├── .gitignore
├── LICENSE
├── Lorenz 1 Attractors.ipynb
├── Lorenz 2 Three Dimensions.ipynb
├── Lorenz 3 FFT.ipynb
├── Lorenz 4 Wavelets.ipynb
├── README.md
├── Statistical Mechanics 1 Quantum Dice.ipynb
├── Statistical Mechanics 2 Probability Distributions.ipynb
├── Statistical Mechanics 3 Six Degrees of Separation.ipynb
├── Statistical Mechanics 4 Random Walks.ipynb
├── Statistical Mechanics 5 Brownian Diffusion and Langevin Equation.ipynb
├── Statistical Mechanics 6 MSD to Diffusion Coefficient.ipynb
├── brownian.py
└── lorenz_rhs.py


/.gitignore:
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102 | 


--------------------------------------------------------------------------------
/LICENSE:
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 1 | MIT License
 2 | 
 3 | Copyright (c) 2017 Chaz Reid
 4 | 
 5 | Permission is hereby granted, free of charge, to any person obtaining a copy
 6 | of this software and associated documentation files (the "Software"), to deal
 7 | in the Software without restriction, including without limitation the rights
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10 | furnished to do so, subject to the following conditions:
11 | 
12 | The above copyright notice and this permission notice shall be included in all
13 | copies or substantial portions of the Software.
14 | 
15 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
16 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
17 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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21 | SOFTWARE.
22 | 


--------------------------------------------------------------------------------
/README.md:
--------------------------------------------------------------------------------
1 | # statistical-mechanics
2 | 
3 |  Lorenz attractors, statistical mechanics, nonlinear dynamical systems, computational physics. 
4 | 


--------------------------------------------------------------------------------
/Statistical Mechanics 1 Quantum Dice.ipynb:
--------------------------------------------------------------------------------
  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "markdown",
  5 |    "metadata": {},
  6 |    "source": [
  7 |     "# Quantum Dice\n",
  8 |     "\n",
  9 |     "You are given several unusual 3-sided dice which, when rolled, show either 1, 2, or 3 spots.\n",
 10 |     "\n",
 11 |     "There are three games played iwth these dice: *Distinguishable*, *Bosons*, and *Fermioons*. In each of htese games, the player rolls one die at a time, starting over if requird by the rules, until a legal combination occurs. In *Distinguishable*, all rolls are legal. In *Bosons*, a roll is legal only if the new number is larger or equal to the preceding number. In *Fermions*, a roll is legal only if hte new number is strictly larger than the preceding number.\n",
 12 |     "\n",
 13 |     "See figure for table of possibilities after rolling two dice.\n",
 14 |     "\n",
 15 |     "Our dice rules ar ethe same ones that govern quantum statistics of identical particles."
 16 |    ]
 17 |   },
 18 |   {
 19 |    "cell_type": "code",
 20 |    "execution_count": 1,
 21 |    "metadata": {},
 22 |    "outputs": [],
 23 |    "source": [
 24 |     "import numpy as np\n",
 25 |     "import matplotlib.pyplot as plt\n",
 26 |     "import seaborn as sns\n",
 27 |     "import pandas as pd"
 28 |    ]
 29 |   },
 30 |   {
 31 |    "cell_type": "code",
 32 |    "execution_count": 2,
 33 |    "metadata": {},
 34 |    "outputs": [],
 35 |    "source": [
 36 |     "from numpy.random import randint\n",
 37 |     "from numpy.random import seed"
 38 |    ]
 39 |   },
 40 |   {
 41 |    "cell_type": "code",
 42 |    "execution_count": 3,
 43 |    "metadata": {},
 44 |    "outputs": [
 45 |     {
 46 |      "name": "stdout",
 47 |      "output_type": "stream",
 48 |      "text": [
 49 |       "Should see 3 True values:\n",
 50 |       "True\n",
 51 |       "True\n",
 52 |       "True\n"
 53 |      ]
 54 |     }
 55 |    ],
 56 |    "source": [
 57 |     "# Define classification functions\n",
 58 |     "\n",
 59 |     "def roll_is_dist(roll):\n",
 60 |     "    \"\"\"Return a boolean: can this roll outcome be classified as 'distinguishable'\n",
 61 |     "    \"\"\"\n",
 62 |     "    return True\n",
 63 |     "\n",
 64 |     "def roll_is_ferm(roll):\n",
 65 |     "    \"\"\"Return a boolean: can this roll outcome be classified as 'fermi'\n",
 66 |     "    \"\"\"\n",
 67 |     "    ndice = len(roll)\n",
 68 |     "    is_fermi = True\n",
 69 |     "    # Ensure that roll j+1 >= roll j\n",
 70 |     "    for j in range(len(roll)-1):\n",
 71 |     "        if(roll[j+1] <= roll[j]):\n",
 72 |     "            is_fermi = False\n",
 73 |     "            break\n",
 74 |     "    return is_fermi\n",
 75 |     "\n",
 76 |     "def roll_is_bos(roll):\n",
 77 |     "    \"\"\"Return a boolean: can this roll outcome be classified as 'boson'\n",
 78 |     "    \"\"\"\n",
 79 |     "    ndice = len(roll)\n",
 80 |     "    is_bos = True\n",
 81 |     "    # Ensure that roll j+1 > roll j\n",
 82 |     "    for j in range(len(roll)-1):\n",
 83 |     "        if(roll[j+1] < roll[j]):\n",
 84 |     "            is_bos = False\n",
 85 |     "            break\n",
 86 |     "    return is_bos\n",
 87 |     "\n",
 88 |     "print(\"Should see 3 True values:\")\n",
 89 |     "print(roll_is_dist(np.array([3,2,1])))\n",
 90 |     "print(roll_is_bos(np.array([1,1,2])))\n",
 91 |     "print(roll_is_ferm(np.array([1,2,3])))"
 92 |    ]
