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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 <a href="https
Step1:  64k Particle LJ System
This benchmark is designed to closely follow the HOOMD-BLUE LJ benchmark here.
Step2:  Prepare the system
Step3:  Benchmark using fixed size neighbor list.
Step4:  On an A100 this comes out to 22.4 s / loop which is 2.24 ms / step.
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Python Code:
#@title Imports
!pip install jax_md
import jax.numpy as np
import numpy as onp
from jax import jit
from jax import random
from jax import lax
from jax.config import config
config.update('jax_enable_x64', True)
from jax_md import space
from jax_md import energy
from jax_md import simulate
from jax_md import quantity
from jax_md import partition
from jax_md.colab_tools import renderer
Explanation: <a href="https://colab.research.google.com/github/google/jax-md/blob/main/notebooks/lj_benchmark.ipynb" target="_parent"><img src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open In Colab"/></a>
End of explanation
!nvidia-smi
Explanation: 64k Particle LJ System
This benchmark is designed to closely follow the HOOMD-BLUE LJ benchmark here.
End of explanation
lattice_constant = 1.37820
N_rep = 40
box_size = N_rep * lattice_constant
# Using float32 for positions / velocities, but float64 for reductions.
dtype = np.float32
# Specify the format of the neighbor list. 
# Options are Dense, Sparse, or OrderedSparse. 
format = partition.OrderedSparse
displacement, shift = space.periodic(box_size)
R = []
for i in range(N_rep):
  for j in range(N_rep):
    for k in range(N_rep):
      R += [[i, j, k]]
R = np.array(R, dtype=dtype) * lattice_constant
N = R.shape[0]
phi = N / (lattice_constant * N_rep) ** 3
print(f'Created a system of {N} LJ particles with number density {phi:.3f}')
Explanation: Prepare the system
End of explanation
neighbor_fn, energy_fn = energy.lennard_jones_neighbor_list(displacement,
                                                            box_size, 
                                                            r_cutoff=3.0,
                                                            dr_threshold=1.,
                                                            format=format)
init, apply = simulate.nvt_nose_hoover(energy_fn, shift, 5e-3, kT=1.2)
key = random.PRNGKey(0)
# We pick an "extra capacity" to ensure ahead of time that the neighbor
# list will have enough capacity. Since sparse neighbor lists are more 
# robust to changes in the number of particles, in this case we only 
# need to actually add more capacity for dense neighbor lists.
if format is partition.Dense: 
  nbrs = neighbor_fn.allocate(R, extra_capacity=55)
else:
  nbrs = neighbor_fn.allocate(R)
state = init(key, R, neighbor=nbrs)
def step(i, state_and_nbrs):
  state, nbrs = state_and_nbrs
  nbrs = nbrs.update(state.position)
  return apply(state, neighbor=nbrs), nbrs
# Run once to make sure the JIT cache is occupied.
new_state, new_nbrs = lax.fori_loop(0, 10000, step, (state, nbrs))
new_state.position.block_until_ready()
# Check to make sure the neighbor list didn't overflow.
new_nbrs.did_buffer_overflow
%%timeit
new_state, new_nbrs = lax.fori_loop(0, 10000, step, (state, nbrs))
new_state.position.block_until_ready()
Explanation: Benchmark using fixed size neighbor list.
End of explanation
renderer.render(
    box_size,
    {'particles': renderer.Sphere(new_state.position)}
)
Explanation: On an A100 this comes out to 22.4 s / loop which is 2.24 ms / step.
End of explanation
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