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Application in Statistical Mechanics: Study of Random Numbers and Time Series
Application in Statistical Mechanics
Study of Random Numbers and Time Series
Generate Different Variate from Uniform Variate
Coin Tossing Simulation
Nuclear Decay Simulation
1D Random Walk Simulation
2D Random Walk Simulation
Monte Carlo Method
Specific Heat of Solids (Einstein’s Theory)
Specific Heat of Solids (Debye’s Theory)
Plots of Different Distribution Functions
Study of Random Numbers and Time Series: Program 1
Introduction to numpy.random
import numpy as np import matplotlib.pyplot as plt # Generate a single random float number between 0 and 1 random_float = np.random.random() print("Random float between 0 and 1:", random_float, "\n") # Generate an array of 5 random floats between 0 and 1 random_floats_array = np.random.random(5) print("Array of 5 random floats between 0 and 1:", random_floats_array, "\n") # Generate a 2x3 array of random floats between 0 and 1 random_floats_2x3 = np.random.random((2, 3)) print("2x3 array of random floats between 0 and 1:", random_floats_2x3, "\n") # Generate a random integer between 0 and 10 (inclusive) random_integer = np.random.randint(0, 11) print("Random integer between 0 and 10:", random_integer, "\n") # Generate an array of 5 random integers between 0 and 10 (inclusive) random_integers_array = np.random.randint(0, 11, size=5) print("Array of 5 random integers between 0 and 10:", random_integers_array, "\n") # Generate a 2x3 array of random integers between 0 and 10 (inclusive) random_integers_2x3 = np.random.randint(0, 11, size=(2, 3)) print("2x3 array of random integers between 0 and 10:", random_integers_2x3, "\n") # Generate a random float from a normal distribution with mean 0 and standard deviation 1 normal_float = np.random.randn() print("Random float from normal distribution (mean 0, std 1):", normal_float, "\n") # Generate an array of 5 random floats from a normal distribution with mean 0 and standard deviation 1 normal_floats_array = np.random.randn(5) print("Array of 5 random floats from normal distribution (mean 0, std 1):", normal_floats_array, "\n") # Generate a 2x3 array of random floats from a normal distribution with mean 0 and standard deviation 1 normal_floats_2x3 = np.random.randn(2, 3) print("2x3 array of random floats from normal distribution (mean 0, std 1):", normal_floats_2x3, "\n") # Generate random samples from a given list sample_list = ['apple', 'banana', 'cherry'] random_sample = np.random.choice(sample_list) print("Random sample from list:", random_sample, "\n") # Generate 3 random samples from a given list with replacement random_samples_with_replacement = np.random.choice(sample_list, size=3, replace=True) print("3 random samples from list with replacement:", random_samples_with_replacement, "\n") # Generate 3 random samples from a given list without replacement random_samples_without_replacement = np.random.choice(sample_list, size=3, replace=False) print("3 random samples from list without replacement:", random_samples_without_replacement, "\n") # Shuffle a list in place arr = np.array([1, 2, 3, 4, 5]) np.random.shuffle(arr) shuffled_list = arr print("Shuffled list:", shuffled_list, "\n") # Generate a random float between a specified range (e.g., 5 to 10) random_float_range = np.random.uniform(5, 10) print("Random float between 5 and 10:", random_float_range, "\n") # Generate an array of 5 random floats between a specified range (e.g., 5 to 10) random_floats_range_array = np.random.uniform(5, 10, size=5) print("Array of 5 random floats between 5 and 10:", random_floats_range_array, "\n") # Generate a random integer from a Poisson distribution poisson_integer = np.random.poisson(lam=5) print("Random integer from Poisson distribution:", poisson_integer, "\n") # Generate an array of 5 random integers from a Poisson distribution poisson_integers_array = np.random.poisson(lam=5, size=5) print("Array of 5 random integers from Poisson distribution:", poisson_integers_array, "\n") # Generate random samples from a binomial distribution binomial_sample = np.random.binomial(n=10, p=0.5) print("Random sample from binomial distribution:", binomial_sample, "\n") # Generate an array of 5 random samples from a binomial distribution binomial_samples_array = np.random.binomial(n=10, p=0.5, size=5) print("Array of 5 random samples from binomial distribution:", binomial_samples_array, "\n") # Generate random numbers from a beta distribution beta_sample = np.random.beta(a=2, b=5) print("Random number from beta distribution:", beta_sample, "\n") # Generate an array of 5 random numbers from a beta distribution