One-dimensional System Of Hard Rods Research Articles

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.

Density fluctuations in many-body systems

The characterization of density fluctuations in systems of interacting particles is of fundamental importance in the physical sciences. We present a formalism for studying local density fluctuations in two special subvolumes (centered around either a reference particle or some arbitrary point in the system) termed {ital particle} and {ital void} regions, respectively. We present formal expressions for the probability, as well as the moments, associated with finding exactly {ital n} particles inside of either of these subvolumes. Furthermore, we derive the relationship between the probability functions and closely related quantities of interest, such as the nth nearest-neighbor distribution functions and the {ital n}-particle conditional pair distribution functions associated with each region. We solve for these quantities exactly in the one-dimensional hard-rod system. The methods developed for studying the hard-rod fluid are applicable for studying a wide class of one-dimensional systems. {copyright} {ital 1998} {ital The American Physical Society}

One-dimensional surface diffusion: Density dependence in a smooth potential

A kinetic treatment of the transport of mobile molecules in nanometer channels is presented, that explores ideas of the Enskog theory of dense gases combined with molecular relaxation to the substrate. Transport diffusivity is shown to increase strongly with density, as an effect of effective transfer of mass in the nonlocal intermolecular collisions. The effect is partially compensated by the additional collisional relaxation, that would be absent in the truly one-dimensional system of hard rods, but is shown to occur in the realistic cases of quasi-one-dimensional channels and/or “soft” particles. A resulting behavior of diffusivity as a function of density close to 1/(1−θ), where θ is the degree of occupancy, is expected. This suggests an explanation for the puzzling universal behavior observed earlier in zeolites.

Hard spheres in the isobaric-isoenthalpic ensemble

We derive the rules for collisions between hard spheres in arbitrary dimensions in the isobaric-isoenthalpic ensemble. The static and time-dependent properties of the one-dimensional hard rod system are investigated in detail, both theoretically and numerically.

Long time behaviour for a chain of colliding rotators

2014 A simple model of a plastic crystal is constructed to study the long term effects of a cog-wheel mechanism. We present some arguments which lead us to conjecture that an initial distribution of rotators’ angular velocity diffuses according to a normal law with a t-1/2 decay of the angular velocity autocorrelation function of the rotators. This has been confirmed by a molecular dynamic calculation performed on a chain of 1 000 rotators with periodic boundary conditions. LE JOURNAL DE PHYSIQUE LETTRES TOME 39, ler JUILLET 1978, P Classification Physics Abstracts 66.90 Recent molecular dynamic studies on two dimensional fluid systems of diatomics [1] have shown at the longest calculated time a t-3 decay of the angular velocity autocorrelation function T~(~). The asymptotic t 2 predicted by hydrodynamics [2] can occur at longer times with very small amplitude, and it was argued in [1] ] that the observed t 3 was possibly due to the transfer of the intrinsic angular momentum by some kind of cog-wheel mechanism. Here, an attempt to understand the long-term effect of cogwhell coupling in one dimension is reported. Further work [3] will be devoted to two dimensional systems. To isolate the cog-wheel coupling, we shall consider a line of colliding rotators with fixed centres. In the simplest case i) no multiple collisions occur and, ii) a couple «(1)1’ w2) of angular velocities transform to ( (1)2’ (1)1) when molecules 1 and 2 collide. This corresponds to two adjacent sectors of angle less than n interacting as soon as they touch each other (Fig. 1). The exchange of angular momentum is then instantaneous and the two sectors have the new momenta until the next contact. This model is inspired by the one-dimensional system of hard rods solved by Jepsen and LebowitzPercus [4]. In both, the dynamics consist of neighbouring particles interchanging velocities at each collision. This forbids ergodicity and relaxation towards an equilibrium. FIG. 1. Collision (a) corresponds to W cu2 > 0 and is compatible with a rough disk model. Collision (b) corresponds to WI c~2 0 and can only take place by virtue of the rules defined in the main

Generalized Bernoulli Construction—A Heuristic Approach to the Equation of State

The equation of state (ES) is a subject of the greatest interest in Statistical Mechanics and Chemical Physics. In this paper we present a simple and rigorous method by which the ES of simple systems can be obtained. The derivation requires at most the knowledge of the Maxwell distribution. The well known formula for the ES of a one-dimensional system of hard rods is derived exactly, as an illustration. The method can be generalized to higher dimensionality and to include other types of potential.

Collapse of a one-dimensional spin fluid

This is a study of the classical equilibrium statistical mechanics of a one-dimensional system of hard rods each of which has an Ising spin variable associated with it. Nearest-neighbor rods attract one another with a force that is inversely proportional to their center-of-mass separation. The proportionality factor contains a nearest-neighbor Ising interaction, and the entire system is taken to be in a uniform magnetic field. The equation of state is given and has the property that for a certain domain of interaction-parameter space the pressure goes continuously to zero along any isotherm as the density ϱ approaches a critical density ϱ c associated with the isotherm. Within a certain subdomain the compressibility diverges as ϱ c is approached along any isotherm. The explicit form of the divergence is presented as is the explicit dependence of ϱ c on the magnetic field.

