Thermodynamics and Statistical Mechanics
description
Transcript of Thermodynamics and Statistical Mechanics
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Thermodynamics and Statistical MechanicsProbabilities
Thermo & Stat Mech - Spring 2006 Class 16
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Pair of DiceFor one die, the probability of any face coming up is the same, 1/6. Therefore, it is equally probable that any number from one to six will come up.
For two dice, what is the probability that the total will come up 2, 3, 4, etc up to 12?
Thermo & Stat Mech - Spring 2006 Class 16
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ProbabilityTo calculate the probability of a particular outcome, count the number of all possible results. Then count the number that give the desired outcome. The probability of the desired outcome is equal to the number that gives the desired outcome divided by the total number of outcomes. Hence, 1/6 for one die.
Thermo & Stat Mech - Spring 2006 Class 16
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Pair of DiceList all possible outcomes (36) for a pair of dice.Total CombinationsHow Many21+1131+2, 2+1241+3, 3+1, 2+2351+4, 4+1, 2+3, 3+2461+5, 5+1, 2+4, 4+2, 3+35
Thermo & Stat Mech - Spring 2006 Class 16
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Pair of DiceTotal CombinationsHow Many71+6, 6+1, 2+5, 5+2, 3+4, 4+3682+6, 6+2, 3+5, 5+3, 4+4593+6, 6+3, 4+5, 5+44104+6, 6+4, 5+53115+6, 6+52126+61 Sum = 36
Thermo & Stat Mech - Spring 2006 Class 16
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Probabilities for Two Dice
Thermo & Stat Mech - Spring 2006 Class 16
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Probabilities for Two Dice
Thermo & Stat Mech - Spring 2006 Class 16
Chart1
0.0277777778
0.0555555556
0.0833333333
0.1111111111
0.1388888889
0.1666666667
0.1388888889
0.1111111111
0.0833333333
0.0555555556
0.0277777778
Number
Probability
Dice
Sheet1
Distributions
Dice Pair
nprob
20.0277777778
30.0555555556
40.0833333333
50.1111111111
60.1388888889
70.1666666667
80.1388888889
90.1111111111
100.0833333333
110.0555555556
120.0277777778
Binomial Distribution
p =0.5
q =0.5
N =6120
npCalcExp
00.0156251.8753110100026.55187513.05
10.0937511.25143312231454.6087511.0712142643
20.23437528.125211366324219.9631251
30.312537.53846821261140.023752
40.23437528.1253465563913635.721252
50.0937511.2595202004536.9056253
60.0156251.875100010069.1506253
3.32.852.8532.83.05Mean =2.975pN =33
1.51986841541.35185058350.85293610551.51657508880.81240384051.0712142643SD =1.23465582251.22474487143
0.33985290940.302282980.19072231120.33911649920.18165902120.23953079133
0.17017148213
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Number
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Dice
Sheet2
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Successes
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Binomial Distribution
Sheet3
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Calc
Exp
No of Heads
No of occurences
Experimental Coin Toss
Sheet4
Binomial Distribution
p =0.5Sqrt(Npq) =2.23606797752.5066282746
q =0.5Mean =Np =105.6049912164
N =200.17841241160.1784124116
nBinomialGaussian
00.00000095370.0000080999
10.00001907350.0000541551
20.00018119810.0002964424
30.00108718870.0013285628
40.00462055210.0048748912
50.01478576660.0146449826
60.03696441650.0360208447
70.0739288330.0725370735
80.12013435360.119593416
90.16017913820.1614342259
100.1761970520.1784124116
110.16017913820.1614342259
120.12013435360.119593416
130.0739288330.0725370735
140.03696441650.0360208447
150.01478576660.0146449826
160.00462055210.0048748912
170.00108718870.0013285628
180.00018119810.0002964424
190.00001907350.0000541551
200.00000095370.0000080999
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Binomial
Gaussian
No. of Heads
Probability
Twenty Coins
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No. of Heads
Probability
Gaussian
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No. of Heads
Probability
Binomial
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No. of Heads
Probability
Gaussian
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Binomial
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Probability
Twenty Coins
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Sheet12
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Microstates and MacrostatesEach possible outcome is called a microstate.The combination of all microstates that give the same number of spots is called a macrostate.The macrostate that contains the most microstates is the most probable to occur.
