Molecular dynamics help - Lennard Jones potention

[Swarby]

Homework Statement

The Lennard-Jones potential function [itex] u(r) = 4ε[(σ/r)^12 - (σ/r)^6] [/itex]

If intermolecular interactions are ignored in this calculation for [itex] r > r_c [/itex], show that in the mean field approximation in the long range correction [itex] P^*_LR to the measured pressure is given by:

P^*_LR / ρ*T* = -16∏ρ*/3T*(r*_c)^3 [1- 2/3(r_c)^6]

* asterisk denotes MD units

Homework Equations

I'm not exactly sure on the equations to be used. I know the Lennard-Jones potential is the starting point, but I need some help beyond this.

The Attempt at a Solution

Unfortunately I don't know where to go from here, if I just have a little nudge I can probably go from there. Many thanks

 

[mark726]

in advance for any help!
Thank you for your question. In order to solve this problem, we will need to use some fundamental equations from statistical mechanics and thermodynamics. Let's start by defining some of the terms in the problem:

- u(r): This is the Lennard-Jones potential function, which describes the potential energy between two particles at a distance r apart. It is dependent on two parameters, ε and σ, which represent the strength and size of the interactions between the particles, respectively.
- r: This is the distance between two particles.
- rc: This is the cutoff distance, beyond which intermolecular interactions are ignored.
- P*: This is the pressure in MD units.
- ρ*: This is the number density in MD units.
- T*: This is the temperature in MD units.

Now, in order to solve this problem, we will need to use the mean field approximation, which is a common method used in statistical mechanics to approximate the behavior of a system. In this approximation, we assume that the interactions between particles can be approximated by an average or mean interaction, rather than considering the individual interactions between each pair of particles.

In the long range correction, we are interested in calculating the difference between the pressure calculated using the Lennard-Jones potential (which includes all interactions) and the pressure calculated by ignoring intermolecular interactions beyond the cutoff distance rc. This can be expressed as:

[itex] P*LR = P* - P*LR [/itex]

where P* is the pressure calculated using the full Lennard-Jones potential and P*LR is the pressure calculated using only the short range interactions (r < rc).

To calculate P*, we can use the following equation from statistical mechanics:

[itex] P* = \frac{\rho*k_B*T*}{1 - \rho*b} [/itex]

where ρ is the number density, k_B is the Boltzmann constant, and b is the second virial coefficient, which is related to the Lennard-Jones potential by:

[itex] b = \frac{2\pi}{3} \int_0^\infty r^3 (e^{-u(r)/k_B*T*} - 1) dr [/itex]

Now, to calculate P*LR, we can use the same equation, but with a modified second virial coefficient:

[itex] b_{LR} = \