lattice, and collide elastically with fixed scatterers. The second class is periodic systems, where N particles interact with each other, and one of them is a tracer particle that diffuses among the cells of the lattice. We assume that, in either case, the dynamics of the system are deterministic and hyperbolic, with positive Lyapunov exponents. This work extends methods originally developed for a chaotic two-dimensional model of diffusion, the multi-baker map, to higher-dimensional, continuous-time dynamical systems appropriate for systems with one or more moving particles. Here we express the rate of entropy production in terms of hydrodynamic measures that are determined by the fractal properties of microscopic hydrodynamic modes that describe the slowest decay of the system to an equilibrium state.

Field-dependent collision frequency of the two-dimensional driven random Lorentz gas.

In the field-driven, thermostated Lorentz gas the collision frequency increases with the magnitude of the applied field due to long-time correlations. We study this effect with computer simulations and confirm the presence of nonanalytic terms in the field dependence of the collision rate as predicted by kinetic theory.

Fractality of the hydrodynamic modes of diffusion.

Transport by normal diffusion can be decomposed into hydrodynamic modes which relax exponentially toward the equilibrium state. In chaotic systems with 2 degrees of freedom, the fine scale structures of these modes are singular and fractal, characterized by a Hausdorff dimension given in terms of Ruelle's topological pressure. For long-wavelength modes, we relate the Hausdorff dimension to the diffusion coefficient and the Lyapunov exponent. This relationship is tested numerically on two Lorentz gases, one with hard repulsive forces, the other with attractive, Yukawa forces. The agreement with theory is excellent.

Chaotic properties of dilute two- and three-dimensional random Lorentz gases. II. Open systems.

We calculate the spectrum of Lyapunov exponents for a point particle moving in a random array of fixed hard disk or hard sphere scatterers, i.e., the disordered Lorentz gas, in a generic nonequilibrium situation. In a large system which is finite in at least some directions, and with absorbing boundary conditions, the moving particle escapes the system with probability one. However, there is a set of zero Lebesgue measure of initial phase points for the moving particle, such that escape never occurs. Typically, this set of points forms a fractal repeller, and the Lyapunov spectrum is calculated here for trajectories on this repeller. For this calculation, we need the solution of the recently introduced extended Boltzmann equation for the nonequilibrium distribution of the radius of curvature matrix and the solution of the standard Boltzmann equation. The escape-rate formalism then gives an explicit result for the Kolmogorov Sinai entropy on the repeller.

CD4+ T cell survival is not directly linked to self-MHC-induced TCR signaling.

T cell receptor (TCR) signaling triggered by recognition of self-major histocompatibility complex (MHC) ligands has been proposed to maintain the viability of naïve T cells and to provoke their proliferation in T cell-deficient hosts. Consistent with this, the partially phosphorylated state of TCR zeta chains in naïve CD4+ and CD8+ T cells in vivo was found to be actively maintained by TCR interactions with specific peptide-containing MHC molecules. TCR ligand-dependent phosphorylation of TCR zeta was lost within one day of cell transfer into MHC-deficient hosts, yet the survival of transferred CD4+ lymphocytes was the same in recipients with or without MHC class II expression for one month. Thus, despite clear evidence for TCR signaling in nonactivated naïve T cells, these data argue against the concept that such signaling plays a predominant role in determining lymphocyte lifespan.

Simple deterministic dynamical systems with fractal diffusion coefficients.

We analyze a simple model of deterministic diffusion. The model consists of a one-dimensional array of scatterers with moving point particles. The particles move from one scatterer to the next according to a piecewise linear, expanding, deterministic map on unit intervals. The microscopic chaotic scattering process of the map can be changed by a control parameter. The macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter. We calculate the diffusion coefficent and the largest eigenmodes of the system by using Markov partitions and by solving the eigenvalue problems of respective topological transition matrices. For different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation. Our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter. We provide qualitative and quantitative arguments to explain features of this fractal structure.

Acquisition of Ly49 receptor expression by developing natural killer cells.

The formation of the repertoire of mouse natural killer (NK) cell receptors for major histocompatibility complex (MHC) class I molecules was investigated by determining the developmental pattern of Ly49 receptor expression. During the first days after birth, few or no splenic NK cells express Ly49A, Ly49C, Ly49G2, or Ly49I receptors. The proportion of Ly49+ splenic NK cells gradually rises to adult levels during the first 6-8 wk of life. The appearance of appreciable numbers of splenic Ly49+ NK cells coincides with the appearance of NK activity at 3-4 wk. After in vivo transfer, NK cells not expressing specific Ly49 receptors can give rise to NK cells that do, and cells expressing one of these four Ly49 receptors can give rise to cells expressing others. Once initiated, expression of a Ly49 receptor is stable for at least 10 d after in vivo transfer. Hence, initiation of Ly49 receptor expression occurs successively. Interestingly, expression of one of the receptors tested, Ly49A, did not occur after in vivo transfer of Ly49A- cells. One possible explanation for these data is that the order of Ly49 receptor expression by NK cells is nonrandom. The results provide a framework for evaluating models of NK cell repertoire formation, and how the repertoire is molded by host class I MHC molecules.

