finally accurate mesh forces

nbody/presentation/main.typ

@@ -31,13 +31,13 @@
 // Finally - The real content
 = N-body forces and analytical solutions
 
-== Objective
-Implement naive N-body force computation and get an intuition of the challenges:
-- accuracy
-- computation time
-- stability
+// == Objective
+// Implement naive N-body force computation and get an intuition of the challenges:
+// - accuracy
+// - computation time
+// - stability
 
-$==>$ still useful to compute basic quantities of the system, but too limited for large systems or the dynamical evolution of the system
+// $==>$ still useful to compute basic quantities of the system, but too limited for large systems or the dynamical evolution of the system
 
 
 == Overview - the system
@@ -47,7 +47,7 @@ Get a feel for the particles and their distribution. [#link(<task1:plot_particle
 
 #columns(2)[
   #helpers.image_cell(t1, "plot_particle_distribution")
-  Note: for visibility the outer particles are not shown.
+  // Note: for visibility the outer particles are not shown.
   #colbreak()
   The system at hand is characterized by:
   - $N ~ 10^4$ stars
@@ -81,12 +81,12 @@ We compare the computed density with the analytical model provided by the _Hernq
     #helpers.image_cell(t1, "plot_density_distribution")
   ]
 )
+// Note that by construction, the first shell contains no particles
+// => the numerical density is zero there
+// Having more bins means to have shells that are nearly empty
+// => the error is large, NBINS = 30 is a good compromise
 
 
-#block(
-  height: 1fr,
-)
-
 
 == Force computation
 // N Body and variations
@@ -110,13 +110,48 @@ We compare the computed density with the analytical model provided by the _Hernq
   ]
 )
 
+// basic $N^2$ matches analytical solution without dropoff. but: noisy data from "bad" samples
+// $N^2$ with softening matches analytical solution but has a dropoff. No noisy data.
+
+// => softening $\approx 1 \varepsilon$ is a sweet spot since the dropoff is "late"
+
 
 == Relaxation
-Relaxation [#link(<task1:compute_relaxation_time>)[code]]:
-// #helpers.code_cell(t1, "compute_relaxation_time")
+
+We express system relaxation in terms of the dynamical time of the system.
+$
+  t_"relax" = overbrace(N / (8 log N), n_"relax") dot t_"crossing"
+$
+where the crossing time of the system can be estimated through the half-mass velocity $t_"crossing" = v(r_"hm")/r_"hm"$.
+
+We find a relaxation of [#link(<task1:compute_relaxation_time>)[code]].
 
 
-Discussion!
+// === Discussion
+#grid(
+  columns: (1fr, 1fr),
+  inset: 0.5em,
+  block[
+    #image("relaxation.png")
+  ],
+  block[
+    - Each star-star interaction contributes $delta v approx (2 G m )/b$
+    - Shifting by $epsilon$ *dampens* each contribution
+    - $=>$ relaxation time increases
+  ]
+)
+// The estimate for $n_{relax}$ comes from the contribution of each star-star encounter to the velocity dispersion. This depends on the perpendicular force
+
+// $\implies$ a bigger softening length leads to a smaller $\delta v$.
+
+// 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 $\varepsilon$.
+
+// === Effect
+// - The relaxation time **increases** with increasing softening length
+// - 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
+
+// **In other words:**
+// The softening dampens the change of velocity => time to relax is longer
 
 
 
@@ -128,10 +163,13 @@ Discussion!
   columns: 2
 )[
   #helpers.image_cell(t2, "plot_particle_distribution")
+
+  $=>$ use $M_"sys" approx 10^4 M_"sol" + M_"BH"$
 ]
 
 
 
+
 == Force computation
 #helpers.code_reference_cell(t2, "function_mesh_force")
 
@@ -156,9 +194,16 @@ Discussion!
     - very large grids have issues with overdiscretization
 
     $==> 75 times 75 times 75$ as a good compromise
+    // Some other comments:
+    // - see the artifacts because of the even grid numbers (hence the switch to 75)
+    // overdiscretization for large grids -> vertical spread even though r is constant
+    // this becomes even more apparent when looking at the data without noise - the artifacts remain
   ]
 )
 
+
+#helpers.image_cell(t2, "plot_force_computation_time")
+
 == Time integration
 === Runge-Kutta
 #helpers.code_reference_cell(t2, "function_runge_kutta")