A two-dimensional vesicle of N monomers of diameter a (or a closed, planar, self-avoiding walk of N links) subject to an internal pressure increment Δp≥0 is analyzed. In the large-inflation scaling limit x∝Δ${\mathit{pa}}^2$${\mathit{N}}^{2\nu }$/${\mathit{k}}_{\mathit{B}}$T≫1, the mean area and mean-square radius of gyration vary as Δ${\mathit{p}}^{2\mathrm{\omega }}$${\mathit{N}}^{2\nu +}$ with ω=(1-ν)/(2ν-1)=1/2 and ${\nu }^{+}$=ν/(2ν-1)=3/2, where ν=3/4 is the self-avoiding-walk size exponent. The results are related to Pincus’s expression for the size of a stretched polymer chain. Monte Carlo simulations confirm the behavior and yield the corresponding scaling functions.

We study the decay of an unstable state when the control parameter is sweeping as a(t)=-${\mathit{a}}_0$+${\mathit{bt}}^{\mathrm{\delta }}$. We introduce an approximation for the cumulative distribution. This allows, in the limit of small noise, the calculation of the first-passage-time distribution and the moment generating function analytically. The transient moments and anomalous fluctuation for the full nonlinear process are calculated using stochastic instantonlike trajectories centered on the first-passage time.

Stochastic relaxation in a one-dimensional, bistable potential driven by colored noise ε is investigated by means of both numerical codes and analog simulation. By embedding the process under study x into a suitable two-dimensional phase space (x,ε), a striking topological effect, observable in the two-dimensional probability density P(x,ε), appears at a critical value of the noise correlation time ${\mathrm{\tau }}_{\mathit{c}}$. The change of the topology in the neighborhood of ${\mathrm{\tau }}_{\mathit{c}}$ can be characterized as a critical transition. In particular, the top of the potential barrier is shown to give rise to a single saddle in the trajectory space of the embedding process so long as τ<${\mathrm{\tau }}_{\mathit{c}}$. Where τ=${\mathrm{\tau }}_{\mathit{c}}$, however, the single saddle disappears and is replaced by a pair of saddles, which move away from the location of the top of the potential barrier when τ>${\mathrm{\tau }}_{\mathit{c}}$, leaving behind a concave depression or ‘‘hole’’ in the probability density. The relevance of this observation to the problem of calculating the mean first-passage time for particle escape from one well to the other at large τ is discussed in detail. A complete analytical description of this transition has defied all attempts thus far. Consequently, many interesting open questions remain.

We propose a unified treatment of the roughening of manifolds by impurities in quenched correlated random media. Our perspective incorporates such apparently distinct problems as domain walls in dirty Ising magnets, biased walks upon random lattices, and flux creep in high-${\mathit{T}}_{\mathit{c}}$ materials. By means of generalized Imry-Ma arguments and a functional renormalization group (RG), we find new results, including the random-bond interfacial roughening exponent, ${\mathrm{\zeta }}_{\mathrm{RB}}$=2ε/9, as well as estimates for the many-dimensional directed-polymer wandering index. This last quantity is also investigated via real-space RG methods, where we find, for example, ${\mathrm{\zeta }}_{2+1}$=0.602, in reasonable agreement with the functional RG value 3/5. Finally, since the Burgers equation permits translation of our directed-polymer results to the Eden cluster and ballistic deposition problems in higher dimensions, we can compare to the most recent computer simulations of these stochastic growth models. In particular, we address issues regarding the exponent conjectures that have been made and suggest the possibility of a finite upper critical dimension.

Anisotropic diffusion on a Sierpriaanski gasket is studied using renormalization equations for the probability densities of waiting times. The effect of an external field on a random walk is described in terms of the dependence of hopping probabilities, mean waiting times, and standard deviations on the hopping distance and on the intensity of the field. Our procedure can also be applied to other deterministic fractals.

The statistical mechanics of a nonuniform nearest-neighbor interaction lattice gas with four states per site is carried out and converted to that of a two-state second-neighbor gas. The latter is presented in a mixed density-potential representation and the appropriate generating functional constructed. Application is made to the site representation of a caricature of a hydrogen-molecule–water mixture.