Dynamics of a Single-File Pore: Non-Fickian Behavior

The dynamics of an infinite one-dimensional system of hard rods is generalized to include the effects of a random background. Each rod follows a trajectory which is described by a generalized random function and when two rods collide they interchange velocities (and trajectories). An exact solution is obtained for the distribution ${f}_{s}(x,v,\frac{t}{{v}^{\ensuremath{'}}})$ which is the probability of finding a particle at $x$ with velocity $v$ at time $t$ that was at $x=0$ with velocity ${v}^{\ensuremath{'}}$ at $t=0$. The most interesting result is that in the long-time limit $p(x,t)$, which is the probability of finding a particle at $x$ at time $t$ that was at $x=0$ at $t=0$, is of the form ${t}^{\ensuremath{-}\frac{1}{4}}\mathrm{exp}(\ensuremath{-}\frac{{x}^{2}}{{t}^{\frac{1}{2}}})$. Thus, the spatial distribution does not become Gaussian and Fick's law is not valid. It is suggested that this qualitative behavior might be expected whenever single-file effects become important and that it is not dependent on the details of the one-dimensional hard-rod collisions which have been used in the derivation.

On the relationship between bifurcation points of the Kirkwood-Salsburg equation and phase transitions

In this paper we explore the Vlasov conjecture on the relationship between bifurcation points and phase transitions. Because of the availability of the exact results of Ruelle, we focus our attention on the Kirkwood-Salsburg hierarchy, and recognizing that the first equation in this hierarchy is of the form of a Lichtenstein-Lyapunov nonlinear operator equation, we use a fundamental theorem of Krasnosel'skii to determine, under a suitable closure, bifurcation points for the same system considered by Ruelle. A special example is treated—that of a one-dimensional system of hard rods— and our main conclusion follows from the results of this study: namely, that ``in this one-dimensional system'' the bifurcation point does not seem to be related to the onset of a phase transition.

Time Evolution in the Hard-Rod System

A mathematical treatment of time correlations in a one-dimensional system of hard rods has been given by Lebowitz, Percus, and Sykes. In this paper, the problem is reexamined from a simple viewpoint which seems to render the physics of the system more transparent, and some of the peculiarities of the system are discussed.

Kinetic-Equation Approach to Time-Dependent Correlation Functions

We have developed and, to some extent, solved a number of kinetic equations for displaced correlation functions in a classical fluid. These functions, of which the Van Hove neutron scattering function is a special example, are one-particle distribution functions obtained from a Gibbs ensemble which is initially, at $t=0$, in equilibrium except for one labeled particle whose distribution $W(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ at $t=0$ differs from its equilibrium value $\ensuremath{\rho}{h}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$, where $\ensuremath{\rho}$ is the average fluid density, and ${h}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ is the Maxwellian velocity distribution function. We investigate the time evolution of the (self-) distribution function of this labeled particle, ${f}_{s}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}},t)$, as well as the deviation from equilibrium, $\ensuremath{\eta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}},t)$, of the total one-particle distribution function. The latter represents the density of fluid particles, labeled and unlabeled, at position $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ and velocity $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$. Since both ${f}_{s}$ and $\ensuremath{\eta}$ are linear functionals of $W$, they will satisfy exactly a linear non-Markovian kinetic equation of the form $f=\mathrm{B}f+\ensuremath{\int}{0}^{t}d{t}^{\ensuremath{'}}\mathrm{M}({t}^{\ensuremath{'}})f(t\ensuremath{-}{t}^{\ensuremath{'}})$. B is a time-in-dependent and M a time-dependent (memory) operator (nonsingular in $t$). Our kinetic equations (first- and higher-order) are based on neglecting or approximating M in such a way that the short-time behavior of ${f}_{s}$ and $\ensuremath{\eta}$ is described exactly. The rationale behind this scheme is that our choice of initial ensemble is precisely of the type generally assumed in the "derivation" of kinetic equations. The calculation of B is straightforward and depends in a very important way on whether the interparticle potential in the fluid is smooth or contains a hard core. In the former case, the first-order kinetic equation is of the Vlasov type with an effective potential given by the equilibrium direct correlation function, while in the latter case, B contains, in addition, a linear Enskog-type collision term. We show that this Vlasov equation (also derived previously by many authors) gives a damping linear in the wave number $k$ for small $k$ instead of the hydrodynamic ${k}^{2}$ dependence. The kinetic equation for systems with hard cores does not give correct hydrodynamic behavior. (For a one-dimensional system of hard rods, the first-order kinetic equation is exact.) We also obtain and solve a second-order kinetic equation, which is a generalized Vlasov-Fokker-Planck-type equation, for systems with continuous potentials.

