We devise an algorithm to enumerate the microstates of a system comprising N independent, distinguishable particles. The algorithm is applicable to a wide class of systems such as harmonic oscillators, free particles, spins, and other models for which there are no analytical solutions, for example, a system with single particle energy spectrum given by ɛ(p,q) = ɛ0(p2 + q4), where p and q are non-negative integers. Our algorithm enables us to determine the approach to the limit N → ∞ within the microcanonical ensemble, and makes manifest the equivalence with the canonical ensemble. Various thermodynamic quantities as a function of N can be computed using our methods.
REFERENCES
1.
J. R.
Drugowich de Felicio
and V. L.
Libero
, “Updating Monte Carlo algorithms
,” Am. J. Phys.
64
, 1281
–1285
(1996
).2.
D. P.
Landau
, S -H
Tsai
, and M.
Exler
, “A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling
,” Am. J. Phys.
72
, 1294
–1301
(2004
).3.
J.
Tobochnik
, H.
Gould
, and J.
Machta
, “Understanding temperature and chemical potential using computer simulations
,” Am. J. Phys.
73
, 708
–716
(2004
).4.
J.
Tobochnik
and H.
Gould
, “Teaching statistical physics by thinking about models and algorithms
,” Am. J. Phys.
76
, 353
–359
(2008
).5.
Ashley H.
Carter
, Classical and Statistical Thermodynamics
(Prentice Hall, Upper Saddle River
, NJ
, 2001
), pp. 251
–241
;6.
T. A.
Moore
and D. V.
Schroeder
, “A different approach to introducing statistical mechanics
,” Am. J. Phys.
65
, 26
–36
(1997
).7.
J. J.
Prentis
and T.
Zainiev
, “The art of statistical mechanics: Looking at microscopic spectra and seeing macroscopic phenomena
,” Am. J. Phys.
67
, 1123
–1128
(1999
).8.
We used a computer with an Intel Pentium 4 CPU 3.00 GHz and 1 GB of RAM.
9.
For our 32-bit computer, the largest real number addressable by the computer is 1.797693134862316 × 10308, and the smallest real number is 2.225073858507201 × 10−308.
© 2011 American Association of Physics Teachers.
2011
American Association of Physics Teachers
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