302 lines
7.9 KiB
Typst
302 lines
7.9 KiB
Typst
#import "@preview/diatypst:0.2.0": *
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#show: slides.with(
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title: "N-Body project ",
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subtitle: "Computational Astrophysics, HS24",
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date: "04.02.2024",
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authors: ("Rémy Moll"),
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toc: false,
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// layout: "large",
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// ratio: 16/9,
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)
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#show footnote.entry: set text(size: 0.6em)
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#set footnote.entry(gap: 3pt)
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#set align(horizon)
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#import "helpers.typ"
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// Setup of code location
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#let t1 = json("../task1.ipynb")
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#let t2 = json("../task2-particle-mesh.ipynb")
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// Finally - The real content
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= N-body forces and analytical solutions
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// == Objective
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// Implement naive N-body force computation and get an intuition of the challenges:
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// - accuracy
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// - computation time
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// - stability
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// $==>$ still useful to compute basic quantities of the system, but too limited for large systems or the dynamical evolution of the system
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== Overview - the system
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Get a feel for the particles and their distribution
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#columns(2)[
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#helpers.image_cell(t1, "plot_particle_distribution")
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// Note: for visibility the outer particles are not shown.
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#colbreak()
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The system at hand is characterized by:
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- $N ~ 10^4$ stars
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- a _spherical_ distribution
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$==>$ treat the system as a *globular cluster*
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#footnote[Unit handling [#link(<task1:function_apply_units>)[code]]]
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]
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// It is a small globular cluster with
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// - 5*10^4 stars => m in terms of msol
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// - radius - 10 pc
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// Densities are now expressed in M_sol / pc^3
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// Forces are now expressed
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== Density
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Compare the computed density
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#footnote[Density sampling [#link(<task1:function_density_distribution>)[code]]]
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with the analytical model provided by the _Hernquist_ model:
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#grid(
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columns: (3fr, 4fr),
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inset: 0.5em,
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block[
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$
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rho(r) = M/(2 pi) a / (r dot (r + a)^3)
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$
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where we infer $a$ from the half-mass radius:
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$
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r_"hm" = (1 + sqrt(2)) dot a
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$
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],
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block[
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#helpers.image_cell(t1, "plot_density_distribution")
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]
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)
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// Note that by construction, the first shell contains no particles
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// => the numerical density is zero there
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// Having more bins means to have shells that are nearly empty
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// => the error is large, NBINS = 30 is a good compromise
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== Force computation
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#grid(
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columns: (3fr, 2fr),
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inset: 0.5em,
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block[
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#helpers.image_cell(t1, "plot_force_radial")
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],
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block[
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Discussion:
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- the analytical
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#footnote[Analytical force [#link(<task1:function_analytical_forces>)[code]]]
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method replicates the behavior accurately
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- at small softenings the $N^2$
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#footnote[$N^2$ force [#link(<task1:function_n2_forces>)[code]]]
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method has noisy artifacts
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- a $1 dot epsilon$
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#footnote[$epsilon$ computation [#link(<task1:function_interparticle_distance>)[code]]]
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softening is a good compromise between accuracy and stability
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]
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)
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// basic $N^2$ matches analytical solution without dropoff. but: noisy data from "bad" samples
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// $N^2$ with softening matches analytical solution but has a dropoff. No noisy data.
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// => softening $\approx 1 \varepsilon$ is a sweet spot since the dropoff is "late"
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== Relaxation
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We express system relaxation in terms of the dynamical time of the system.
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$
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t_"relax" = overbrace(N / (8 log N), n_"relax") dot t_"crossing"
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$
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where the crossing time of the system can be estimated through the half-mass velocity $t_"crossing" = v(r_"hm")/r_"hm"$.
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We find a relaxation of $approx 30 "Myr"$ ([#link(<task1:compute_relaxation_time>)[code]])
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#grid(
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columns: (1fr, 1fr),
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inset: 0.5em,
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block[
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#image("relaxation.png")
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],
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block[
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- Each star-star interaction contributes $delta v approx (2 G m )/b$
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- Shifting by $epsilon$ *dampens* each contribution
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- $=>$ relaxation time increases
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]
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)
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// The estimate for $n_{relax}$ comes from the contribution of each star-star encounter to the velocity dispersion. This depends on the perpendicular force
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// $\implies$ a bigger softening length leads to a smaller $\delta v$.
