Entropy and Ordering of Hard Rods in One Dimension

Abstract: We revisit the equilibrium properties of a classical one-dimensional system of hard-core particles in the framework provided by the multiparticle correlation expansion of the con-figurational entropy. The vanishing of the cumulative contribution of more-than-two-particlecorrelations to the excess entropy is put in relation with the onset of a solidlike behavior athigh densities.Keywords: Entropy; residual multiparticle entropy; hard rods; one dimension; Tonks gas.1. IntroductionThe entropy of a classical system can be expressed as an infinite sum of contributions associated withspatially integrated n -point density correlations: S =X 1n =0 S n ; (1)where S 0 is the entropy of the corresponding non-interacting gas. The above expansion was separatelyderived for closed [1] and open systems [2], but the two apparently different expressions were later shownto be in fact equivalent [3]. In the absence of external fields, the leading and quantitatively dominant termof the series is the so called “pair entropy”

[1]  P. Giaquinta,et al.  Statistical Entropy of a Lattice-Gas Model: Multiparticle Correlation Expansion , 1999 .

[2]  M. Donato,et al.  Entropy and multi-particle correlations in two-dimensional lattice gases , 1999 .

[3]  Thomas M Truskett,et al.  Residual multiparticle entropy does not generally change sign near freezing. , 2008, The Journal of chemical physics.

[4]  D. Haar,et al.  Correlations and entropy in classical statistical mechanics , 1969 .

[5]  P. Whitlock,et al.  The structure of hyperspherical fluids in various dimensions. , 2005, The Journal of chemical physics.

[6]  L. Peliti,et al.  Harmonic properties of hard-sphere crystals : a one-dimensional study , 1993 .

[7]  Statistical entropy and clustering in absence of attractive terms in the interparticle potential , 2009 .

[8]  D. Kofke,et al.  Exact solution for the singlet density distributions and second-order correlations of normal-mode coordinates for hard rods in one dimension , 1999 .

[9]  H. Reiss,et al.  One-dimensional rigorous hole theory of fluids: Internally constrained ensembles , 1985 .

[10]  F. Saija,et al.  On entropy and ordering in binary hard-sphere mixtures , 1994 .

[11]  Evans,et al.  Direct entropy calculation from computer simulation of liquids. , 1989, Physical review. A, General physics.

[12]  Torquato,et al.  Nearest-neighbor distribution functions in many-body systems. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[13]  L. Tonks.  The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres , 1936 .

[14]  A. Baranyai,et al.  On the convergence of Green's entropy expansion , 1992 .

[15]  R. E. Nettleton,et al.  Expression in Terms of Molecular Distribution Functions for the Entropy Density in an Infinite System , 1958 .

[16]  ErratumErratum to: “Angular correlations and statistical entropy of hard spherocylinders: the isotropic–nematic transition”: [Chem. Phys. Lett. 283 (1998) 86–90]1 , 1998 .

[17]  J. Oldroyd.  Molecular Theory of Fluids , 1953, Nature.

[18]  P. Giaquinta,et al.  Errata: Statistical Entropy of a Lattice-Gas Model: Multiparticle Correlation Expansion , 2000 .

[19]  F. Saija,et al.  Statistical entropy of a binary hard-sphere mixture: the low-density limit , 1996 .

[20]  J. Kirkwood.  Statistical Mechanics of Fluid Mixtures , 1935 .

[21]  E. Guth,et al.  The Pair Distribution Function for a One‐Dimensional Gas , 1953 .

[22]  T. George,et al.  States of Aggregation , 1998 .

[23]  D. Kofke,et al.  A comparison of some variational formulas for the free energy as applied to hard-sphere crystals , 2002 .

[24]  F. Saija,et al.  Monte Carlo simulation and phase behavior of nonadditive hard-core mixtures in two dimensions , 2002 .

[25]  W. G. Hoover.  Computational Statistical Mechanics , 1991 .

[26]  A. Marco Saitta,et al.  Statistical entropy and density maximum anomaly in liquid water , 2003 .

[27]  H. Frisch,et al.  Comment on Kirkwood Instability , 1969 .

[28]  D. Costa,et al.  Angular correlations and statistical entropy of hard spherocylinders: the isotropic–nematic transition , 1998 .

[29]  Giaquinta,et al.  Entropy and the freezing of simple liquids. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[30]  Phase transitions in simple fluids: An application of a one-phase entropic criterion to Lennard-Jones and point Yukawa fluids. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  F. Saija,et al.  Re-entrant melting in the Gaussian-core model: the entropy imprint. , 2005, Chemphyschem : a European journal of chemical physics and physical chemistry.

[32]  P. Giaquinta,et al.  The entropy multiparticle-correlation expansion for a mixture of spherical and elongated particles , 2004, cond-mat/0412428.

[33]  Henry Eyring,et al.  A Theory of Liquid Structure , 1937 .

[34]  Evans,et al.  Three-particle contribution to the configurational entropy of simple fluids. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[35]  Rayleigh.  On the Virial of a System of Hard Colliding Bodies , 1891, Nature.

[36]  R. Lovett.  On the stability of a fluid toward solid formation , 1977 .

[37]  B. Alder,et al.  Cell Theories for Hard Particles , 1966 .

[38]  Entropy and Correlations in a Fluid of Hard Spherocylinders: The Onset of Nematic and Smectic Order , 2002, cond-mat/0204423.

[39]  E. Lomba,et al.  Role of the interaction range in the shaping of phase diagrams in simple fluids. The hard sphere Yukawa fluid as a case study , 1994 .

[40]  Cheng,et al.  Prediction of the phase diagram of rigid C60 molecules. , 1993, Physical Review Letters.

[41]  Smectic Ordering of Parallel Hard Spherocylinders: An Entropy-Based Monte Carlo Study , 2003, cond-mat/0304086.

[42]  F. Saija,et al.  Scaling of local density correlations in a fluid close to freezing , 2001 .

[43]  F. Saija,et al.  ENTROPY AND FLUID-FLUID SEPARATION IN NONADDITIVE HARD-SPHERE MIXTURES , 1998 .

[44]  G. Malescio,et al.  Entropy versus correlations in simple fluids: The gas–liquid and freezing transitions , 1998 .

[45]  C. Caccamo,et al.  Phase diagram of simple fluids: a comprehensive theoretical approach , 1993 .

[46]  J. J. Kozak,et al.  The Kirkwood superposition approximation for hard rods at high pressure , 1984 .

[47]  J. Haus,et al.  Computer studies of dynamics in one dimension: Hard rods , 1978 .

[48]  F. Saija,et al.  Entropy, correlations, and ordering in two dimensions , 2000 .

[49]  J. Kirkwood,et al.  The Radial Distribution Function in Liquids , 1942 .

[50]  A. Baranyai,et al.  On the Entropy of the Hard Sphere Fluid , 1991 .

[51]  G. Cole.  Classical fluids and the superposition approximation , 1968 .

[52]  P. Giaquinta,et al.  About entropy and correlations in a fluid of hard spheres , 1992 .

[53]  An entropy-based approach to the freezing of the generalized exponential model. , 2008, The Journal of chemical physics.

[54]  J. J. Kozak,et al.  The influence of closure on the behaviour of the Yvon-Born-Green equation for a system of hard rods , 1982 .