 93 |   },
 94 |   {
 95 |    "cell_type": "code",
 96 |    "execution_count": 4,
 97 |    "metadata": {},
 98 |    "outputs": [],
 99 |    "source": [
100 |     "# Define roll functions\n",
101 |     "\n",
102 |     "def quantum_rolls(n, ndice, rolltype):\n",
103 |     "    \"\"\"Roll ndice dice n times the game rolltype\n",
104 |     "    \"\"\"\n",
105 |     "    if(rolltype is 'dist' or rolltype is 'distinguishable'):\n",
106 |     "        return dist_rolls(n, ndice)\n",
107 |     "    elif(rolltype is 'ferm' or rolltype is 'fermion'):\n",
108 |     "        return ferm_rolls(n, ndice)\n",
109 |     "    elif(rolltype is 'bos' or rolltype is 'boson'):\n",
110 |     "        return bos_rolls(n, ndice)\n",
111 |     "    else:\n",
112 |     "        raise Exception(\"Roll type not recognized: %s\"%(rolltype))\n",
113 |     "    \n",
114 |     "def dist_rolls(n, ndice):\n",
115 |     "    \"\"\"Roll ndice dice n times for the game 'distinguishable'\n",
116 |     "    \"\"\"\n",
117 |     "    return randint(1, 3+1, size=(n,ndice))\n",
118 |     "\n",
119 |     "def ferm_rolls(n, ndice):\n",
120 |     "    \"\"\"Roll ndice dice n times for the game 'fermi'\n",
121 |     "    \"\"\"\n",
122 |     "    if(ndice>3):\n",
123 |     "        raise Exception(\"Error: max 3 dice, you specified %d\"%(ndice))\n",
124 |     "    rolls = []\n",
125 |     "    for i in range(n):\n",
126 |     "        # Keep rolling until we get a 'fermi' roll\n",
127 |     "        roll = randint(1, 3+1, size=(ndice,))\n",
128 |     "        while(roll_is_ferm(roll) is False):\n",
129 |     "            roll = randint(1, 3+1, size=(ndice,))\n",
130 |     "        # Add it to list of rolls results\n",
131 |     "        rolls.append(roll)\n",
132 |     "    rolls = np.array(rolls)\n",
133 |     "    return rolls\n",
134 |     "\n",
135 |     "def bos_rolls(n, ndice):\n",
136 |     "    \"\"\"Roll ndice dice n times for the game 'boson'\n",
137 |     "    \"\"\"\n",
138 |     "    rolls = []\n",
139 |     "    for i in range(n):\n",
140 |     "        # Keep rolling until we get a 'boson' roll\n",
141 |     "        roll = randint(1, 3+1, size=(ndice,))\n",
142 |     "        while(roll_is_bos(roll) is False):\n",
143 |     "            roll = randint(1, 3+1, size=(ndice,))\n",
144 |     "        # Add it to the list of rolls results\n",
145 |     "        rolls.append(roll)\n",
146 |     "    rolls = np.array(rolls)\n",
147 |     "    return rolls"
148 |    ]
149 |   },
150 |   {
151 |    "cell_type": "code",
152 |    "execution_count": 5,
153 |    "metadata": {
154 |     "scrolled": true
155 |    },
156 |    "outputs": [
157 |     {
158 |      "data": {
159 |       "text/plain": [
160 |        "array([[2, 2, 3],\n",
161 |        "       [1, 2, 1],\n",
162 |        "       [3, 3, 1],\n",
163 |        "       [1, 2, 1],\n",
164 |        "       [3, 2, 1]])"
165 |       ]
166 |      },
167 |      "execution_count": 5,
168 |      "metadata": {},
169 |      "output_type": "execute_result"
170 |     }
171 |    ],
172 |    "source": [
173 |     "quantum_rolls(5, 3, 'dist')"
174 |    ]
175 |   },
176 |   {
177 |    "cell_type": "code",
178 |    "execution_count": 6,
179 |    "metadata": {},
180 |    "outputs": [
181 |     {
182 |      "data": {
183 |       "text/plain": [
184 |        "array([[2, 2, 2],\n",
185 |        "       [1, 2, 2],\n",
186 |        "       [1, 1, 3],\n",
187 |        "       [1, 1, 2],\n",
188 |        "       [3, 3, 3]])"
189 |       ]
190 |      },
191 |      "execution_count": 6,
192 |      "metadata": {},
193 |      "output_type": "execute_result"
194 |     }
195 |    ],
196 |    "source": [
197 |     "quantum_rolls(5, 3, 'bos')"
198 |    ]
199 |   },
200 |   {
201 |    "cell_type": "code",
202 |    "execution_count": 7,
203 |    "metadata": {},
204 |    "outputs": [
205 |     {
206 |      "data": {
207 |       "text/plain": [
208 |        "array([[2, 3],\n",
209 |        "       [1, 3],\n",
210 |        "       [1, 2],\n",
211 |        "       [2, 3],\n",
212 |        "       [1, 2]])"
213 |       ]
214 |      },
215 |      "execution_count": 7,
216 |      "metadata": {},
217 |      "output_type": "execute_result"
218 |     }
219 |    ],
220 |    "source": [
221 |     "quantum_rolls(5, 2, 'ferm')"
222 |    ]
223 |   },
224 |   {
225 |    "cell_type": "code",
226 |    "execution_count": 33,
227 |    "metadata": {},
228 |    "outputs": [],
229 |    "source": [
230 |     "# Define score functions\n",
231 |     "\n",
232 |     "def get_outcomes():\n",
233 |     "    return np.array([[2,3,4],[3,4,5],[4,5,6]])\n",
234 |     "\n",
235 |     "def roll_dice(n):\n",
236 |     "    return randint(1,3+1,size=(n,3))\n",
237 |     "\n",
238 |     "def dist_score_outcomes(roll1,roll2):\n",
239 |     "    return get_outcomes()[roll1-1][roll2-1]\n",
240 |     "\n",
241 |     "def bos_score_outcomes(roll1,roll2):\n",
242 |     "    if(roll1 <= roll2):\n",
243 |     "        return get_outcomes()[roll1-1][roll2-1]\n",
244 |     "    else:\n",
245 |     "        return 0\n",
246 |     "\n",
247 |     "def ferm_score_outcomes(roll1,roll2):\n",
248 |     "    if(roll1 < roll2):\n",
249 |     "        return get_outcomes()[roll1-1][roll2-1]\n",
250 |     "    else:\n",
251 |     "        return 0\n",
252 |     "\n",
253 |     "def score_rolls(rolls, rolltype):\n",
254 |     "    if(rolltype is 'dist' or rolltype is 'distinguishable'):\n",
255 |     "        score_outcomes = dist_score_outcomes\n",
256 |     "    elif(rolltype is 'bos' or rolltype is 'boson'):\n",
