دانلود کتاب Probability in physics. An introductory guide

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Aims of the Book......Page 7
An Overview of the Book......Page 8
Exercises......Page 9
Acknowledgements......Page 10
Contents......Page 11
Part I The Basics......Page 23
1.2 Where Does Unpredictability Come From?......Page 24
1.2.4 Open Systems......Page 25
1.3 Randomness and Probability......Page 26
1.4.1 Frequencies......Page 27
1.4.3 Continuous Random Variables: Frequency Densities......Page 29
1.4.4 The Importance of Relative and Integrated Probabilities......Page 30
1.5.1 Probability of A A A A and B B B B......Page 31
1.5.4 Probability of A A A A or B B B B......Page 32
1.6 Probability as Degree of Belief......Page 34
1.7 Look Ahead......Page 35
1.8 Key Concepts......Page 36
1.10 Exercises......Page 37
References......Page 39
2.2 Sample Versus Population Distributions......Page 40
2.2.2 Sampling a Continuous Population Distribution: Binning......Page 42
2.3 Multi-variate Distributions......Page 43
2.3.1 Conditional Distributions......Page 44
2.3.2 Marginal Distributions......Page 46
2.4 Summarising Quantities for Distributions......Page 47
2.4.2 Measures of Dispersion......Page 48
2.4.4 Summarising Quantities for Multivariate Distributions......Page 49
2.5.1 Expectation Values......Page 50
2.5.3 Moments of a Distribution......Page 51
2.5.5 Higher Moments: Skewness and Kurtosis......Page 52
2.6.1 Transformation of Variance: Univariate Case......Page 53
2.6.2 Transformation of Variance: Bivariate Case......Page 54
2.7.1 Types of Error......Page 55
2.7.2 Evaluating Errors......Page 56
2.8 Key Concepts......Page 57
2.10 Exercises......Page 58
References......Page 59
Part II Frequency Distributions in the Physical World......Page 60
3.2 Balls, Slots, Boxes, and Labels......Page 61
3.4 Arrangements or Permutations......Page 62
3.4.1 Arranging r r r r Things Out of n n n n......Page 63
3.5 Subsets or Combinations......Page 64
3.6.1 Two-Box Problems......Page 65
3.6.3 A First Look at the Second Law of Thermodynamics......Page 66
3.7 Multi-box Partitioning......Page 67
3.7.1 Maximising Multiplicity for a Multi-box System......Page 68
3.8.1 Particle Energy Distributions......Page 69
3.9 Key Concepts......Page 71
3.11 Exercises......Page 72
References......Page 73
4.2.1 The Simplest Hit-and-Miss Problem......Page 74
4.2.2 Varying the Hit-Probability......Page 75
4.3.1 Mean Value of the Binomial Distribution......Page 76
4.3.2 Variance of the Binomial Distribution......Page 77
4.3.3 Shape of the Binomial Distribution......Page 78
4.4.2 The Hypergeometric Distribution: Binomial Without Relacement......Page 79
4.5 Counting Rare Things: The Poisson Distribution......Page 80
4.6 Derivation of the Poisson Distribution......Page 81
4.7.2 Shape of the Poisson Distribution......Page 82
4.8.1 Why Large Samples Are Better......Page 83
4.8.3 Cautions......Page 84
4.10 Further Reading......Page 85
References......Page 86
5.1 Outline of Content......Page 88
5.2 Rescaling to Reveal a Universal Distribution......Page 90
5.3.1 Moments of the Gaussian......Page 92
5.3.2 Integrated Probability Within Standard Regions......Page 93
5.5 Mathematical Origin of the Gaussian: Adding Random Variables......Page 94
5.5.1 Integration over the Half-Plane......Page 95
5.5.2 Understanding Convolutions......Page 96
5.6.1 Convolving the Gaussian......Page 97
5.7.1 Limiting Form of the Binomial Distribution......Page 99
5.7.2 Limiting Form of the Poisson Distribution......Page 100
5.8 Multi-variate Gaussians......Page 101
5.8.1 Bivariate or 2D Gaussian......Page 102
5.8.3 2D Gaussian: More General Case......Page 103
5.9 Gaussians in Disguise......Page 104
5.9.2 Stellar Masses: An Example of a Log-Gaussian......Page 105
5.10.1 Poisson Errors......Page 107
5.10.4 Propagation of Gaussian Errors......Page 108
5.12 Further Reading......Page 109
5.13 Exercises......Page 110
References......Page 111
6.2 Random Walks......Page 112
6.2.1 1D Random Walk......Page 113
6.2.2 3D Random Walk......Page 114
6.2.4 Random Walks in Momentum Space......Page 115
6.3.1 Distribution of Events Versus Time......Page 116
6.3.3 Waiting Time Distribution......Page 117
6.4.1 Moments of the Cauchy Distribution......Page 118
