We introduce a class of deterministic ultrametric fractal models in d=2, which are expected to mimic some dynamic properties of random walks. The relative diffusion and dc conduction problems are solved exactly, showing both universal and nonuniversal regimes, as already found in simpler d=1 hierarchical structures. For a natural choice of parameters, the model’s spectral dimension takes the Alexander-Orbach value (4/3, which was also conjectured for random walks in d=2. The problem of self-avoiding walks on these structures is also briefly discussed.

We study two types of S-unimodal one-dimensional asymmetric maps. The asymmetries concern the amplitude and the exponent of the map 1-a‖x${\Vert }^z$. In both cases the well-known metric universality of the symmetric map is lost: the behavior of the scaling factors δ and α and of the multifractal function $f_k$(a) is oscillatory. Nevertheless, the road to chaos remains that of doubling-period bifurcations, and all its topological properties are preserved.

Island tori of coupled standard maps are investigated. The study of invariants of the tangent map shows that the tori are embedded in higher-dimensional topological spheres. A new numerical method, the ‘‘quasisurface of sections’’ construction, reveals the torus structure in the nonlinear domain. Scaling laws and estimates for island boundaries are obtained. Arnold diffusion out of the island is found to be nonexistent or exceedingly slow. The measure of phase space occupied by islands versus that occupied by chaotic trajectories is shown to decline rapidly as the number of degrees of freedom increases.

Morriss [Phys. Rev. A 39, 4811 (1989)] recently published a stimulating study of a nonequilibrium Lorentz gas. He measured a multifractal ‘‘spectrum of singularities’’ f(a) describing the ‘‘coarse-grained’’ phase space-representation of a time-reversible, two-body, space- and time-periodic shear flow. The measured function f(a) is the ‘‘Hausdorff dimension’’ of attractor singularities whose local bin integrals vary as the ath power of the bin length. Morriss found a spectrum of singularities f(a) very different from those familiar to nonlinear dynamical systems theory. Here we consider a closely related, but simpler, two-body time-reversible atomistic system. It is also a Lorentz-gas problem, a nonequilibrium diffusive flow, periodic in space but stationary in time. This system appears to be both mixing and ergodic, even far from equilibrium. We use the Chhabra-Jensen technique to show that the phase-space singularity spectrum f(a) for this nonequilibrium flow more closely resembles those of dynamical systems theory.

The existence of ordered phases in stochastic Ising dynamical systems with short-range ‘‘ferromagnetic’’ interactions is discussed with the emphasis on open questions. Various examples of models are presented; some of these do and some others do not undergo positive temperature (i.e., nonzero noise) phase transitions in two and three dimensions. The examples suggest that such equilibrium notions as lower critical dimension may not be uniquely definable for these nonequilibrium systems. The existence of temporally periodic states is also briefly discussed.

Topological entanglements play an important role in the physical properties, such as viscosity, of macromolecular structures. We investigate the likelihood of the appearance of entanglements in the bond percolation problem. We show that below the percolation threshold $p_c$ (but extremely close to it) there exists an entanglement threshold $p_e$. Between $p_e$ and $p_c$, there exists an infinite spanning group of interlocked (linked) clusters. Thus, in a strong gelation process, the sol-gel transition appears at $p_e$ rather than at $p_c$. We define the problem and discuss the possible approaches to its resolution. We apply the Monte Carlo renormalization-group approach to find the distance between the thresholds, and investigate some numerical characteristics of the entangled clusters. The achieved numerical resolution of $p_c$-$p_e$≊2.3×${10}^{\mathrm{-}7}$ required averaging over an extremely large numbers of configurations.

We investigate a new criterion that can be used to locate the transition temperatures of walk models and apply the criterion to locate the transition temperatures of trails and silhouettes on a square, a triangular, a simple cubic, and a face-centered-cubic lattice. As the temperature is varied, it is observed that the odd moments or the reduced moments of the persistence lengths undergo a drastic change in a narrow temperature range. The criterion exploits this interesting observation and identifies the drastic change with a collapse of the walk configurations from the swollen phase to the compact phase. The transition temperatures obtained from this criterion for trails on the square, triangular, simple cubic, and face-centered-cubic lattice are, respectively, 1.1, 0.4, 0.6, and 0.3; and for the silhouettes they are, respectively, 1.9, 1.2, 1.4, and 0.9. The results for trails on the square and simple cubic lattices are very close to those of existing computer-simulation results of much longer chains (∼250).

The Saffman-Taylor instability is extended to diffusive systems with an imposed drift. Under uniform motion of the interface with velocity V, the modified Saffman-Taylor instability is found to occur in the growth process. When the strength of the attractive drift becomes larger than a critical value, the flat interface is stabilized. The structural phase transition from dense aggregation, through diffusion-limited aggregation, to the needle pattern, which occurs when the direction of the drift changes, gives evidence for the presence of a hydrodynamic instability. The morphology diagram is obtained to describe distinct patterns in the diffusion-limited aggregation with drift. The similarity to viscous fingering under a gravitational field is discussed.

A recent method used to optimize biased neural networks with low levels of activity is applied to a hierarchical model. As a consequence, the performance of the system is strongly enhanced. The steps to achieve optimization are analyzed in detail.

