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 TMF: Year: Volume: Issue: Page: Find

 TMF, 2002, Volume 132, Number 1, Pages 141–149 (Mi tmf352)

A Step-Function Approximation in the Theory of Critical Fluctuations

P. L. Rubin

P. N. Lebedev Physical Institute, Russian Academy of Sciences

Abstract: We consider fluctuations near the critical point using the step-function approximation, i.e. the approximation of the order parameter field $f(x)$ by a sequence of step functions converging to $f(x)$. We show that the systematic application of this method leads to a trivial result in the case where the fluctuation probability is defined by the Landau Hamiltonian: the fluctuations disappear because the measure in the space of functions that describe the fluctuations proves to be supported on the single function $f\equiv 0$. This can imply that the approximation of the initial smooth functions by the step functions fails as a method for evaluating the functional integral and for defining the corresponding measure, although the step-function approximation proves to be effective in the Gaussian case and yields the same result as alternative methods do.

Keywords: critical fluctuations, non-Gaussian functional integral, Landau Hamiltonian, step-function approximation

DOI: https://doi.org/10.4213/tmf352

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English version:
Theoretical and Mathematical Physics, 2002, 132:1, 1012–1018

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Revised: 26.02.2002

Citation: P. L. Rubin, “A Step-Function Approximation in the Theory of Critical Fluctuations”, TMF, 132:1 (2002), 141–149; Theoret. and Math. Phys., 132:1 (2002), 1012–1018

Citation in format AMSBIB
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