6.4.3 Angular-to-Linear Transformations: The Rotating Gun Problem......Page 120
6.4.4 Resonance and the Lorentzian......Page 121
6.4.5 Comparison to Gaussian......Page 122
6.5 Power Law Tails......Page 123
6.5.1 Normalising a Power Law Distribution: Power Law Tail Approximations......Page 124
6.5.2 Moments of a Power Law Distribution......Page 125
6.5.4 Power Laws from Inverse Quantitities......Page 126
6.5.5 Power Laws from Growth and Survival Processes......Page 127
6.6 Key Concepts......Page 128
6.8 Exercises......Page 129
References......Page 131
Part III Probabilistic Inference: Reasoning in the Presence of Uncertainty......Page 132
7.1 Outline of Content......Page 134
7.2.1 Comparing Hypotheses Using Relative Likelihoods......Page 135
7.2.2 Assessing a Single Hypothesis Using Absolute Likelihood......Page 136
7.3.1 Bayes's Formula for Hypothesis Probabilities (Credibilities)......Page 137
7.3.2 Assigning Priors......Page 138
7.4.1 Testing Spin State Theories......Page 139
7.4.2 Particle Mass Test......Page 140
7.6 Confidence/Significance Testing: P-Values......Page 141
7.6.2 One-Tailed Versus Two Tailed Tests......Page 142
7.7.2 The Flaring Star......Page 144
7.8 Pitfalls of Using P-Values......Page 145
7.9 Comparing Bayesian and Confidence Testing Results......Page 146
7.10.1 Bayesian Inference with Compound Datasets......Page 147
7.10.2 Significance Testing with Compound Datasets: Test Statistics......Page 148
7.11 The χ2 Test Statistic......Page 149
7.11.2 Distribution of 2......Page 150
7.12 Using χ2......Page 151
7.13 Key Concepts......Page 152
7.15 Exercises......Page 153
References......Page 155
8.1 Outline of Content......Page 156
8.3 Methods Using Sample Mean and Variance......Page 157
8.3.1 Bias in Sample Variance......Page 158
8.3.2 Error on Mean from Sampling Distribution......Page 159
8.3.4 Confidence Intervals for the Mean: Unknown σ Case......Page 160
8.4 Maximum Likelihood Estimates......Page 161
8.4.1 Maximum Likelihood Solution for Mean and Variance......Page 162
8.4.2 Weighted Mean Estimate......Page 163
8.5 Maximum Posterior Estimates......Page 164
8.5.2 Gaussian Approximation to P(θ)......Page 165
8.5.4 Maximum Posterior Worked Example......Page 167
8.6.1 Relation Between 2 and Likelihood......Page 170
8.6.4 Confidence Intervals Based on 2 and Δ2......Page 171
8.6.5 Minimum 2 Worked Example......Page 172
8.7 Key Concepts......Page 173
8.8 Further Reading......Page 174
8.9 Exercises......Page 175
References......Page 176
9.1 Outline of Content......Page 178
9.2 Bivariate Data Sets and Their Origins......Page 179
9.3.1 Dependence: Shape Change......Page 180
9.3.2 Correlation: Systematic Drift......Page 181
9.4 Quantifying Correlation: Covariance......Page 182
9.4.1 Correlation Coefficient......Page 183
9.5 Testing for Correlation......Page 184
9.5.2 Bayesian Correlation Testing......Page 186
9.5.3 Rank Correlation Tests......Page 187
9.6 Worked Example......Page 188
9.7 Correlation Test Pitfalls......Page 189
9.8.1 The Method of Least Squares......Page 190
9.9 Least Squares Fit to a Straight Line......Page 191
9.9.1 Least Squares for Data Points with Individual Errors......Page 192
9.10.1 Errors on Fitted Parameters......Page 193
9.11 Curvilinear Line Fitting......Page 195
9.13 Key Concepts......Page 197
9.14 Further Reading......Page 198
9.15 Exercises......Page 199
References......Page 200
10.1 Outline of Content......Page 201
10.2.2 The Model......Page 202
10.2.3 The Experiment Model......Page 203
10.3.1 The Data and Its Errors......Page 204
10.3.2 The Physical Model......Page 206
10.3.3 The Calculation......Page 207
10.4 Techniques for Finding the Best Parameter Set......Page 208
10.4.3 Sub-grid Maximum Location......Page 209
10.5.1 The Gaussian Approximation......Page 210
10.5.3 The Case of Two Interesting But Uncorrelated Parameters......Page 211
10.5.5 Finding 2 Confidence Intervals......Page 212
10.5.6 Modelling with Many Parameters......Page 213
10.5.7 Limits of the Gaussian Approximation......Page 214
10.6 Goodness of Fit......Page 215
10.7.1 Model Comparison Using Δ2......Page 216
10.7.3 Likelihood Ratio Test......Page 217
10.7.4 Bayesian Model Comparison: The Bayes Factor......Page 218
