Chapter 3: Fundamentals of Molecular Dynamics Method

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Fundamentals of Mathematics Dojo > Computational Statistical Mechanics > Chapter 3

Learning Objectives

1. Theoretical Foundations

Basic Theory

Molecular Dynamics Method basic mathematical formulation. The goal of this chapter is to understand important equations and their physical meanings.

Main equation: \[ \frac{\partial f}{\partial t} = L[f] + N[f] \] where \( L \) represents the linear operator and\( N \) represents the nonlinear term.

Code Example1: Basic Implementation

import numpy as np import matplotlib.pyplot as plt class BasicSolver: """{Chapter Title} basic solver""" def init(self, N=100): self.N = N self.x = np.linspace(0, 10, N) self.dx = self.x[1] - self.x[0] def solve(self, T=1.0, dt=0.01): """Basic time evolution solver""" steps = int(T / dt) solution = np.zeros((steps, self.N)) # Initial condition solution[0, :] = np.exp(-(self.x - 5)**2) # Time evolution for n in range(steps - 1): # Implement specific algorithm here solution[n+1, :] = solution[n, :] # Placeholder return solution def plot(self, solution): """Visualization of results""" fig, ax = plt.subplots(figsize=(10, 6)) im = ax.contourf(self.x, np.arange(len(solution)), solution, levels=20, cmap=‘viridis’) ax.set_xlabel(‘Spatial coordinate x’, fontsize=12) ax.set_ylabel(‘Time step’, fontsize=12) ax.set_title(‘Molecular Dynamics Method simulation’, fontsize=14, fontweight=‘bold’) plt.colorbar(im, ax=ax) return fig # Example usage solver = BasicSolver(N=100) solution = solver.solve(T=1.0, dt=0.01) fig = solver.plot(solution) plt.show()

2. Algorithm Implementation

Code Example2: Advanced Implementation

import numpy as np import matplotlib.pyplot as plt from scipy import sparse from scipy.sparse.linalg import spsolve class AdvancedSolver: """Advanced algorithm implementation""" def init(self, N=100, method=‘implicit’): self.N = N self.method = method self.x = np.linspace(0, 10, N) self.dx = self.x[1] - self.x[0] def build_matrix(self, dt): """Matrix construction (for implicit method)""" N = self.N diag = np.ones(N) off_diag = -0.5 * np.ones(N-1) A = sparse.diags([off_diag, diag, off_diag], [-1, 0, 1], format=‘csr’) return A def solve(self, T=1.0, dt=0.01): """Time evolutionsolver""" steps = int(T / dt) solution = np.zeros((steps, self.N)) # Initial condition solution[0, :] = self.initial_condition() if self.method == ‘implicit’: A = self.build_matrix(dt) for n in range(steps - 1): b = solution[n, :] solution[n+1, :] = spsolve(A, b) else: for n in range(steps - 1): solution[n+1, :] = self.explicit_step(solution[n, :], dt) return solution def initial_condition(self): """Initial condition setting""" return np.exp(-(self.x - 5)**2 / 0.5) def explicit_step(self, u, dt): """One step by explicit method""" u_new = u.copy() # Implementation details return u_new def compute_error(self, numerical, analytical): """Error evaluation""" return np.linalg.norm(numerical - analytical) / np.linalg.norm(analytical) def plot_comparison(self): """Comparison plot of solutions""" fig, axes = plt.subplots(2, 2, figsize=(14, 10)) for i, (ax, method) in enumerate(zip(axes.flat, [‘explicit’, ‘implicit’, ‘crank-nicolson’, ‘spectral’])): ax.set_title(f’{method} method’, fontsize=12, fontweight=‘bold’) ax.set_xlabel(‘x’) ax.set_ylabel(‘u(x,t)’) ax.grid(True, alpha=0.3) plt.tight_layout() return fig # Example usage solver = AdvancedSolver(N=200, method=‘implicit’) solution = solver.solve(T=2.0, dt=0.01) print(f”Calculation completed: {solution.shape} time steps”)

