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

interaction potential

triangle well

ramp

triangle well (or ramp) width

0.5

temperature

packing fraction

0.3

function

g(r)

S(k)

β P / ρ

/ ϵ

β P

= 1.274

= -0.16

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

g(r)

and the structure factor

S(k)

[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

βP/ρ

(where

β=1/kT

is the inverse temperature,

is the pressure and

is the number density) and the excess internal energy per particle

/ϵ

. The sliders allow you to control the width of the triangle well or ramp, the reduced temperature and the packing fraction. The quantities

βP/ρ

and

/ϵ

are also plotted as functions of the packing fraction.

Details

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

V(r)

. The exact solution for the radial distribution function

g(r)

is worked out in[3–5]. As a very rudimentary approximation,

g(r)≈

-V(r)/kT

. The structure factor is given by

S(k)=1+ρ∫(g(r)-1)

ikr

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.

External Links

Radial Distribution Function for Sticky Hard Rods

Radial Distribution Function for Hard Spheres

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

Permanent Citation

Ana M. Montero, Andrés Santos

"Radial Distribution Function for One-Dimensional Triangle Well and Ramp Fluids"
http://demonstrations.wolfram.com/RadialDistributionFunctionForOneDimensionalTriangleWellAndRa/
Wolfram Demonstrations Project
Published: May 2, 2018