The influence of additive white noise on the delay of a bifurcation point in the presence of a swept control parameter has been studied by analog experiment and digital simulation. Measurements of the time taken for the second moment 〈$x^2$(t)〉 to reach a given threshold are in excellent agreement with the calculations of Zeghlache, Mandel, and Van den Broeck [Phys. Rev. A 40, 286 (1989)]. It is demonstrated, however, that the mean first-passage time (MFPT) for $x^2$(t) to attain the same threshold, corresponding to the quantity that is usually determined in laser experiments, can be markedly different. It may be either larger or smaller, depending on the conditions under which the measurements are made. A calculation of the MFPT is presented and shown to be in excellent agreement with the experimental measurements and to reduce, in the relevant limit, to the theoretical results previously published by Torrent and San Miguel [Phys. Rev. A 38, 245 (1988)].

In this paper we make the first careful comparison between a computer simulation and an experimental measurement of the conductance of a two-dimensional continuum random conducting medium. In the experiment horizontal and vertical slits are cut in a conducting sheet. The centers of the slits are randomly positioned, and the conductance is measured all the way to percolation. The measurements are consistent with the expected critical exponent for the conductance of t=1.3. The experimental results are compared with computer simulations of ants that are parachuted to random starting points and then diffuse with a Brownian motion. From the behavior at large times, the diffusion constant can be found and hence the conductance, using the Einstein relation. The agreement with experiment is good except near the critical point. We conclude that the analog experiment is superior to the digital computations in this continuum system. This is the reverse of the situation in discrete lattice systems.

A simple method to compute void distributions of random Cantor sets is developed. In particular, a detailed calculation is presented for the void distribution of the random Cantor set used in a recent theory of fractal growth. The exact results allow us to clarify and rediscuss some of the implicit assumptions made previously in this theory.

Hopf bifurcations in two-dimensional maps give rise to closed invariant curves and circle maps induced on these curves. It is not obvious whether the induced maps will exhibit the full array of scaling phenomena familiar from the study of one-dimensional maps. In the present work we numerically study a variable Jacobian map (the coupled logistic map). A detailed investigation along its critical line shows an excellent agreement of the map with the critical scaling predictions for a circle map with a smooth cubic inflection point. This occurs in spite of the fact that within mode-locking intervals on the critical line, which together occupy a set of full measure, the induced map has no cubic inflection point.

Any object in thermodynamic equilibrium (TEQ) with a heat bath executes thermal (Brownian) motion. This motion is completely random; i.e., the object is equally likely to move in any direction. This motion is completely random because TEQ is the state of maximum randomness (maximum entropy). In this paper, we prove that if the value of an appropriate fluctuating property of our object functionally depends upon the object’s velocity of thermal motion (i.e., if velocity-dependent fluctuations exist), then the randomness of our object’s thermal motion at TEQ will be broken—so that, at TEQ, our object will have a net systematic nonrandom drift-velocity component in a preferred direction. This decrease in randomness (in entropy) is shown to occur without any compensating increase either within the object or elsewhere; i.e., it is shown to occur in violation of the second law of thermodynamics. It is also shown how this systematic drift-velocity component can, in turn, be utilized in a process whose only systematic net effect is the conversion, at TEQ, of heat from our heat bath into work, again in violation of the second law of thermodynamics. Both the systematic drift velocity and the time rate of (i.e., the power output associated with) the conversion of heat into work are calculated quantitatively for a simple example.

Generalizing the concept of the Berry phase, I show that an additional gauge structure associated with a time-component vector potential $A_0$ is present in the adiabatic evolution of a quantum system. The gauge structure, which generates the usual dynamical phase, is revealed by a formalism in which the time variable is treated on an equal footing with other parameters on which the Hamiltonian depends. The invariant curvature associated with $A_0$, an electriclike field, determines the phase difference between low- and high-energy paths in a generalized phase-space and time picture. The covariance of the Born-Oppenheimer approximation illustrates the results.

For the classical point electron with Zitterbewegung (hence spin) we derive, after regularization, the radiation reaction force and covariant equations for the dynamical variables ($x_{\mu }$, ${\pi }_{\mu }$, $v_{\mu }$, and $S_{\mu \nu }$), which reduce to the Lorentz-Dirac equation in the spinless limit.

We measured the total electron scattering cross section for ${\mathrm{CO}}_2$ in a time-of-flight gas-target transmission experiment at gas temperatures of 250 and 520 K in the energy range 0.5–9 eV. From measurements at both temperatures we determined the cross section for molecules in vibrationally excited states ${\sigma }_{\mathrm{exc}}$ and the ground-state cross section ${\sigma }_0$. $sigma sub roman exc— exceeds ${\sigma }_0$ at all energies. A qualitative theoretical interpretation of this shift is possible on the basis of the calculated potentials from the ${\mathrm{CO}}_2^{\mathrm{-}}$ compound state formed in the resonance scattering.

An angular distribution of the emitted photons in the 3$P_{1/2}$,3/2–3$S_{1/2}$ state transition of sodium atoms resulting from fast-electron impact was measured. It is shown that the direction of the emitted photons is obviously not isotropic, but is centralized in a narrow angular range around 27.5°, and there is no emission in both the larger and smaller angular ranges.

Simple expressions are derived for all multipole, finite-mass, and energy-dependent corrections to the long-range effective potential due to two-Coulomb-photon exchange between Coulombic systems by using dispersion-theory methods. The effective potential is obtained as the Fourier transform in the momentum transfer of the scattering amplitude, which is in turn obtained from its discontinuity via a dispersion relation. The contribution to this potential from each intermediate state can be expressed in closed form in terms of the Struve and Neuman functions.

The Bethe stopping-power theory is considered in the context of heavy target atoms in which the relativistic effects of the target electrons are partially accounted for within the semirelativistic and independent-particle description. Numerical results show that, for target elements with large atomic numbers, these relativistic effects can become comparable with other corrections such as the shell correction, the Barkas effect, and the Bloch term.