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// Using $n_{relax} = \frac{v^2}{\delta v^2}$, and knowing that the value of $v^2$ is derived from the Virial theorem (i.e. unaffected by the softening length), we can see that $n_{relax}$ should increase with $\varepsilon$.
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// === Effect
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// - The relaxation time **increases** with increasing softening length
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// - From the integration over all impact parameters $b$ even $b_{min}$ is chosen to be larger than $\varepsilon$ $\implies$ expect only a small effect on the relaxation time
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// **In other words:**
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// The softening dampens the change of velocity => time to relax is longer
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= Particle Mesh
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== Overview - the system
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#page(
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columns: 2
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)[
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#helpers.image_cell(t2, "plot_particle_distribution")
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$==>$ use $M_"sys" approx 10^4 M_"sol" + M_"BH"$
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]
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== Force computation
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#helpers.code_reference_cell(t2, "function_mesh_force")
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#helpers.image_cell(t2, "plot_force_radial")
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#grid(
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columns: (2fr, 1fr),
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inset: 0.5em,
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block[
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#helpers.image_cell(t2, "plot_force_radial_single")
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],
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block[
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- using the (established) baseline of $N^2$
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#footnote[$N^2$ force [#link(<task1:function_n2_forces>)[code]]]
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with $1 dot epsilon$
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#footnote[$epsilon$ computation [#link(<task1:function_interparticle_distance>)[code]]]
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softening
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- small grids
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#footnote[Mesh force [#link(<task2:function_mesh_force>)[code]]]
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are stable but inaccurate at the center
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- very large grids have issues with overdiscretization
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$==> 75 times 75 times 75$ as a good compromise
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]
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)
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// Some other comments:
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// - see the artifacts because of the even grid numbers (hence the switch to 75)
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// overdiscretization for large grids -> vertical spread even though r is constant
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// this becomes even more apparent when looking at the data without noise - the artifacts remain
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//
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// We can not rely on the interparticle distance computation for a disk!
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// Given softening length 0.037 does not match the mean interparticle distance 0.0262396757880128
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//
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// Discussion of the discrepancies
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// TODO
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#helpers.image_cell(t2, "plot_force_computation_time")
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// Computed for 10^4 particles => mesh will scale better for larger systems
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== Time integration
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*Integration step*
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#helpers.code_reference_cell(t2, "function_runge_kutta")
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*Timesteps*
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Chosen such that displacement is small (compared to the inter-particle distance) [#link(<task2:integration_timestep>)[code]]:
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$
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op(d)t = 10^(-4) dot S / v_"part"
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$
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// too large timesteps lead to instable systems <=> integration not accurate enough
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*Full integration*
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[#link(<task2:function_time_integration>)[code]]
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#pagebreak()
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== First results
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#helpers.image_cell(t2, "plot_system_evolution")
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== Varying the softening
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#helpers.image_cell(t2, "plot_second_system_evolution")
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== Stability [#link("../task2_nsquare_integration.gif")[1 epsilon]]
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#page(
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columns: 2
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)[
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#helpers.image_cell(t2, "plot_integration_stability")
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]
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== Particle mesh solver
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#helpers.image_cell(t2, "plot_pm_solver_integration")
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#helpers.image_cell(t2, "plot_pm_solver_stability")
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= Appendix - Code <appendix>
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== Code
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#helpers.code_reference_cell(t1, "function_apply_units")
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<task1:function_apply_units>
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#pagebreak(weak: true)
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#helpers.code_reference_cell(t1, "function_density_distribution")
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<task1:function_density_distribution>
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#pagebreak(weak: true)
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#helpers.code_reference_cell(t1, "function_analytical_forces")
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<task1:function_analytical_forces>
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#pagebreak(weak: true)
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#helpers.code_reference_cell(t1, "function_n2_forces")
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<task1:function_n2_forces>
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#pagebreak(weak: true)
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#helpers.code_reference_cell(t1, "function_interparticle_distance")
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<task1:function_interparticle_distance>
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#pagebreak(weak: true)
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#helpers.code_cell(t1, "compute_relaxation_time")
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<task1:compute_relaxation_time>
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#pagebreak(weak: true)
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#helpers.code_reference_cell(t2, "function_mesh_force")
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<task2:function_mesh_force>
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#pagebreak(weak: true)
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#helpers.code_cell(t2, "integration_timestep")
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<task2:integration_timestep>
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#pagebreak(weak: true)
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#helpers.code_cell(t2, "function_time_integration")
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<task2:function_time_integration>
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#context {
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counter(page).update(locate(<appendix>).page())
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}
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