Corrected code to more accurately depict the distribution of the order parameter
This commit is contained in:
Karthik
parent
4a4ddf0e9c
commit
24392f2e35
@ -1,174 +1,166 @@
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%% Evolve to skewed to single gaussian with lower mean
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%% Evolve across a second-order-like transition
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clear; clc;
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N_params = 50;
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N_reps = 500;
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N_reps = 50;
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alpha_values = linspace(0, 45, N_params);
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all_data = cell(1, N_params);
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% Transition control
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alpha_start = 5; % where transition begins
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alpha_end = 40; % where transition ends
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alpha_start = 5; % where sigma starts changing
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alpha_widen_end = 15; % when sigma finishes first change
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alpha_shift_start = 15; % when mean starts shifting
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alpha_end = 40; % when mean finishes shifting and sigma narrows
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mu_start = 1.2; % high initial mean
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mu_end = 0.8; % low final mean
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mu_start = 1.2; % high initial mean
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mu_end = 0.2; % low final mean
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sigma_start = 0.2; % wide initial std
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sigma_end = 0.07; % narrow final std
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sigma_start = 0.25; % wide std at start
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sigma_mid = 0.15; % mid-range std in middle
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sigma_end = 0.07; % narrow std at end
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max_skew = 5; % peak skew strength
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max_skew = 5; % peak skew strength
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% Loop through alpha
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for i = 1:N_params
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alpha = alpha_values(i);
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alpha = alpha_values(i);
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% Normalized transition variable t ∈ [0, 1]
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t = min(max((alpha - alpha_start) / (alpha_end - alpha_start), 0), 1);
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% Use cosine-based smooth interpolation
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smooth_t = (1 - cos(pi * t)) / 2; % ease-in-out
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% Mean and sigma interpolation (smoothly decrease)
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mu = mu_start * (1 - smooth_t) + mu_end * smooth_t;
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sigma = sigma_start * (1 - smooth_t) + sigma_end * smooth_t;
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% Skewness: sinusoidal profile, max at middle
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skew_strength = max_skew * sin(t * pi);
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% Generate data
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if abs(skew_strength) < 1e-2 % near-zero skew, use normal
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data = normrnd(mu, sigma, [N_reps, 1]);
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% === Sigma evolution (variance large -> small) ===
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if alpha < alpha_start
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sigma = sigma_start; % wide at start
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elseif alpha < alpha_widen_end
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% Smooth transition from wide to mid
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t_sigma = (alpha - alpha_start) / (alpha_widen_end - alpha_start);
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sigma = sigma_start * (1 - t_sigma) + sigma_mid * t_sigma;
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elseif alpha < alpha_end
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% Smooth transition from mid to narrow
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t_sigma = (alpha - alpha_widen_end) / (alpha_end - alpha_widen_end);
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sigma = sigma_mid * (1 - t_sigma) + sigma_end * t_sigma;
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else
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data = skewnormrnd(mu, sigma, skew_strength, N_reps);
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sigma = sigma_end; % narrow at end
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end
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all_data{i} = data;
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% === Mean evolution ===
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if alpha < alpha_shift_start
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mu = mu_start; % fixed at high initially
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elseif alpha <= alpha_end
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% Smooth cosine shift
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t_mu = (alpha - alpha_shift_start) / (alpha_end - alpha_shift_start);
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smooth_t_mu = (1 - cos(pi * t_mu)) / 2;
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mu = mu_start * (1 - smooth_t_mu) + mu_end * smooth_t_mu;
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else
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mu = mu_end;
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end
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% === Skew evolution ===
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if alpha < alpha_end
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t_skew = (alpha - alpha_start) / (alpha_end - alpha_start);
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skew_strength = max_skew * (1 - t_skew); % fade out
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else
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skew_strength = 0;
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end
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% Generate data
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if abs(skew_strength) < 1e-2
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data = normrnd(mu, sigma, [N_reps, 1]);
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else
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data = skewnormrnd(mu, sigma, skew_strength, N_reps);
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end
