An important result of statistical mechanics is the Boltzmann equation, which describes the evolution of the velocity distribution of a gas towards the equilibrium Maxwell distribution. We introduce the Boltzmann equation by considering a dynamical model of a two-dimensional gas consisting of hard disks. We derive the Boltzmann equation for the model and compare the behavior predicted by this equation against the actual behavior of the system as observed in computer simulations. A puzzling feature of the Boltzmann equation is that although the dynamical laws governing the gas are time-reversal invariant, the behavior predicted by the Boltzmann equation is time asymmetric. We show that this time asymmetry arises from assumptions made in the derivation of the Boltzmann equation, and we use computer simulations of the model system to investigate the circumstances under which these assumptions hold.

1.
A standard textbook that derives and discusses the Boltzmann equation for a three-dimensional gas is
K.
Huang
,
Statistical Mechanics
, 2nd ed. (
Wiley
,
New York
,
1987
).
2.

The Boltzmann equation can also be introduced using one-dimensional models; see, for example, Ref. 3, which derives a Boltzmann equation for the fraction of right-moving molecules in a one-dimensional single-species gas, and Refs. 4 and 5, which derive Boltzmann equations for the velocity distributions of molecules in one-dimensional two-species gases. These models are very instructive, but must be used with caution, since they have unique properties that do not carry over to higher dimensions.

3.
G. L.
Baker
, “
A simple model of irreversibility
,”
Am. J. Phys.
54
(
8
),
704
708
(
1986
).
4.
P.
Mohazzabi
and
J. R.
Scmidt
, “
Maxwellian relaxation of elastic particles in one dimension
,”
Am. J. Phys.
79
(
8
),
861
866
(
2011
).
5.
A. D.
Boozer
, “
Boltzmann's H-theorem and the assumption of molecular chaos
,”
Eur. J. Phys.
32
,
1391
1403
(
2011
).
6.
The evolution algorithm we describe here is a standard event-driven molecular dynamics algorithm. Such algorithms date back to the 1950s; see, for example,
B. J.
Alder
and
T. E.
Wainwright
, “
Studies in molecular dynamics. I. General method
,”
J. Chem. Phys.
31
(
2
),
459
466
(
1959
).
For a modern introduction to molecular dynamics simulations, see
J. M.
Haile
,
Molecular Dynamics Simulation: Elementary Methods
(
Wiley
,
New York
,
1997
).
7.

Indeed, the Boltzmann assumption is not recognized as an assumption in standard treatments. For example, in Ref. 1, the analog of the Boltzmann assumption is the pair of equations (3.29) and (3.30), which are taken to be exact and always valid, rather than as assumptions that hold only for certain states of the gas.

8.

The H-function for a three-dimensional gas is discussed in Sec. 4.1 of Ref. 1.

9.

The proofs of these statements are straightforward adaptations of the corresponding proofs for a three-dimensional gas that are presented in Sec. 4.1 of Ref. 1.

10.
Numerical integration of the Boltzmann equation, as we are performing here, is computationally intensive, and alternative methods for numerically solving the Boltzmann equation have been developed. One such method, known as the direct simulation Monte Carlo method (DSMC), is described in
H.
Karabulut
, “
Direct simulation for a homogeneous gas
,”
Am. J. Phys.
75
(
1
),
62
66
(
2007
). The method relies on a stochastic algorithm for time-evolving the molecule velocity distribution; this type of algorithm is described for the case of a two-dimensional gas in Refs. 11 and 12, and for the case of a three-dimensional gas in Ref. 13.
11.
J.
Novak
and
A. B.
Bortz
, “
Evolution of the two-dimensional Maxwell-Boltzmann distribution
,”
Am. J. Phys.
38
(
12
),
1402
1406
(
1970
).
12.
R. P.
Bonomo
and
F.
Riggi
, “
The evolution of the speed distribution for a two-dimensional ideal gas: A computer simulation
,”
Am. J. Phys.
52
(
1
),
54
55
(
1984
).
13.
M.
Eger
and
M.
Kress
, “
Simulation of the Boltzmann process: An energy space model
,”
Am. J. Phys.
50
(
2
),
120
124
(
1982
).
14.
The computer programs used to numerically integrate the Boltzmann equation and to simulate the model system are provided as an online supplement to this article at http://dx.doi.org/10.1119/1.4898433.
15.

Using energy conservation, one can show that the equilibrium temperature is given by T=(m/6kB)(Wx2+Wy2).

16.

Using energy conservation, one can show that the equilibrium temperature is given by T=(m/2kB)(Wx2+Wy2).

17.

From Eq. (22), we find that the predicted equilibrium value the H-function is HE=log(4/5π)12.37.

18.
Our explanation for the origin of the time asymmetry of the Boltzmann equation differs from the explanation given in Ref. 1, which is criticized in
T. P.
Eggarter
, “
A comment on Boltzmann's H-theorem and time reversal
,”
Am. J. Phys.
41
(
7
),
874
877
(
1973
).
19.
The problem of rigorously proving that these assumptions hold for our two-dimensional gas model is open. For the one-dimensional gas model described in Ref. 5, however, one can rigorously prove that for a large class of initial states the Boltzmann assumption holds if and only if t ≥ 0 and the anti-Boltzmann assumption holds if and only if t ≤ 0; see
A. D.
Boozer
, “
Boltzmann equations for a binary one-dimensional ideal gas
,”
Phys. Rev. E
84
,
031127
(
2011
).

Supplementary Material

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