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Exact Solution for the Singlet Density Distributions and Second-Order Correlations of Normal-Mode Coordinates for Hard Rods in One Dimension C

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Exact Solution for the Singlet Density Distributions and Second-Order Correlations of Normal-Mode Coordinates for Hard Rods in One Dimension C JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 23 15 JUNE 1999 Exact solution for the singlet density distributions and second-order correlations of normal-mode coordinates for hard rods in one dimension C. Daniel Barnes and David A. Kofke Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260-4200 ~Received 8 December 1998; accepted 25 March 1999! We examine the distribution of normal-mode coordinates ~defined via the eigenvectors of a chain of harmonic oscillators! for a system of purely repulsive hard rods in one dimension. We obtain an exact solution for the singlet density distribution, and separately for the covariances of the normal-mode coordinates. The hard-rod behavior is examined in terms of its deviation from the corresponding distributions for the system of harmonic oscillators. All off-diagonal covariances are zero in the hard-rod system, and the ~on-diagonal! variances vary with the normal-mode wave number exactly as in the harmonic system. The detailed singlet normal-mode density distributions are very smooth but nonanalytic, and they differ from the ~Gaussian! distributions of the corresponding harmonic system. However, all of the normal-mode coordinate distributions differ in roughly the same way when properly scaled by the distribution variance, and the differences vanish as 1/N in the thermodynamic limit of an infinite number of particles N.©1999 American Institute of Physics. @S0021-9606~99!51523-7# I. INTRODUCTION of this system can be solved analytically by diagonalizing the quadratic form of the internal energy, which effectively con- The system of hard rods in one dimension, also known verts the system into a collection of independent harmonic 1 as the Tonks gas, is of interest as a prototype for studying oscillators. This is the basis of lattice dynamics.15 Knowl- the effect of steric exclusion on the behavior of fluids. The edge of the distribution of lattice vibration frequencies for a model exhibits no nontrivial behaviors—like all systems in harmonic system is sufficient to determine the free energy one dimension, it does not undergo any phase transition—but and thus most other thermodynamic properties of interest, so many aspects of its behavior can be solved in closed form, the focus of study in lattice dynamics calculations and mea- and these solutions have provided some insight on the be- 2 surements is this quantity. havior of its higher-dimensional counterparts. Usually the The usual approach taken in applying lattice dynamics solutions do not invoke the thermodynamic limit to yield a calculations to estimate the properties of realistic systems result, so it is possible to examine the hard-rod model to gain involves an expansion of the intermolecular potential in some understanding of finite-size effects. Tonks was among powers of the molecular separations. Crystal symmetry the earliest to present formulas for the basic properties such causes the first-order terms to vanish, leaving the harmonic as the equation of state1 but, as pointed out by Robledo and system as a natural reference. Corrections are then applied by Rowlinson,3 the earliest solution was presented by considering higher-order terms in the expansion. This is not Rayleigh.4 It is now widely known that the van der Waals a viable route to the study of hard potentials such as hard equation of state is exact in one dimension, in the sense that ~ the van der Waals contribution to the pressure due to repul- rods!, because these potentials are not analytic. We deal in- sion is exactly that of the hard-rod system.1,5 Other proper- stead with an approach that is couched more in the language ties known analytically for the hard-rod model include the of fluid-state theory. We are concerned with the singlet, pair, pair and higher-order correlation functions for the pure and higher-order correlations of the ‘‘normal-mode coordi- system,6 and mixtures,7,8,17 the behavior in confined nates’’ occupied by the phonons. For the harmonic system, spaces,3,9–11 and in the presence of other external fields,12 the singlet distribution is simply a Gaussian, and there are no and various dynamical properties.13 Most recently, Corti and higher-order correlations between the other modes. To the Debenedetti14 as part of their studies of thermodynamic extent that these distributions are similar in a harmonic sys- metastability turned to the hard-rod model for insight on the tem and a system of interest, we have a viable