assignment19_diffusion_experiment_ and_2D_random_wolks.ipynb

Kernel: Python 3 (Anaconda 2019)

ניתוח תוצאות ניסוי בדיפוזיה

מטרה:

חקירת התפשטותו של KMnO4 במים כתלות בזמן.

רקע:

דיפוזיה, או בשמה העברי פעפוע, היא תופעה של השוואת ריכוזים. חומר עובר ממקום שבו ריכוזו גבוה למקום שבו ריכוזו נמוך, עד שהריכוזים משתווים. קטע הקוד שבהמשך טוען למחשב סידרה של 10 תמונות שצולמו בפרקי זמן של דקה אחת, האחת מהשניה. התמונות הן של ניסוי בו לקחו גביש של $KMnO_4$ והניחו אותו במרכזה של צלוחית פטרי מלאה במים. בתמונות ניתן לראות כיצד החומר ממנו עשוי הגביש מתמוסס במים ומתפזר. בפעילות תחקרו את קוטר הכתם כתלות בזמן. התמונות נטענות על ידי קטע הקוד שבהמשך לרשימה בשם images באופן שבו כל איבר ברשימה הוא אחת מ- 10 התמונות.

שימו לב! חוברת העבודה בפעילות זו צריכה להיות מסוג של classic notebook כדי לעבור לזה עליכם להכנס ללשונית file ולרדת עד הסוף, שם תמצאו אופציה switch to classical notebook. כדי לחזור ל cocalc עליכם ללחוץ למעלה ליד הRefresh על בסרגל על CoCalc Jupyter Notebook...

%matplotlib notebook
import matplotlib.pyplot as plt
images = []
for i in range(1,11):
    file_name='../../images/diffusion/'+str(i)+'.jpg'
    im = plt.imread(file_name)
    images.append(im)

הפונקציה ParseError: KaTeX parse error: Expected 'EOF', got '_' at position 11: \text{show_̲image(k)} מציגה את התמונה ה-k.

def show_image(k):
    plt.figure( figsize=(8, 6))
    plt.imshow(images[k])

1. כאשר תצביעו בעזרת הסמן על נקודה בתמונה, המחשב יציג מתחת לתמונה בצידה הימני את שיעורי הנקודה x,y ורשימה המכילה שלושה מספרים בתחום שבין 0 ל- 255. רשימה זו מכילה את מרכיבי הצבע (כמות האדום הירוק והכחול) של הנקודה. בשלב ראשון עבור כל אחת מעשרת התמונות אתם צריכם למצוא את קוטר הכתם. לשם כך מצאו בעזרת הסמן, בכל תמונה, את שעוריהן של שתי נקודות הנמצאות על קוטר הכתם. צרו טבלה של קוטר הכתם כתלות בזמן

לאחר שלב זה חיזרו לקוקלק

show_image(9)

WARNING: 2 intermediate output messages were discarded.

#image 1: 10
#image 2: 14
#image 3: 35
#image 4: 47
#image 5: 50
#image 6: 57
#image 7: 63
#image 8: 73
#image 9: 74
#image 10: 81
l = [10,14,35,47,50,57,63,73,74,81] 
ls = [0,1,2,3,4,5,6,7,8,9]
dicton = {}
for i in range (0,10):
    dicton[ls[i]] = [l[i]]
print(dicton)

{0: [10], 1: [14], 2: [35], 3: [47], 4: [50], 5: [57], 6: [63], 7: [73], 8: [74], 9: [81]}

2.שרטטו גרף של ריבוע קוטר הכתם כתלות בזמן.

import matplotlib.pyplot as plt
l = [0,14,35,47,50,57,63,73,74,81] 
x = 0
y = 0
for k in range(0,10):
    t = l[k]**2
    plt.plot([x,k],[y,t])
    x = k
    y = t

5. מהו הקשר הנובע מהגרף בין ריבוע קוטר הכתם והזמן? (האם ישנו קשר קווי, פרבולי וכו) ? הניחו שב t=0 גודל הכתם היה זניח (אפס)

רשמו משוואה מתמטית המתארת קשר.

#קשר ישר
#y = 650*t

הלך אקראי במישור והשוואתו לניסוי הדיפוזיה

בחלק קודם ניתחתם תוצאות של ניסוי בדיפוזיה. אותם מצאתם (בהנחה שבזמן t=0 קוטר הכתם זניח) כי
$$d^2 = k\cdot t$$
בנוסחא זו d הוא קןטר הכתם, t הזמן ו-k הוא מקדם פרופורציה. בתרגיל זה תראו כי תוצאה זו מתיישבת עם המודל החלקיקי. נניח כי בזמן t=0 נמצאים כל המולקולות של הקאלי באיזור מעגלי קטן שרדיוסו זניח. כתוצאה מהתנגשיות בין מולקולות הקאלי למולקולות המים שבצלחת הפטרי מבצעת כל אחת ממולקולות הקאלי צעד בכיוון אקראי.
גודלם הממוצע של צעדים אלו מכונה root mean square radius ויסומן כ- $r_{r.m.s}$ או בקיצור ב -r.

