PART%20TWO%20Statistical%20Physics%20Chapter%20III:Statistic%20Distributions%20for%20ideal%20gases

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PART TWO Statistical Physics Chapter
IIIStatistic Distributions for ideal gases

• 32 Statistics Regularities. Distributions, Most
  Probable Distributions
• (???????,??????)
• The main objective of statistical physics
• 1) to establish the behavior laws for macroscopic
  quantities of a substance.
• 2) to offer a theoretical substantiation(??) of
  thermodynamic laws on the basis of atomic and
  molecular ideas.

2
Basic Methods

• Condition a system consisting of a large number
  N of molecules(????????????).
• Classical using Newtons classical
  mechanics(????) to describe the state of the
  system
• Quantum using quantum-mechanical description,
  the ideal of wave mechanics.

3
Newtons classical mechanics(????)

• Ignoring intramolecular(???) structure, to
  visualize a molecule as a point or particle.
• The equation of Newtons motion for each of the N
  particles.

• Fihith?hth??????vi velocity.
• ???????1)???????????????
• 2)???6N??????????????????
• 3)????????,??????????

4
? ? ? ? ? ? ?

• ???????????,??????????????????1025m-3?
• ???????????????,?????????????????????
• In a system consisting of a great number of
  particles new purely statistical or probability
  laws take effect that are foreign to (????) a
  system containing a small number of particles.

5
Statistical Method

• Assumption It is possible to measure rapidly the
  energy of each molecule of a gas. The results of
  such measurement are present graphically in the
  Figure.

• The axis of the abscissa(????) is subdivided into
  equal sections each ?0 long, and ?0 is
  sufficiently small enough. All energies in
  l?0(l1)?0 is assumed to be equal to l?0.

6
Energy Distribution Function and Boxes

• The relative number of molecules in the range
  l?0(l1)?0 is denoted by n(l)

• N is the total number. n(l) is energy
  distribution function for molecules or energy
  distribution
• To divide the x-axis into longer unequal segments
• Boxes
• The number of molecules with lower or higher
  energies is very small.(?????????????)

7
Cell

• A box is a larger unit which contains several
  cells molecules have the fully same energy.
• A box which energy l?0(lm)?0,
• m?0 is the length of a box, which varies little.
• Actually, all the histograms(????) will be close
  to some averaged histogram and large deviations
  from it will be rare.

8
Microstate ????

• To describe the state of a gas at some moment of
  time
• Microstate
• Classical mechanics coordinates(??) and
  velocity
• ???????????,????????

• ??????????????????????????,?????N????????????,
  ??????-???????????,????????????,N??????(?????)??
  ???,???????????

9

• ????????????????

• ??????????????

• It is possible to find the mean values of any
  energy function if the momentum distribution
  are know.

• ???????????P?V?T?S?????????(??????),??????????????
  ????????????????????,?????????????

10
Basic physical postulate of statistical physics

• the greatest number of microstates of the most
  probable distribution and is equivalent to the
  equilibrium state of thermodynamics.------?????

• Thermodynamics assumes that a system remains in a
  state of equilibrium indefinitely long, but
  statistical physics predicts there existence of
  fluctuations(??) spontaneous(??) and rare
  deviations from the equilibrium state. ------?????

11
??????????

• The problem of finding the most probable
  distribution for ensembles of non-interacting
  particles or for ideal gases.

12
33. ?-Space. Boxes and Cells

• ?????-Space(???????)??????????
• ?-Space x,y,z, ?(ksai),?(eta),?(zita). System
  has N points.
• This six-dimensional surface is specified by the
  equation

• The concept of the phase volume(???) in the
  ?-Space is introduced by the expression

• Subdivided into(??) the volume in the
  configurational space and in the momentum space.
  (?????????)

13

• It might be convenient to select the spherical
  layers(???) dV4?rdr2, dVp4?pdp2.
• ?????????

• ???????dxd? ??.

?????????????????????, ???????????,???????
14
Boxes in the ?-Space

• The qi and pi are applied to represent coordinate
  and momentum. It is not homogeneous (???) in all
  the space.
• In phase volume d?, the number of representative
  points is dN. The density is ?(qi,pi)
  dN/d?.
• A postulate is introduced the distribution
  function for the ?-Space, ?(qi,pi), depends only
  on the particle energy ? and not on qi and pi
  individually.

