Radial Distribution Function for One-Dimensional Square-Well and Square-Shoulder Fluids

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In statistical mechanics, the radial distribution function represents the distribution of interparticle separations [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 square-well and square-shoulder potentials [3, 4]. We also find the values of the ratio (where is the inverse temperature, is the pressure, and is the number density), the scaled inverse isothermal compressibility , and the excess internal energy per particle . Sliders let you control the width of the square well or shoulder, the reduced temperature, and the packing fraction. The quantities and are also plotted as functions of the packing fraction.


Contributed by: Andrés Santos (April 2015)
Open content licensed under CC BY-NC-SA



In this one-dimensional system, the particles interact with an impenetrable hard core of length plus either an attractive square-well or a repulsive square-shoulder 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 references [3] and [4]. As a very rudimentary approximation, . The structure factor is given by .


[1] Wikipedia. "Radial Distribution Function." (Apr 10, 2015) en.wikipedia.org/wiki/Radial_distribution_function.

[2] Wikipedia. "Structure Factor." (Apr 10, 2015) 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] S. B. Yuste and A. Santos, "A Model for the Structure of Square-Well Fluids," The Journal of Statistical Physics, 101(3), 1994 pp. 2355–2364. doi:10.1063/1.467676.

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