257 |     "        score_outcomes = bos_score_outcomes\n",
258 |     "    elif(rolltype is 'ferm' or rolltype is 'fermion'):\n",
259 |     "        score_outcomes = ferm_score_outcomes\n",
260 |     "    else:\n",
261 |     "        raise Exception(\"Roll type not recognized: %s\"%(rolltype))\n",
262 |     "    \n",
263 |     "    total_score = np.zeros(len(rolls),)\n",
264 |     "    for i in range(len(rolls)):\n",
265 |     "        roll = rolls[i,:]\n",
266 |     "        for j in range(len(roll)-1):\n",
267 |     "            total_score[i] += score_outcomes(roll[j], roll[j+1])\n",
268 |     "\n",
269 |     "    return total_score"
270 |    ]
271 |   },
272 |   {
273 |    "cell_type": "code",
274 |    "execution_count": 34,
275 |    "metadata": {},
276 |    "outputs": [
277 |     {
278 |      "name": "stdout",
279 |      "output_type": "stream",
280 |      "text": [
281 |       "[ 11.  10.   5.  10.  12.]\n"
282 |      ]
283 |     }
284 |    ],
285 |    "source": [
286 |     "t = 'dist'\n",
287 |     "r = quantum_rolls(5, 3, t)\n",
288 |     "print(score_rolls(r, t))"
289 |    ]
290 |   },
291 |   {
292 |    "cell_type": "code",
293 |    "execution_count": 35,
294 |    "metadata": {},
295 |    "outputs": [
296 |     {
297 |      "name": "stdout",
298 |      "output_type": "stream",
299 |      "text": [
300 |       "[  8.   6.   5.   6.  10.]\n"
301 |      ]
302 |     }
303 |    ],
304 |    "source": [
305 |     "t = 'bos'\n",
306 |     "r = quantum_rolls(5, 3, t)\n",
307 |     "print(score_rolls(r, t))"
308 |    ]
309 |   },
310 |   {
311 |    "cell_type": "code",
312 |    "execution_count": 36,
313 |    "metadata": {},
314 |    "outputs": [
315 |     {
316 |      "name": "stdout",
317 |      "output_type": "stream",
318 |      "text": [
319 |       "[ 8.  8.  8.  8.  8.]\n"
320 |      ]
321 |     }
322 |    ],
323 |    "source": [
324 |     "t = 'ferm'\n",
325 |     "r = quantum_rolls(5, 3, t)\n",
326 |     "print(score_rolls(r, t))"
327 |    ]
328 |   },
329 |   {
330 |    "cell_type": "code",
331 |    "execution_count": 37,
332 |    "metadata": {},
333 |    "outputs": [
334 |     {
335 |      "name": "stdout",
336 |      "output_type": "stream",
337 |      "text": [
338 |       "[ 3.  4.  5.  4.  5.]\n"
339 |      ]
340 |     }
341 |    ],
342 |    "source": [
343 |     "t = 'dist'\n",
344 |     "r = quantum_rolls(5, 2, t)\n",
345 |     "print(score_rolls(r, t))"
346 |    ]
347 |   },
348 |   {
349 |    "cell_type": "markdown",
350 |    "metadata": {},
351 |    "source": [
352 |     "## Part A\n",
353 |     "\n",
354 |     "Presuming fair dice (each of the faces has a 1/3 chance of appearing)...\n",
355 |     "\n",
356 |     "For a legal turn rolling two dice in Bosons, what is probability $\\rho(4)$ of rolling a 4?\n",
357 |     "\n",
358 |     "For a legal turn rolling two dice in Fermion, what is probability $\\rho(4)$ of rolling a 4?"
359 |    ]
360 |   },
361 |   {
362 |    "cell_type": "code",
363 |    "execution_count": 38,
364 |    "metadata": {},
365 |    "outputs": [
366 |     {
367 |      "name": "stdout",
368 |      "output_type": "stream",
369 |      "text": [
370 |       "Rho(4) for Boson: 2 / 6 = 0.3333\n",
371 |       "Rho(4) for Fermion: 1 / 3 = 0.3333\n"
372 |      ]
373 |     }
374 |    ],
375 |    "source": [
376 |     "# Analytical calculation:\n",
377 |     "\n",
378 |     "o = get_outcomes()\n",
379 |     "\n",
380 |     "bos_n4 = 2\n",
381 |     "ferm_n4 = 1\n",
382 |     "\n",
383 |     "bos_sum = 0\n",
384 |     "ferm_sum = 0\n",
385 |     "for i in range(o.shape[0]):\n",
386 |     "    for j in range(o.shape[1]):\n",
387 |     "        if(i <= j):\n",
388 |     "            bos_sum += 1\n",
389 |     "        if(i < j):\n",
390 |     "            ferm_sum += 1\n",
391 |     "\n",
392 |     "print(\"Rho(4) for Boson: %d / %d = %0.4f\"%(bos_n4, bos_sum, bos_n4/bos_sum))\n",
393 |     "print(\"Rho(4) for Fermion: %d / %d = %0.4f\"%(ferm_n4, ferm_sum, ferm_n4/ferm_sum))"
394 |    ]
395 |   },
396 |   {
397 |    "cell_type": "code",
398 |    "execution_count": 39,
399 |    "metadata": {},
400 |    "outputs": [
401 |     {
402 |      "name": "stdout",
403 |      "output_type": "stream",
404 |      "text": [
405 |       "Rho(4) for Boson: 3409 / 10000 = 0.3409\n",
406 |       "Rho(4) for Boson: 3403 / 10000 = 0.3403\n"
407 |      ]
408 |     }
409 |    ],
410 |    "source": [
411 |     "# Computational verification:\n",
412 |     "\n",
413 |     "total = 10000\n",
414 |     "\n",
415 |     "rf = quantum_rolls(total, 2, 'ferm')\n",
416 |     "rb = quantum_rolls(total, 2, 'bos')\n",
417 |     "\n",
418 |     "def is_four(roll):\n",
419 |     "    if(roll[0]+roll[1]==4):\n",
420 |     "        return True\n",
421 |     "    return False\n",
422 |     "\n",
423 |     "bos_n4_comp = np.sum(np.apply_along_axis(arr=rb, func1d=is_four, axis=1))\n",
424 |     "print(\"Rho(4) for Boson: %d / %d = %0.4f\"%(bos_n4_comp, total, bos_n4_comp/total))\n",
425 |     "\n",
426 |     "ferm_n4_comp = np.sum(np.apply_along_axis(arr=rf, func1d=is_four, axis=1))\n",
427 |     "print(\"Rho(4) for Boson: %d / %d = %0.4f\"%(ferm_n4_comp, total, ferm_n4_comp/total))"
428 |    ]
429 |   },
430 |   {
431 |    "cell_type": "markdown",
432 |    "metadata": {},
433 |    "source": [
434 |     "## Part B\n",
435 |     "\n",
436 |     "For a legal turn rolling 3 different 3-sided dice in Fermions, what is probability $\\rho(6)$ of rolling a 6?\n",
437 |     "\n",
438 |     "(The answer is trivial: the only 3-dice outcome for \"Fermions\" is a 6.)"