The dynamics of quasiparticle transport in a degenerate adiabatic molecular dimer with dissipative as well as nonlinear local site coupling has been analyzed. The dissipative part of the model is based on the Haken-Strobl type of Liouville equation for a density matrix that includes multichannel coupling to a dissipative environment and, moreover, the additional nonlinear term of polaronic type. The theory of positive semigroups has been applied in deriving equations of motion for components of the Bloch vector that satisfy requirements of a relaxing semigroup. The presence of only one decay channel, represented by local dissipative coupling, always leads to the long-time equalization of site-occupation probabilities. We have shown that whether the nonsymmetrical steady state is reached or not depends exclusively on the properties of the Haken-Strobl dissipative part of Liouvillian, regardless of the nonlinear coupling. Clearly, the magnitude of the asymmetry, if it occurs, is influenced by the strength of the nonlinear local site coupling.

A study is reported of a simple dynamical system with two degrees of freedom having discontinuities due to collisions. It consists of two point masses or balls constrained to move in one dimension above a floor in a constant gravitational field. All collisions are assumed to be elastic. When the ratio r of the upper mass to the lower mass is less than unity the motion is chaotic almost everywhere. On the other hand, when r>1 the motion shows typical Kolmogorov-Arnol’d-Moser behavior with quasiperiodic and chaotic trajectories coexisting in the phase space. It is shown that for particular values of the mass ratio, denoted by ${\mathit{r}}_{\mathit{n}}$, a sequence of n rapid ball-ball collisions close to the floor has the net effect of reversing the velocities of both balls. This demonstration leads to the identification of families of stable and unstable fixed points of the Poincaré section, which to a considerable extent determine the overall structure of the map. By means of a method due to Lorenz, the largest Lyapunov exponent ${\lambda }_1$ has been calculated for many values of the mass ratio and for a variety of trajectories. For chaotic trajectories, a plot of ${\lambda }_1$ as a function of r is found to have local minima at the values ${\mathit{r}}_{\mathit{n}}$ corresponding to velocity-reversing collision sequences. This is thought to result from the fact that when r=${\mathit{r}}_{\mathit{n}}$ the chaotic trajectories lie in many isolated regions of the phase space, whereas when r is different from any of the ${\mathit{r}}_{\mathit{n}}$, the chaotic regions merge to form a single region of global chaos.

We explore the underlying connection between period-doubling bifurcations and diminished fluctuations in a phase-sensitive measurement, observed in recent experiments by Bocko and Battiato [Phys. Rev. Lett. 60, 1763 (1988)]. A simple model of the phase-space dynamics shows that proper choice of a phase-sensitive quantity leads to noise suppression. This effect occurs generically for oscillators having constant dissipation. The model predicts the point at which suppression begins, and that noise attenuation is maximized at the bifurcation, at which point the fluctuations are diminished by a factor of 2.

The violation of particle-number conservation by some spherical solutions of the Boltzmann equation is known to exist. The reason stems from the fact that the Boltzmann equation itself includes a contradiction: The velocity of a particle is independent of the position of the particle, but the acceleration, namely the time derivative of velocity, is related to the position. Considering the velocity as a function of position and time, we introduce a new equation. Under the same initial condition as for the Boltzmann equation, a strict solution of the new equation does not violate the particle-number conservation law.

A general relation between the statistical entropy, given by a functional of the distribution function, and the thermodynamic entropies, defined by the heat exchange of the system, is established for models described by master equations with time-dependent transition probabilities. The result is consistent with that derived by Langer and Sethna [Phys. Rev. Lett. 61, 570 (1988)] by using thermodynamic arguments. The asymptotic form of this relation for a single two-level system, in the limit of a very slow cooling rate, is presented and discussed.

We have performed computer simulations in order to investigate the effect of anisotropy in a continuous-energy model recently proposed by Y. C. Zhang [Phys. Rev. Lett. 63, 470 (1989)]. We have found that the most anisotropy-sensitive characteristic function is the peaked energy distribution, and the other quantities (critical exponents, etc.) are affected by the anisotropy in the extreme cases only. We present some quantitative connections between the anisotropy parameter and the characteristic quantities up to five dimensions.

In the extended standard map, the critical line for breakup of an invariant circle contains, for almost all rotation numbers, a Cantor set of cusps in parameter space. The Farey route to a given rotation number illustrates how the cusps appear and allows the characterization of their pattern. The fact that the cusp-splitting rules seem attainable by the Farey route rather than by the convergents must be incorporated in the renormalization theory.