Thermo & Stat Mech - Spring 2006 Class 16
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Combining ProbabilitiesIf a given outcome can be reached in two (or more) mutually exclusive ways whose probabilities are pA and pB, then the probability of that outcome is: pA + pB.
This is the probability of having either A or B.
Thermo & Stat Mech - Spring 2006 Class 16
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Combining ProbabilitiesIf a given outcome represents the combination of two independent events, whose individual probabilities are pA and pB, then the probability of that outcome is: pA pB.
This is the probability of having both A and B.
Thermo & Stat Mech - Spring 2006 Class 16
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ExamplePaint two faces of a die red. When the die is thrown, what is the probability of a red face coming up?
Thermo & Stat Mech - Spring 2006 Class 16
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Another ExampleThrow two normal dice. What is the probability of two sixes coming up?
Thermo & Stat Mech - Spring 2006 Class 16
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Complicationsp is the probability of success. (1/6 for one die)q is the probability of failure. (5/6 for one die)
p + q = 1, or q = 1 p
When two dice are thrown, what is the probability of getting only one six?
Thermo & Stat Mech - Spring 2006 Class 16
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ComplicationsProbability of the six on the first die and not the second is:
Probability of the six on the second die and not the first is the same, so:
Thermo & Stat Mech - Spring 2006 Class 16
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SimplificationProbability of no sixes coming up is:
The sum of all three probabilities is:p(2) + p(1) + p(0) = 1
Thermo & Stat Mech - Spring 2006 Class 16
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Simplificationp(2) + p(1) + p(0) = 1p + 2pq + q =1(p + q) = 1
The exponent is the number of dice (or tries).Is this general?
Thermo & Stat Mech - Spring 2006 Class 16
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Three Dice(p + q) = 1p + 3pq + 3pq + q = 1p(3) + p(2) + p(1) + p(0) = 1
It works! It must be general!(p + q)N = 1
Thermo & Stat Mech - Spring 2006 Class 16
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Binomial DistributionProbability of n successes in N attempts(p + q)N = 1
where, q = 1 p.
Thermo & Stat Mech - Spring 2006 Class 16
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Thermodynamic ProbabilityThe term with all the factorials in the previous equation is the number of microstates that will lead to the particular macrostate. It is called the thermodynamic probability, wn.
Thermo & Stat Mech - Spring 2006 Class 16
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MicrostatesThe total number of microstates is:
For a very large number of particles
Thermo & Stat Mech - Spring 2006 Class 16
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Mean of Binomial Distribution
Thermo & Stat Mech - Spring 2006 Class 16
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Mean of Binomial Distribution
Thermo & Stat Mech - Spring 2006 Class 16
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Standard Deviation (s)
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Standard Deviation
Thermo & Stat Mech - Spring 2006 Class 16
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Standard Deviation
Thermo & Stat Mech - Spring 2006 Class 16
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For a Binomial Distribution
Thermo & Stat Mech - Spring 2006 Class 16
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CoinsToss 6 coins. Probability of n heads:
Thermo & Stat Mech - Spring 2006 Class 16
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For Six Coins
Thermo & Stat Mech - Spring 2006 Class 16
Chart2
0.015625
0.09375
0.234375
0.3125
0.234375
0.09375
0.015625
Successes
Probabilty
Binomial Distribution
Sheet1
Distributions
Dice Pair
nprob
20.0277777778
30.0555555556
40.0833333333
50.1111111111
60.1388888889
70.1666666667
80.1388888889
90.1111111111
100.0833333333
110.0555555556
120.0277777778
Binomial Distribution
p =0.5
q =0.5
N =6150
npCalcExp
00.0156252.343753100101029.328133333313.05