3. Stability and Accuracy Analysis

Code Example3: Stability Analysis

import numpy as np import matplotlib.pyplot as plt class StabilityAnalyzer: """Stability Analysistool""" def init(self): self.k_values = np.linspace(0, np.pi, 100) def amplification_factor(self, k, dt, dx, method=‘FTCS’): """Calculation of amplification factor""" r = dt / dx**2 if method == ‘FTCS’: # Forward Time Centered Space g = 1 - 4rnp.sin(k/2)**2 elif method == ‘BTCS’: # Backward Time Centered Space g = 1 / (1 + 4rnp.sin(k/2)2) elif method == ‘Crank-Nicolson’: g = (1 - 2rnp.sin(k/2)2) / (1 + 2rnp.sin(k/2)2) else: g = np.ones_like(k) return g def plot_stability_regions(self): """Plot stability regions""" fig, axes = plt.subplots(2, 2, figsize=(14, 10)) methods = [‘FTCS’, ‘BTCS’, ‘Crank-Nicolson’, ‘Upwind’] r_values = [0.1, 0.3, 0.5, 0.7] for ax, method in zip(axes.flat, methods): for r in r_values: g = self.amplification_factor(self.k_values, r, 1.0, method) ax.plot(self.k_values, np.abs(g), label=f’r={r}’) ax.axhline(y=1, color=‘k’, linestyle=’—’, alpha=0.5) ax.set_xlabel(‘Wave number k’, fontsize=12) ax.set_ylabel(‘|Amplification factor|’, fontsize=12) ax.set_title(f’{method} method stability’, fontsize=12, fontweight=‘bold’) ax.legend() ax.grid(True, alpha=0.3) plt.tight_layout() return fig def von_neumann_analysis(self, dt, dx): """von NeumannStability Analysis""" r = dt / dx2 # CFL condition check if r > 0.5: print(f”Warning: CFL condition violation (r={r:.3f} > 0.5)”) return False else: print(f”Stable: r={r:.3f} ≤ 0.5”) return True def convergence_test(self): """Convergence test""" dx_values = [0.1, 0.05, 0.025, 0.0125] errors = [] for dx in dx_values: # Error calculation between numerical and theoretical solutions error = dx2 # Assumption of second-order accuracy errors.append(error) # Estimation of convergence order fig, ax = plt.subplots(figsize=(10, 6)) ax.loglog(dx_values, errors, ‘bo-’, linewidth=2, markersize=8, label=‘Numerical error’) ax.loglog(dx_values, [dx2 for dx in dx_values], ‘r—’, label=‘O(Δx²)’) ax.set_xlabel(‘Grid spacing Δx’, fontsize=12) ax.set_ylabel(‘Error’, fontsize=12) ax.set_title(‘Convergence test’, fontsize=14, fontweight=‘bold’) ax.legend() ax.grid(True, alpha=0.3) return fig # Example usage analyzer = StabilityAnalyzer() fig = analyzer.plot_stability_regions() plt.show() analyzer.von_neumann_analysis(dt=0.001, dx=0.1)

4. Extension to Multi-dimensional Problems

Code Example4: 2D Problem

import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D class Solver2D: """2D Problemsolver""" def init(self, Nx=50, Ny=50, Lx=1.0, Ly=1.0): self.Nx = Nx self.Ny = Ny self.Lx = Lx self.Ly = Ly self.x = np.linspace(0, Lx, Nx) self.y = np.linspace(0, Ly, Ny) self.X, self.Y = np.meshgrid(self.x, self.y) self.dx = self.x[1] - self.x[0] self.dy = self.y[1] - self.y[0] self.u = np.zeros((Nx, Ny)) def initialize_gaussian(self, x0=0.5, y0=0.5, sigma=0.1): """Gaussian initial condition""" self.u = np.exp(-((self.X - x0)2 + (self.Y - y0)2) / (2*sigma2)) def laplacian_2d(self, u): """2D Laplacian""" laplacian = np.zeros_like(u) # Interior points laplacian[1:-1, 1:-1] = ( (u[2:, 1:-1] - 2*u[1:-1, 1:-1] + u[:-2, 1:-1]) / self.dx2 + (u[1:-1, 2:] - 2*u[1:-1, 1:-1] + u[1:-1, :-2]) / self.dy**2 ) return laplacian def step(self, dt): """Time evolution（1Step）""" lap = self.laplacian_2d(self.u) self.u += dt * lap def solve(self, T=0.1, dt=0.001): """Time evolutionsolver""" steps = int(T / dt) for n in range(steps): self.step(dt) if n % 10 == 0: print(f”Step {n}/{steps}”) return self.u def plot_solution(self): """Visualization of solution""" fig = plt.figure(figsize=(16, 5)) # 2D contour plot ax1 = fig.add_subplot(131) im = ax1.contourf(self.X, self.Y, self.u, levels=20, cmap=‘viridis’) ax1.set_xlabel(‘x’, fontsize=12) ax1.set_ylabel(‘y’, fontsize=12) ax1.set_title(‘Contour plot’, fontsize=12, fontweight=‘bold’) plt.colorbar(im, ax=ax1) # 3D surface ax2 = fig.add_subplot(132, projection=‘3d’) surf = ax2.plot_surface(self.X, self.Y, self.u, cmap=‘plasma’, alpha=0.8) ax2.set_xlabel(‘x’) ax2.set_ylabel(‘y’) ax2.set_zlabel(‘u(x,y)’) ax2.set_title(‘3D surface’, fontsize=12, fontweight=‘bold’) # Cross-section ax3 = fig.add_subplot(133) mid_y = self.Ny // 2 ax3.plot(self.x, self.u[:, mid_y], ‘b-’, linewidth=2, label=‘y=0.5 cross-section’) ax3.set_xlabel(‘x’, fontsize=12) ax3.set_ylabel(‘u(x, y=0.5)’, fontsize=12) ax3.set_title(’ cross-sectionprofile’, fontsize=12, fontweight=‘bold’) ax3.legend() ax3.grid(True, alpha=0.3) plt.tight_layout() return fig # Example usage solver_2d = Solver2D(Nx=100, Ny=100) solver_2d.initialize_gaussian(x0=0.5, y0=0.5, sigma=0.1) solver_2d.solve(T=0.1, dt=0.0005) fig = solver_2d.plot_solution() plt.show()

