# Monte Carlo Simulations in Statistical Physics

• D. Stauffer

## Abstract

In Statistical Physics one mostly deals with thermal motion of a system of particles at nonzero temperatures. For example, in a classical ideal gas of point-like molecules each particle has an average kinetic energy equal to dk B T/2 in d dimensions. Here T is the absolute temperature and k B = 1.6 x 1023 Joule per Kelvin is Boltzmann’s constant. Statistical Physics is used try to explain such laws and to predict the properties of materials consisting of many such particles; therefore, in this example the specific heat is 3Nk B /2 in three dimensions if the gas consists of N particles. In most applications, the number N of particles is very large, and they influence each other by their intermolecular forces. For example, a glass of beer contains about 1025 water molecules, and if these molecules did not interact with each other, the beer would vanish by evaporation, not by drinking. These interactions are also unhealthy for theoretical physics since with interactions usually one cannot solve exactly the problem of how the molecules move and what their average energy is, because even on a computer it is not possible to store the positions and velocities of 1025 point-like molecules. (The Cray-2 supercomputer has only two Gigabytes of main memory.) Instead, one is forced to work with a much smaller number of molecules, below 106, and solve numerically the equations of motion arising from Newton’s law: force equals mass times acceleration. This method is called molecular dynamics and has already been used in the first chapter of this book by Zabolitzky. We will not deal with this technique here; readers who want to know more are referred to the book of D.W.Heermann [1].

### Keywords

Nickel Anisotropy Cobalt Agglomeration Librium

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### References

1. [1]
D.W. Heermann, Computer Simulation Methods in Statistical Physics, Springer Verlag, Berlin, Heidelberg, New York 1986Google Scholar
2. [2]
S.Kirkpatrick, E.P.Stoll, J.Comp.Phys. 40, 517 (1980)
3. [3]
K. Binder (ed.), Applications of the Monte Carlo Method in Statistical Physics, Springer Verlag, Berlin, Heidelberg, New York 1984, Chap.1 . See also the textbook: K.Binder and D.W.Heermann, Monte Carlo Simulation in Statistical Physics: An Introduction, Springer Verlag, Berlin, Heidelberg, New York, in press, which is suited as a follow-up on the present chapter for the more advanced student.
4. [4]
N.Metropolis, A.W.Rosenbluth, M.N.Rosenbluth, A.H.Teler, E.Teller, J.Chem.Phys. 21, 1087 (1953)
5. [5]
G.Y.Vichniac, Physica D 10, 96 (1984)
6. [6]
M.Creutz, Phys.Rev.Letters 50,1411 (1983)
7. [7]
D.Stauffer, Introduction to Percolation Theory, Taylor and Francis, London 1985
8. [8]
M.Eden, in: Proc. 4th Berkeley Symp. Math.Statist, and Probability, ed. F.Neyman, vol.IV, University of California, Berkeley 1961Google Scholar
9. [9]
S.A.Kauffman, J.Theor.Biol. 22, 437 (1969)
10. [10]
G.Weisbuch, H.Atlan, J.Phys. A 21, L 189 (1988)