439 |    ]
440 |   },
441 |   {
442 |    "cell_type": "markdown",
443 |    "metadata": {},
444 |    "source": [
445 |     "## Part C\n",
446 |     "\n",
447 |     "In a turn of three rolls, what is the factor by which the probability of getting triples in \"Bosons\" is enhanced over that in \"Distinguishable\"?"
448 |    ]
449 |   },
450 |   {
451 |    "cell_type": "code",
452 |    "execution_count": 54,
453 |    "metadata": {},
454 |    "outputs": [],
455 |    "source": [
456 |     "def count_identical(rolll):\n",
457 |     "    identical = 0\n",
458 |     "    for i in range(len(rolll)):\n",
459 |     "        if(len(np.unique(rolll[i][:]))==1):\n",
460 |     "            identical += 1\n",
461 |     "    return identical"
462 |    ]
463 |   },
464 |   {
465 |    "cell_type": "code",
466 |    "execution_count": 55,
467 |    "metadata": {},
468 |    "outputs": [
469 |     {
470 |      "name": "stdout",
471 |      "output_type": "stream",
472 |      "text": [
473 |       "Rho(triples) for Dist: 1161 / 10000 = 0.1161\n",
474 |       "Rho(triples) for Boson: 3014 / 10000 = 0.3014\n"
475 |      ]
476 |     }
477 |    ],
478 |    "source": [
479 |     "n = 10000\n",
480 |     "\n",
481 |     "t = 'dist'\n",
482 |     "r = quantum_rolls(n, 3, t)\n",
483 |     "dist_nt_comp = count_identical(r)\n",
484 |     "print(\"Rho(triples) for Dist: %d / %d = %0.4f\"%(dist_nt_comp, n, dist_nt_comp/n))\n",
485 |     "\n",
486 |     "t = 'bos'\n",
487 |     "r = quantum_rolls(n, 3, t)\n",
488 |     "bos_nt_comp = count_identical(r)\n",
489 |     "print(\"Rho(triples) for Boson: %d / %d = %0.4f\"%(bos_nt_comp, n, bos_nt_comp/n))"
490 |    ]
491 |   },
492 |   {
493 |    "cell_type": "markdown",
494 |    "metadata": {},
495 |    "source": [
496 |     "In a turn of M rolls, what is the enhancement factor for generating an M-tuple (all rolls having same number of dots showing)?"
497 |    ]
498 |   },
499 |   {
500 |    "cell_type": "code",
501 |    "execution_count": 76,
502 |    "metadata": {},
503 |    "outputs": [
504 |     {
505 |      "name": "stdout",
506 |      "output_type": "stream",
507 |      "text": [
508 |       "Number of dice = 2\n",
509 |       "Rho(triples) for Dist: 3309 / 10000 = 0.3309\n",
510 |       "Rho(triples) for Boson: 4926 / 10000 = 0.4926\n",
511 |       "\n",
512 |       "Number of dice = 3\n",
513 |       "Rho(triples) for Dist: 1127 / 10000 = 0.1127\n",
514 |       "Rho(triples) for Boson: 2932 / 10000 = 0.2932\n",
515 |       "\n",
516 |       "Number of dice = 4\n",
517 |       "Rho(triples) for Dist: 367 / 10000 = 0.0367\n",
518 |       "Rho(triples) for Boson: 1963 / 10000 = 0.1963\n",
519 |       "\n",
520 |       "Number of dice = 5\n",
521 |       "Rho(triples) for Dist: 132 / 10000 = 0.0132\n",
522 |       "Rho(triples) for Boson: 1412 / 10000 = 0.1412\n",
523 |       "\n",
524 |       "Number of dice = 6\n",
525 |       "Rho(triples) for Dist: 31 / 10000 = 0.0031\n",
526 |       "Rho(triples) for Boson: 1103 / 10000 = 0.1103\n",
527 |       "\n",
528 |       "Number of dice = 7\n",
529 |       "Rho(triples) for Dist: 12 / 10000 = 0.0012\n",
530 |       "Rho(triples) for Boson: 835 / 10000 = 0.0835\n",
531 |       "\n"
532 |      ]
533 |     }
534 |    ],
535 |    "source": [
536 |     "xx = []\n",
537 |     "dist_yy = []\n",
538 |     "bos_yy = []\n",
539 |     "for m in range(2,8):\n",
540 |     "    n = 10000\n",
541 |     "    \n",
542 |     "    xx.append(m)\n",
543 |     "    print(\"Number of dice = %d\"%(m))\n",
544 |     "\n",
545 |     "    t = 'dist'\n",
546 |     "    r = quantum_rolls(n, m, t)\n",
547 |     "    dist_nt_comp = count_identical(r)\n",
548 |     "    ratio = dist_nt_comp/n\n",
549 |     "    print(\"Rho(triples) for Dist: %d / %d = %0.4f\"%(dist_nt_comp, n, ratio))\n",
550 |     "    dist_yy.append(ratio)\n",
551 |     "    \n",
552 |     "    t = 'bos'\n",
553 |     "    r = quantum_rolls(n, m, t)\n",
554 |     "    bos_nt_comp = count_identical(r)\n",
555 |     "    ratio = bos_nt_comp/n\n",
556 |     "    print(\"Rho(triples) for Boson: %d / %d = %0.4f\"%(bos_nt_comp, n, ratio))\n",
557 |     "    bos_yy.append(ratio)\n",
558 |     "    \n",
559 |     "    print()\n",
560 |     "\n",
561 |     "xx = np.array(xx)\n",
562 |     "dist_yy = np.array(dist_yy)\n",
563 |     "bos_yy = np.array(bos_yy)"
564 |    ]
565 |   },
566 |   {
567 |    "cell_type": "code",
568 |    "execution_count": 77,
569 |    "metadata": {},
570 |    "outputs": [
571 |     {
572 |      "data": {
573 |       "image/png": 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tO5DCe8sOc/lavtJRDFKr1IxvMQZLjSVrE7dw/WaO4UZCCFFJeXl5JCUlKR1D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\nEEIIIYQQzfD/AX9yN1GxclftAAAAAElFTkSuQmCC\n",
574 |       "text/plain": [
575 |        "<matplotlib.figure.Figure at 0x1138e28d0>"
576 |       ]
577 |      },
578 |      "metadata": {},
579 |      "output_type": "display_data"
580 |     }
581 |    ],
582 |    "source": [
583 |     "fig = plt.figure(figsize=(12,4))\n",
584 |     "ax1, ax2 = [fig.add_subplot(1,2,i+1) for i in range(2)]\n",
585 |     "\n",
586 |     "ax1.plot(xx, dist_yy)\n",
587 |     "ax1.plot(xx, bos_yy)\n",