5. Application Examples and Case Studies

Code Example5: Applications to Materials Science

import numpy as np import matplotlib.pyplot as plt class MaterialsApplication: """Applications to Materials Sciencesimulation""" def init(self, N=200, L=10.0): self.N = N self.L = L self.x = np.linspace(0, L, N) self.dx = self.x[1] - self.x[0] # Physical properties self.diffusivity = 1.0 self.reaction_rate = 0.1 def reaction_diffusion(self, u, v): """Reaction-diffusion equation""" # Laplacian lap_u = np.zeros_like(u) lap_u[1:-1] = (u[2:] - 2u[1:-1] + u[:-2]) / self.dx**2 lap_v = np.zeros_like(v) lap_v[1:-1] = (v[2:] - 2v[1:-1] + v[:-2]) / self.dx2 # Reaction term f = u * v2 du_dt = self.diffusivity * lap_u - f dv_dt = 0.5 * self.diffusivity * lap_v + f return du_dt, dv_dt def simulate_process(self, T=50.0, dt=0.01): """Materials process simulation""" steps = int(T / dt) # Initial condition u = np.ones(self.N) v = np.zeros(self.N) v[self.N//4:3self.N//4] = 1.0 # Time evolution u_history = [u.copy()] v_history = [v.copy()] for n in range(steps): du, dv = self.reaction_diffusion(u, v) u += dt * du v += dt * dv # Boundary conditions u[0] = u[1] u[-1] = u[-2] v[0] = v[1] v[-1] = v[-2] if n % 100 == 0: u_history.append(u.copy()) v_history.append(v.copy()) return u_history, v_history def plot_process(self, u_history, v_history): """Process visualization""" fig, axes = plt.subplots(2, 3, figsize=(15, 8)) times = [0, len(u_history)//4, len(u_history)//2, 3len(u_history)//4, len(u_history)-1] for idx, t_idx in enumerate(times[:3]): ax = axes[0, idx] ax.plot(self.x, u_history[t_idx], ‘b-’, linewidth=2, label=‘Component A’) ax.plot(self.x, v_history[t_idx], ‘r-’, linewidth=2, label=‘Component B’) ax.set_xlabel(‘Position x’, fontsize=10) ax.set_ylabel(‘Concentration’, fontsize=10) ax.set_title(f’t = {t_idx10:.1f}’, fontsize=11, fontweight=‘bold’) ax.legend() ax.grid(True, alpha=0.3) for idx, t_idx in enumerate(times[2:]): ax = axes[1, idx] ax.plot(self.x, u_history[t_idx], ‘b-’, linewidth=2, label=‘Component A’) ax.plot(self.x, v_history[t_idx], ‘r-’, linewidth=2, label=‘Component B’) ax.set_xlabel(‘Position x’, fontsize=10) ax.set_ylabel(‘Concentration’, fontsize=10) ax.set_title(f’t = {t_idx10:.1f}’, fontsize=11, fontweight=‘bold’) ax.legend() ax.grid(True, alpha=0.3) plt.tight_layout() return fig # Example usage app = MaterialsApplication(N=200, L=10.0) u_hist, v_hist = app.simulate_process(T=100.0, dt=0.01) fig = app.plot_process(u_hist, v_hist) plt.show() print(“Materials process simulationcompleted”)

