second set of exercises

.gitignore

+# I keep the slides/books here on my PC
+course_notes
\ No newline at end of file

notebooks/particle_systems.jl

 ### A Pluto.jl notebook ###
 # v0.19.27
 
+#> [frontmatter]
+#> Author = "Alexander Fleming"
+#> title = "Particle Systems and Ensembles"
+#> date = "2023-07-31"
+
 using Markdown
 using InteractiveUtils
 
@@ -18,38 +23,102 @@ md"""
 """
 
 # ╔═╡ 97fc9341-c7b8-4ab7-b5f3-c1253b9f1c0a
-@doc """
-	Ensemble{T, D}
-"""
-mutable struct Ensemble{T, D}
+begin
+
+mutable struct PSystem{T, D}
 	n::Int
 	m_i::Vector{T}
+	m::T
 	x_i::Array{T, 2}
 	p_i::Array{T, 2}
-	q_i::Vector{T}
+end
+
+function PSystem(ndims, n::Int, m_i::Vector{T}, x_i::Matrix{T}, p_i::Matrix{T}) where {T}
+	mass = sum(m_i)
+
+	return PSystem{T, ndims}(n, m_i, mass, x_i, p_i)
+end
+
+@doc """
+	PSystem{T, D}
+Represents a collection of particles in ``D`` space dimensions represented by numerical type `T`.
+
+Type Parameters
+---
+- `T`: Numerical type of the underlying representation.
+- `D`: Number of space dimensions.
+
+Fields
+---
+- `n::Int` Number of particles in the system.
+- `m_i::Vector{T}` Mass of each particle.
+- `m_t::T` Total mass of the system.
+- `p_i::Matrix{T}` Momentum of each particle.
+- `
+
+Methods
+---
+
+""" PSystem
+
 end
 
 # ╔═╡ 03c1f366-320f-4715-bf7b-22661c2914db
 begin
-	mass(e::Ensemble) = sum(e.m_i)
-	center_of_mass(e::Ensemble) = (e.x_i * e.m_i)/mass(e)
+	mass(e::PSystem) = e.m
+	center_of_mass(e::PSystem) = (e.x_i * e.m_i)/mass(e)
+end
+
+# ╔═╡ dfbe8230-7c71-4b35-86ae-543ec4fa4d2b
+begin
+	momentum(e::PSystem) = reduce(+, e.p_i, dims=2)[:,1]
 end
 
 # ╔═╡ e90ac67c-e4c8-4937-9507-8b9372b4e371
 begin
-	r_ab(e::Ensemble, a::Int, b::Int) = e.x_i[:, b] - e.x_i[:, a]
+	r_ab(e::PSystem, a::Int, b::Int) = e.x_i[:, b] - e.x_i[:, a]
 	
-	function dist_ab(e::Ensemble, a::Int, b::Int) 
+	function dist_ab(e::PSystem, a::Int, b::Int) 
 		r = r_ab(e, a, b)
 		return sqrt(r⋅r)
 	end
 end
 
+# ╔═╡ 5ce46eeb-95e2-4963-9dda-ede887e74847
+
+
 # ╔═╡ 3cb9c3a7-a354-427e-aff5-dafdbeadd47a
-system = Ensemble{Float64, 2}(4, [2.0, 2.0, 1,1], randn((2,4)), zeros(2,4), [1, 2, 2, 4])
+system = PSystem(2, 4, [2.0, 2.0, 1,1], randn((2,4)), zeros(2,4))
+
+# ╔═╡ cf042dda-92d0-4998-9f8b-ab3593943085
+md"""
+# Exercise: Stirling's Approximation
+
+```math
+	\begin{align}
+		n!&=\sqrt{2\pi n}\left(\frac{n}{\mathrm e}\right)^n\left(1+\mathrm O\left(\frac{1}{n}\right)\right)\\
+\ln n! &= \frac{1}{2}\ln 2\pi n + n\ln\frac{n}{\mathrm e} + \ln \left(1+\mathrm O\left(\frac{1}{n}\right)\right)\\
+\ln n! &\approx \frac{1}{2}\ln 2\pi n + n\ln n - n\ln\mathrm e\\
+\ln n! &\approx \frac{1}{2}\ln 2\pi n + n\ln n - n
+	\end{align}
+```
+"""
+
+# ╔═╡ c1716a64-e971-48a9-a85e-a8ebfc5603a9
+begin
+	# ln n! = sum 1 to n ln n
+	logn!(n) = sum(log(n) for n=1:n)
+	logn!_appx(n) = 0.5*log(2*π*n)+n*log(n)-n
+	rel_err(n) = abs(logn!_appx(n)-logn!(n))/logn!(n)
+end;
 
-# ╔═╡ 1a06e6ff-bb13-4fbd-9ae1-6cbf9b5bbfbe
-center_of_mass(system)
+# ╔═╡ 535bb6a0-4ad5-4ce2-a792-7f432a5032ac
+plot(1:1000, rel_err, yscale=:log10, ylabel="Relative Error", xlabel=L"n", label=L"\ln n! - \mathrm{Stirling}(n!)", dpi=600)
+
+# ╔═╡ 51eac4fc-6e19-43c8-a755-165f8c38ca19
+md"""
+# Exercise: Ideal Gas
+"""
 
 # ╔═╡ 00000000-0000-0000-0000-000000000001
 PLUTO_PROJECT_TOML_CONTENTS = """
@@ -1093,10 +1162,15 @@ version = "1.4.1+0"
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-# ╠═97fc9341-c7b8-4ab7-b5f3-c1253b9f1c0a
+# ╟─97fc9341-c7b8-4ab7-b5f3-c1253b9f1c0a
 # ╠═03c1f366-320f-4715-bf7b-22661c2914db
+# ╠═dfbe8230-7c71-4b35-86ae-543ec4fa4d2b
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+# ╠═5ce46eeb-95e2-4963-9dda-ede887e74847
 # ╠═3cb9c3a7-a354-427e-aff5-dafdbeadd47a
-# ╠═1a06e6ff-bb13-4fbd-9ae1-6cbf9b5bbfbe
+# ╟─cf042dda-92d0-4998-9f8b-ab3593943085
+# ╠═c1716a64-e971-48a9-a85e-a8ebfc5603a9
+# ╟─535bb6a0-4ad5-4ce2-a792-7f432a5032ac
+# ╠═51eac4fc-6e19-43c8-a755-165f8c38ca19
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