588 |     "ax1.set_xlabel(\"Number of dice M\")\n",
589 |     "ax1.set_ylabel(\"Percentage of Triplets\")\n",
590 |     "\n",
591 |     "ax2.plot(xx, bos_yy/dist_yy)\n",
592 |     "ax2.set_xlabel(\"Number of dice M\")\n",
593 |     "ax2.set_ylabel(\"Ratio: Triplets in Bosons/Triplets in Distinguishable\")\n",
594 |     "\n",
595 |     "plt.show()"
596 |    ]
597 |   },
598 |   {
599 |    "cell_type": "code",
600 |    "execution_count": null,
601 |    "metadata": {},
602 |    "outputs": [],
603 |    "source": []
604 |   },
605 |   {
606 |    "cell_type": "code",
607 |    "execution_count": null,
608 |    "metadata": {},
609 |    "outputs": [],
610 |    "source": []
611 |   }
612 |  ],
613 |  "metadata": {
614 |   "kernelspec": {
615 |    "display_name": "Python 3",
616 |    "language": "python",
617 |    "name": "python3"
618 |   },
619 |   "language_info": {
620 |    "codemirror_mode": {
621 |     "name": "ipython",
622 |     "version": 3
623 |    },
624 |    "file_extension": ".py",
625 |    "mimetype": "text/x-python",
626 |    "name": "python",
627 |    "nbconvert_exporter": "python",
628 |    "pygments_lexer": "ipython3",
629 |    "version": "3.6.3"
630 |   }
631 |  },
632 |  "nbformat": 4,
633 |  "nbformat_minor": 2
634 | }
635 | 


--------------------------------------------------------------------------------
/Statistical Mechanics 2 Probability Distributions.ipynb:
--------------------------------------------------------------------------------
  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "markdown",
  5 |    "metadata": {},
  6 |    "source": [
  7 |     "# Probability Distributions\n",
  8 |     "\n",
  9 |     "Three most common probaiblity distributions are:\n",
 10 |     "* Uniform\n",
 11 |     "* Exponential\n",
 12 |     "* Gaussian\n",
 13 |     "\n",
 14 |     "For uniform distribution:\n",
 15 |     "\n",
 16 |     "$$\n",
 17 |     "\\rho_{uniform}(x) = \\dfrac{1}{b-a} \\quad \\mbox{ for } a \\leq x < b\n",
 18 |     "$$\n",
 19 |     "\n",
 20 |     "For exponential distribution:\n",
 21 |     "\n",
 22 |     "$$\n",
 23 |     "\\rho_{exponential}(t) = \\dfrac{ e^{-\\frac{t}{\\tau}} }{ \\tau } \\mbox{ for } t \\geq 0\n",
 24 |     "$$\n",
 25 |     "\n",
 26 |     "For Gaussian distribution:\n",
 27 |     "\n",
 28 |     "$$\n",
 29 |     "\\rho_{Gaussian}(v) = \\dfrac{1}{\\sqrt{2 \\pi \\sigma}} e^{-\\frac{v^2}{2\\sigma^2}}\n",
 30 |     "$$\n",
 31 |     "\n",
 32 |     "where $\\rho$ denotes the probability density function (PDF)."
 33 |    ]
 34 |   },
 35 |   {
 36 |    "cell_type": "code",
 37 |    "execution_count": 50,
 38 |    "metadata": {},
 39 |    "outputs": [],
 40 |    "source": [
 41 |     "import numpy as np\n",
 42 |     "import matplotlib.pyplot as plt"
 43 |    ]
 44 |   },
 45 |   {
 46 |    "cell_type": "markdown",
 47 |    "metadata": {},
 48 |    "source": [
 49 |     "## Part A: Likelihoods\n",
 50 |     "\n",
 51 |     "What is the probability that a random number uniform on $[0,1)$ will happen to lie between x = 0.7 and x = 0.75?\n",
 52 |     "\n",
 53 |     "**Answer:**\n",
 54 |     "\n",
 55 |     "$$\n",
 56 |     "P(0.7 \\leq 0.75) = \\dfrac{ \\int_{0.7}^{0.75} \\rho(x) dx }{ \\int_{0}^{1} \\rho(x) dx } = \\dfrac{ 0.75-0.7 }{ 1 - 0 } = 0.05\n",
 57 |     "$$"
 58 |    ]
 59 |   },
 60 |   {
 61 |    "cell_type": "code",
 62 |    "execution_count": 56,
 63 |    "metadata": {},
 64 |    "outputs": [
 65 |     {
 66 |      "name": "stdout",
 67 |      "output_type": "stream",
 68 |      "text": [
 69 |       "Percentage = 493 / 10000 = 0.0493\n"
 70 |      ]
 71 |     }
 72 |    ],
 73 |    "source": [
 74 |     "n = 10000\n",
 75 |     "sim = np.random.rand(n,)\n",
 76 |     "inside = len(sim[(sim>0.7)&(sim<0.75)])\n",
 77 |     "print(\"Percentage = %d / %d = %0.4f\"%(inside, n, inside/n))"
 78 |    ]
 79 |   },
 80 |   {
 81 |    "cell_type": "markdown",
 82 |    "metadata": {},
 83 |    "source": [
 84 |     "What is probability that waiting time for radioactive decay of nucleus is more than twice the exponential decay time $\\tau$?\n",
 85 |     "\n",
 86 |     "**Answer:**\n",
 87 |     "\n",
 88 |     "$$\n",
 89 |     "P(t > 2 \\tau) = \n",
 90 |     "\\dfrac{  \\int_{2 \\tau}^{\\infty} \\exp(-t/\\tau) / \\tau dt  }{  \\int_{0}^{\\infty} \\exp(-t/\\tau) / \\tau dt  }\n",
 91 |     "$$"
 92 |    ]
 93 |   },
 94 |   {
 95 |    "cell_type": "markdown",
 96 |    "metadata": {},
 97 |    "source": [
 98 |     "Denominator evaluates to:\n",
 99 |     "\n",
100 |     "$$\n",
101 |     "\\int_{0}^{\\infty} \\exp(-t/\\tau) / \\tau dt = 0 + e^0 = 1\n",
102 |     "$$\n",
103 |     "\n",
104 |     "Numerator evaluates to:\n",
105 |     "\n",
106 |     "$$\n",
107 |     "\\int_{2 \\tau}^{\\infty} \\exp(-t / \\tau) / \\tau dt \n",
108 |     "= \\left[ -e^{-t/\\tau} \\right]_{2\\tau}^{\\infty} = 0 + e^{-2}\n",
109 |     "$$\n",
110 |     "\n",
111 |     "for an overall probability of:\n",
112 |     "\n",
113 |     "$$\n",
114 |     "P(t > 2 \\tau) = \\dfrac{1}{e^{2}} \\sim 0.135\n",
115 |     "$$"
116 |    ]
117 |   },
118 |   {
119 |    "cell_type": "code",
120 |    "execution_count": 57,