10.0937514.06259114201940.704411.0712142643
20.23437535.15625356956547044.42822222221
30.312546.875448876961320.70595555562
40.23437535.156254184686916431.27115555562
50.0937514.0625141331247049.13128888893
60.0156252.3437540001302433.02417777783
702.962.962.9633.563.32Mean =3.1266666667pN =33
801.11283421951.07628992381.24835892271.32664991611.23547561691.2560254774SD =1.23448594791.22474487143
900.22256684390.21525798480.24967178450.26532998320.24709512340.25120509553
1000.23257018063
1104
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1304
1404
1504
1604
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Sheet1
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Number
Probability
Dice
Sheet2
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Successes
Probabilty
Binomial Distribution
Sheet3
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Calc
Exp
No of Heads
No of occurences
Experimental Coin Toss
Sheet4
Binomial Distribution
p =0.1666666667Sqrt(Npq) =1.66666666672.5066282746
q =0.8333333333Mean =Np =3.33333333334.1777137911
N =200.23936536820.2393653682
nBinomialGaussian
00.02608405330.0323945799
10.10433621320.0898364794
20.19823880510.1738149317
30.23788656610.2346256164
40.20220358120.2209620842
50.1294102920.1451824347
60.0647051460.0665525008
70.02588205840.0212847557
80.0084116690.0047492709
90.00224311170.0007393315
100.00049348460.0000802981
110.00008972450.0000060845
120.00001345870.0000003217
130.00000165650.0000000119
140.00000016560.0000000003
150.00000001330
160.00000000080
1705.99902724909831E-16
1803.65745487773748E-18
1901.55571882066023E-20
202.73511122779126E-164.61675917602385E-23
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Sheet4
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Binomial
Gaussian
No. of Heads
Probability
Twenty Coins
Sheet5
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Page &P
No. of Heads
Probability
Gaussian
Sheet6
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Page &P
No. of Heads
Probability
Binomial
Sheet7
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Page &P
No. of Heads
Probability
Gaussian
Sheet8
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Binomial
Gaussian
No. of Heads
Probability
Twenty Coins
Sheet9
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Sheet10
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Sheet11
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Sheet12
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Sheet13
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Sheet14
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Sheet15
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For 100 Coins
Thermo & Stat Mech - Spring 2006 Class 16
Chart3
7.8886090522101E-31
7.88860905221012E-29
3.90486148084402E-27
1.27558808374238E-25
3.09330110307528E-24
5.93913811790455E-23
9.40363535334876E-22
1.26277389030683E-20
1.4679746474817E-19
1.50059630631463E-18
1.36554263874631E-17
1.11726215897426E-16
8.28636101239238E-16
0
0
0
0
0
0
0.0000000001
0.0000000004
0.0000000016
0.0000000058
0.0000000196
0.0000000629
0.0000001913
0.0000005519
0.0000015125
0.0000039434
0.0000097904
0.0000231707
0.0000523209
0.000112817
0.0002324713
0.0004581053
0.0008638557
0.0015597394
0.0026979276
0.00447288
0.0071107323
0.0108438667
0.0158690732
0.0222922695
0.0300686426
0.0389525598
0.0484742966
0.0579583981
0.0665905
0.0735270104
0.0780286641
0.0795892374
0.0780286641
0.0735270104
0.0665905
0.0579583981
0.0484742966
0.0389525598
0.0300686426
0.0222922695
0.0158690732
0.0108438667
0.0071107323
0.00447288
0.0026979276
0.0015597394
0.0008638557
0.0004581053
0.0002324713
0.000112817
0.0000523209
0.0000231707
0.0000097904
0.0000039434
0.0000015125
0.0000005519
0.0000001913
0.0000000629
0.0000000196
0.0000000058
0.0000000016
0.0000000004
0.0000000001
0
0
0
0
0
0
8.28636101239238E-16
1.11726215897426E-16
1.36554263874631E-17
1.50059630631463E-18
1.4679746474817E-19
1.26277389030683E-20
9.40363535334876E-22
5.93913811790455E-23
3.09330110307528E-24
1.27558808374238E-25
3.90486148084402E-27
7.88860905221012E-29
7.8886090522101E-31
Successes
Probabilty
Binomial Distribution
Sheet1
Distributions
Dice Pair
nprob
20.0277777778
30.0555555556
40.0833333333
50.1111111111
60.1388888889
70.1666666667
80.1388888889
90.1111111111