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all_data{i} = data;
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% Cumulants
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kappa = computeCumulants(data, 6);
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mean_vals(i) = kappa(1);
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var_vals(i) = kappa(2);
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skew_vals(i) = kappa(3);
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kurt_vals(i) = kappa(4);
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kappa4_vals(i) = kappa(4);
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kappa5_vals(i) = kappa(5);
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kappa6_vals(i) = kappa(6);
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end
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%% Evolve to bimodal
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%% Evolve across a first-order-like transition
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% First-order-like distribution evolution with significant bimodality
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clear; clc;
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N_params = 50;
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N_reps = 500;
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alpha_values = linspace(0, 45, N_params);
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N_params = 50;
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N_reps = 50;
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alpha_values = linspace(0, 45, N_params);
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all_data = cell(1, N_params);
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all_data = cell(1, N_params);
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bimodal_start = 20; % start earlier
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transition_width = 5; % wider window for smoothness
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% Define transition regions
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skewed_start = 10;
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bimodal_start = 20;
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bimodal_end = 35;
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final_narrow_start = 40;
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% Peak positions and widths
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mu_high = 1.2; % Initial metastable peak
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mu_low = 0.2; % Final stable peak
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mu_new_peak = 0.8; % New peak appears slightly lower
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sigma_initial = 0.08;
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for i = 1:N_params
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alpha = alpha_values(i);
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if alpha < (bimodal_start - transition_width)
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% Pure skewed unimodal, smaller max skewness for subtlety
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skew_strength = 3 * (alpha / bimodal_start);
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data = skewnormrnd(1, 0.1, skew_strength, N_reps);
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elseif alpha <= (bimodal_start + transition_width)
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% Smooth transition window
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t = (alpha - (bimodal_start - transition_width)) / (2 * transition_width);
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% Weights transition 1 -> 0.7
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w1 = 1 - 0.3 * t;
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w2 = 1 - w1;
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% Peak separation smaller (max delta = 0.3)
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delta_max = 0.3;
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delta = delta_max * t;
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mu1 = 1 - delta;
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mu2 = 1 + delta;
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sigma1 = 0.1;
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sigma2 = 0.1;
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N1 = round(N_reps * w1);
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N2 = N_reps - N1;
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mode1_samples = normrnd(mu1, sigma1, [N1, 1]);
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mode2_samples = normrnd(mu2, sigma2, [N2, 1]);
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data = [mode1_samples; mode2_samples];
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data = data(randperm(length(data)));
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alpha = alpha_values(i);
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if alpha < skewed_start
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% Region I: Narrow unimodal at high mean
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data = normrnd(mu_high, sigma_initial, [N_reps, 1]);
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elseif alpha < bimodal_start
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% Region II: Slightly skewed
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t_skew = (alpha - skewed_start) / (bimodal_start - skewed_start);
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mu = mu_high - 0.15 * t_skew;
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sigma = sigma_initial + 0.02 * t_skew;
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skew_strength = 3 * t_skew;
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data = skewnormrnd(mu, sigma, skew_strength, N_reps);
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elseif alpha < bimodal_end
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% Region III: Bimodal with fixed or slowly drifting peak positions
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t = (alpha - bimodal_start) / (bimodal_end - bimodal_start); % t in [0, 1]
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% Increased separation between peaks
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drift_amount = 0.3; % larger = more drift toward final mean
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sep_offset = 0.25; % larger = more initial separation between peaks
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% Peaks start separated and move toward mu_low
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mu1 = mu_high * (1 - t)^drift_amount + mu_low * (1 - (1 - t)^drift_amount); % Right peak drifts to left
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mu2 = (mu_new_peak - sep_offset) * (1 - t)^drift_amount + mu_low * (1 - (1 - t)^drift_amount); % Left peak moves slightly
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sigma1 = sigma_initial + 0.02 * (1 - abs(0.5 - t) * 2);
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sigma2 = sigma1;
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% Weight shift: right peak dies out, left peak grows
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w2 = 0.5 + 0.5 * t; % left peak grows: 0.5 → 1