1. התא שבהמשך מכיל שתי פונקציות step ו- make_n_steps. הוסיפו לכל פונקציה doc string וליד כל משפט בתוכנית כתבו הערות המסבירות אותו.

שימו לב! המשתנה הפורמלי particle צריך להיות dicionary בעל שני מפתחות האחד 'x' והשני 'y'. הערכים המשויכים למפתחות אלו הם שיעורי שיעורי ה- x וה-y של חלקיק מסוים.

import random,math
import matplotlib.pyplot as plt


def step():
    dx,dy=random.uniform(-1,1),random.uniform(-1,1) #מגריל את שיעור האיקס והוואי של כל צעד
    L=math.sqrt(dx**2+dy**2)#מחשב את האורך הכולל של הצעד
    return (dx/L,dy/L)# הפונקציה מחזירה את שיעור האיקס והוואי של כל צעד חלקי אורכו הכולל

def make_n_steps(particle,n):#אן מבטא את מספר הצעדים
    for j in range(n):
        dx, dy = step()
        particle['x'].append(particle['x'][-1] + dx)
        particle['y'].append(particle['y'][-1] + dy)
    return particle

2. הפונקציה שבהמשך מקבלת את מספר הצעדים אותו אמור חלקיק לבצע, את שיעורי x ו-y של מקומו ההתחלית ומחזירה שתי רשימות, האחת של שיעורי ה- x של החלקיק לאורך n הצעדים שביצע והשנייה את שיעורי ה- y המתאימים. השלימו את קיטעי הקוד החסרים:

def particle_path(N, x0 = 0, y0 = 0):
    x, y = [x0],[y0]
    for i in range(1,N+1):
        dx, dy = step()
        x.append(x[i-1]+dx)
        y.append(y[i-1]+dy)
    return (x,y)
particle_path(10000, x0 = 0, y0 = 0)

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3. צרו גרף המתאר את מקומו של חלקיק בודד כתלות בזמן אחרי 10000 צעדים.

l =particle_path(10000, x0 = 0, y0 = 0)
x_list = l[0]
y_list = l[1]
plt.plot(x_list, y_list, 'b.')

[<matplotlib.lines.Line2D at 0x7f506da23d50>]

4.מתוך הנחה שגודל מהירותם הממוצע של החלקיקים קבוע אזי מספר הצעדים ביחס ישר לזמן. הקוד שבתא שבהמשך, מתאר מערכת המכילה 2000 חלקיקים אחרי 1000 צעדים. העזרו בקוד זה ליצירת פונקציה המקבלת את מספר החלקיקים, מספר הצעדים ומחזירה שתי רשימות האחת מכילה את שיעור ה-x הסופי עבור כל אחד מהחלקיקים והשניה את שיעור ה- y. על תשכחו להוסיף לפונקציה doc string והערות המסבירות את משפטי הקוד השונים.

import matplotlib.pyplot as plt
N = 1000
particles=2000
fig = plt.figure(figsize = (6, 6))
plt.xlim(-100,100)
plt.ylim(-100,100)
pl = [{'x':0,'y':0} for i in range(particles)]
x_pos = []
y_pos = []
for p in pl:
    for i in range(N):
        dx, dy = step()
        p['x'] += dx
        p['y'] += dy
    x_pos.append(p['x'])
    y_pos.append(p['y'])
plt.plot(x_pos,y_pos,'b.') 

[<matplotlib.lines.Line2D at 0x7f29d90c2a90>]

def final_position(particles, N):
    """(int number, int number) -- > tupul of 2 lists
    the function return tupul of two lists. final position on x and final position on y"""
    pl = [{'x':0,'y':0} for i in range(particles)] #יוצר מילון
    x_pos = [] #יוצר רשימה לציר האופקי
    y_pos = [] #יוצר רשימה לציר האנכי
    for p in pl: #מתחיל לולאה על כל המילונים
        for i in range(N): # מתחיל לולאה כמספר הצעדים
            dx, dy = step() #יוצר צעד על שני הצירים
            p['x'] += dx #מגדיל את ערך x
            p['y'] += dy #מגדיל את ערך y
        x_pos.append(p['x']) #מוסיף לרשימה של הציר האופקי
        y_pos.append(p['y']) #מוסיף לרשימה של הציר האנכי
    return(x_pos, y_pos) #מחזיר את הרשימה

final_position(2000, 1000)