121 |    "metadata": {},
122 |    "outputs": [
123 |     {
124 |      "name": "stdout",
125 |      "output_type": "stream",
126 |      "text": [
127 |       "0.135335283237\n"
128 |      ]
129 |     }
130 |    ],
131 |    "source": [
132 |     "print(1.0/np.exp(2))"
133 |    ]
134 |   },
135 |   {
136 |    "cell_type": "code",
137 |    "execution_count": 58,
138 |    "metadata": {},
139 |    "outputs": [
140 |     {
141 |      "name": "stdout",
142 |      "output_type": "stream",
143 |      "text": [
144 |       "Percentage = 1332 / 10000 = 0.1332\n"
145 |      ]
146 |     }
147 |    ],
148 |    "source": [
149 |     "tau = 1\n",
150 |     "n = 10000\n",
151 |     "sim = np.random.exponential(scale=tau, size=(n,))\n",
152 |     "inside = len(sim[sim>2*tau])\n",
153 |     "print(\"Percentage = %d / %d = %0.4f\"%(inside, n, inside/n))"
154 |    ]
155 |   },
156 |   {
157 |    "cell_type": "markdown",
158 |    "metadata": {},
159 |    "source": [
160 |     "What is probability that your score on an exam with a Gaussian distribution is greater than $2 \\sigma$ above the mean?\n",
161 |     "\n",
162 |     "Note:\n",
163 |     "\n",
164 |     "$$\n",
165 |     "\\int_{2}^{\\infty} \\frac{1}{\\sqrt{2 \\pi}} \\exp(-v^2/2) dv = \\dfrac{1 - \\mbox{erf}(\\sqrt{2}) }{ 2 } \\sim 0.023\n",
166 |     "$$"
167 |    ]
168 |   },
169 |   {
170 |    "cell_type": "markdown",
171 |    "metadata": {},
172 |    "source": [
173 |     "**Answer:**\n",
174 |     "\n",
175 |     "The probability of a score greater than $2 \\sigma$ above the mean is given by:\n",
176 |     "\n",
177 |     "$$\n",
178 |     "P(x > 2 \\sigma) = \\dfrac{  \\int_{2 \\sigma}^{\\infty} \\rho_{Gaussian}(x) dx  }{  \\int_{-\\infty}^{\\infty} \\rho_{Gaussian}(x) dx  }\n",
179 |     "$$"
180 |    ]
181 |   },
182 |   {
183 |    "cell_type": "markdown",
184 |    "metadata": {},
185 |    "source": [
186 |     "The expression in the denominator is equal to 1.\n",
187 |     "\n",
188 |     "The expression in the numerator is equal to:\n",
189 |     "\n",
190 |     "$$\n",
191 |     "\\dfrac{1}{ \\sqrt{2 \\pi \\sigma^2}} \\int_{2 \\sigma}^{\\infty} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp = \n",
192 |     "\\dfrac{1}{ \\sqrt{2 \\pi \\sigma^2}} \\left[ \\int_{-\\infty}^{\\infty} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp \n",
193 |     "- \\int_{-\\infty}^{2 \\sigma} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp \\right] \\\\\n",
194 |     "= \\dfrac{1}{ \\sqrt{2 \\pi \\sigma^2}} \\left[ \\int_{0}^{\\infty} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp \n",
195 |     "- \\int_{0}^{2\\sigma} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp \\right]\n",
196 |     "$$\n",
197 |     "\n",
198 |     "Now the first term becomes:\n",
199 |     "\n",
200 |     "$$\n",
201 |     "\\dfrac{1}{ \\sqrt{2 \\pi \\sigma^2}} \\int_{0}^{\\infty} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp = \\frac{1}{2}\n",
202 |     "$$\n",
203 |     "\n",
204 |     "and the second integral can be evaluated:\n",
205 |     "\n",
206 |     "$$\n",
207 |     "\\int_{-\\infty}^{2 \\sigma} \\dfrac{1}{2 \\pi \\sigma^2} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp = \\dfrac{1}{\\sqrt{\\pi}} \\int_{0}^{\\sqrt{2}} \\exp( -t^2 ) dt \\\\\n",
208 |     "= \\dfrac{\\mbox{erf}(\\sqrt{2})}{2}\n",
209 |     "$$\n",
210 |     "\n",
211 |     "Together, these make the overall integral evaluate to:\n",
212 |     "\n",
213 |     "$$\n",
214 |     "\\dfrac{1}{ \\sqrt{2 \\pi \\sigma^2}} \\int_{2 \\sigma}^{\\infty} \\exp \\left( -\\dfrac{p^2}{2\\sigma^2} \\right) dp = \\dfrac{1 - \\mbox{erf}(\\sqrt{2})}{2}\n",
215 |     "$$"
216 |    ]
217 |   },
218 |   {
219 |    "cell_type": "code",
220 |    "execution_count": 59,
221 |    "metadata": {},
222 |    "outputs": [
223 |     {
224 |      "name": "stdout",
225 |      "output_type": "stream",
226 |      "text": [
227 |       "0.0227501319482\n"
228 |      ]
229 |     }
230 |    ],
231 |    "source": [
232 |     "from scipy.special import erf\n",
233 |     "print((1-erf(np.sqrt(2)))/2)"
234 |    ]
235 |   },
236 |   {
237 |    "cell_type": "code",
238 |    "execution_count": 60,
239 |    "metadata": {},
240 |    "outputs": [
241 |     {
242 |      "name": "stdout",
243 |      "output_type": "stream",
244 |      "text": [
245 |       "Percentage = 238 / 10000 = 0.0238\n"
246 |      ]
247 |     }
248 |    ],
249 |    "source": [
250 |     "n = 10000\n",
251 |     "mu = 0\n",
252 |     "sigma = 3\n",
253 |     "sim = sigma*np.random.randn(n,) + mu\n",
254 |     "inside = len(sim[sim>2*sigma])\n",
255 |     "print(\"Percentage = %d / %d = %0.4f\"%(inside, n, inside/n))"
256 |    ]
257 |   },
258 |   {
259 |    "cell_type": "markdown",
260 |    "metadata": {},
261 |    "source": [
262 |     "## Part B: Normalization, Mean, and Standard Deviation\n",
263 |     "\n",
264 |     "Show that these probability distributions are normalized: $\\int \\rho(x) dx = 1$.\n",
265 |     "\n",
266 |     "What is the mean $x_0$ of each distribution?\n",
267 |     "\n",
268 |     "What is the standard deviation $\\sigma$, where $\\sigma^2 = \\int \\left(x-x_0\\right)^2 \\rho(x) dx$?"