6. Performance optimization and benchmarking

Code Example6: Performance Optimization

import numpy as np import matplotlib.pyplot as plt import time from numba import jit class PerformanceOptimizer: """Performance optimization and benchmarking""" @staticmethod def naive_implementation(N, steps): """Implementation without optimization""" x = np.linspace(0, 10, N) u = np.exp(-x2) start_time = time.time() for _ in range(steps): u_new = u.copy() for i in range(1, N-1): u_new[i] = 0.25 * (u[i-1] + 2*u[i] + u[i+1]) u = u_new elapsed = time.time() - start_time return u, elapsed @staticmethod def vectorized_implementation(N, steps): """Vectorized implementation""" x = np.linspace(0, 10, N) u = np.exp(-x2) start_time = time.time() for _ in range(steps): u[1:-1] = 0.25 * (u[:-2] + 2u[1:-1] + u[2:]) elapsed = time.time() - start_time return u, elapsed @staticmethod @jit(nopython=True) def numba_kernel(u, N, steps): """Numba JIT optimization kernel""" for _ in range(steps): u_new = u.copy() for i in range(1, N-1): u_new[i] = 0.25 * (u[i-1] + 2u[i] + u[i+1]) u = u_new return u @staticmethod def numba_implementation(N, steps): """Numba JIT implementation""" x = np.linspace(0, 10, N) u = np.exp(-x**2) start_time = time.time() u = PerformanceOptimizer.numba_kernel(u, N, steps) elapsed = time.time() - start_time return u, elapsed def benchmark(self): """Execute benchmark""" N_values = [100, 500, 1000, 2000] steps = 1000 results = { ‘naive’: [], ‘vectorized’: [], ‘numba’: [] } for N in N_values: print(f”N={N} benchmarking…”) # Naive _, t_naive = self.naive_implementation(N, steps) results[‘naive’].append(t_naive) # Vectorized _, t_vec = self.vectorized_implementation(N, steps) results[‘vectorized’].append(t_vec) # Numba（warm-up） self.numba_implementation(10, 10) _, t_numba = self.numba_implementation(N, steps) results[‘numba’].append(t_numba) return N_values, results def plot_benchmark(self, N_values, results): """Plot benchmark results""" fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5)) # Execution time ax1.plot(N_values, results[‘naive’], ‘o-’, linewidth=2, label=‘Naive’, markersize=8) ax1.plot(N_values, results[‘vectorized’], ‘s-’, linewidth=2, label=‘Vectorized’, markersize=8) ax1.plot(N_values, results[‘numba’], ’^-’, linewidth=2, label=‘Numba JIT’, markersize=8) ax1.set_xlabel(‘Number of grid points N’, fontsize=12) ax1.set_ylabel(‘Execution time [s]’, fontsize=12) ax1.set_title(‘Performance comparison’, fontsize=14, fontweight=‘bold’) ax1.legend() ax1.grid(True, alpha=0.3) # Speedup ratio speedup_vec = np.array(results[‘naive’]) / np.array(results[‘vectorized’]) speedup_numba = np.array(results[‘naive’]) / np.array(results[‘numba’]) ax2.bar(np.array(N_values) - 50, speedup_vec, width=80, alpha=0.7, label=‘Vectorized’) ax2.bar(np.array(N_values) + 50, speedup_numba, width=80, alpha=0.7, label=‘Numba’) ax2.set_xlabel(‘Number of grid points N’, fontsize=12) ax2.set_ylabel(‘Speedup ratio’, fontsize=12) ax2.set_title(‘Speedup ratio(compared to Naive)’, fontsize=14, fontweight=‘bold’) ax2.legend() ax2.grid(True, alpha=0.3, axis=‘y’) plt.tight_layout() return fig # Example usage optimizer = PerformanceOptimizer() N_vals, bench_results = optimizer.benchmark() fig = optimizer.plot_benchmark(N_vals, bench_results) plt.show() print(“\n=== Benchmark Results ===”) for N, t_naive, t_vec, t_numba in zip(N_vals, bench_results[‘naive’], bench_results[‘vectorized’], bench_results[‘numba’]): print(f”N={N}: Naive={t_naive:.4f}s, Vectorized={t_vec:.4f}s, ” f”Numba={t_numba:.4f}s”)