100.0833333333
110.0555555556
120.0277777778
Binomial Distribution
p =0.5
q =0.5
N =100150
npCalcExp
07.8886090522101E-311.18329135783152E-283100101029.328133333313.05
17.88860905221012E-291.18329135783152E-269114201940.704411.0712142643
23.90486148084402E-275.85729222126602E-25356956547044.42822222221
31.27558808374238E-251.91338212561357E-23448876961320.70595555562
43.09330110307528E-244.63995165461292E-224184686916431.27115555562
55.93913811790455E-238.90870717685682E-21141331247049.13128888893
69.40363535334876E-221.41054530300231E-1940001302433.02417777783
71.26277389030683E-202.962.962.9633.563.32Mean =3.1266666667pN =503
81.4679746474817E-191.11283421951.07628992381.24835892271.32664991611.23547561691.2560254774SD =1.234485947953
91.50059630631463E-180.22256684390.21525798480.24967178450.26532998320.24709512340.25120509553
101.36554263874631E-170.23257018063
111.11726215897426E-164
128.28636101239238E-164
1304
1404
1504
1604
1704
1804
190.00000000014
200.0000000004
210.0000000016
220.0000000058
230.0000000196
240.0000000629
250.0000001913
260.0000005519
270.0000015125
280.0000039434
290.0000097904
300.0000231707
310.0000523209
320.000112817
330.0002324713
340.0004581053
350.0008638557
360.0015597394
370.0026979276
380.00447288
390.0071107323
400.0108438667
410.0158690732
420.0222922695
430.0300686426
440.0389525598
450.0484742966
460.0579583981
470.0665905
480.0735270104
490.0780286641
500.0795892374
510.0780286641
520.0735270104
530.0665905
540.0579583981
550.0484742966
560.0389525598
570.0300686426
580.0222922695
590.0158690732
600.0108438667
610.0071107323
620.00447288
630.0026979276
640.0015597394
650.0008638557
660.0004581053
670.0002324713
680.000112817
690.0000523209
700.0000231707
710.0000097904
720.0000039434
730.0000015125
740.0000005519
750.0000001913
760.0000000629
770.0000000196
780.0000000058
790.0000000016
800.0000000004
810.0000000001
820
830
840
850
860
870
888.28636101239238E-16
891.11726215897426E-16
901.36554263874631E-17
911.50059630631463E-18
921.4679746474817E-19
931.26277389030683E-20
949.40363535334876E-22
955.93913811790455E-23
963.09330110307528E-24
971.27558808374238E-25
983.90486148084402E-27
997.88860905221012E-29
1007.8886090522101E-31
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Page &P
Sheet1
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Number
Probability
Dice
Sheet2
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Page &P
Successes
Probabilty
Binomial Distribution
Sheet3
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Page &P
Calc
Exp
No of Heads
No of occurences
Experimental Coin Toss
Sheet4
Binomial Distribution
p =0.1666666667Sqrt(Npq) =1.66666666672.5066282746
q =0.8333333333Mean =Np =3.33333333334.1777137911
N =200.23936536820.2393653682
nBinomialGaussian
00.02608405330.0323945799
10.10433621320.0898364794
20.19823880510.1738149317
30.23788656610.2346256164
40.20220358120.2209620842
50.1294102920.1451824347
60.0647051460.0665525008
70.02588205840.0212847557
80.0084116690.0047492709
90.00224311170.0007393315
100.00049348460.0000802981
110.00008972450.0000060845
120.00001345870.0000003217
130.00000165650.0000000119
140.00000016560.0000000003
150.00000001330
160.00000000080
1705.99902724909831E-16
1803.65745487773748E-18
1901.55571882066023E-20
202.73511122779126E-164.61675917602385E-23
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Sheet4
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Binomial
Gaussian
No. of Heads
Probability
Twenty Coins
Sheet5
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No. of Heads
Probability
Gaussian
Sheet6
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Probability
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No. of Heads
Probability
Gaussian
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Binomial
Gaussian
No. of Heads
Probability
Twenty Coins
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For 1000 Coins
Thermo & Stat Mech - Spring 2006 Class 16
Chart1
9.33263618503283E-302
9.33263618503232E-299
4.66165177442391E-296
1.55077615695827E-293
3.86530957121872E-291
7.69969666586726E-289
1.2768663637564E-286
1.81315023653398E-284
2.25057273109768E-282
2.48063127694337E-280
2.45830559545074E-278
2.2124750359058E-276
1.8234481754256E-274
1.38582061332353E-272