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w1 = 1 - w2; % right peak fades: 0.5 → 0
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N1 = round(N_reps * w1);
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N2 = N_reps - N1;
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mode1 = normrnd(mu1, sigma1, [N1, 1]);
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mode2 = normrnd(mu2, sigma2, [N2, 1]);
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data = [mode1; mode2];
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data = data(randperm(length(data)));
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else
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% After transition: bimodal, but not strongly balanced
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w1 = 0.5;
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w2 = 0.5;
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mu1 = 1 - 0.3;
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mu2 = 1 + 0.3;
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sigma1 = 0.1;
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sigma2 = 0.1;
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N1 = round(N_reps * w1);
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N2 = N_reps - N1;
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mode1_samples = normrnd(mu1, sigma1, [N1, 1]);
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mode2_samples = normrnd(mu2, sigma2, [N2, 1]);
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data = [mode1_samples; mode2_samples];
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data = data(randperm(length(data)));
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% Region IV: Final stable low-mean Gaussian
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data = normrnd(mu_low, sigma_initial, [N_reps, 1]);
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end
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all_data{i} = data;
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kappa = computeCumulants(data,6);
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mean_vals(i) = kappa(1);
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var_vals(i) = kappa(2);
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skew_vals(i) = kappa(3);
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kurt_vals(i) = kappa(4);
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kappa5_vals(i) = kappa(5);
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kappa6_vals(i) = kappa(6);
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end
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%% Animate evolving distribution and cumulant value
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figure(1); clf;
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set(gcf, 'Color', 'w', 'Position',[100 100 1300 750])
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for i = 1:N_params
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clf;
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% PDF
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subplot(1,2,1); cla; hold on;
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data = all_data{i};
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% Plot histogram with normalized PDF
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histogram(data, 'Normalization', 'pdf', 'BinWidth', 0.03, ...
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'FaceColor', [0.3 0.5 0.8], 'EdgeColor', 'k', 'FaceAlpha', 0.6);
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title(sprintf('Histogram at $\\alpha = %.1f^\\circ$', alpha_values(i)), ...
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'Interpreter', 'latex', 'FontSize', 16);
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xlabel('$\mathrm{max}[g^{(2)}]$', 'Interpreter', 'latex', 'FontSize', 14);
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ylabel('PDF', 'FontSize', 14);
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set(gca, 'FontSize', 12); grid on;
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xlim([0.5, 1.8]);
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% Cumulant evolution (e.g., Variance)
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subplot(1,2,2); hold on;
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plot(alpha_values(1:i), var_vals(1:i), 'bo-', 'LineWidth', 2);
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title('Cumulant Tracking', 'Interpreter', 'latex', 'FontSize', 16);
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', 14);
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ylabel('$\kappa_2$', 'Interpreter', 'latex', 'FontSize', 14);
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xlim([0, 45]); grid on;
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set(gca, 'FontSize', 12);
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pause(0.3);
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% Store data and compute cumulants
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all_data{i} = data;
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kappa = computeCumulants(data, 6);
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mean_vals(i) = kappa(1);
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var_vals(i) = kappa(2);
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skew_vals(i) = kappa(3);
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kappa4_vals(i) = kappa(4);
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kappa5_vals(i) = kappa(5);
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kappa6_vals(i) = kappa(6);
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end
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%% === Compute 2D PDF heatmap: f(x, alpha) ===
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x_grid = linspace(0.5, 1.8, 200); % max[g²] values on y-axis
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x_grid = linspace(0.0, 1.8, 200); % max[g²] values on y-axis
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pdf_matrix = zeros(numel(x_grid), N_params); % Now: rows = y, columns = alpha
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for i = 1:N_params
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@ -194,9 +186,45 @@ c = colorbar;
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ylabel(c, 'PDF', 'FontSize', 14, 'Interpreter', 'latex');
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set(gca, 'FontSize', 14);
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%% Animate evolving distribution and cumulant value
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figure(1); clf;
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set(gcf, 'Color', 'w', 'Position',[100 100 1300 750])
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for i = 1:N_params
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clf;
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% PDF
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subplot(1,2,1); cla; hold on;
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data = all_data{i};
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% Plot histogram with normalized PDF
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histogram(data, 'Normalization', 'pdf', 'BinWidth', 0.03, ...