269 |    ]
270 |   },
271 |   {
272 |    "cell_type": "markdown",
273 |    "metadata": {},
274 |    "source": [
275 |     "### Uniform Distribution\n",
276 |     "\n",
277 |     "Mean of uniform distribution is $\\dfrac{b+a}{2}$, and the variance is $V = \\dfrac{(b-a)^2}{12}$, making the standard deviation $\\sigma = \\dfrac{b - a}{2 \\sqrt{3}}$."
278 |    ]
279 |   },
280 |   {
281 |    "cell_type": "markdown",
282 |    "metadata": {},
283 |    "source": [
284 |     "### Exponential Distribution\n",
285 |     "\n",
286 |     "The mean of the exponential distribution is $\\tau$, and the standard deviation $\\sigma = \\tau$ too.\n",
287 |     "\n",
288 |     "The variance is $\\tau^2$."
289 |    ]
290 |   },
291 |   {
292 |    "cell_type": "markdown",
293 |    "metadata": {},
294 |    "source": [
295 |     "### Gaussian Distribution\n",
296 |     "\n",
297 |     "Mean of Gaussian distribution is $\\mu$, standard deviation is $\\sigma$."
298 |    ]
299 |   },
300 |   {
301 |    "cell_type": "markdown",
302 |    "metadata": {},
303 |    "source": [
304 |     "## Sum of Uniform Random Variables"
305 |    ]
306 |   },
307 |   {
308 |    "cell_type": "code",
309 |    "execution_count": 105,
310 |    "metadata": {},
311 |    "outputs": [],
312 |    "source": [
313 |     "import seaborn as sns"
314 |    ]
315 |   },
316 |   {
317 |    "cell_type": "code",
318 |    "execution_count": 106,
319 |    "metadata": {
320 |     "scrolled": true
321 |    },
322 |    "outputs": [],
323 |    "source": [
324 |     "x1 = np.random.rand(10000,)\n",
325 |     "x2 = np.random.rand(10000,)"
326 |    ]
327 |   },
328 |   {
329 |    "cell_type": "code",
330 |    "execution_count": 113,
331 |    "metadata": {},
332 |    "outputs": [
333 |     {
334 |      "data": {
335 |       "image/png": 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AAQAA4CgUYAAAADgKBRgAAACOQgEGAACAo1CAAQAA4CimF8IAAACwQ7QLNgCTwRFgAAAA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\n19NsagAAAABJRU5ErkJggg==\n",
336 |       "text/plain": [
337 |        "<matplotlib.figure.Figure at 0x115a38c88>"
338 |       ]
339 |      },
340 |      "metadata": {},
341 |      "output_type": "display_data"
342 |     }
343 |    ],
344 |    "source": [
345 |     "x = x1 + x2\n",
346 |     "\n",
347 |     "fig = plt.figure(figsize=(12,4))\n",
348 |     "ax1, ax2, ax3 = [fig.add_subplot(1,3,i+1) for i in range(3)]\n",
349 |     "\n",
350 |     "ax1.hist(x1, normed=True, bins=25)\n",
351 |     "ax2.hist(x2, normed=True, bins=25)\n",
352 |     "ax3.hist(x,  normed=True, bins=25)\n",
353 |     "\n",
354 |     "ax1.set_xlabel(r'$x_1$')\n",
355 |     "ax2.set_xlabel(r'$x_2$')\n",
356 |     "ax3.set_xlabel(r'$x_1 + x_2$')\n",
357 |     "\n",
358 |     "plt.show()"
359 |    ]
360 |   },
361 |   {
362 |    "cell_type": "code",
363 |    "execution_count": 135,
364 |    "metadata": {},
365 |    "outputs": [],
366 |    "source": [
367 |     "def draw_uniform_sums(nx):\n",
368 |     "\n",
369 |     "    tot = 10000\n",
370 |     "\n",
371 |     "    xs = []\n",
372 |     "    xsum = np.zeros(tot,)\n",
373 |     "    for x in range(nx):\n",
374 |     "        x = np.random.rand(tot,)\n",
375 |     "        xsum += x\n",
376 |     "        xs.append(x)\n",
377 |     "\n",
378 |     "    fig = plt.figure(figsize=(14,2))\n",
379 |     "    axes = [fig.add_subplot(1,nx,i+1) for i in range(nx)]\n",
380 |     "\n",
381 |     "    for i, (ax, x) in enumerate(zip(axes,xs)):\n",
382 |     "        ax.hist(x, normed=True, bins=25)\n",
383 |     "        ax.set_xlabel(r'$x_%d$'%(i+1))\n",
384 |     "\n",
385 |     "    fig2 = plt.figure(figsize=(4,4))\n",
386 |     "    ax2 = fig2.add_subplot(1,1,1)\n",
387 |     "    ax2.hist(xsum, normed=True, bins=25)\n",
388 |     "    ax2.set_xlabel(r'$\\Sigma x_i$')\n",
389 |     "    \n",
390 |     "    plt.show()"
391 |    ]
392 |   },
393 |   {
394 |    "cell_type": "code",
395 |    "execution_count": 136,
396 |    "metadata": {},
397 |    "outputs": [
398 |     {
399 |      "data": {
400 |       "image/png": 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AnJKenq7Vq1f32DZq1KjYx3//+9+/9vM2btwY11wA3GM71FAcAJxAlwAAgHhJdzsAAAAA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\n0RhqAAAAABiNoQYAAACA0f4/WgonwRXbPJsAAAAASUVORK5CYII=\n",
401 |       "text/plain": [
402 |        "<matplotlib.figure.Figure at 0x117446160>"
403 |       ]
404 |      },
405 |      "metadata": {},
406 |      "output_type": "display_data"
407 |     },
408 |     {
409 |      "data": {
410 |       "image/png": 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411 |       "text/plain": [
412 |        "<matplotlib.figure.Figure at 0x1177cf978>"
413 |       ]
414 |      },
415 |      "metadata": {},
416 |      "output_type": "display_data"
417 |     }
418 |    ],
419 |    "source": [
420 |     "draw_uniform_sums(3)"
421 |    ]
422 |   },
423 |   {
424 |    "cell_type": "code",
425 |    "execution_count": 137,