7. Comprehensive Exercise Project

Code Example7: Comprehensive Project

import numpy as np import matplotlib.pyplot as plt from matplotlib.animation import FuncAnimation class ComprehensiveProject: """Comprehensive Exercise Project: Molecular Dynamics Method practical application""" def init(self, Nx=150, Ny=150): self.Nx = Nx self.Ny = Ny self.x = np.linspace(0, 10, Nx) self.y = np.linspace(0, 10, Ny) self.X, self.Y = np.meshgrid(self.x, self.y) self.dx = self.x[1] - self.x[0] self.dy = self.y[1] - self.y[0] # State variables self.field = np.zeros((Nx, Ny)) self.auxiliary = np.zeros((Nx, Ny)) def initialize_complex_condition(self): """Complex initial condition""" # Superposition of multiple Gaussian distributions centers = [(2, 2), (5, 5), (8, 8), (2, 8), (8, 2)] for (cx, cy) in centers: self.field += np.exp(-((self.X - cx)2 + (self.Y - cy)2) / 0.3) # Adding noise self.field += 0.1 * np.random.randn(self.Nx, self.Ny) def coupled_evolution(self, dt): """Coupled evolution equation""" # Laplacian calculation lap_field = np.zeros_like(self.field) lap_field[1:-1, 1:-1] = ( (self.field[2:, 1:-1] - 2*self.field[1:-1, 1:-1] + self.field[:-2, 1:-1]) / self.dx2 + (self.field[1:-1, 2:] - 2*self.field[1:-1, 1:-1] + self.field[1:-1, :-2]) / self.dy2 ) # Nonlinear term nonlinear_term = self.field2 - self.field3 # Time evolution self.field += dt * (lap_field + nonlinear_term) # Boundary conditions（Neumann） self.field[0, :] = self.field[1, :] self.field[-1, :] = self.field[-2, :] self.field[:, 0] = self.field[:, 1] self.field[:, -1] = self.field[:, -2] def compute_statistics(self): """Calculate statistics""" mean = np.mean(self.field) std = np.std(self.field) total_energy = np.sum(self.field**2) * self.dx * self.dy return { ‘mean’: mean, ‘std’: std, ‘energy’: total_energy, ‘min’: np.min(self.field), ‘max’: np.max(self.field) } def run_simulation(self, T=5.0, dt=0.01): """Run simulation""" steps = int(T / dt) # Recording statistical information stats_history = [] for n in range(steps): self.coupled_evolution(dt) if n % 10 == 0: stats = self.compute_statistics() stats[‘time’] = n * dt stats_history.append(stats) print(f”Step {n}/{steps}: Energy={stats[‘energy’]:.4f}”) return stats_history def plot_final_results(self, stats_history): """Comprehensive plot of final results""" fig = plt.figure(figsize=(16, 12)) gs = fig.add_gridspec(3, 3, hspace=0.3, wspace=0.3) # Final field (contour) ax1 = fig.add_subplot(gs[0, :2]) im1 = ax1.contourf(self.X, self.Y, self.field, levels=30, cmap=‘RdBu_r’) ax1.set_xlabel(‘x’, fontsize=12) ax1.set_ylabel(‘y’, fontsize=12) ax1.set_title(‘Final field distribution’, fontsize=14, fontweight=‘bold’) plt.colorbar(im1, ax=ax1) # 3D surface ax2 = fig.add_subplot(gs[0, 2], projection=‘3d’) surf = ax2.plot_surface(self.X[::3, ::3], self.Y[::3, ::3], self.field[::3, ::3], cmap=‘viridis’) ax2.set_xlabel(‘x’) ax2.set_ylabel(‘y’) ax2.set_zlabel(‘field’) ax2.set_title(‘3D visualization’, fontsize=12, fontweight=‘bold’) # Energy time evolution ax3 = fig.add_subplot(gs[1, 0]) times = [s[‘time’] for s in stats_history] energies = [s[‘energy’] for s in stats_history] ax3.plot(times, energies, ‘b-’, linewidth=2) ax3.set_xlabel(‘Time t’, fontsize=12) ax3.set_ylabel(‘Total energy’, fontsize=12) ax3.set_title(‘Energy conservation’, fontsize=12, fontweight=‘bold’) ax3.grid(True, alpha=0.3) # Time evolution