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'FaceColor', [0.3 0.5 0.8], 'EdgeColor', 'k', 'FaceAlpha', 0.6);
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title(sprintf('Histogram at $\\alpha = %.1f^\\circ$', alpha_values(i)), ...
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'Interpreter', 'latex', 'FontSize', 16);
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xlabel('$\mathrm{max}[g^{(2)}]$', 'Interpreter', 'latex', 'FontSize', 14);
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ylabel('PDF', 'FontSize', 14);
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set(gca, 'FontSize', 12); grid on;
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xlim([0.0, 2.0]);
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% Cumulant evolution
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subplot(1,2,2); hold on;
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plot(alpha_values(1:i), kappa4_vals(1:i), 'bo-', 'LineWidth', 2);
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title('Binder Cumulant Tracking', 'Interpreter', 'latex', 'FontSize', 16);
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', 14);
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ylabel('$\kappa_4$', 'Interpreter', 'latex', 'FontSize', 14);
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xlim([0, 45]); grid on;
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set(gca, 'FontSize', 12);
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pause(0.3);
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end
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%% === Plotting ===
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figure(1)
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set(gcf, 'Color', 'w', 'Position', [100 100 950 750])
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t = tiledlayout(2, 2, 'TileSpacing', 'compact', 'Padding', 'compact');
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scan_vals = alpha_values; % your parameter sweep values
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@ -208,59 +236,41 @@ title_fontsize = 16;
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% 1. Mean with error bars (if you have error data, else just plot)
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% If no error, replace errorbar with plot or omit error data
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% For now, no error bars assumed
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subplot(3,2,1);
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nexttile;
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plot(scan_vals, mean_vals, 'o-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Mean of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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title('Mean', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_1$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% 2. Variance
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subplot(3,2,2);
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nexttile;
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plot(scan_vals, var_vals, 's-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Variance of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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title('Variance', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_2$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% 3. Skewness
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subplot(3,2,3);
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nexttile;
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plot(scan_vals, skew_vals, 'd-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Skewness of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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title('Skewness', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_3$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% 4. Kurtosis
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subplot(3,2,4);
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plot(scan_vals, kurt_vals, '^-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Fourth-order Cumulant of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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% 4. Binder Cumulant
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nexttile;
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plot(scan_vals, kappa4_vals, '^-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Binder Cumulant', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_4$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% 5. 5th-order cumulant
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subplot(3,2,5);
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plot(scan_vals, kappa5_vals, 'v-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Fifth-order Cumulant of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_5$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% 6. 6th-order cumulant
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subplot(3,2,6);
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plot(scan_vals, kappa6_vals, '>-', 'LineWidth', 1.5, 'MarkerSize', 6);
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title('Sixth-order Cumulant of Distribution', 'FontSize', title_fontsize, 'Interpreter', 'latex');
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xlabel('$\alpha$ (degrees)', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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ylabel('$\kappa_6$', 'Interpreter', 'latex', 'FontSize', label_fontsize);
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set(gca, 'FontSize', axis_fontsize);
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grid on;
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% Super title (you can customize the string)
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sgtitle('Cumulants of a simulated evolving distribution', ...
|
||||
'FontWeight', 'bold', 'FontSize', 18, 'Interpreter', 'latex');
|
||||
|
||||
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Reference in New Issue
Block a user