15

• The ?-Space is subdivided into boxes by
  carefully drawing the hypersurfaces of constant
  energy. This energy layer is sufficiently thin
  that the representative points???confined in the
  layer have the same energy ? .

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??????????

• ???????,?0, 2?0, 3?0?,????cells
  (??)?6???,??6????????,????12?0,?????????,???????
• ????????????

????????????,???????,??????????????1,6156, 153,
202?
17
34 Bose-Einstein and Fermi-Dirac Distributions

• Subdivision non-equidimensional energy boxes and
  equidimensional cells.
• The ith energy box having an energy ?i ,gi
  cells, Ni representative points.
• How do these representative points distribute
  among the cells.?
• Principle any arrangement of representative
  points in the cells to be equiprobable. ????
• The distribution is realized by the most probable
  distribution,--- the equilibrium state.

18
Two Hypotheses ????

• 1. All particles of one kind are absolutely
  identical to one another (???????).
• 2. These particles differ slightly just as
  producting-line(???) identical parts produced in
  a factory differ from one another.
• Both of above
• particles of one kind are identical

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????????????

• N??????????,???????????????,??????????????????????
  ????????????????????????????
• ??????,?????????,?????????????????????????,???????
  ????????
• ????????????????????,---????????????,??????????
  ????????????,???????

20
??????

• ???????????fermion and boson
• Fermions follow an important law
• the Pauli exclusion principle
• in a system of N identical fermions one cell in
  the ?-space can contain no more than one
  representative point.
• in a system of N identical bosons one cell in the
  ?-space can contain any number representative
  points from zero to N.

21
Statistical properties of the different particles

• To illustrate the difference in the statistical
  properties of the different particles by a simple
  example
• Arrange two particles on three cells 1, 2, 3
• For the classic particles, they are
  distinguishable

22
??
???

• ???

23

• Classical 9 arrangements
• Bosons 6 arrangementsFermion 3 arrangements.
• How about Ni particles in gi cells?

For the Boson
How to express?
The ith box
24
Analyses
25
Calculate Wi

• Wi denoted as the number of different ways of
  arranging Ni particles in gi cells.
• Two classes of objects Particles partitions
• ?? Ni ????gi-1
• ???????????????
• (1)??????(2)??????
• ??,?????????????,??????
• (Ni gi-1)! ?????????Ni !
• ??????(gi-1)!

26
Boson??????

• ????????????,???W??????????,??lnW???????? lnW

???????
27
?? ????

• ???????the total number of gas particles and the
  total energy of the gas are fixed.
• ????????????????????????

??????????????,???? ??N - ? U????????
The most probable number of particles in a cell
is
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Fermi-Dirac distribution, Fermion (???)

• Bose-Einstein distribution are specified by

• Fermions are considered. For the ith energy box
  with a number of cells gi, and a number of
  particles Ni (Ni lt gi), the different ways of
  distribution differ from each other only in that
  some cells are occupied by one particle and some
  cells are empty---permutations(??) of empty cells

29

• ??,???????????,???????????????????????(gi-Ni),
  ????gi!,????????Ni!,??????(gi-Ni)!?????

N?????????
?????W??????????
30
???????

• ????Lagrangian??,????????

?????Box?????
?????????????????????
31
35. The Boltzmann Principle

• Two statistical distributions are known, but the
  meaning to be imparted(??) .
• The important is to know the meaning of two
  parameters ? and ?. Make physical postulate

• 1) WW1W2Wn
• ? ?1?2?n
• ? is a extensive quantity

32

• 2) In an isolated system, and in ???
• Thermodynamics
• Statistical Physics

• ?????S?????????????
• ??????? ??S ?????????
• These arguments make it reasonable to postulate
  that with a degree of accuracy up to a constant
  multiplier thermodynamically defined entropy
  coincides(??) with the quantity ?.

33
? ?
Boltzmann consider this situation and thought
that there must be some internal connection
between them. He applied a multiple constant to
establish an equation
------The Boltzmann Principle

• Boltzmann endue(??) the entropy an statistical
  meaning . ????????????
• It is convenient to use another definition form.

34

• ??????,?????????????

• Here , the entropy is assumed to be a
  dimensionless quantity. Since the product TdS
  must have the dimension of energy, then the
  temperature must be in the energy units.
• ??S??????,??????????
• The entropy of a system in a state of equilibrium
  is

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??????

