On the kinetic equations of a system of one-dimensional hard rods

The hydrodynamical approximation to an infinite system of one-dimensional identical hard rods interacting through elastic collisions, is shown to be an integrable system possessing a one-parameter family of nonlinear Hamiltonian structures.     

Dynamics of hard rods in one dimension

We examine a system consisting ofN classical, Newtonian, perfectly elastic hard rods constrained to move on a line. The mass and length of each rod are arbitrary. We develop an algorithm which gives, after any given possible sequence of collisions, the new velocities of theN rods and a necessary condition for any given pair of rods to be involved in the next collision, all in terms of the initial velocities of the rods. These results are then used to prove that for the case where there are exactly three rods on the line, the maximum possible number of collisions among them is the largest integern such that , wherem ₂ is the mass of the central particle and ₁₂ and ₂₃ are the reduced masses of the left and right particle pairs. We further derive for this three-particle case a condition on the initial velocities which is necessary and sufficient fork collisions, 1<kn, to occur, as well as explicit expressions for the velocities after each collision in terms of the initial velocities.     

Ergodic properties of a semi-infinite hard rods system

A semi-infinite hard rods system in thermodynamic equilibrium is proved to be aK-system.     

Ground-state properties of a one-dimensional system of hard rods

A quantum Monte Carlo simulation of a system of bosonic hard rods in one dimension is presented and discussed. The calculation is exact since the analytical form of the wave function is known and is in excellent agreement with predictions obtained from asymptotic expansions valid at large distances. The analysis of the static structure factor and the pair distribution function indicates that a solidlike and a gaslike phases exist at high and low densities, respectively. The one-body density matrix decays following a power law at large distances and produces a divergence in the low density momentum distribution at k=0 which can be identified as a quasicondensate.     

On anomalous diffusion of fractal clusters under certain realistic physical conditions

Summary Diffusion of a fractal cluster of dimensiond _f in a three-dimensional space is investigated. The diffusion process is assumed to be modelled by a standard parabolic diffusion equation, although a more general case represented by the Fokker-Planck-Kolmogoroff equation is also introduced. The mean-square displacement of the cluster mass centre is analysed and its anomalous behaviour is presented and critically discussed. The results obtained can be applied to describe some effects which may occur during the diffusion-limited cluster-cluster aggregation process, especially when the viscosity of the solvent is changed in time and/or a directed transport of clusters is present in the system. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Homogenization of two-dimensional clusters of rigid rods in air

The scattering of sound waves by circular-shaped clusters consisting of two-dimensional distributions of rigid cylinders in air is studied in the low-frequency limit (homogenization) both theoretically and experimentally. Analytical formulas for the effective density and sound speed are obtained in the framework of multiple scattering. Here, an experimental demonstration is reported in which a cluster of wooden rods acoustically behaves as a cylinder of argon gas. Moreover, evidence is presented indicating the validity of the homogenization in this cluster at frequencies lower than 3 kHz, which corresponds to a wavelength that is only 4 times the parameter of the embedded lattice and is a quarter of the cluster's diameter.     

Hexagonal order in crystalline and columnar phases of hard rods

We report a study of colloidal suspensions of highly monodisperse semiflexible chiral rodlike viruses, denoted fd, in the range of high concentrations. Small angle x-ray scattering experiments reveal the existence of two hexagonal phases: the first one is crystalline and the second one is hexatic columnar, as shown by its short-range positional order. The suspension of rodlike viruses is the first experimental system showing the whole phase sequence with increasing particle concentration theoretically predicted for systems of hard rods, ranging from the chiral nematic via the smectic to columnar and crystalline phases.     

Stability and diffusion of surface clusters

Using kinetic Monte Carlo simulations and a bond-counting ansatz, thermal stability and diffusion of an adatom island on a crystal surface are studied. At low temperatures, the diffusion constant D is found to decrease for a wide range of island sizes like , where is close to one, N being the number of adatoms in the cluster. By heating up the surface, the system undergoes a phase transition above which the island disappears. Characteristics of that transition are discussed. Received 20 January 1999     

On the relation between conduction and diffusion in a random walk along self-similar clusters

The relation between diffusion and conduction in the random walk of a particle by means of Lévy hops is investigated. It is shown that on account of the unusual character of Lévy hops, the mobility of a particle is a nonlinear function of the electric field for arbitrarily weak fields. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 7, 518–520 (10 April 1998)     

Approach to field-induced stationary state in a gas of hard rods

The Boltzmann equation describing one-dimensional motion of a charged hard rod in a neutral hard rod gas at temperatureT = 0 is solved. Under the action of a constant and uniform field the charged particle attains a stationary state. In the long time limit the velocity autocorrelation function decays via damped oscillations. In the reference system moving with the mean particle velocity the decay of fluctuations in the position space is governed (in the hydrodynamic limit) by the diffusion equation. Both the stationary current and the diffusion coefficient are proportional to the square root of the field. It is conjectured that this result also holds forT > 0 in a strong field limit.On leave from the Institute of Theoretical Physics, University of Warsaw, Hoza 69, 00-081 Warsaw, Poland.     

