A New, Exact Perturbation Theory for Classical Statistical Mechanics
ORAL
Abstract
We have developed a new, exact mathematical technique that we call statistical physics perturbation theory [1]. Our method supersedes the high-temperature series expansion and is also competitive with Monte Carlo simulations. Indeed, we do not require the assumption of a phase transition to predict one!
Our perturbation theory is for operators related by the Baker-Campbell-Hausdorff (BCH) identity, exp(C)=exp(A)exp(B) [2], rather than by addition, C=A+B, as in quantum mechanics. We use the transfer matrix approach to classical statistical mechanics [3] which leads directly to the BCH identity and we must find the operator C. Our perturbation theory is almost verbatim the quantum case, save that the denominators, 1/x, are replaced by hyperbolic functions, coth(x), and corrections!
We also present predictions for the correlation length as a function of temperature and the associated critical exponent for the Ising model on the square, cubic and hypercubic lattices. Surprisingly, for the square lattice, we find the exact correlation length at first order with all higher order contributions vanishing!
[1] J. M. Jones and M. W. Long, in preparation
[2] J. C. Moodie and M. W. Long, 2021 J. Phys. A: Math. Theor. 54 015208
[3] T. D. Schultz, D. C. Mattis and E. H. Lieb, Rev. Mod. Phys. 36, 85
Our perturbation theory is for operators related by the Baker-Campbell-Hausdorff (BCH) identity, exp(C)=exp(A)exp(B) [2], rather than by addition, C=A+B, as in quantum mechanics. We use the transfer matrix approach to classical statistical mechanics [3] which leads directly to the BCH identity and we must find the operator C. Our perturbation theory is almost verbatim the quantum case, save that the denominators, 1/x, are replaced by hyperbolic functions, coth(x), and corrections!
We also present predictions for the correlation length as a function of temperature and the associated critical exponent for the Ising model on the square, cubic and hypercubic lattices. Surprisingly, for the square lattice, we find the exact correlation length at first order with all higher order contributions vanishing!
[1] J. M. Jones and M. W. Long, in preparation
[2] J. C. Moodie and M. W. Long, 2021 J. Phys. A: Math. Theor. 54 015208
[3] T. D. Schultz, D. C. Mattis and E. H. Lieb, Rev. Mod. Phys. 36, 85
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Publication: J. M. Jones and M. W. Long, Statistical Physics Perturbation Theory Applied to the Ising Model on the Square, Cubic and Hypercubic Lattices, in preparation
Presenters
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Joseph M Jones
University of Birmingham
Authors
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Joseph M Jones
University of Birmingham
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Martin W Long
University of Birmingham