426 |    "metadata": {},
427 |    "outputs": [
428 |     {
429 |      "data": {
430 |       "image/png": 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ZmZldmphqaxsiPgbxoaamTi5Xsj755Ixqaurk9/sUCoXkdp+Pk8fj0cyZM8OXhUeNGqWD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\nxhzNjcW++uor3X3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431 |       "text/plain": [
432 |        "<matplotlib.figure.Figure at 0x1159283c8>"
433 |       ]
434 |      },
435 |      "metadata": {},
436 |      "output_type": "display_data"
437 |     },
438 |     {
439 |      "data": {
440 |       "image/png": 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9/PDD8R6nT3V1dWpvb9fSpUuVlpampKQkJScPvb8nO3fuND/2eDyqqKiQ0+mM40S927t3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\n",
441 |       "text/plain": [
442 |        "<matplotlib.figure.Figure at 0x113465278>"
443 |       ]
444 |      },
445 |      "metadata": {},
446 |      "output_type": "display_data"
447 |     }
448 |    ],
449 |    "source": [
450 |     "draw_uniform_sums(6)"
451 |    ]
452 |   },
453 |   {
454 |    "cell_type": "code",
455 |    "execution_count": 138,
456 |    "metadata": {},
457 |    "outputs": [
458 |     {
459 |      "data": {
460 |       "image/png": 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pQVFREebMmYNly5bJt9XV1WHz5s3YunUr3nnnHaxYsQItLS1Ys2YNxowZgy1b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\npGtsaoiIiIiISNf+P38xTkxAuFvyAAAAAElFTkSuQmCC\n",
461 |       "text/plain": [
462 |        "<matplotlib.figure.Figure at 0x10f1bf8d0>"
463 |       ]
464 |      },
465 |      "metadata": {},
466 |      "output_type": "display_data"
467 |     },
468 |     {
469 |      "data": {
470 |       "image/png": 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471 |       "text/plain": [
472 |        "<matplotlib.figure.Figure at 0x115a0e4e0>"
473 |       ]
474 |      },
475 |      "metadata": {},
476 |      "output_type": "display_data"
477 |     }
478 |    ],
479 |    "source": [
480 |     "draw_uniform_sums(8)"
481 |    ]
482 |   },
483 |   {
484 |    "cell_type": "code",
485 |    "execution_count": null,
486 |    "metadata": {},
487 |    "outputs": [],
488 |    "source": []
489 |   }
490 |  ],
491 |  "metadata": {
492 |   "kernelspec": {
493 |    "display_name": "Python 3",
494 |    "language": "python",
495 |    "name": "python3"
496 |   },
497 |   "language_info": {
498 |    "codemirror_mode": {
499 |     "name": "ipython",
500 |     "version": 3
501 |    },
502 |    "file_extension": ".py",
503 |    "mimetype": "text/x-python",
504 |    "name": "python",
505 |    "nbconvert_exporter": "python",
506 |    "pygments_lexer": "ipython3",
507 |    "version": "3.6.3"
508 |   }
509 |  },
510 |  "nbformat": 4,
511 |  "nbformat_minor": 2
512 | }
513 | 


--------------------------------------------------------------------------------
/brownian.py:
--------------------------------------------------------------------------------
 1 | import numpy as np
 2 | 
 3 | def brownian(delta, dt, N, d):
 4 |     """Returns a d-dimensional Brownian walk 
 5 |     with length scale delta
 6 |     and random step size drawn from 
 7 |     zero-mean normal distribution 
 8 |     with standard deviation delta^2*dt.
 9 |     
10 |     N steps in d dimensional space.
11 | 
12 |     Parameters:
13 |         
14 |         delta : length scale of diffusion step
15 |         dt : time scale of diffusion step
16 |         N : number of diffusion steps
17 |         d : dimensionality of Brownian walk
18 |     """
19 |     mean = 0.0
20 |     stdev = np.sqrt(delta*delta*dt)
21 |     
22 |     # Get N-1 random steps
23 |     r = np.random.normal(loc=mean, scale=stdev, size=(N-1,d))
24 |     
25 |     # Make sure the first location is 0,0
26 |     r2 = np.zeros((N, d))
27 |     r2[1:N,:] = r[0:(N-1),:]
28 |     
29 |     return r2.cumsum(axis=0)
30 | 


--------------------------------------------------------------------------------
/lorenz_rhs.py:
--------------------------------------------------------------------------------
 1 | import numpy as np
 2 | from scipy.integrate import odeint
 3 | 
 4 | def get_lorenz_solution(in_0, tmax, nt, args_tuple):
 5 |     t = np.linspace(0, tmax, nt)
 6 |     soln = odeint(lorenz, in_0, t, args=args_tuple).T
 7 |     return t, soln
 8 | 
 9 | def lorenz(in_, t, sigma, b, r):
10 |     """Evaluates the RHS of the 3 
11 |     Lorenz attractor differential equations.
12 | 
13 |     in_ : initial vector of [x_0, y_0, z_0]
14 |     t : time vector (not used, but present for odeint() call)
15 |     sigma : numerical parameter 1
16 |     b :     numerical parameter 2
17 |     r :     numerical parameter 3
18 |     """
19 |     x = in_[0]
20 |     y = in_[1]
21 |     z = in_[2]
22 |     return [sigma*(y-x),
23 |             r*x - y - x*z,
24 |             x*y - b*z]
25 | 
26 | 


--------------------------------------------------------------------------------