beta_samples_array = np.random.beta(a=2, b=5, size=5) print("Array of 5 random numbers from beta distribution:", beta_samples_array, "\n") # Generate random numbers from a gamma distribution gamma_sample = np.random.gamma(shape=2, scale=2) print("Random number from gamma distribution:", gamma_sample, "\n") # Generate an array of 5 random numbers from a gamma distribution gamma_samples_array = np.random.gamma(shape=2, scale=2, size=5) print("Array of 5 random numbers from gamma distribution:", gamma_samples_array, "\n") # Generate random samples from a chi-square distribution chi_square_sample = np.random.chisquare(df=2) print("Random sample from chi-square distribution:", chi_square_sample, "\n") # Generate an array of 5 random samples from a chi-square distribution chi_square_samples_array = np.random.chisquare(df=2, size=5) print("Array of 5 random samples from chi-square distribution:", chi_square_samples_array, "\n") # Generate random numbers from an exponential distribution exponential_sample = np.random.exponential(scale=1) print("Random number from exponential distribution:", exponential_sample, "\n") # Generate an array of 5 random numbers from an exponential distribution exponential_samples_array = np.random.exponential(scale=1, size=5) print("Array of 5 random numbers from exponential distribution:", exponential_samples_array, "\n") # Generate a random float from a log-normal distribution lognormal_sample = np.random.lognormal(mean=0, sigma=1) print("Random float from log-normal distribution:", lognormal_sample, "\n") # Generate an array of 5 random floats from a log-normal distribution lognormal_samples_array = np.random.lognormal(mean=0, sigma=1, size=5) print("Array of 5 random floats from log-normal distribution:", lognormal_samples_array, "\n") # Generate random samples from a standard normal distribution and compute their mean and variance normal_samples = np.random.randn(1000) mean_normal_samples = np.mean(normal_samples) variance_normal_samples = np.var(normal_samples) print(f"Mean of normal samples: {mean_normal_samples}, Variance of normal samples: {variance_normal_samples}", "\n") # Generate random integers from a specified range (inclusive) and size random_integers_range_size = np.random.randint(low=1, high=10, size=(3, 4)) print("Random integers from specified range and size:", random_integers_range_size, "\n") # Generate random floats from a uniform distribution in the range [0, 1) with size uniform_floats_array = np.random.random_sample(size=(2, 2)) print("Array of random floats from uniform distribution [0, 1):", uniform_floats_array, "\n") # Generate a random permutation of a given sequence random_permutation = np.random.permutation([1, 2, 3, 4, 5]) print("Random permutation of sequence:", random_permutation, "\n") # Generate random numbers from a multivariate normal distribution mean = [0, 0] cov = [[1, 0.5], [0.5, 1]] # covariance matrix multivariate_normal_samples = np.random.multivariate_normal(mean, cov, size=3) print("Random numbers from multivariate normal distribution:", multivariate_normal_samples, "\n") # Generate random numbers from a Dirichlet distribution alpha = [1.0, 1.0, 1.0] dirichlet_samples = np.random.dirichlet(alpha) print("Random numbers from Dirichlet distribution:", dirichlet_samples, "\n") # Generate random numbers from a Weibull distribution weibull_samples = np.random.weibull(a=1.5, size=5) print("Random numbers from Weibull distribution:", weibull_samples, "\n") # Generate random integers from a binomial distribution with varying number of trials binomial_integer_varying_trials = np.random.binomial(n=20, p=0.3) print("Random integer from binomial distribution with varying trials:", binomial_integer_varying_trials, "\n") # Generate a random float from a standard Cauchy distribution cauchy_float = np.random.standard_cauchy() print("Random float from standard Cauchy distribution:", cauchy_float, "\n") # Generate an array of random floats from a standard Cauchy distribution cauchy_floats_array = np.random.standard_cauchy(size=5) print("Array of random floats from standard Cauchy distribution:", cauchy_floats_array, "\n") # Generate random samples from a log-normal distribution lognormal_samples = np.random.lognormal(mean=0, sigma=1, size=5) print("Random samples from log-normal distribution:", lognormal_samples, "\n") # Generate random samples from a geometric distribution geometric_sample = np.random.geometric(p=0.5) print("Random sample from geometric distribution:", geometric_sample, "\n") # Generate an array of random samples from a geometric distribution geometric_samples_array = np.random.geometric(p=0.5, size=5) print("Array of random samples from geometric