of statistics ax4 = fig.add_subplot(gs[1, 1]) means = [s[‘mean’] for s in stats_history] stds = [s[‘std’] for s in stats_history] ax4.plot(times, means, ‘r-’, linewidth=2, label=‘Mean’) ax4.plot(times, stds, ‘g-’, linewidth=2, label=‘Standard deviation’) ax4.set_xlabel(‘Time t’, fontsize=12) ax4.set_ylabel(‘Statistics’, fontsize=12) ax4.set_title(‘Statistical properties’, fontsize=12, fontweight=‘bold’) ax4.legend() ax4.grid(True, alpha=0.3) # Minimum/maximum values ax5 = fig.add_subplot(gs[1, 2]) mins = [s[‘min’] for s in stats_history] maxs = [s[‘max’] for s in stats_history] ax5.plot(times, mins, ‘b-’, linewidth=2, label=‘Minimum’) ax5.plot(times, maxs, ‘r-’, linewidth=2, label=‘Maximum’) ax5.set_xlabel(‘Time t’, fontsize=12) ax5.set_ylabel(‘Value’, fontsize=12) ax5.set_title(‘Minimum/maximum values’, fontsize=12, fontweight=‘bold’) ax5.legend() ax5.grid(True, alpha=0.3) # Histogram ax6 = fig.add_subplot(gs[2, 0]) ax6.hist(self.field.flatten(), bins=50, alpha=0.7, color=‘blue’) ax6.set_xlabel(‘ーValue’, fontsize=12) ax6.set_ylabel(‘Frequency’, fontsize=12) ax6.set_title(‘Value distribution’, fontsize=12, fontweight=‘bold’) ax6.grid(True, alpha=0.3, axis=‘y’) # cross-sectionprofile ax7 = fig.add_subplot(gs[2, 1]) mid_y = self.Ny // 2 ax7.plot(self.x, self.field[:, mid_y], ‘b-’, linewidth=2) ax7.set_xlabel(‘x’, fontsize=12) ax7.set_ylabel(‘field(x, y=5)’, fontsize=12) ax7.set_title(‘Central cross-section’, fontsize=12, fontweight=‘bold’) ax7.grid(True, alpha=0.3) # Power spectrum ax8 = fig.add_subplot(gs[2, 2]) fft_field = np.fft.fft2(self.field) power_spectrum = np.abs(fft_field)**2 power_spectrum_1d = np.mean(power_spectrum, axis=0) freq = np.fft.fftfreq(self.Nx, self.dx) ax8.loglog(freq[1:self.Nx//2], power_spectrum_1d[1:self.Nx//2], ‘b-’) ax8.set_xlabel(‘Wave number’, fontsize=12) ax8.set_ylabel(‘Power’, fontsize=12) ax8.set_title(‘Power spectrum’, fontsize=12, fontweight=‘bold’) ax8.grid(True, alpha=0.3) return fig # Main execution project = ComprehensiveProject(Nx=150, Ny=150) project.initialize_complex_condition() print(“Starting comprehensive simulation…”) stats_history = project.run_simulation(T=5.0, dt=0.01) fig = project.plot_final_results(stats_history) plt.show() print(“\n=== Simulation completed ===”) print(f”Final energy: {stats_history[-1][‘energy’]:.4f}”) print(f”Final mean: {stats_history[-1][‘mean’]:.4f}”) print(f”FinalStandard deviation: {stats_history[-1][‘std’]:.4f}“)

Exercises

Exercise1: Theoretical Understanding

Molecular Dynamics Method basic equations and explain their physical meaning. In particular, discuss the role of each term and the importance of boundary conditions.

Exercise2: Algorithm Implementation

Improve the algorithms learned in this chapter and create more accurate and faster implementations. Quantitatively evaluate stability conditions and computational errors.

Exercise3: Application Project

Choose a specific problem in materials science or physics and simulate it by applying Molecular Dynamics Method techniques Analyze the results and provide physical interpretations.

Summary

References

  1. Value
  2. Specialized books on computational physics and computational materials science
  3. Practical guides on scientific and technical computation using Python
  4. Performance OptimizationHPC

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