diff --git a/Resources/Week1/curie_weiss.jl b/Resources/Week1/curie_weiss.jl deleted file mode 100644 index 1a0f5ed..0000000 --- a/Resources/Week1/curie_weiss.jl +++ /dev/null @@ -1,1989 +0,0 @@ -### A Pluto.jl notebook ### -# v0.19.22 - -using Markdown -using InteractiveUtils - -# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). -macro bind(def, element) - quote - local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end - local el = $(esc(element)) - global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) - el - end -end - -# ╔═╡ 90361bb7-075b-49eb-b788-ee99b5c64ea0 -begin - using LaTeXStrings - using OrdinaryDiffEq - using Plots - using PlutoTeachingTools - using PlutoUI - using Random -end - -# ╔═╡ 34387fe2-61bc-4ee4-a669-d38447ced424 -md""" -# Curie-Weiss model -""" - -# ╔═╡ 867dc158-2dff-4e59-9bca-cee709a11b2c -TableOfContents() - -# ╔═╡ 23aaee39-6a7b-4f6a-8388-8f734aa0d220 -md""" -## Model -""" - -# ╔═╡ bad021ba-3b83-4c63-a29b-53fb7cea5b5c -md""" -The Hamiltonian for $N$ spins $S_1, ..., S_N$ with external field $h$ is given by -```math -\mathcal{H}(\mathbf{S}) = -\frac{1}{2N} \sum_{i,j} S_i S_j - h \sum_i S_i -``` -The partition function at inverse temperature $\beta$ is -```math -Z = \sum_{\mathbf{S}} \exp(-\beta \mathcal{H}(\mathbf{S})) -``` -The Boltzmann distribution is -```math -\mathbb{P}(\mathbf{S}) = \frac{\exp(-\beta \mathcal{H}(\mathbf{S}))}{Z} -``` -""" - -# ╔═╡ dce6c18b-aa8e-470d-b574-6dc2584ab8c2 -H(m) = -((1+m)/2) * log((1+m)/2) - ((1-m)/2) * log((1-m)/2) - -# ╔═╡ 19098e4b-7e1a-41fe-9d3c-106d82016e92 -ϕ(m; β, h) = H(m) + β*m^2/2 + β*h*m - -# ╔═╡ e8cafb9d-cf47-4da1-b314-c70e83416154 -function fixed_point_iteration(m_start; β, h, tol) - error = Inf - m = m_start - while error > tol - m_new = tanh(β * (h + m)) - error = abs(m_new - m) - m = m_new - end - return m -end - -# ╔═╡ 1d3eb626-ee0d-4d3d-a799-7a28f9ffc9cd -function mean_field_solutions(; β, h, tol=1e-7) - m1 = fixed_point_iteration(1; β, h, tol) # right solution - m2 = fixed_point_iteration(0; β, h, tol) # middle solution - m3 = fixed_point_iteration(-1; β, h, tol) # left solution - return (m1, m2, m3) -end - -# ╔═╡ 222e54ae-b3a7-4119-a5e0-6de8e173e4ad -md""" -## Mean field equation -""" - -# ╔═╡ 3f0aa97e-2c81-4faa-807c-7638b4f99c84 -md""" -## MCMC -""" - -# ╔═╡ 26b02b7a-fdbf-40db-b633-ddbbe16020ba -md""" -### Metropolis-Hastings -""" - -# ╔═╡ f8e5a350-f135-464d-b068-9d0f757fa094 -md""" -Let $\mathbf{S}_{\text{flip}}^{(k)}$ denote the vector of spins $\mathbf{S}$ in which $S_k$ has been flipped to $-S_k$. -The variation in energy from the sign switch is given by: -```math -\begin{aligned} -\mathcal{H}(\mathbf{S}_{\text{flip}}^{(k)}) - \mathcal{H}(\mathbf{S}) -= & -\frac{1}{2N} \left(\sum_{i \neq k,j \neq k} S_i S_j + \sum_{i \neq k} S_i (-S_k) + \sum_{j \neq k} (-S_k) S_j + (-S_k) (-S_k) \right) \\ -& + \frac{1}{2N} \left(\sum_{i \neq k,j \neq k} S_i S_j + \sum_{i \neq k} S_i S_k + \sum_{j \neq k} S_k S_j + S_k S_k \right) \\ -& - h \left(\sum_{i \neq k} S_i + (-S_k)\right) + h \left(\sum_{i \neq k} S_i + S_k\right) \\ -= & + \frac{2}{N} S_k \sum_{i \neq k} S_i + 2 h S_k -\end{aligned} -``` -""" - -# ╔═╡ 451a09ff-e1ef-4109-b237-e0d36be1b870 -function metropolis_hastings(N; β, h, tmax=100N) - S = ones(Int, N) - S_sum = sum(S) - m_history = Vector{Float64}(undef, tmax + 1) - m_history[1] = S_sum / N - for t in 1:tmax - k = rand(1:N) - Sₖ = S[k] - ΔE = (2/N) * Sₖ * (S_sum - Sₖ) + 2 * h * Sₖ - r = rand() - if r < exp(-β * ΔE) - S_sum = S_sum - 2Sₖ - S[k] = -Sₖ - end - m_history[t + 1] = S_sum / N - end - return m_history -end - -# ╔═╡ 046646bd-11c9-40eb-a550-f6aaac5d8e5d -md""" -### Glauber -""" - -# ╔═╡ 0a46b881-cad9-4187-99cf-d45cfb59dab4 -md""" -Let $\mathbf{S}_{\text{rem}}^{(k)}$ denote the vector of spins $\mathbf{S}$ in which $S_k$ has been removed. -The conditional probability that $S_k = 1$ given $\mathbf{S}_{\text{rem}}^{(k)}$ is given by -```math -\begin{aligned} -P_+ & = \mathbb{P}(S_k = 1 \mid \mathbf{S}_{\text{rem}}^{(k)}) = \frac{\mathbb{P}(S_k = 1, \mathbf{S}_{\text{rem}}^{(k)})}{\mathbb{P}(\mathbf{S}_{\text{rem}}^{(k)})} \\ -& = \frac{\exp \left[-\beta \mathcal{H}(S_k = 1, \mathbf{S}_{\text{rem}}^{(k)})\right]}{\exp \left[-\beta \mathcal{H}(S_k = 1, \mathbf{S}_{\text{rem}}^{(k)})\right] + \exp \left[-\beta \mathcal{H}(S_k = -1, \mathbf{S}_{\text{rem}}^{(k)})\right]} \\ -& = \frac{1}{1 + \exp \left[\beta \mathcal{H}(S_k = 1, \mathbf{S}_{\text{rem}}^{(k)}) -\beta \mathcal{H}(S_k = -1, \mathbf{S}_{\text{rem}}^{(k)})\right]} -\end{aligned} -``` -We can reuse the computation from Metropolis-Hastings to find: -```math -P_+ = \frac{1}{1 + \exp \left[\beta \left(-\frac{2}{N} \sum_{i \neq k} S_i - 2h\right)\right]} -``` -Now is a good time to remember that -```math -1 + \tanh(x) = 1 + \frac{e^{2x} - 1}{e^{2x} + 1} = \frac{2 e^{2x}}{e^{2x} + 1} = \frac{2}{1 + e^{-2x}} -``` -Let's conclude: -```math -P_+ = \frac{1}{2} \tanh\left[\beta \left(\frac{1}{N}\sum_{i \neq k} S_i + h \right)\right] -``` -""" - -# ╔═╡ 07a0292a-da08-411c-a54c-a46d5a64ab91 -function glauber(N; β, h, tmax=100N) - S = ones(Int, N) - S_sum = sum(S) - m_history = Vector{Float64}(undef, tmax + 1) - m_history[1] = S_sum / N - for t in 1:tmax - k = rand(1:N) - Sₖ = S[k] - P₊ = (1 + tanh(β * ((S_sum - Sₖ) / N + h))) / 2 - r = rand() - if r < P₊ - S_sum = S_sum - Sₖ + 1 - S[k] = 1 - else - S_sum = S_sum - Sₖ - 1 - S[k] = -1 - end - m_history[t + 1] = S_sum / N - end - return m_history -end - -# ╔═╡ 38891968-1f3a-4d46-b303-cff1933a17ed -md""" -## ODE -""" - -# ╔═╡ 0518f532-2970-4712-92d1-9aa7640b4b23 -ode_update(m, (β, h), t) = -m + tanh(β * (m + h)) - -# ╔═╡ 40411c7e-8f45-4b05-808e-a377ef76c5e8 -md""" -## Utilities -""" - -# ╔═╡ ee24e4fb-1ce9-4b31-8bc7-08047ce8ddde -md""" -### Sliders -""" - -# ╔═╡ b71d9e03-cf14-4cc5-8082-2224b15d5793 -slider_β_mf = md""" -``\log_{10}\beta =`` $(@bind logβ_mf Slider(-0.5:0.01:0.5; default=0, show_value=true)) -"""; - -# ╔═╡ 842868eb-3133-46c9-b52e-bb329e1f4868 -slider_h_mf = md""" -``h =`` $(@bind h_mf Slider(-1.5:0.01:1.5; default=0, show_value=true)) -"""; - -# ╔═╡ 2b40a185-2bc8-4bea-820d-d777849b5d78 -TwoColumn(slider_β_mf, slider_h_mf) - -# ╔═╡ f36c6552-6374-4124-af16-a5a891c78409 -TwoColumn(slider_β_mf, slider_h_mf) - -# ╔═╡ 2e9d5534-888b-4a92-b8b5-acd3f25ce2df - β_mf = 10^logβ_mf; - -# ╔═╡ 30b49685-fac0-4410-b292-4bfd9d87df28 -slider_N_mh = md""" -``N=`` $(@bind N_mh Select([10, 20, 50, 100, 200, 1000])) -"""; - -# ╔═╡ 35e6727c-4e39-4d0b-b68b-0ec8a303b029 -slider_N_mh - -# ╔═╡ fb145d3d-077e-483a-bc02-ecb4f359d185 -slider_N_mh - -# ╔═╡ 52a9f1fc-5f38-422c-9a75-cd1adc7d7ff6 -slider_N_g = md""" -``N=`` $(@bind N_g Select([10, 20, 50, 100, 200, 1000])) -"""; - -# ╔═╡ a59b959c-165f-498e-acab-ac05db0076c5 -slider_N_g - -# ╔═╡ f4ff7e4d-ad52-4f5f-b558-b0144951a8fc -slider_N_g - -# ╔═╡ fa3f9c48-2c95-4366-a5d4-c04414025c7b -slider_β_ode = md""" -``\log_{10}\beta =`` $(@bind logβ_ode Slider(-0.5:0.01:0.5; default=0, show_value=true)) -"""; - -# ╔═╡ f917bcc8-8bf4-46be-8781-daa927670245 -slider_h_ode = md""" -``h =`` $(@bind h_ode Slider(-1.5:0.01:1.5; default=0, show_value=true)) -"""; - -# ╔═╡ 575748ac-b605-483c-950a-359870aeb6f1 -TwoColumn(slider_β_ode, slider_h_ode) - -# ╔═╡ f465d047-9029-426f-af35-2f4a69e71453 -β_ode = 10^logβ_ode - -# ╔═╡ 382c2f3d-6727-4e58-b750-3f4087142ed5 -slider_tmax_ode = md""" -``t_{\max} =`` $(@bind tmax_ode Slider(10:10:1000; default=100, show_value=true)) -"""; - -# ╔═╡ d6822bc4-97af-4295-8d8a-3c41bed0d72e -slider_tmax_ode - -# ╔═╡ deeb99aa-4468-4976-ba4f-06d03d04acfe -md""" -### Plotting -""" - -# ╔═╡ e7ce7842-5a8a-4069-adbb-deabd47d8cd1 -function plot_potential(; β, h) - m_values = -1:0.01:1 - plot(xlabel=L"m", ylabel=L"\phi(m)") - plot!(m_values, ϕ.(m_values; β, h), linewidth=2, label=nothing) - m_extr = collect(mean_field_solutions(; β, h)) - scatter!(m_extr, ϕ.(m_extr; β, h); label=L"m^*") - plot!(title="Potential function\n" * L"\beta = %$(round(β; digits=2)) \qquad h = %$(round(h; digits=2))") -end - -# ╔═╡ 5b54818a-5c4b-4ad2-a5dc-690b64c0b3ce -plot_potential(β=β_mf, h=h_mf) - -# ╔═╡ 0f59fdaa-fe9b-47d2-84a5-bf624fc2214a -function plot_fixed_point(; β, h) - m_values = -1:0.01:1 - plot(xlabel=L"m") - plot!(m_values, m_values; linewidth=2, label=L"m") - plot!(m_values, tanh.(β .* (h .+ m_values)); linewidth=2, label=L"\tanh(\beta(h+m))") - m_fixed = collect(mean_field_solutions(; β, h)) - scatter!(m_fixed, m_fixed; label=L"m^*") - plot!(title="Mean field equation\n" * L"\beta = %$(round(β; digits=2)) \qquad h = %$(round(h; digits=2))") -end - -# ╔═╡ cb9e39f6-119d-411a-950f-4766353174fb -plot_fixed_point(β=β_mf, h=h_mf) - -# ╔═╡ 15813162-829c-456c-9fa7-66f5f9f6a698 -function plot_metropolis_hastings(N; β, h) - m_history = metropolis_hastings(N; β=β, h=h) - plot(xlabel=L"t", ylabel=L"m") - plot!