• Boltzmann the increase in entropy in an
  equalization process is the result of the system
  passing from a less probable states to the most
  probable state.

???????,??????????????,??????????????????????????,
??????
??????????????,?????????????,????????????????????
?????,??,??
????
??,???????????????.
???????,???,??????(??S1)
36
About an irreversible process

• What is the fundamental difference between the
  statistical interpretation and the thermodynamic
  interpretation?
• From thermodynamics a reverse process is
  impossible by definition.
• From statistical physics a transition from the
  most probable distribution to the less probable.
• (?????????,???????)

37
The discuss of distribution

• The Boltzmann principle can be used to find the
  meaning of two parameters. By the formulae

The upper sign pertains to the Bose distribution,
and the lower to the Fermi distribution. The
entropy S is related with both N and U.

• ???BOX????CELLS?????,????????cell????????Cells
  ??,?????box??????????????Cells???,???

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Occupation Number ?????

• ?????cell???
• ????????

• ???????????????????,??????????????1 ???????ni
  Ni / g i ,??????????????????

• ??,????????????????????????,??????

39
??????(????)

• ??????ni????

• ??

?????????,??
40
??????(????)

• ???????ni????

??
41
???????

• Here, minus ? bosons positive ? fermions.
• Important emphasize
• 1) ?(35.5)??(35.8) ???????????
• 2)?(35.7) ??(35.10)???????,???????????????????????
  ????n1,n2,,ni????????????????------????????

42
The meaning of the parameters ? and ?

• Compare Eq(35.11) with the expression dS in
  thermodynamics.
• ?????gi, ?i????
• 1) ???,S????
• 2) ???,S?? ???
• gi??I???????????,??V???
• ??????????, ??(35.11)???????,????V??,?

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and ? ????
44
and ? ??

• ??(35.12)??(35.13)????

??
45

• ??????????k????,???N????,?????????????
• Nak 6.02310231.3810 23 8.31 (J K-1mol-1)
• ????R 8.314 (J K-1mol-1) k R/NA.
• ?????????PV nRT NkT.
• ??,N?????????n??????
• Accordingly, in all following sections the
  chemical potential ? does not refer to one mole
  of substance, as in thermodynamics, but to one
  particle, so that ?therm NA ?stat is true. (
  ?s ,T)??(?,?)????

46
???????????

• From the Eq.(35.11), the entropy is

????i????????,????????????????????????????????????
?,????????????????,??,??????????????????????

• G ? N(??????????)?
• F U TS G PV,F GU TS G U -TS- ? N

47

• ????

• ??,????????

?????????,???????????? ????????,
?????????,???????N???T???,?????,?????????????????
??????N???T???? ------??????????
48

• ?????,???????U/N?????????,????????V/N?????
• ??????????,???????

?i is the function of intensive parameters, i.e.
field intensities.
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In Conclusion

• The B.E. distribution and F.D. distribution are
  derived by the box-cell method presupposing
  (????) that thermodynamic equilibrium state sets
  in.
• The initial non-equilibrium particle distribution
  ? equilibrium distribution ? particles change
  their boxes to a equilibrium state.
• Reason Ncons. and particles interact with
  surrounding walls (thermostat).
• Indeviation both N and U are fixed, only T.

50
36. The Maxwell-Boltzmann Distribution

• Question How does classical particles
  distribute?
• Let Box1 for N1, Box2 for N2,
• Box n for Nn, .
• ??????????N! ?
• ???N1????BOX1????
• ???N2????BOX2????
• ??,?N??????BOX ?????

51
?????

• ???BOX?,?gi?cells,Ni??????????????????,??????????,
  ?????????????????
• ???BOX???????????

giNi
??????????????
??Stirling??,??
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• ??????????

?Ni????????
???????
53

• If for any ?i the condition exp(?i -?)/kT gtgt1 is
  satisfied, the unity in the denominator can be
  ignored and we obtain the Maxwell-Boltzmann
  distribution

• ????????????????,??

In this rarefied gas, the average interparticle
distances are large, so they cannot be
confused---distinguishable. ????????,?????????,???
???????????????????????
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??????????????????????

• The Boso-Einstein and the Fermi-Dirac
  distributions are valid for all particles, thile
  the Maxwell-Boltzmann distribution is
  approximately true in the limiting case of small
  occupation number.
• The entropy of a gas in an arbitrary equilibrium
  or non-equilibrium state can be obtained in two
  ways

55
????

