Understanding Delta Function Approximations: Gaussian Delta Sequence (Heat Kernel)

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In the previous blog, we understood the Lorentzian Delta Sequence (Cauchy Kernel). Let's take another step and explore the Gaussian Delta Sequence (Heat Kernel).

Why Study These Approximations?

Delta functions are central in many fields:

• Signal Processing: Ideal impulse, filter response
• Physics: Point charges/masses, Green's functions
• Spectral Theory: Lorentzian profiles in resonance
• Diffusion Models: Gaussians arise from the heat equation
• Numerics: Regularizing singular integrals

Each kernel has a story to tell.

Gaussian Delta Sequence (Heat Kernel)

Formula \[ f_t(x) = \frac{1}{2\sqrt{\pi t}} .e^{-\frac{x^2}{4t}} \]

• Smooth and fast-decaying
• Bell-shaped
• Natural from the heat equation

#Define the Function

var('x t')
f_gauss(x, t) = (1/(2*sqrt(pi*t))) * exp(-x^2 / (4*t))
f_gauss(x, t)

# Check Symbolic Integration
var('xi')
assume(t > 0)  # Ensure t is positive
integral(f_gauss(xi, t), xi, -oo, oo).simplify_full()

#Limit Evaluation at x → 0
limit(f_gauss(0, t), t=0)

#Alternative Approach: Use Numerical Evaluation
#If the symbolic engine struggles, try evaluating the function numerically at progressively smaller values of ( t ):

t_values = [0.1, 0.01, 0.001, 0.0001]
[f_gauss(0, t).n() for t in t_values]

#Integral Test (Distributional Behavior)
var('a b')
assume(a < 0, b > 0)  # Ensure a < 0 < b to match delta behavior
integral(f_gauss(xi, t), xi, a, b).simplify_full()

# Numerical Evaluation
# To see how the integral behaves for small ( t ):

import numpy as np
import matplotlib.pyplot as plt
import sage.all as sage

def gaussian_integral(t, a=-1, b=1):
    from math import erf, sqrt, pi
    return (1/2) * (erf(b / sqrt(4*t)) - erf(a / sqrt(4*t)))

# Test for different t values
t_values = np.logspace(-3, 0, 50)  # Log-spaced values from 0.001 to 1
integral_values = [gaussian_integral(t) for t in t_values]

# Plotting
plt.figure(figsize=(8, 5))
plt.plot(t_values, integral_values, marker='o', linestyle='-', color='blue')
plt.axhline(y=1, color='r', linestyle='--', label="Expected Limit (1)")
plt.xscale("log")
plt.xlabel(r"$t$")
plt.ylabel(r"Integral Value")
plt.title("Numerical Verification: Gaussian Integral Convergence")
plt.legend()
plt.grid(True)
plt.show()

# Plot the Gaussian Sequence
p1 = plot(f_gauss(x, 0.5), (x, -5, 5), color='red', legend_label='t = 0.5') + \
     plot(f_gauss(x, 0.2), (x, -5, 5), color='blue', legend_label='t = 0.2') + \
     plot(f_gauss(x, 0.05), (x, -5, 5), color='green', legend_label='t = 0.05')

p1.show(title='Gaussian Approximation to δ(x)', ymin=0, ymax=3)

# Compute First and Second Derivatives

var('x t')
f_gauss(x, t) = (1/(2*sqrt(pi*t))) * exp(-x^2 / (4*t))

# First derivative (approximating δ'(x))
f_gauss_prime(x, t) = diff(f_gauss(x, t), x)

# Second derivative (approximating δ''(x))
f_gauss_double_prime(x, t) = diff(f_gauss_prime(x, t), x)

f_gauss_prime(x, t), f_gauss_double_prime(x, t)

#Plot the Derivatives

p1 = plot(f_gauss_prime(x, 0.5), (x, -5, 5), color='red', legend_label="t=0.5") + \
     plot(f_gauss_prime(x, 0.2), (x, -5, 5), color='blue', legend_label="t=0.2") + \
     plot(f_gauss_prime(x, 0.05), (x, -5, 5), color='green', legend_label="t=0.05")

p1.show(title="First Derivative of Gaussian Delta Approximation")

p2 = plot(f_gauss_double_prime(x, 0.5), (x, -5, 5), color='red', legend_label="t=0.5") + \
     plot(f_gauss_double_prime(x, 0.2), (x, -5, 5), color='blue', legend_label="t=0.2") + \
     plot(f_gauss_double_prime(x, 0.05), (x, -5, 5), color='green', legend_label="t=0.05")

p2.show(title="Second Derivative of Gaussian Delta Approximation")

#Plotting the Integrated Sequences
p1=plot(integral(f_gauss(x, 0.1), x, -5, 5), (x, -5, 5), color='green', legend_label="Gaussian")
p1.show(title="Integrated Delta Approximations")

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Physics Note
Appears in diffusion, heat kernels, quantum mechanics (e.g., wavepacket spreading).