Content """ Ising Model for the MCH2022 Badge Implements the [[https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm][Metropolis-Hastings algorithm]]. And probably in a bad way. See Also -------- [[https://en.wikipedia.org/wiki/Square_lattice_Ising_model]]: Background info on the Ising model. """ import math import display import urandom import buttons import mch22 import framebuf # the initial temperature tau = 0.4 # Critical temperature for 2d Ising # this is 2 / Arsinh(1) tau_curie = 2.269185314213022 # Note: max size is 320×240 size = (64, 48) # FrameBuffer needs 2 bytes for every RGB565 pixel spins = framebuf.FrameBuffer( bytearray(size[0] * size[1] // 8), size[0], size[1], framebuf.MONO_HMSB) # scale up size_pixel = (display.width() // size[0], display.height() // size[1]) border = ((display.width() - size_pixel[0] * size[0]) // 2, (display.height() - size_pixel[1] * size[1]) // 2) display.translate(border[0], border[1]) color_down = 0x000000 color_up = 0xffffff def get_spin_value(i, j): """ return the value of the spin at (i, j) If i, j are too large or negative, use periodic boundary conditions. """ return 2 * spins.pixel( (i + size[0]) % size[0], (j + size[1]) % size[1]) - 1 def set_spin_value(i, j, value): """ set the value of a spin """ spins.pixel(i, j, (value + 1) // 2) color = get_color_from_value(value) display.drawRect( i * size_pixel[0], j * size_pixel[1], size_pixel[0], size_pixel[1], True, color) def reboot(pressed): """ reboot on button press """ if pressed: mch22.exit_python() def create_random_state(): """ start with a random configuration """ # Okay, we start with the most stupid setup first: for i in range(size[0]): for j in range(size[1]): set_spin_value(i, j, 2 * urandom.randint(0, 1) - 1) display_info(0) display.flush() def display_info(step): """ Print info about temperature on screen """ m = get_magnetization() text_color = 0xff0000 font = "roboto_regular12" display.drawRect(5, 5, 297, 25, True, 0x9f9f9f) display.drawText( 10, 10, f"Ising Magnet ({size[0]}x{size[1]}) , T = {tau:.1f}, M = {m:.3f} (step {step})", text_color, font) def get_magnetization(): """ calculate the average magnetization """ m = 0 for i in range(size[0]): for j in range(size[1]): m += get_spin_value(i, j) return m / (size[0] * size[1]) def get_color_from_value(value): """ The inverse of the above """ value = max(-1, min(1, value)) if value > 0.0: return color_up else: return color_down def metropolis(): """ Perform n·m Metropolis steps """ for _ in range(size[0] * size[1]): # pick a random pixel i = urandom.randint(0, size[0] - 1) j = urandom.randint(0, size[1] - 1) ij = get_spin_value(i, j) ij_left = get_spin_value(i - 1, j) ij_right = get_spin_value(i + 1, j) ij_up = get_spin_value(i, j - 1) ij_down = get_spin_value(i, j + 1) # difference in energy if spin is flipped deltaE = 2 * ij * (ij_left + ij_right + ij_up + ij_down) if deltaE < 0 or ( # if energy is lower - flip spin # else only flip with exponentially small probability urandom.uniform(0, 1) < math.exp(- deltaE / (tau * tau_curie))): new_value = -ij set_spin_value(i, j, new_value) global step display_info(step) display.flush() def increase_temp(pressed): """ increase the temperature by 0.1 """ if not pressed: return global tau tau += 0.1 display_info(step) def decrease_temp(pressed): """ decrease temperature by 0.1 """ if not pressed: return global tau tau -= 0.1 # make sure we are not running into negative or zero temperatures tau = max(0.1, tau) display_info(step) buttons.attach(buttons.BTN_MENU, reboot) buttons.attach(buttons.BTN_UP, increase_temp) buttons.attach(buttons.BTN_DOWN, decrease_temp) # start with a grey screen as background color_bg = 0x7f7f7f display.drawFill(color_bg) create_random_state() step = 0 while True: step += 1 metropolis()