The classical fluid of associating hard rods in an external field

Two models of one-dimensional fluids of associating hard rods in an arbitrary external field are investigated. In the first model particles can only form dimers, while in the second model, which has been solved previously by Percus, aggregates of any size coexist. In both cases the grand canonical potential and the external potential are found exactly as functionals of the density. It is shown that Wertheim's thermodynamic perturbation theory of polymerization provides a straightforward route to the exact solution by expanding the functional space to include more density parameters. This suggests that Wertheim's theory should be used also for studying the structure (and not only the thermo-dynamics) of real associating fluids.     

Microcanonical ensemble study of a classical fluid of hard rods under gravity

The study of a system of hard rods in a box of finite length in the presence of a uniform gravitational field is made by means of the microcanonical ensemble. Explicit expressions are derived for the phase volume and the density of states, the primary functions of this ensemble. Related statistical quantities are reported, such as the entropy, the temperature, the heat capacity and the forces exerted on the fluid by the bottom and top walls. The microcanonical number density and higher order molecular distribution functions are also derived. Received: 7 April 1998 / Received in final form: 20 July 1998 / Accepted: 28 July 1998     

Kinetics of a finite one-dimensional mixture of hard rods with different masses

A one-dimensional binary mixture of impenetrable (hard core) particles with different mass ratios,m ₂/m ₁=1, 1.05, 1.2, 2, 3, and 4, was simulated to evolve in a computer by the molecular dynamics method. The systems withm ₂>m ₁ and initial velocity distribution ±v ₀ show a clear tendency to the equipartition of energy and relaxation toward a Maxwellian velocity distribution unlike the nonergodic system withm ₂=m ₁. Several quantities have been monitored during the evolution to investigate its dependence on the mass ratiom ₂/m ₁.     

Equilibrium state of a classical fluid of hard rods in an external field

The external field required to produce a given density pattern is obtained explicitly for a classical fluid of hard rods. All direct correlation functions are shown to be of finite range in all pairs of variables. The one-sided factors of the pair direct correlation are also found to be of finite range.Supported in part by U.S. ERDA, Contract No. E(11-1)-3077.     

Enhancement by polydispersity of the biaxial nematic phase in a mixture of hard rods and plates

The phase diagram of a polydisperse mixture of uniaxial rodlike and platelike hard parallelepipeds is determined for aspect ratios kappa=5 and 15. All particles have equal volume, and polydispersity is introduced in a highly symmetric way. The corresponding binary mixture is known to have a biaxial phase for kappa=15, but to be unstable against demixing into two uniaxial nematics for kappa=5. The phase diagram for kappa=15 is qualitatively similar to that of the binary mixture, regardless of the amount of polydispersity, while for kappa=5 a sufficient amount of polydispersity stabilizes the biaxial phase. This provides clues for designing an experiment to observe this long searched biaxial phase.     

A diffusion Monte Carlo study of small para-hydrogen clusters

An improved Monte Carlo diffusion model is used to calculate the ground state energies and chemical potentials of parahydrogen clusters of three to forty molecules, using two different p-H₂-p-H₂ interactions. The improvement is due to three-body correlations in the importance sampling, to the time step adjustment and to a better estimation of statistical errors. In contrast to path-integral Monte Carlo results, this method predicts no magic clusters other than that with thirteen molecules.      

Time-displaced correlation functions in an infinite one-dimensional mixture of hard rods with different diameters

Time-displaced conditional distribution functions are calculated for an infinite, one-dimensional mixture of equal-mass hard rods of different diameters. The kinetic equation that describes the time dependence of the one-particle total distribution function is found to be non-Markovian, in contrast with the situation in systems of identical rods. The correlation function does not contain any isolated damped oscillation, except for systems of equal-diameter rods with discrete velocities. Thus, we generalize the one-component results of Lebowitz, Perçus, and Sykes, removing some nontypical features of that system.Supported by NSF grant No. MCS 75-21684 A01 (M. A.), NSF grant No. MPS 75-20638 (J. L.), and USAFOSR grant No. 73-2430 B (J. M.)John Guggenheim Fellow on sabbatical leave from Belfer Graduate School of Science, Yeshiva University, New York.