Time Evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods

We continue our investigation of the time evolution of a one-dimensional system of hard rods. At $t=0$ there is one particle with a specified position ${r}^{\ensuremath{'}}$ and velocity ${v}^{\ensuremath{'}}$, and the remainder are in "equilibrium." Since in this system collisions merely interchange velocities, the "equilibrium" velocity distribution ${h}_{0}(v)$ need not be Maxwellian. Exact solutions are obtained for the time-dependent one-particle position-velocity distribution function $f(r\ensuremath{-}{r}^{\ensuremath{'}}, v, \frac{t}{{v}^{\ensuremath{'}}})$. We investigate in particular the averaged positional part of $f$, viz., $G(r\ensuremath{-}{r}^{\ensuremath{'}}, t)$, which is the time-dependent pair correlation function whose space-time Fourier transform $S(k,\ensuremath{\omega})$ describes coherent neutron scattering in realistic systems. It is shown that $S(k,\ensuremath{\omega})$ does not generally contain modes corresponding to sound propagation. The exceptions are systems with discrete velocity distributions. In the latter case the space Fourier transform $\ensuremath{\chi}(k,t)$ of $G(r,t)$ is rigorously a sum of simple damped oscillations. An exact kinetic equation for the time evolution of $f$ is derived and investigated. Also found is an approximate kinetic equation which, however, gives exact values of $S(k,\ensuremath{\omega})$.

Kinetic Equations and Density Expansions: Exactly Solvable One-Dimensional System

We have made a detailed study of the time evolution of the distribution function $f(q,v,t)$ of a labeled (test) particle in a one-dimensional system of hard rods of diameter $a$. The system has a density $\ensuremath{\rho}$ and is in equilibrium at $t=0$. (Some properties of this system were studied earlier by Jepsen.) When the distribution function $f$ at $t=0$ corresponds to a delta function in position and velocity, then $f(q,v,t)$ is essentially the time-displaced self-distribution function ${f}_{s}$. This function ${f}_{s}$ (which can be found in an explicit closed form) and all of the system properties which can be derived from it depend on $\ensuremath{\rho}$ and $a$ only through the combination $n=\frac{\ensuremath{\rho}}{(1\ensuremath{-}\ensuremath{\rho}a)}$. In particular, the diffusion constant $D$ is given by ${D}^{\ensuremath{-}1}={\mathrm{lim}}_{s\ensuremath{\rightarrow}0}{[\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(s)]}^{\ensuremath{-}1}={(2\ensuremath{\pi}\ensuremath{\beta}m)}^{\frac{1}{2}}n$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(s)$ is the Laplace transform of the velocity autocorrelation function $\ensuremath{\psi}(t)=〈v(t)v〉$. An expansion of ${[\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(s)]}^{\ensuremath{-}1}$ in powers of $n$, on the other hand, has the form $\ensuremath{\Sigma}\frac{{B}_{l}{n}^{l}}{{s}^{l\ensuremath{-}1}}$, leading to divergence of the density coefficients for $l\ensuremath{\ge}2$ when $s\ensuremath{\rightarrow}0$. This is similar to the divergences found in higher dimensional systems. Similar results are found as well in the expansion of the collision operator describing the time evolution of $f(q,v,t)$. The lowest-order term in the expansion is the ordinary (linear) Boltzmann equation, while higher terms are $O({\ensuremath{\rho}}^{l}{t}^{l\ensuremath{-}1})$. Thus any attempt to write a Bogoliubov, Choh-Uhlenbeck-type Markoffian kinetic equation as a power series in the density leads to divergence in the terms beyond the Boltzmann equation. A Markoffian collision operator can, however, be constructed, without using a density expansion, which, e.g., describes the stationary distribution of a charged test particle in the system in the presence of a constant electric field. The distribution of the test particle in the presence of an oscillating external field is also found. Finally, the short- and long-time behavior of the self-distribution is examined.

New Series Expansion in Statistical Mechanics

A new approximation scheme is applied to thermodynamic variables which have power-series expansions in terms of real parameters. This method is based upon a previously discussed formula designed to overcome the convergence difficulties of the quantum-mechanical Born (perturbation) series. Some properties of the approximation scheme are obtained, and then the method is used to obtain information from power-series expansions such as those met with in thermodynamics and statistical mechanics. In particular, the virial coefficients for several molecular models are used to obtain numerical approximations to the equations of state. The solution for a one-dimensional system of hard rods is obtained exactly in this scheme. For the two-dimensional hard-square lattice gas, the results agree quite well with recent low- and high-density Pad\'e approximants. The qualitative behavior of the solution for a system of hard spheres resembles that of the Mayer theory of condensation. It is noteworthy that the approximation scheme has the property of generating solutions which are monotonically increasing in the density. Thus, no van der Waals loop is obtained.