Program 1

Specific Heat of Solids (Debye’s Theory)

Python 3.11 Ready

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Debye Specific Heat Theory

Introduction

The Debye model describes the vibrational modes in a solid by treating them as acoustic phonons. Unlike the Einstein model, which assumes a single frequency for all oscillators, the Debye model considers a continuous spectrum of vibrational frequencies.

Phonon Density of States

The phonon density of states $ g(\omega) $ is proportional to $ \omega^2 $ up to a maximum frequency $ \omega_D $:

$ g(\omega) = \frac{V \omega^2}{2 \pi^2 \left( \frac{\hbar}{v} \right)^3} $

where $ V $ is the volume, $ \hbar $ is Planck's constant, and $ v $ is the velocity of sound.

Debye Temperature

The Debye temperature $ \theta_D $ is given by:

$ \theta_D = \frac{\hbar \omega_D}{k_B} $

where $ k_B $ is Boltzmann's constant.

Heat Capacity Calculation

The heat capacity at constant volume $ C_V $ is given by:

$ C_V = \int_0^{\omega_D} \frac{\partial \langle E \rangle}{\partial T} g(\omega) \, d\omega $

where $ \langle E \rangle $ is the average energy of a phonon mode:

$ \langle E \rangle = \frac{\hbar \omega}{e^{\frac{\hbar \omega}{k_B T}} - 1} $

Low-Temperature Limit

At low temperatures, $ T \ll \theta_D $, the heat capacity $ C_V $ follows a $ T^3 $ dependence:

$ C_V \approx \frac{12 \pi^4}{5} \frac{N k_B}{\left( \frac{\theta_D}{T} \right)^3} T^3 $

where $ N $ is the number of atoms.

High-Temperature Limit

At high temperatures, $ T \gg \theta_D $, the heat capacity approaches:

$ C_V \approx 3Nk_B $

Program 2

Specific Heat of Solids (Debye’s Theory)

Python 3.11 Ready

"""Variation of Einstein and Debye specific heat with temperature"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad

h, k, V = 6.262e-34, 1.38e-23, 1.0
thD = 125   #taking thetaD = thetaE
nD = k*thD/h    # h*nD = k*thD
e0 = h*nD
T0, Tn, dT = 0.001, 400, 1.0
Trange = np.arange(T0, Tn, dT)
N = 10000  #No. of atoms

##Debye specific heat
def D(n, T):
    return (9*N/nD**3)*(n**2)*(h*n)/(np.exp(h*n/(k*T))-1)

UTD = []
for T in Trange:
    U = quad(D, 0, nD, args = (T))
    UTD.append(U[0])
    
CVD = np.diff(UTD)/np.diff(Trange)

## Einstein specific heat
def UE(T):
    E = []
    Z = []
    for i in range(1000):
        u = (i*e0*np.exp(-(i*e0)/(k*T)))
        z = np.exp(-(i*e0)/(k*T))
        E.append(u)
        Z.append(z)
    UE = 3*N*np.sum(E)/np.sum(Z)
    return UE

UTE = []
for T in Trange:
    UTE.append(UE(T))

CVE = np.diff(UTE)/np.diff(Trange)

##Plotting
plt.plot(Trange[1:]/thD, CVD/(3*N*k), color = 'green', label = 'Debye')
plt.plot(Trange[1:]/thD, CVE/(3*N*k) , color = 'red', label = 'Einstein')
plt.xlabel('T / ${\Theta}_D$', fontsize = 12)
plt.ylabel('Cv / 3NK', fontsize =12)
plt.legend()
plt.savefig('plot.png')
plt.show()

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Comparisn of Einstein and Debye Theory

Theoretical Principles & Computational Physics Guide

Specific Heat of Solids (Debye’s Theory): Statistical Physics & Stochastic Modeling

Dr. Alorika Chatterjee & Dr. Anirban Shaw Science & Physics Learners Python 3 / NumPy / SciPy

1. Core Principles & Theoretical Framework

Stochastic Processes: Simulates random physical phenomena including Brownian diffusion, radioactive decay, and thermal fluctuations.

Ensemble Averages: Computes macroscopic thermodynamic observables as statistical ensemble averages over microscopic configurations.

Monte Carlo Convergence: Statistical error scales inversely with the square root of trial count: $\sigma \propto 1/\sqrt{N}$.

2. Mathematical Formulation & Algorithmic Pipeline

Random walk diffusive spreading and Monte Carlo error scaling:

$$ \langle x(t) \rangle = 0, \quad \langle x^2(t) \rangle = 2 D t, \quad \text{Statistical Uncertainty: } \sigma_N = \frac{\sigma}{\sqrt{N}} $$

3. Numerical Mechanics & Implementation Logic

Random Number Seeding: Initialize pseudo-random generators with fixed seeds to ensure reproducible simulation results.
Ensemble Sampling: Execute $M$ independent trials of $N$ steps each to build accurate probability distributions.
Histogramming: Bin particle final positions to verify convergence toward Gaussian/Poisson distributions.

4. Relevant Computational Practice Exercises

Exercise 1

Simulate $M = 1000$ independent random walk trajectories and confirm that root-mean-square displacement scales as $\sqrt{t}$.

Exercise 2

Apply Monte Carlo integration to estimate $\pi$ using hit-or-miss circle sampling and plot error vs trial count $N$.

Exercise 3

Model radioactive nuclear decay for $N_0 = 10^4$ nuclei and compare against the continuous decay law $N(t) = N_0 e^{-\lambda t}$.

Standard Literature & References:

F. Reif, "Fundamentals of Statistical and Thermal Physics", Waveland Press (2009).
Mark Newman, "Computational Physics", University of Michigan (2013).

Ch. 21.9 Specific Heat of Solids (Debye’s Theory)

Specific Heat of Solids (Debye’s Theory)

Mathematical Problem Formulation

Debye Specific Heat Theory

Introduction

Phonon Density of States

Debye Temperature

Heat Capacity Calculation

Low-Temperature Limit

High-Temperature Limit

Specific Heat of Solids (Debye’s Theory)

Mathematical Problem Formulation

1. Core Principles & Theoretical Framework

2. Mathematical Formulation & Algorithmic Pipeline

3. Numerical Mechanics & Implementation Logic

4. Relevant Computational Practice Exercises