Excerpt

One of the triumphs of statistical mechanics in the second half of the twentieth century was the progress in understanding phase transitions (Stanley 1971), where the bulk properties of a substance can suddenly change as a physical parameter is varied. For example, ice melts when the temperature is increased above 0°C. Another example, more relevant for this essay, is the transition between paramagnetism, or no magnetism, and ferromagnetism in a magnetic material (such as iron or nickel) when the temperature is decreased below a critical, material-dependent temperature.

We can visualize what happens at this transition by the following picture: each atom in the material can be viewed as a tiny bar magnet or “spin”—a fixed-length arrow that points in some direction. When a large number of these atomistic bar magnets are placed in fixed locations and in proximity, as in a real material, the interaction between these spins tends to align them as illustrated here in one dimension: ↑↑↑↑↑↑. This alignment corresponds to ferromagnetism, in which the orientational order of the spins leads to the bulk material becoming a macroscopic magnet that can attract iron filings, nails, and other magnetic materials. On the other hand, temperature acts in the opposite way. Pictorially, the effect of temperature is to agitate each spin, with the agitation increasing as the temperature is increased. This thermal “shaking” tends to randomize the directions of the spins and thereby destroy the macroscopic magnetic ordering—this corresponds to paramagnetism. Thus, in a magnetic material, such as iron, ferromagnetism occurs at low temperatures and paramagnetism occurs at high temperatures.

Bibliography

Dotsenko, V. S. 1995. “Critical Phenomena and Quenched Disorder.” Physics-Uspekhi 38 (5): 457. https://doi.org/10.1070/PU1995v038n05ABEH000084.

Hubbard, J. 1959. “Calculation of Partition Functions.” Physical Review Letters 3 (2): 77. https://doi.org/10.1103/PhysRevLett.3.77.

Ising, E. 1924. “Beitrag zur Theorie des Ferro- und Paramagnetismus.” PhD diss., Hamburg Universität.

Parisi, G. 1979. “Infinite Number of Order Parameters for Spin-Glasses.” Physical Review Letters 43 (23):1754. https://doi.org/10.1103/PhysRevLett.43.1754.

—. 1983. “Order Parameter for Spin-Glasses.” Physical Review Letters 50 (24): 1946. https://doi.org/10.1103/PhysRevLett.50.1946.

Pathria, R. K. 2016. Statistical Mechanics. Oxford, UK: Butterworth–Heinemann Ltd.

Stanley, H. E. 1971. Phase Transitions and Critical Phenomena. Vol. 7. Oxford, UK: Clarendon Press.

Stein, D. L. 1989. “Spin Glasses.” Scientific American 261 (1): 52–61. https://doi.org/10.1038/scientificamerican0789-52.

Stratonovich, R. L. 1958. “On a Method for the Computation of Quantum Distribution Functions.” Soviet Physics Doklady 2:416.

van Hemmen, J. L., and R. G. Palmer. 1979. “The Replica Method and Solvable Spin Glass Model.” Journal of Physics A: Mathematical and General 12 (4): 563. https://doi.org/10.1088/0305-4470/12/4/016.

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