Equation of State Calculations by Fast Computing Machines
1953 · paper · Physics
Describes a modified Monte Carlo method for computing properties, such as equations of state, of substances made of interacting molecules on fast computers. Results for a two-dimensional rigid-sphere system are compared with free-volume and virial-expansion predictions.
Checked: PDF from bayes.wustl.edu, fetched 5 Oct 2026. Page numbers are those of that PDF.
2 findings
Substance Table I prints (A/A0) = "2.41751" for ν = 6.5, but Eq. (11b), "1/0.98974329(1−2^{ν−8})²", gives 2.41775 at ν = 6.5.
- p. 6 Table I, row ν = 6.5, column (A/A0)
2.41751
- p. 5 Eq. (11b)
1/0.98974329(1
Only the authors can say what is right here.
Judge's note: Read the value in the Table I crop (2.41751) and Eq. (11b) in the page 5 crop. With 2^(-1.5) = 0.35355339 the formula gives 2.41775. I recomputed ν = 6 (1.79620, printed 1.7962) and ν = 7 (4.04145, printed 4.04145), which agree with the formula, so the ν = 6.5 entry stands alone. The constant 0.98974329 is itself right (√3/2 × 224/196). The difference is in the fourth decimal and does not affect the other columns at their printed precision; whether it is a misprint or a machine value only the authors could say.
Second check: another model agreed
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Substance C4 is printed as "C4=8π³(0.585)/135" (page 7) and with the prefactor "(8π³/135)" in Eq. (13), which evaluates to about 1.07, not the "+3.38(A0/A)⁴" of Eq. (14); π⁴ in place of π³ gives 3.38.
- p. 7 Sec. V, displayed value of C4 before Eq. (14)
(0.585)/135
- p. 6 Eq. (13), definition of C4
/135)
- p. 7 Eq. (14)
+3.38
- p. 7 Eq. (14)
+3.38(A0/A)
Judge's note: Viewed Eq. (13) on page 6 (crop) and the C4 line and Eq. (14) on page 7: the exponent of π is 3 in both places. 8π³(0.585)/135 = 1.075; 8π⁴(0.585)/135 = 3.377, which rounds to the printed 3.38. The other coefficients go as π, π² and π³ for C1, C2 and C3, so π⁴ is the expected power for C4, and Table I's X3 column follows Eq. (14) with 3.38. The printed power of π in the C4 expressions is therefore a misprint, repeated in both places; the numerical coefficient used is unaffected.
Second check: another model agreed
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4 minor (typos and the like; most checked by the judge only)
Wording Sec. V defines a bond by "f(r_ij)=1 if r_ij<d" and normalises A3,3 by π²d⁴, using d for the sphere diameter, although Eq. (11a) defines "d=(1/14)" as the lattice spacing and d0 as the forbidden distance.
- p. 7 Sec. V, definition of f(r_ij)
<d,
- p. 5 Eq. (11a)
d= (1/14)
Judge's note: Viewed Eq. (11a) on page 5 and the definition of f(r_ij) and A3,3 on page 7. Sec. V says a bond exists "if the two particles overlap", which in Sec. III happens at distances below d0, not d. The symbol d is thus used with two meanings. The cluster integrals are dimensionless, so no number is affected, and Sec. V can be read as a self-contained derivation; a minor notational point.
Second check: another model agreed
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Typo The sentence after Eq. (2) prints the volume element as "(d^{2n}p d^{2n}q)" with a lower-case n, while Eq. (2) itself has d^{2N}p d^{2N}q and the same sentence says "4N-dimensional phase space".
- p. 3 Sentence after Eq. (2)
is a volume element in the 4N-dimen
- p. 3 Eq. (2)
exp(−E/kT)
- p. 3 text after Eq. (2)
where (d2npd2nq) is a volume element in the 4N-dimen-
- p. 3 Eq. (2)
d2NPd2Nq
Judge's note: Viewed the page 3 image and the crop. Both lines of Eq. (2) carry a capital N in the exponents; the explanatory sentence clearly has lower-case n in both exponents. N is the particle number and n is not defined, so this is a typographical slip with an evident fix.
Second check: none (judge only)
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Typo Fig. 5 caption reads "The average of the extrapolated values of N½ in N̄½=6301"; "in" should be "is".
- p. 6 Fig. 5 caption
The average of the extrapolated values of N1* in N1*=6301
Judge's note: Enlarged the caption region of the page 6 image: the word is "in", clearly different from the "is" printed later in the same caption line. The value itself is right: the mean of 6367, 6160 and 6377 is 6301.3.
Second check: none (judge only)
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Typo "It is seen in Fig. 4 that the curves agrees very well with our calculated equation of state": subject and verb disagree ("the curve agrees", referring to curve C, is evidently meant).
- p. 7 Sec. V, paragraph after Eq. (14)
the curves agrees very well
- p. 7
the curves agrees very well with
Judge's note: Viewed the paragraph after Eq. (14) on the page 7 image; it reads "the curves agrees". The preceding sentence is about curve C alone.
Second check: none (judge only)