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5. כתבו פונקציה בשם getR המקבלת שתי רשימות, האחת של שיעורי ה-x של החלקיקים והשניה של שיעורי ה- y שלהם. הפונקציה צריכה להחזיר רשימה של המרחק הסופי של החלקיקים מהראשית.

def distance(p1, p2):
    distance = math.sqrt(((p1[0]-p2[0])**2)+((p1[1]-p2[1])**2))
    return(distance)
x_pos = final_position(2000, 1000)[0]
y_pos = final_position(2000, 1000)[1]

def getR(x_pos, y_pos):
    ls = []
    start = (0, 0)
    for i in range(len(x_pos)):
        p = (x_pos[i], y_pos[i])
        dist_from_start = distance(p, start)
        ls.append(dist_from_start)
    return(ls)

getR(x_pos, y_pos)

[31.98982122683903,
 47.53351669354256,
 16.151685994870775,
 43.18227000492883,
 35.84485854232272,
 20.273655727855434,
 66.14762081312026,
 25.227698174304585,
 19.891256105598142,
 8.278330396717836,
 29.326108097849342,
 37.34810299805729,
 42.16353308527869,
 16.45479048315367,
 13.390009221884618,
 16.579032229946893,
 40.06026002393504,
 15.207359745577756,
 21.04125936010086,
 61.98471844095026,
 58.035574920149344,
 27.44505092672236,
 37.67166558033936,
 29.28271058089555,
 18.423231565042997,
 25.420916444343913,
 7.499141455191513,
 42.405308863430704,
 25.34989025594796,
 15.67643360401509,
 21.246834669612443,
 9.731720896263123,
 18.278137874783067,
 28.418372089358932,
 9.930726870786108,
 16.974605387922026,
 18.218205570154414,
 35.49545987619872,
 7.280580667086242,
 9.735205037682983,
 40.85633394495079,
 18.14467688114201,
 15.659866061396514,
 23.442200129470823,
 7.29046979726751,
 11.871315704784115,
 25.631027219372886,
 40.09089002952534,
 15.941557511401674,
 20.716462696441365,
 31.628211988054826,
 20.29877119985691,
 26.548240868037396,
 19.218930382018886,
 20.21406347529946,
 4.776879392932124,
 27.517023258240744,
 15.719994050512897,
 11.504527444667623,
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6.כתבו קוד שיפיק גרף של המרחק הממוצע של כל החלקיקים כתלות בזמן. מה הקשר המתמטי של המרחק הממוצע של החלקיקים מהראשית לזמן?
השוו את הקשר המתמטי הזה לתוצאות הניסוי של הדיפוזיה. הסבירו מה דומה ומה שונה

import math,random
import  matplotlib.pyplot as plt
particles = 2
N= 101
def step(): 
    dx,dy=random.uniform(-1,1),random.uniform(-1,1) 
    L=math.sqrt(dx**2+dy**2) #חישוב המרחק הסופי לאחר הצעד 
    return (dx/L,dy/L)#יחזיר את שיעורי הנקודה הסופיים 

def finishedPoint(particles, N):
    
    avgs = []
    N = 101
    distan = []
    x_pos = [0] 
    y_pos = [0] 
    for i in range(N):
        
        
        for j in range(particles):
            dx, dy = step()
            x_pos.append(dx+x_pos[-1])
            y_pos.append(dy+y_pos[-1])
            dist = math.sqrt(((x_pos[-1] - 0)**2 + (y_pos[-1] - 0)**2))
            distan.append(dist)
        avg=(sum(distan)/len(distan))
        avgs.append(avg)
    return (avgs)


time = []
for i in range(N):
    time.append(i)
plt.plot(time,finishedPoint(particles, N ))

[<matplotlib.lines.Line2D at 0x7f506b3c7410>]

7$^*$ (לא חובה) - נתון מכל המלא מים. מכניסים לתוך המכל כפית של חומר העשוי גרגרים קטנים הזהים בצורתם ובוחשים באופן כזה שהחלקיקים מפוזרים באופן אחיד בחלל הכלי. החלקיקים נעים לכל הכיוונים באופן אקראי, אבל כתוצאה מפעולת כח הכובד ההסתברות שחלקיק ינוע מטה גבוהה יותר. צרו הדמיה של התרחיש המתואר ובידקו כיצד משתנה צפיפות החלקיקים כתלות בגובה. הניחו כי החלקיקים יכולים לנוע אך ורק בתוך המיכל.