means for nature of voids ~regions of space with no particles present! estimating the target system’s properties via molecular simu- that arise as part of the natural fluctuations in fluid systems. lation. We reserve the details of this method for a separate One of our research interests lies in the development of publication. Instead we present here our exact solution for methods for computing free energies of solids by molecular the singlet normal-mode coordinate distribution for the hard- simulation. One of the avenues that we have explored in- rod model system. volves the use of a harmonic reference system, in which all The harmonic and hard-rod systems are very much un- interactions between the particles in the system take the form like one another, and in fact they lie at the extreme limits of of simple harmonic springs. As is well known, the properties the Toda lattice,1 which is an integrable one-dimensional 0021-9606/99/110(23)/11390/9/$15.0011390 © 1999 American Institute of Physics Downloaded 15 Jul 2002 to 128.205.114.91. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp J. Chem. Phys., Vol. 110, No. 23, 15 June 1999 C. D. Barnes and D. A. Kofke 11391 model that has been very helpful in understanding nonlinear lattice dynamics. We do not pursue the more general Toda lattice in our work because our ultimate aim is to apply the insights from the present study to develop molecular simula- tion methods for three-dimensional systems. The harmonic system retains its simplicity when extended this way, but the Toda lattice does not. FIG. 1. System of N hard rods of equal length d and total system length L. We begin in Sec. II by reviewing the treatment of har- The system is bounded by walls of infinite potential at 0 and L. monic systems, doing this mainly to establish the notation we use in subsequent sections. Then in Sec. III we review the hard-rod model and the solution for its partition function, where lm is the mth diagonal element of L. The normal- and we consider a way to obtain simple statistical measures mode coordinates h are simple linear transformations from of the normal-mode correlations in the hard-rod system. the real-space deviations g and are obtained from the eigen- Then in Sec. IV we present the analytic solution for the vectors of H, normal-mode singlet distribution, discuss the results in Sec. 21 V, and provide concluding remarks in Sec. VI. h5F g, ~5! where F is the matrix with columns given by the eigenvec- tors of H and, because H is symmetric F215FT; also F is II. DESCRIPTION OF A SYSTEM OF HARMONIC symmetric with elements OSCILLATORS 2 1/2 kmp f 5 sin , ~6! We consider a system of N harmonic oscillators in one mk SN11D S N11D dimension with only nearest-neighbor interactions. A con- 21 figuration is described by the vector of coordinates xT so F 5F. The normal-mode singlet density distribution pm(hm) 5$x1 ,x2 ,...,xN%, where xk is the coordinate of the kth os- cillator (xT is the transpose of vector x!. The energy for a describes the distribution of values adopted by the normal- given configuration is mode coordinate hm . Formally it may be expressed as an ensemble average N11 2 pm~h!5^d~hm2h!&, ~7! U5U01W ( ~xk2xk21! . ~1! k51 where d is the Dirac delta function. For the simple system of Here we identify W as the strength of the harmonic in- harmonic oscillators, this distribution function is a Gaussian teraction, which is the same for all pairs. Also, we bound the with zero mean system by fixed walls, which for notational simplicity we harm 21/2 2 pm ~h!5~p/lm! exp~2lmh !, ~8! assign coordinates x050 and xN115L. It is much more con- venient to work with the coordinates that describe the devia- so the variance in the distribution of the coordinate hm is 2 21 tion of each oscillator from its mean ~or minimum-energy! given directly by its eigenvalue ^hm&5(2lm) ; the 0 0 normal-mode coordinates for the harmonic system are uncor- position xk 5kL/(N11): gk5xk2xk. This step introduces new constant terms that can be lumped with U0 , but since all related, i.e., ^hmhl&50, mÞl, which is why they are desig- these contributions are not important to the present develop- nated ‘‘normal modes.’’ ment we will drop them entirely. Then the energy may be written III. HARD-ROD MODEL AND SECOND-ORDER U5 TH , 2 g g ~ ! CORRELATIONS where g is the vector of deviation coordinates and H is an We consider a system of N hard rods of equal length s, N3N tridiagonal matrix with diagonal elements equal to bounded by hard walls ~of infinite potential! separated by a 12W and secondary diagonal elements equal to 2W. distance L as pictured Fig. 1. Diagonalization of the above quadratic form results in an We could just as easily have worked within periodic expression for the energy in terms of the N ‘‘normal-mode’’ T boundaries.
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