9.77003532393038E-271
6.42216988626395E-269
3.95364833623102E-267
2.28846468403033E-265
1.24975599133427E-263
6.45926517626487E-262
3.16826956895775E-260
1.47852579884703E-258
6.57943980486894E-257
2.79769223007052E-255
1.13889387865781E-253
4.44624170228036E-252
1.66734063835505E-250
6.01477696947373E-249
2.09013499689201E-247
7.00555592061776E-246
2.26746493297316E-244
7.09497091930352E-243
2.14844588150148E-241
6.30210791907139E-240
1.79239363462991E-238
4.94700643157885E-237
1.3260725573537E-235
3.45495660888891E-234
8.75558740621108E-233
2.15971156019862E-231
5.18870702337748E-230
1.2149167664493E-228
2.77405995005938E-227
6.1803475166436E-226
1.34422558487006E-224
2.85573257585713E-223
5.92874915205155E-222
1.20340993426742E-220
2.38927014032692E-219
4.64201055834918E-218
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1.36166868889373E-89
3.27340428309043E-90
7.81889884451013E-91
1.85565955271449E-91
4.37569104405492E-92
1.02513107074042E-92
2.38608094051629E-93
5.51762874313763E-94
1.26756335991006E-94
2.89284398703385E-95
6.55853109805506E-96
1.4770743231849E-96
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7.34323828152231E-98
1.62088552311661E-98
3.55370760244789E-99
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1.67372102581788E-100
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1.62497186972616E-102
3.41892509470781E-103
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1.48210601093987E-104
3.05349551651484E-105
6.24662139359189E-106
1.26884497057343E-106
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1.01869664525809E-108
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1.25353346589304E-119
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3.86406395164891E-121
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1.15402828606121E-122
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3.3380853167071E-124
5.60891815176019E-125
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1.76839859634159E-188
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3.00146258865374E-214
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2.38927014032692E-219
1.20340993426742E-220
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1.34422558487006E-224
6.1803475166436E-226
2.77405995005938E-227
1.2149167664493E-228
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2.15971156019862E-231
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3.45495660888891E-234
1.3260725573537E-235
4.94700643157885E-237
1.79239363462991E-238
6.30210791907139E-240
2.14844588150148E-241
7.09497091930352E-243
2.26746493297316E-244
7.00555592061776E-246
2.09013499689201E-247
6.01477696947373E-249
1.66734063835505E-250
4.44624170228036E-252
1.13889387865781E-253
2.79769223007052E-255
6.57943980486894E-257
1.47852579884703E-258
3.16826956895775E-260
6.45926517626487E-262
1.24975599133427E-263
2.28846468403033E-265
3.95364833623102E-267
6.42216988626395E-269
9.77003532393038E-271
1.38582061332353E-272
1.8234481754256E-274
2.2124750359058E-276
2.45830559545074E-278
2.48063127694337E-280
2.25057273109768E-282
1.81315023653398E-284
1.2768663637564E-286
7.69969666586726E-289
3.86530957121872E-291
1.55077615695827E-293
4.66165177442391E-296
9.33263618503232E-299
9.33263618503283E-302
Successes
Probabilty
Binomial Distribution
Sheet1
Distributions
Dice Pair
nprob
20.0277777778
30.0555555556
40.0833333333
50.1111111111
60.1388888889
70.1666666667
80.1388888889
90.1111111111
100.0833333333
110.0555555556
120.0277777778
Binomial Distribution
p =0.5
q =0.5
N =1000275271
npCalcExp
09.33263618503283E-3022.56647495088403E-299200020000000018.17496859513.05
19.33263618503232E-2992.56647495088389E-296281214521244228113.63501487611.0712142643
24.66165177442391E-2961.28195423796658E-293633113321135610612664.84605619831
31.55077615695827E-2934.26463443163524E-2917192107526610952130.01502148762
43.86530957121872E-2911.06296013208515E-28877989611867211030874.77629090912
57.69969666586726E-2892.1174165831135E-2862642131235212130102.49277355373
61.2768663637564E-2863.51138250033009E-2844001010101002435.65175537193
71.81315023653398E-2843.62.883.362.83.162.882.83.322.82.43.16Mean =3.0145454545pN =5003
82.25057273109768E-2821.03846039891.17711511761.01508620321.46969384571.34699665921.17711511761.24579292021.22376468331.26491106410.9380831521.12SD =1.220420018615.81138830083