10.8 Artificial Intelligence Techniques......Page 219
10.10 Further Reading......Page 221
10.11 Exercises......Page 222
References......Page 223
Part IV Selected Topics......Page 225
11.1 Outline of Content......Page 226
11.2.1 A Measure of Uncertainty/Information......Page 227
11.2.3 The Units of Uncertainty/Information......Page 228
11.2.5 Information Theory Terminology......Page 229
11.3 Uncertainty Inherent in Probability Distributions......Page 230
11.3.2 Multi-box Experiments......Page 231
11.3.4 Poisson Distribution......Page 233
11.3.5 Bivariate Distributions......Page 234
11.3.6 Mutual Information: Testing Dependence......Page 235
11.3.7 Continuous Distributions......Page 236
11.3.8 Comparing Distributions......Page 237
11.4 Messages, Alphabets, and Data Compression......Page 238
11.4.2 Codes, Channels, Compression and Efficiency......Page 239
11.4.3 Messages with Mutual Information......Page 240
11.5.1 Shannon Entropy and Macrostate Multiplicity......Page 241
11.5.2 Using Uncertainty/Entropy to Choose Priors......Page 242
11.5.3 Image Restoration......Page 243
11.6 Key Concepts......Page 245
11.7 Further Reading......Page 246
11.8 Exercises......Page 247
References......Page 248
12.1 Outline of Content......Page 249
12.2 Characterising Time Series......Page 250
12.2.1 Stationary and Non-stationary Processes......Page 251
12.2.2 Growth of Variance and the Structure Function......Page 252
12.2.3 Autocorrelation......Page 253
12.2.4 The Periodogram......Page 256
12.3.1 Purely Random Processes......Page 257
12.3.2 Moving Average (MA) Processes......Page 258
12.3.3 Useful Filters......Page 259
12.3.4 Autoregressive (AR) Processes......Page 260
12.3.6 Wiener, Cauchy and Poisson Processes: Shot Noise......Page 263
12.3.7 Other Variations......Page 264
12.4.1 The Wiener Process as an Approximation......Page 265
12.4.2 Solving Difference Equations......Page 267
12.4.3 The Ornstein–Uhlenbeck Process......Page 268
12.4.4 Other SDEs......Page 269
12.5 Markov Chains......Page 270
12.7 Further Reading......Page 272
12.8 Exercises......Page 274
References......Page 275
13.2 Unpredictability in the Classical and Quantum Worlds......Page 276
13.2.1 Incomplete Knowledge and Observer Dependence......Page 277
13.3.2 Unpredictability in Radioactivity......Page 278
13.3.3 Random Number Applications......Page 279
13.4.1 The Bullet Version of the Two-Slit Experiment......Page 281
13.4.2 The Wave Version of the Two-Slit Experiment......Page 282
13.4.3 The Quantum Version of the Two-Slit Experiment......Page 283
13.5 The Strange Behaviour of Measurement......Page 284
13.5.2 The Uncertainty Principle......Page 285
13.5.4 Indeterminacy and State Mixtures......Page 286
13.6.1 Normal Entanglement and Quantum Entanglement......Page 287
13.6.3 The EPR Paradox and Hidden Variables......Page 288
13.7 Statistical Tests of Hidden Variable Theories......Page 289
13.7.1 Consistency Tests for Trivariate Yes/No Distributions......Page 290
13.7.2 Application to Entangled Spin Experiments......Page 292
13.8.2 Bohmian Mechanics......Page 294
13.8.4 Quantum Bayesianism......Page 295
13.10 Further Reading......Page 296
13.11 Exercises......Page 297
References......Page 299
14.1 Outline of Content......Page 300
14.2.1 Entropy as a Thermodynamic State Variable......Page 301
14.2.2 Example Entropy Calculation......Page 302
14.2.4 The Idea of Free Energy......Page 303
14.3.1 Boltzmann Entropy......Page 304
14.3.3 Counting Microstates......Page 306
14.3.4 Equivalence of Statistical and Thermodynamic Entropy......Page 307
14.3.5 Equilibrium and Non-equilibrium States......Page 308
14.4.1 The Ergodic Assumption and Relaxation Times......Page 309
14.4.2 Subjectivity of Entropy Values......Page 311
14.4.3 Order, Disorder, and Structure......Page 313
14.4.4 Complexity......Page 314
14.4.5 Entropy in Gravitating Systems......Page 315
14.4.6 Black Hole Thermodynamics......Page 317
14.5.2 Monotonicity of Time......Page 318
14.5.4 The Importance of Initial Conditions......Page 319
14.5.5 Cosmological Arrow of Time......Page 320
14.6 Key Concepts......Page 321
14.7 Further Reading......Page 322
14.8 Exercises......Page 323
References......Page 324
Solutions......Page 325
Index......Page 357