Title: PART%20TWO%20Statistical%20Physics%20Chapter%20III:Statistic%20Distributions%20for%20ideal%20gases
1PART 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.
2Basic 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.
3Newtons 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.
5Statistical 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.
6Energy 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.(?????????????)
7Cell
- 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.
8Microstate ????
- 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?????????(??????),??????????????
????????????????????,?????????????
10Basic 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.
1233. ?-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. - ?????????
?????????????????????, ???????????,???????
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 ? .
16??????????
- ???????,?0, 2?0, 3?0?,????cells
(??)?6???,??6????????,????12?0,?????????,??????? - ????????????
????????????,???????,??????????????1,6156, 153,
202?
1734 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.
18Two 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
19????????????
- 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.
21Statistical 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
24Analyses
25Calculate 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)!
26Boson??????
- ????????????,???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
28Fermi-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?????
?????????????????????
3135. 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
35??????
- 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)
36About 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. - (?????????,???????)
37The 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???,???
38Occupation Number ?????
- ???????????????????,??????????????1 ???????ni
Ni / g i ,??????????????????
- ??,????????????????????????,??????
39 ??????(????)
?????????,??
40??????(????)
??
41???????
- Here, minus ? bosons positive ? fermions.
- Important emphasize
- 1) ?(35.5)??(35.8) ???????????
- 2)?(35.7) ??(35.10)???????,???????????????????????
????n1,n2,,ni????????????????------????????
42The 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??,?
43 and ? ????
44 and ? ??
??
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.
49In 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.
5036. 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??,??
52?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. ????????,?????????,???
???????????????????????
54??????????????????????
- 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.
56Gibbs ???
- 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)????
60????? k ?
- ???????,???????
- p hk mv
- ???????????
- ????????????
- ??????????,
- ???????????
- ?????(??)???????????(??)?,?????????
61What 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 ? .
6237. 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
63Discussion
- 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
64The 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.
65Three distributions
- The important distinction(??)?
- For the boson and fermion, a can be solved by the
comparison with the experiment results of Cv.
66The 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
67Distinction
- 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.
?????,??????????????????????????,?????
68The 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 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.
71Examples
- 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?