92.48063127694337E-2800.20769207980.23542302350.20301724060.29393876910.26939933180.23542302350.2491585840.24475293670.25298221280.18761663040.224M of Ms =3.01454545453
102.45830559545074E-278SD of Ms =0.32328888393
112.2124750359058E-2764
121.8234481754256E-2744
131.38582061332353E-2724
149.77003532393038E-2714
156.42216988626395E-2694
163.95364833623102E-2674
172.28846468403033E-2654
181.24975599133427E-2634
196.45926517626487E-2624
203.16826956895775E-260
211.47852579884703E-258
226.57943980486894E-257
232.79769223007052E-255
241.13889387865781E-253
254.44624170228036E-252
261.66734063835505E-250
276.01477696947373E-249
282.09013499689201E-247
297.00555592061776E-246
302.26746493297316E-244
317.09497091930352E-243
322.14844588150148E-241
336.30210791907139E-240
341.79239363462991E-238
354.94700643157885E-237
361.3260725573537E-235
373.45495660888891E-234
388.75558740621108E-233
392.15971156019862E-231
405.18870702337748E-230
411.2149167664493E-228
422.77405995005938E-227
436.1803475166436E-226
441.34422558487006E-224
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471.20340993426742E-220
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523.00146258865374E-214
535.36865383781807E-213
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562.73271191162254E-209
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601.84251890180376E-204
612.83929142245154E-203
624.30015265432604E-202
636.40244950755179E-201
649.37358623215057E-200
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661.91221159135873E-197
672.66568003929694E-196
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706.57051201213456E-193
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1911.85565955271449E-91
1927.81889884451013E-91
1933.27340428309043E-90
1941.36166868889373E-89
1955.62823058076105E-89
1962.31159470281244E-88
1979.43412254345842E-88
1983.82606080929125E-87
1991.54196018545315E-86
2006.1755505427395E-86
2012.45793056427457E-85
2029.72221049928351E-85
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&A
Page &P
Sheet1
&A
Page &P
Number
Probability
Dice
Sheet2
&A
Page &P
Successes
Probabilty
Binomial Distribution
Sheet3
&A
Page &P
Calc
Exp
No of Heads
No of occurences
Experimental Coin Toss
Sheet4
Binomial Distribution
p =0.4Sqrt(Npq) =4.381780462.5066282746
q =0.6Mean =Np =3210.9834947944
N =800.09104570250.0910457025
nBinomialGaussian
01.78689910246017E-180
19.53012854645423E-170
200
300
400.0000000001
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6100
6200
6300
6400
6500
6600
678.96363343417974E-160
681.14242386906213E-162.00364458596945E-16
691.32454941340537E-172.99370347549111E-17
701.3876231949961E-184.24597440189231E-18
711.30293257746112E-195.71645063346486E-19
721.08577714788427E-207.3056025068393E-20
737.93261843203118E-228.86270137272028E-21
745.00255216434398E-231.02060264387352E-21
752.66802782098346E-241.11564945532018E-22
761.17018764078222E-251.15765545400271E-23
774.05259789015486E-271.1402796725712E-24
781.03912766414227E-281.06616381384621E-25
791.7538019647971E-309.46274179539422E-27
801.46150163733092E-327.9724267427055E-28
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Sheet4
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Binomial
Gaussian
No. of Heads
Probability
Both
Sheet5
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No. of Heads
Probability
Gaussian
Sheet6
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No. of Heads
Probability
Binomial
Sheet7
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No. of Heads
Probability
Gaussian
Sheet8
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Binomial
Gaussian
No. of Heads
Probability
Both
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Sheet16
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Multiple Outcomes
Thermo & Stat Mech - Spring 2006 Class 16
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Stirlings Approximation
Thermo & Stat Mech - Spring 2006 Class 16
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Number ExpectedToss 6 coins N times. Probability of n heads:
Number of times n heads is expected is:n = N P(n)
Thermo & Stat Mech - Spring 2006 Class 16
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