S. Bravo Yuste

An accurate and simple equation of state for hard disks

An equation of state for a fluid of hard disks is proposed: Z=[1−2η+(2η0−1)(η/η0)2]−1. The exact fit of the second virial coefficient and the existence of a single pole singularity at the close‐packing fraction η0 are the only requirements imposed on its construction. A comparison of the prediction of virial coefficients and of the values of the compressibility factor Z with those stemming out of other known equations of state is made. The overall performance of this very simple equation of state is quite satisfactory.

Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions

Abstract An improved version of a Krylov-Bogoliubov method that gives the approximate solution of the non-linear cubic oscillator x + c 1 x + c 3 x 3 + ef(x, x dot ) = 0 in terms of Jacobi elliptic functions is described. Compact general expressions are given for the time derivatives of the amplitude and phase similar to those obtained by the usual Krylov-Bogoliubov method (which gives the approximate solution in terms of circular functions). These expressions are especially simple for quasilinear (c3 = 0) and quasi-pure-cubic (c1 = 0) oscillators. Two types of cubic oscillators have been used as examples: the linear damped oscillator f(x, x dot ) = x dot , and the van der Pol oscillator f(x, x dot ) = (α − βx 2 ) x dot . The approximate solutions of these quasilinear and quasi-pure-cubic oscillators are simple and accurate. The influence of the non-linearity on the rate of variation of the amplitude of these two types of cubic oscillators was also studied.

Extension and improvement to the Krylov–Bogoliubov methods using elliptic functions

Abstract It is shown that the Krylov–Bogoliubov methods that give the approximate oscillatory solution of the equation in terms of Jacobi elliptic functions are applicable not only when c 1>0 and c 3>0, but also when c 1>0 and c 3 0. In particular, the most precise of these methods, the Christopher-Brocklehurst method, is discussed in detail. Its accuracy has been improved by utilizing the transformation properties of elliptic functions with respect to their parameters.

A MODEL FOR THE STRUCTURE OF SQUARE-WELL FLUIDS

A simple explicit expression for the Laplace transform of rg(r) for 3D square‐well fluids is proposed. The model is constructed by imposing the following three basic physical requirements: (a) limr→σ+g(r)=finite, (b) limq→0S(q)=finite, and (c) limr→λσ−g(r)/limr→λσ+g(r)= exp(e/kBT). When applied to 1D square‐well fluids, the model yields the exact radial distribution function. Furthermore, in the sticky‐hard‐sphere limit [λ→1, e→∞, (λ−1)exp(e/kBT)=finite] the model reduces to Baxter’s exact solution of the Percus–Yevick equation. Comparison with Monte Carlo simulation data shows that the model is a good extension of Baxter’s solution to ‘‘thin’’ square‐well fluids. For ‘‘wide’’ square‐well fluids the model is still an acceptable approximation even for densities slightly above the critical density and temperatures slightly below the critical temperature.

Generalized fourier series for the study of limit cycles

The approximate solution, to first order, of non-linear differential equations is studied using the method of harmonic balance with generalized Fourier series and Jacobian elliptic functions. As an interesting use of the series, very good analytic approximations to the limit cycles of Lienards ordinary differential equation (ODE), X + g(X) = f(X)X, are presented. Specifically, it is shown that, contrary to an opinion given in a well-known textbook on non-linear oscillations, g(X) not only modifies the period but influences the topology. In the generalized van der Pol equation with f(X) = e(1−X2) and g(X) = AX + 2BX3 for ϵ < 0·1, the presence of zero, one, or three limit cycles is found to depend on the value of AB.

“Cubication” of non-linear oscillators using the principle of harmonic balance

A new method is given of “cubication” of autonomous non-linear oscillators (NLO) of the class ẍ + c1x + c3x3+ ϵ[g(x)+ tf(x)ẍ] i.e.of constructing the cubic oscillator ẍ + λ∗ẍ + c1∗x + c3∗x3= 0 from the NLO. The solution, limit cycles, bifurcations, fixed points, and stability of this NLO are approached by studying its associated cubic oscillator which is equal at least in the largest harmonics (principle of harmonic balance) and by assuming as a first approximation a solution for the NLO problem in terms of Jacobian elliptic functions. When c3= 0, the elliptic functions become circular functions and the present method reduces to the well-studied harmonic-balance method of linearization. The present method is equivalent to a third-order Chebyshev expansion of the NLO force if this is conservative. For a dissipative NLO, it gives the position and features of limit cycles and bifurcations.

COMMENTS ON THE METHOD OF HARMONIC BALANCE IN WHICH JACOBI ELLIPTIC FUNCTIONS ARE USED

Abstract A harmonic balance method is presented in which Jacobi elliptic functions are used in the trial solution instead of circular functions to obtain approximate periodic solutions of the oscillator x + F ( x, dot x ) = 0 . Conditions for the method to work well are the usual ones of the current method of harmonic balance, and that x(t) must pass through zero. The procedure for obtaining a higher order approximation is described, and in particular two criteria for chosing the elliptic function parameter m are discussed. Illustrative examples are presented with F being diverse polynomials of x.

On Duffing oscillators with slowly varying parameters

Abstract A method of Krylov-Bogoliubov type, which gives the approximate solution in terms of Jacobi elliptic functions, is used for the study of perturbed Duffing oscillators with slowly varying parameters: d dt [μ(τ) x ] + c 1 (τ)x + c 3 (τ)x 3 + ef(x, x , τ) = 0 . This method is a natural generalization of the usual Krylov-Bogoliubov method that is only valid when c3(τ)x3 is of e order. Two examples are given. One is a pure cubic oscillator (c1 = 0) with variable mass and linear damping, f(x, x ) = x , for which a simple accurate approximate solution is found. The other is a pendulum with variable length and damping proportional to the velocity for which an approximate analytical expression for the rate of variation of the oscillation amplitude is obtained, and successfully compared with the numerically calculated result and with that obtained using the normal Krylov-Bogoliubov method.

Is there a glass transition for dense hard-sphere systems?

The recent results for the contact value of the radial distribution function obtained from large-scale molecular dynamics and Monte Carlo simulations of systems of dense hard spheres [M. D. Rintoul and S. Torquato, J. Chem. Phys. 105, 9258 (1996)] are compared to those of the Pade equation of state for a hard-sphere fluid (derived from the knowledge of the first eight virial coefficients), including the metastable fluid region up to a packing fraction of ηg≃0.56, and above such packing fraction to the ones corresponding to an equation of state of the free-volume type which presents a simple pole at random close-packing. This latter equation involves the same value for the pressure at ηg as the Pade equation of state, and arises in a consistent way from the application of a rational-function approximation method to the computation of the radial distribution function of a hard-sphere fluid. The substantial improvement of the agreement with the simulation results over the one obtained with the equation of st...

Radial distribution function for sticky hard-core fluids

Following heuristic arguments, analytic expressions for the radial distribution functiong(r) of one- and three-dimensional sticky hard-core fluids (i.e., square-well fluids in a scaled limit of infinite depth and vanishing width) are proposed. The expressions are derived in terms of the simplest Padé approximant of a function defined in the Laplace space that is consistent with the following physicaly requirements:y(r) ≡eϕ(r)/kBTg(r) is finite at the contact point, and the isothermal compressibility is finite. In the case of sticky hard rods the expression obtained is exact, while in the case of sticky hard spheres it coincides with the solution of the Percus-Yevick equation.