Stochastic differential equations describing the Markovian evolution of state vectors in the quantum Hilbert space are studied as possible expressions of a universal dynamical principle. The general features of the considered class of equations as well as their dynamical consequences are investigated in detail. The stochastic evolution is proved to induce continuous dynamical reduction of the state vector onto mutually orthogonal subspaces. A specific choice, expressed in terms of creation and annihilation operators, of the operators defining the Markov process is then proved to be appropriate to describe continuous spontaneous localization of systems of identical particles. The dynamics obtained in such a way leaves practically unaffected the standard quantum evolution of microscopic systems and induces a very rapid suppression of coherence among macroscopically distinguishable states. The classical behavior of macroscopic objects as well as the reduction of the wave packet in a quantum measurement process can be consistently derived from the postulated universal dynamical principle.

The neglect of either mass, momentum, or energy conservation leaves one degree of freedom in the determination of the steady state behind stationary shock waves, the entropy production rate in the shock front becoming a function of this. In all three cases this function has an absolute maximum that is approached by the entropy production rate of real shock waves for a Mach number M close to 6 in the first case and asymptotically approached for M→∞ in the last two cases. These results are shown to hold true also when the shock waves are treated relativistically. The example of shock waves demonstrates that some systems far from equilibrium may be characterized by maximum entropy production similarly as certain systems close to equilibrium are characterized by minimum entropy production.

The self-diffusion, shear viscosity, bulk viscosity, and the thermal conductivity coefficients of the Lennard-Jones argon fluid were calculated by kinetic theory and molecular dynamics computations. Densities between 1.3 and 1.5 g ${\mathrm{cm}}^{\mathrm{-}3}$ and a temperature range of 100–300 K were considered for the comparison. The results indicate that the refined versions of the kinetic variational and reference theories give agreement with the exact data in a certain temperature region, except for the bulk viscosity, which is predicted much too high. Furthermore, the temperature dependence of the transport coefficients at constant density is correctly given solely for the self-diffusion and the thermal conductivity coefficients. In particular, the strong increase of the shear viscosity near the phase transition to the solid state is poorly reflected by the theory. On the other hand, among all the transport coefficients, the latter is theoretically well approximated for a relatively large temperature range, apart from the self-diffusion coefficient which represents, however, a purely kinetic quantity.

Time-dependent electron degradation spectra and yields of initial products are studied for the Ar-${\mathrm{H}}_2$ mixture by using the time-dependent Spencer-Fano (TDSF) theory. An integral form of the TDSF equation is derived and applied in numerical solutions. The present study shows, in the time-dependent context, that addition of a small amount of ${\mathrm{H}}_2$ in Ar greatly enhances the yield of the triplet states of ${\mathrm{H}}_2$ because of the presence of subexcitation electrons whose energies are below the electronic excitation threshold of Ar. The production of the triplet states is slow; it takes a period ten times longer than fast processes such as ion production. Secondary processes such as ion-molecule reactions following the electron degradation process are also discussed.

The concept of macrokinetic distributions is used to investigate the macroscopic dynamics of an assembly of electrons in a weakly ionized gas in the hydrodynamic regime. In this regime, the macrokinetic distribution (MKD) is shown to obey an equation that is equivalent to the Boltzmann equation in the time scale of electron-density transport. Formal, approximate solutions to this equation are obtained whose range of validity depend on the magnitude of the spatial derivatives of the density. Specific conditions on the magnitude of these derivatives have been obtained. Explicit expressions for the MKD are presented for the case of a quasi-Lorentz gas model. They have been used to evaluate the electron current density in the hydrodynamic regime and to obtain expressions for the mobility and diffusion coefficient. In the regime of large electron-density gradient, these coefficients have been found to depend on the normalized gradient. The consequences of these results are illustrated for the case of constant collision frequency.

Using the kinetic theory of monatomic gases of hard-sphere particles as a base, we develop the method of moments of Grad [Commun. Pure Appl. Math 2, 331 (1949)] within the framework of algebraic computing. In this theory the basic fields are characterized by 13+9N moments of the distribution function, with 0≤N≤4. The Navier-Stokes and Fourier laws follow from the use of an iteration method akin to the Maxwellian procedure. Fifth-order approximations to the transport coefficients of shear viscosity and thermal conductivity are given. The approximations of lower orders agree with those obtained through the method of Chapman and Enskog.

The scaling exponents of two-point correlation functions in nonisotropic convective turbulence are evaluated on the basis of the Boussinesq equations. The scaling indices differ significantly from Kolmogorov’s scaling for isotropic turbulence. The physical reason for this difference is the length dependent buoyancy forcing in this case, in contrast to forcing on the largest scale only in Kolmogorov’s picture. The calculation appears to be in agreement with preliminary experimental findings in convective turbulence with helium as the working fluid.