This code goal is to develop a simple microscopic simulation based on the Lennard-Jones potential

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Build

Default build :

$ cmake -S . -B Build -G Ninja
$ cd Build
$ ninja

In order to enable optimizations :

$ cmake -S . -B Build -G Ninja -DCMAKE_BUILD_TYPE=Release
$ cd Build
$ ninja

To run the default configuration

$ ninja run

Other usefull targets are :

$ ninja doc
$ ninja plot
$ ninja vmd

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Input

A file containing cartesian coordinates of a particle system. The format is the following :

> 0    1   
> type x y z  
> type x y z  

First line is the number of input types Type is ignored here

When the input is parsed we need to be checking for duplicates particles as it's physically wrong to have two particles at the same place

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Computation

These are the main formulas used in the simulation

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Lennard Jones Energy :

$$\tilde{U}^{LJ} = \frac{4 \epsilon^*}{2} \sum_{i=1}^{N} \sum_{j > j} \left[ \left( \frac{\sigma}{r} \right)^{12} - 2 \times \left( \frac{\sigma}{r} \right)^{6} \right] = \sum_{i=1}^{N} \sum_{j > j} u_{ij}$$

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Lennard Jones Potential :

$$U^{LJ}(r) = 4 \epsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - 2 \times \left( \frac{\sigma}{r} \right)^{6} \right] + \delta U$$

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Forces computations (gradient of Lennard Jones) :

$$\frac{\partial u_{ij}}{\partial x_i} = -48 \epsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right] \times \frac{x_i - x_j}{r^2_{ij}}$$

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Sigmoid Correction :

$$\left\{ \begin{array}{l} P_5(r) = 1 - 10 \left( \frac{r - r_{min}}{r_{max} - r_{min}} \right)^{3} + 15 \left( \frac{r - r_{min}}{r_{max} - r_{min}} \right)^{4} - 6 \left( \frac{r - r_{min}}{r_{max} - r_{min}} \right)^{5} \\\ \frac{\partial P_5}{\partial x_i} = - \frac{30}{r_{max} - r_{min}} \left( \frac{r - r_{min}}{r_{max} - r_{min}} \right)^{2} + \frac{60}{r_{max} - r_{min}} \left( \frac{r - r_{min}}{r_{max} - r_{min}} \right)^{3} - \frac{30}{r_{max} - r_{min}} \left( \frac{r - x_{min}}{x_{max} - x{min}} \right)^{4} \times \frac{x_j - x_i}{r} \\\ x_{min} \leq x \leq x_{max} \\\ (x_{min} = R_c - \delta_r, x_{max} = R_c + \delta_r) \end{array} \right .$$

Then the correction Lennard Jones and Forces with sigmoid for points between R_c - dr < r_ij < R_c + dr to keep the equation continuous and differentiable (necessary as we are considering a cut off radius)

Lennard-Jones shifted :

$$U^{LJ}(r) = 4 \epsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - 2 \times \left( \frac{\sigma}{r} \right)^{6} \right] \times P_5(r)$$

Forces : (gradient of the new L_J term)

$$\frac{\partial u^s_{ij}}{\partial x_i} = \frac{\partial u_{ij}}{\partial x_i} \times P_5(r) + \frac{\partial P_5}{\partial x_i} \times U_{LJ}$$

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Periodic Conditions and main formula :

$$\tilde{U}^{LJ}(R_{cut}) = \frac{4 \epsilon^*}{2} \sum_{\vec{\eta}}^{N_{sym}} \sum_{i=1}^{N} \sum_{j, r_{ij} \leq R_{cut}} \left[ \left( \frac{\sigma}{r} \right)^{12} - 2 \times \left( \frac{\sigma}{r} \right)^{6} \right]$$

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Angular momentum

Velocity Verlet
-> with Verlet lists (sorted for efficiency)

$$\left\{ \begin{array}{l} m_i \frac{\partial \vec{r_i}}{\partial t}(t + \frac{1}{2} \partial t) = m_i \frac{\partial \vec{r_i}}{\partial t}(t) - \frac{1}{2} \vec{\nabla}_i U(t) \cdot \partial t \\\ \vec{r}_i (t + \partial(t)) = \vec{r}_i (t) + \frac{\partial \vec{r}_i}{\partial t} (t + \frac{1}{2} \partial(t)) \cdot \partial t \\\ m_i \frac{\partial \vec{r_i}}{\partial t} (t + \partial t) = m_i \frac{\partial \vec{r_i}}{\partial t}(t + \frac{1}{2} \partial t) - \frac{1}{2} \vec{\nabla}_i U(t + \partial t) \cdot \partial t \\\ \end{array} \right .$$

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Berendsen thermostat :

$$\frac{dT}{dt} = \frac{T_0 - T}{\tau}$$

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Output

PDB formatted file with the simulation state at different time steps, by default results are dumped every iterations

CSV for the Temperatures, Kinetic Energy, Potential Energy, Sum of the energies by default results are also dumped every iterations

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Usefull ressources :

https://fortranwiki.org/fortran/show/Keywords
https://fortran-lang.org/en/learn/
https://en.wikipedia.org/wiki/Lennard-Jones_potential
https://en.wikipedia.org/wiki/Periodic_boundary_conditions
https://en.wikipedia.org/wiki/Verlet_integration
https://www.biostat.jhsph.edu/~iruczins/teaching/260.655/links/pdbformat.pdf
https://curc.readthedocs.io/en/latest/programming/OpenMP-Fortran.html