We calculate the average momentum distribution and the average deviation from this distribution for a one-dimensional ideal gas of atoms. The calculation is performed using analytical and numerical methods, and the results obtained using the two methods are compared. We use these results to show that in the limit of large , almost all the possible microstates of an ideal gas have momentum distributions that are very close to the Maxwell distribution. We discuss the significance of this fact for understanding why an ideal gas approaches thermal equilibrium.
REFERENCES
1.
The program used to perform the numerical experiment will be provided upon request.
2.
Computer simulations of the evolution of an ideal gas toward thermal equilibrium are discussed in
J.
Novak
and A. B.
Bortz
, “The evolution of the two-dimensional Maxwell–Boltzmann distribution
,” Am. J. Phys.
38
(12
), 1402
–1406
(1970
) andA. D.
Boozer
, “Time asymmetry in a dynamical model of the one-dimensional ideal gas
,” Am. J. Phys.
76
(11
), 1026
–1030
(2008
).3.
The first property is established in
K.
Huang
, Statistical Mechanics
, 2nd ed. (Wiley
, New York
, 1987
), Sec. 4.3.4.
A close analog to the second property is sometimes established by employing the canonical rather than microcanonical ensemble. See, for example,
R. C.
Tolman
, The Principles of Statistical Mechanics
(Oxford U. P.
, Oxford
, 1938
), p. 506
.5.
An overview of various proofs of this result is given in
P. A.
Mello
and T. A.
Brody
, “A different proof of the Maxwell–Boltzmann distribution
,” Am. J. Phys.
40
(9
), 1239
–1245
(1972
).6.
By an “-sphere” we mean a sphere in -dimensional Euclidean space. According to this convention a circle is a 2-sphere.
7.
Here is the step function, defined such that for , for , and for .
9.
One can show that the expected number of trials is , where , the solid angle for the unit -sphere, is given by Eq. (11). The function grows very rapidly with increasing , so this algorithm is practical only for small values of .
10.
For simplicity we compute and only for values of that are integer multiples of . This procedure is equivalent to making a histogram of the atom momenta in which the size of the momentum bins is .
11.
Equation (11) is derived in
M. E.
Peskin
and D. V.
Schroeder
, An Introduction to Quantum Field Theory
(Addison-Wesley
, Reading, MA
, 1995
), p. 249
.12.
Equation (15) is derived in
R.
López-Ruiz
and X.
Calbert
, “Derivation of the Maxwell distribution from the microcanonical ensemble
,” Am. J. Phys.
75
(8
), 752
–753
(2007
), and inJ. R.
Ray
and H. W.
Graben
, “Small systems have non-Maxwellian momentum distributions in the microcanonical ensemble
,” Phys. Rev. A
44
(10
), 6905
–6908
(1991
).
[PubMed]
13.
The analog to Eq. (15) for a two-dimensional ideal gas is derived in
S.
Velasco
, J. A.
White
, and J.
Guemez
, “Single particle energy and velocity distributions for finite simple systems in the microcanonical ensemble
,” Eur. J. Phys.
14
(4
), 166
–170
(1993
).© 2010 American Association of Physics Teachers.
2010
American Association of Physics Teachers
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