(0:(length(m_history) - 1), m_history; label=nothing) - for m_th in mean_field_solutions(; β, h) - plot!([0, (length(m_history) - 1)], [m_th, m_th]; color=:black, linestyle=:dash, label=nothing) - end - plot!(title="Metropolis-Hastings\n" * L"\beta = %$(round(β; digits=2)) \qquad h = %$(round(h; digits=2)) \qquad N = %$N") -end - -# ╔═╡ af2d670f-44be-46f2-9c08-aa7e52a61ced -plot_metropolis_hastings(N_mh; β=1.2, h=0) - -# ╔═╡ 6110e38b-2af2-45f7-984f-69b7fd4e2abd -plot_metropolis_hastings(N_mh; β=1.2, h=-0.1) - -# ╔═╡ 43b4d9f5-f9ee-4890-9346-58b5a5c2068b -function plot_glauber(N; β, h) - m_history = glauber(N; β=β, h=h) - plot(xlabel=L"t", ylabel=L"m") - plot!(0:(length(m_history) - 1), m_history; label=nothing) - for m_th in mean_field_solutions(; β, h) - plot!([0, (length(m_history) - 1)], [m_th, m_th]; color=:black, linestyle=:dash, label=nothing) - end - plot!(title="Glauber\n" * L"\beta = %$(round(β; digits=2)) \qquad h = %$(round(h; digits=2)) \qquad N = %$N") -end - -# ╔═╡ dc6b8367-df8b-42a0-8362-89c567a3961a -plot_glauber(N_g; β=1.2, h=0) - -# ╔═╡ 5d3ae250-0492-405f-8e95-288783a43347 -plot_glauber(N_g; β=1.2, h=-0.1) - -# ╔═╡ e0bef9f5-d610-448e-b306-f5d102614b33 -function plot_ode(; β, h, tmax) - m0 = 1 - tspan = (0.0, tmax) - prob = ODEProblem(ode_update, m0, tspan, (β, h)) - sol = solve(prob, Tsit5()) - plot(xlabel=L"t", ylabel=L"m") - plot(sol; label=nothing) - for m_th in mean_field_solutions(; β, h) - plot!([0, tmax], [m_th, m_th]; color=:black, linestyle=:dash, label=nothing) - end - plot!(title="ODE\n" * L"\beta = %$(round(β; digits=2)) \qquad h = %$(round(h; digits=2))") -end - -# ╔═╡ ef326268-e74e-4733-b089-2ab6f9f744d9 -plot_ode(β=β_ode, h=h_ode, tmax=tmax_ode) - -# ╔═╡ 00000000-0000-0000-0000-000000000001 -PLUTO_PROJECT_TOML_CONTENTS = """ -[deps] -LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" -OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" -Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" -PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69" -PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" -Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" - 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-[[deps.x264_jll]] -deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] -git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" -uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" -version = "2021.5.5+0" - -[[deps.x265_jll]] -deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] -git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" -uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" -version = "3.5.0+0" - -[[deps.xkbcommon_jll]] -deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] -git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" -uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" -version = "1.4.1+0" -""" - -# ╔═╡ Cell order: -# ╟─34387fe2-61bc-4ee4-a669-d38447ced424 -# ╠═90361bb7-075b-49eb-b788-ee99b5c64ea0 -# ╠═867dc158-2dff-4e59-9bca-cee709a11b2c -# ╟─23aaee39-6a7b-4f6a-8388-8f734aa0d220 -# ╟─bad021ba-3b83-4c63-a29b-53fb7cea5b5c -# ╠═dce6c18b-aa8e-470d-b574-6dc2584ab8c2 -# ╠═19098e4b-7e1a-41fe-9d3c-106d82016e92 -# ╠═e8cafb9d-cf47-4da1-b314-c70e83416154 -# ╠═1d3eb626-ee0d-4d3d-a799-7a28f9ffc9cd -# ╟─222e54ae-b3a7-4119-a5e0-6de8e173e4ad -# ╠═2b40a185-2bc8-4bea-820d-d777849b5d78 -# ╠═5b54818a-5c4b-4ad2-a5dc-690b64c0b3ce -# ╠═f36c6552-6374-4124-af16-a5a891c78409 -# ╠═cb9e39f6-119d-411a-950f-4766353174fb -# ╟─3f0aa97e-2c81-4faa-807c-7638b4f99c84 -# ╟─26b02b7a-fdbf-40db-b633-ddbbe16020ba -# ╟─f8e5a350-f135-464d-b068-9d0f757fa094 -# ╠═451a09ff-e1ef-4109-b237-e0d36be1b870 -# ╠═35e6727c-4e39-4d0b-b68b-0ec8a303b029 -# ╠═af2d670f-44be-46f2-9c08-aa7e52a61ced -# ╠═fb145d3d-077e-483a-bc02-ecb4f359d185 -# ╠═6110e38b-2af2-45f7-984f-69b7fd4e2abd -# ╟─046646bd-11c9-40eb-a550-f6aaac5d8e5d -# ╟─0a46b881-cad9-4187-99cf-d45cfb59dab4 -# ╠═07a0292a-da08-411c-a54c-a46d5a64ab91 -# ╠═a59b959c-165f-498e-acab-ac05db0076c5 -# ╠═dc6b8367-df8b-42a0-8362-89c567a3961a -# ╠═f4ff7e4d-ad52-4f5f-b558-b0144951a8fc -# ╠═5d3ae250-0492-405f-8e95-288783a43347 -# ╟─38891968-1f3a-4d46-b303-cff1933a17ed -# ╠═0518f532-2970-4712-92d1-9aa7640b4b23 -# ╠═575748ac-b605-483c-950a-359870aeb6f1 -# ╠═d6822bc4-97af-4295-8d8a-3c41bed0d72e -# ╠═ef326268-e74e-4733-b089-2ab6f9f744d9 -# ╟─40411c7e-8f45-4b05-808e-a377ef76c5e8 -# ╟─ee24e4fb-1ce9-4b31-8bc7-08047ce8ddde -# ╠═b71d9e03-cf14-4cc5-8082-2224b15d5793 -# ╠═842868eb-3133-46c9-b52e-bb329e1f4868 -# ╠═2e9d5534-888b-4a92-b8b5-acd3f25ce2df -# ╠═30b49685-fac0-4410-b292-4bfd9d87df28 -# ╠═52a9f1fc-5f38-422c-9a75-cd1adc7d7ff6 -# ╠═fa3f9c48-2c95-4366-a5d4-c04414025c7b -# ╠═f917bcc8-8bf4-46be-8781-daa927670245 -# ╠═f465d047-9029-426f-af35-2f4a69e71453 -# ╠═382c2f3d-6727-4e58-b750-3f4087142ed5 -# ╟─deeb99aa-4468-4976-ba4f-06d03d04acfe -# ╠═e7ce7842-5a8a-4069-adbb-deabd47d8cd1 -# ╠═0f59fdaa-fe9b-47d2-84a5-bf624fc2214a -# ╠═15813162-829c-456c-9fa7-66f5f9f6a698 -# ╠═43b4d9f5-f9ee-4890-9346-58b5a5c2068b -# ╠═e0bef9f5-d610-448e-b306-f5d102614b33 -# ╟─00000000-0000-0000-0000-000000000001 -# ╟─00000000-0000-0000-0000-000000000002 diff --git a/Resources/Week1/curie_weiss.jl.html b/Resources/Week1/curie_weiss.jl.html deleted file mode 100644 index 18cab18..0000000 --- a/Resources/Week1/curie_weiss.jl.html +++ /dev/null @@ -1,15 +0,0 @@ - - - - - -
\ No newline at end of file diff --git a/index.md b/index.md index 6bec162..06942f7 100644 --- a/index.md +++ b/index.md @@ -56,7 +56,7 @@ Lecture content: Curie-Weiss model in Chap 1 Exercises: 1.1 (Bounds on the binomial) and 1.5 (Glauber dynamics). Due on March 2nd at 10 AM -Julia notebook: [Curie-Weiss model](https://idephics.github.io/EPFLDoctoralLecture2023/Resources/Week1/curie_weiss.jl.html) + **Main lecture**