An analytical construction of the SRB measures for Baker-type maps.

For a class of dynamical systems, called the axiom-A systems, Sinai, Ruelle and Bowen showed the existence of an invariant measure (SRB measure) weakly attracting the temporal average of any initial distribution that is absolutely continuous with respect to the Lebesgue measure. Recently, the SRB measures were found to be related to the nonequilibrium stationary state distribution functions for thermostated or open systems. Inspite of the importance of these SRB measures, it is difficult to handle them analytically because they are often singular functions. In this article, for three kinds of Baker-type maps, the SRB measures are analytically constructed with the aid of a functional equation, which was proposed by de Rham in order to deal with a class of singular functions. We first briefly review the properties of singular functions including those of de Rham. Then, the Baker-type maps are described, one of which is nonconservative but time reversible, the second has a Cantor-like invariant set, and the third is a model of a simple chemical reaction R<-->I<-->P. For the second example, the cases with and without escape are considered. For the last example, we consider the reaction processes in a closed system and in an open system under a flux boundary condition. In all cases, we show that the evolution equation of the distribution functions partially integrated over the unstable direction is very similar to de Rham's functional equation and, employing this analogy, we explicitly construct the SRB measures. (c) 1998 American Institute of Physics.

Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy methods for sums of Lyapunov exponents for dilute gases.

We consider a general method for computing the sum of positive Lyapunov exponents for moderately dense gases. This method is based upon hierarchy techniques used previously to derive the generalized Boltzmann equation for the time-dependent spatial and velocity distribution functions for such systems. We extend the variables in the generalized Boltzmann equation to include a new set of quantities that describe the separation of trajectories in phase space needed for a calculation of the Lyapunov exponents. The method described here is especially suitable for calculating the sum of all of the positive Lyapunov exponents for the system, and may be applied to equilibrium as well as nonequilibrium situations. For low densities we obtain an extended Boltzmann equation, from which, under a simplifying approximation, we recover the sum of positive Lyapunov exponents for hard-disk and hard-sphere systems, obtained before by a simpler method. In addition we indicate how to improve these results by avoiding the simplifying approximation. The restriction to hard-sphere systems in d dimensions is made to keep the somewhat complicated formalism as clear as possible, but the method can be easily generalized to apply to gases of particles that interact with strong short-range forces. (c) 1998 American Institute of Physics.

Specificity, tolerance and developmental regulation of natural killer cells defined by expression of class I-specific Ly49 receptors.

Natural killer cells in the mouse express class I MHC-specific inhibitory receptors of the Ly49 protein family. The receptors mediate inhibition of the lysis of tumor cells and normal cells, and mediate the specificity of bone-marrow graft rejection by NK cells in vivo. The function of these receptors may be to confer upon NK cells the capacity to distinguish normal self cells from cells that have down-regulated expression of some or all self-class I molecules. Ly49 receptors discriminate between different class I molecules, and are distributed in expression to overlapping subsets of NK cells. The receptors appear to interact with class I-MHC residues and associated N-glycans, with little or no discrimination of the class I-bound peptide. The Ly49 receptor repertoire may be initially generated by a stochastic process that distributes receptors randomly to different cells and treats the two alleles of a given Ly49 gene independently. However, class I-MHC-dependent "education" processes shape the functional repertoire. The education processes silence potentially auto-aggressive NK cells, probably by ensuring that each NK cell expresses at least one self-specific Ly49 receptor. In addition, NK cell clones that express multiple self-specific Ly49 receptors are disfavored by the education processes, perhaps to confer greater discrimination on to individual NK cells.

The basis for self-tolerance of natural killer cells in beta2-microglobulin- and TAP-1- mice.

Cells from mice with mutations in the genes for beta2-microglobulin (beta2m) or for TAP-1 express only low levels of MHC class I proteins on their surfaces, and are thus sensitive to attack by normal NK cells. Although NK cells are present in beta2m- mice and TAP-1(-) mice, they are completely self-tolerant. The underlying mechanism for this tolerance is unknown. It has been proposed that education processes render NK cells from these mice hypersensitive to class I-mediated inhibition, so that they can be inhibited even by the low levels of class I expressed on autologous cells. In this study, we present evidence against this hypothesis, by demonstrating that NK cells from beta2m- mice and TAP-1(-) mice fail to attack beta2m(-)TAP-1(-) double-mutant cells in both in vitro and in vivo assays. The latter cells express substantially lower levels of class I than single-mutant cells, based on serologic tests, as well as a significantly diminished sensitivity to attack by class I-specific CTL. Furthermore, the Ly-49 repertoire on NK cells derived from beta2m(-)TAP-1(-) mice is highly similar to that of either single mutant, indicating that the developmental processes that shape the Ly-49 repertoire cannot respond to the differences in class I levels among these mice. We propose that self-tolerance of NK cells in beta2m- mice and TAP-1(-) mice is likely to result from hyporesponsiveness of the cells to activating signals, or alternatively, to induction of inhibitory signaling through receptors specific for non-class I MHC ligands.