• If the Boltzmann formula SlnW is used, the
  classical gas (36.2) would follow

It is not true, otherwise, S will be not an
extensive quantity.------ we return to the Gibbs
paradox.
56
Gibbs ???

• Gibbs foresight is worthy of admiration, for as
  far back as the end of the nineteenth century he
  anticipated the present-day concept of the
  indistinguishability of particles.
• ??????,???????????????,????????????gi???,Ni????or
  close to unity.
• In these conditions, the Stirlings formula
  becomes incorrect for Ni and gi.
• An general Gibbs method can be applicable to
  ideal gses but also the systems of interacting
  particles.
• Problem Page 190

57

• ????????
• ?????L,????N????,????????????????
• ?????????

??????? L/n. n??????????k2?/ ?. ??????????,??
58
?????
59
??P228 6.2, 6.4, 6.5

• ???????????????(???)?????????,????????(Ldp)?h?????
  ?????????h3??????????????????
• ????-????? p2/2m.
• ?????,???????
• ?????????,????
• ??(??---BOX,??---Cell)
• ?????????,???????????????D(E)????

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????? k ?

• ???????,???????
• p hk mv
• ???????????
• ????????????
• ??????????,
• ???????????
• ?????(??)???????????(??)?,?????????

61
What is the concept of Boxes

• Here, D(?) is defined as the density of state.

dN ? is the number of energy in the range of ?
? d?. How about D(?) V, m, ? ?
What is the box? One box is one
state(????,?????), or one line in the figure,
about one value of ? .
62
37. Transition to continuously Varying Energy.
Degeneracy Conditions for Ideal Gases

• Three Statistical Distributions
• 1) the Bose-Einstein distribution
• 2) the Fermi-Dirac distribution
• 3) the Maxwell-Boltzmann distribution

63
Discussion

• In deriving the statistical distributions, the
  energy was a discretely varying quantity
  ------Box.
• If it is suitable? In what degree? Size of the
  cell?
• If the energy layers(boxes) are sufficiently
  thin, we can even replace above summation by
  integration.
• How do we integrate?
• By a new concept phase volume ----d??dqidpi
• In this volume the particle number is dN.
• If the volume of one cell is a, g weight
  factor

64
The meaning of g

• For instance, the spin of a particle is s, the
  projection of the spin in any direction have 2s1
  different values(-s, -s1, s-1,s). In this case
  g 2s1.
• The light quantum, photon, has not spin, but has
  two vibrational directions, g 2.
• The photon is Boson.
• The electron, the Fermion, g 2s1 2.

65
Three distributions

• The important distinction(??)?
• For the boson and fermion, a can be solved by the
  comparison with the experiment results of Cv.

66
The discussion of Maxwell-Boltzmann distribution

• The situation is quite different for the
  Maxwell-Boltzmann distribution. The chemical
  potential and the cell volume are presented in
  the same form exp(?/T)/a.

But two others
67
Distinction

• The energy depends on ? and, consequently, (??)
  on a?For the exact statistical Fermi-Dirac and
  Bose-Einstein distribution, the volume of a cell
  is not arbitrary exactly by laws of nature and
  by experiment.
• Impossible in the case of small occupation
  numbers, because of the phase volume of a cell
  acquires arbitrary value.

?????,??????????????????????????,?????
68
The criterion of validity of the
Maxwell-Boltzmann distribution

• In the case of mono-atomic ideal gas

???????,??????????
??Maxwell-Boltzmann ???????
69

• ?????????,????????a h3,(Plancks constant), and
  g 1
• The criterion of validity of the
  Maxwell-Boltzmann distribution is

The criterion are low density, high temperature,
and large molecular masses m.
70
????

• The reverse criterion

• The MB distribution is inapplicable
• The gas obeys the BE or FD distribution.
• The gas is then said to be degenerate(??)
• Eq.(37.9) is known as the degeneracy criterion
• For a gas at low temperatures shows quantum
• and at high temperatures shows
  classical.

71
Examples

• For the ordinary atoms, N/V1019cm-3, m ( 10-23
  to 10-24 )g, Tltlt10-1K as quantum.
• For ordinary gases, the normal MB distribution is
  a good approximation down to rather low
  temperatures.
• ?????????,????????????????????????????????---????
  ?
• ?????????10-27g, ??????104K?