distribution:", geometric_samples_array, "\n") # Generate random integers from a multinomial distribution multinomial_samples = np.random.multinomial(n=10, pvals=[0.2, 0.3, 0.5]) print("Random integers from multinomial distribution:", multinomial_samples, "\n") # Generate random samples from a beta distribution and compute statistics beta_samples = np.random.beta(a=2, b=5, size=1000) mean_beta_samples = np.mean(beta_samples) variance_beta_samples = np.var(beta_samples) print(f"Mean of beta samples: {mean_beta_samples}, Variance of beta samples: {variance_beta_samples}", "\n") # Generate random samples from a normal distribution and plot a histogram normal_samples_for_histogram = np.random.randn(1000) plt.hist(normal_samples_for_histogram, bins=30, density=True) plt.title('Histogram of Normally Distributed Samples') plt.show() # Generate a 3x4 array of random floats from a normal distribution with a different mean and variance normal_floats_3x4 = np.random.normal(loc=10, scale=5, size=(3, 4)) print("3x4 array of random floats from normal distribution (mean 10, std 5):", normal_floats_3x4, "\n") # Generate random samples from a Pareto distribution pareto_samples = np.random.pareto(a=5, size=5) print("Random samples from Pareto distribution:", pareto_samples, "\n") # Generate random samples from a standard normal distribution and compute their quantiles normal_samples_for_quantiles = np.random.randn(1000) percentile_25 = np.percentile(normal_samples_for_quantiles, 25) percentile_75 = np.percentile(normal_samples_for_quantiles, 75) print(f"25th percentile of normal samples: {percentile_25}") print(f"75th percentile of normal samples: {percentile_75}", "\n") # Generate random numbers from a normal distribution with specified mean and variance mean_normal = 5 std_dev_normal = 2 normal_samples_custom = np.random.normal(loc=mean_normal, scale=std_dev_normal, size=5) print(f"Random numbers from normal distribution (mean {mean_normal}, std {std_dev_normal}):", normal_samples_custom, "\n") # Generate random samples from a uniform distribution over a specific interval uniform_samples_interval = np.random.uniform(low=10, high=20, size=5) print("Random samples from uniform distribution (10 to 20):", uniform_samples_interval, "\n")
Run Code
Output 1
Study of Random Numbers and Time Series: Program 2
Generation of uniform random number between (0,1) using LCG method
from numpy import * import matplotlib.pyplot as plt X = [] def u(x, size): for i in range(size): m, a, c = 17452351, 8547, 563221 x = (a*x + c) % m r = x/m X.append(r) return X plt.subplot(121) plt.hist(u(5421,10000), bins = 10, edgecolor = 'w') plt.subplot(122) plt.hist(u(12371,10000), bins = 10, edgecolor = 'w') plt.savefig('plot.png')
Run Code
Output 2
Study of Random Numbers and Time Series: Program 3
Create a random variate on the interval (a, b)
from numpy import * import matplotlib.pyplot as plt def udev(x): m,a,c= 1771875, 24126, 375641 x= (a*x+c)%m return x, x/m def udevs(x, p, q, size): X=[] for i in range(size): x,r=udev(x) y=p+(q-p)*r X.append(y) return(X) c= udevs(123, 1, 10, 10000) plt.hist(c, bins=50, edgecolor='w') plt.savefig('plot.png')
Run Code
Output 3
Study of Random Numbers and Time Series: Program 4
plot the autocorrelation of a random series
from numpy.random import * from numpy import * import matplotlib.pyplot as plt n = 10000 x = uniform(0,1,10000) xav = mean(x) def c(k): return mean([(x[i]-xav)*(x[i+k]-xav) for i in range(n-k)]) def r(k): return c(k)/c(0) #print(r(2)) corr = [r(k) for k in range(1, 1000)] plt.plot(corr) plt.savefig('plot.png')
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Output 4
Study of Random Numbers and Time Series: Program 5
statistical parameters such as mean, median, mode, and standard deviation for a random time series
import numpy as np from collections import Counter # Generate a random time series np.random.seed(42) # For reproducibility n = 100 # Length of the time series random_series = np.random.randn(n) # Calculate statistics mean = np.mean(random_series) median = np.median(random_series) mode = Counter(random_series).most_common(1)[0][0] # Find the most common value std_dev = np.std(random_series, ddof=0) # Population standard deviation variance = np.var(random_series, ddof=0) # Population variance skewness = np.mean(((random_series - mean) / std_dev) ** 3) # Sample skewness kurtosis = np.mean(((random_series - mean) / std_dev) ** 4) - 3 # Sample kurtosis (excess kurtosis) # Print results print("Mean:", mean) print("Median:", median) print("Mode:", mode) print("Standard Deviation:", std_dev) print("Variance:", variance) print("Skewness:", skewness) print("Kurtosis:", kurtosis)
Run Code
Output 5