Radial Distribution Function for One-Dimensional Triangle Well and Ramp Fluids

In statistical mechanics, the radial distribution function describes how density varies as a function of distance from a reference particle [1]. This Demonstration shows the results of exact statistical-mechanical computations of the radial distribution function and the structure factor [2] for a one-dimensional system of particles interacting via triangle well or ramp potentials [3–5]. We also find the values of the ratio (where is the inverse temperature, is the pressure and is the number density) and the excess internal energy per particle . The sliders allow you to control the width of the triangle well or ramp, the reduced temperature and the packing fraction. The quantities and are also plotted as functions of the packing fraction.

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In this one-dimensional system, the particles interact with an impenetrable hard core of length plus either an attractive triangle well or a repulsive ramp potential of energy and width . This can be represented by a pairwise potential energy function . The exact solution for the radial distribution function is worked out in [3–5]. As a very rudimentary approximation, . The structure factor is given by .
References
[1] Wikipedia. "Radial Distribution Function." (Apr 11, 2018) en.wikipedia.org/wiki/Radial_distribution_function.
[2] Wikipedia. "Structure Factor." (Apr 11, 2018) en.wikipedia.org/wiki/Structure_factor.
[3] Z. W. Salsburg, R. W. Zwanzig and J. G. Kirkwood, "Molecular Distribution Functions in a One-Dimensional Fluid," The Journal of Chemical Physics, 21(6), 1953 pp. 1098–1107. doi:10.1063/1.1699116.
[4] A. Santos, A Concise Course on the Theory of Classical Liquids: Basics and Selected Topics, Switzerland: Springer International Publishing, 2016. link.springer.com/978-3-319-29668-5.
[5] A. M. Montero, "Correlation Functions and Thermophysical Properties of One-Dimensional Liquids." arxiv.org/abs/1710.01118.
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