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Thermodynamical averages for the Ising model and spectral invariants of Toeplitz matrices
V. M. Kaplitskii Institute of Mathematics, Mechanics and Computer Sciences, Southern Federal University, Rostov-on-Don, Russia
Abstract:
We derive a general formula giving a representation of the partition function of the one-dimensional Ising model of a system of $N$ particles in the form of an explicitly defined functional of the spectral invariants of finite submatrices of a certain infinite Toeplitz matrix. We obtain an asymptotic representation of the partition function for large $N$, which can be a base for explicitly calculating some thermodynamic averages, for example, the specific free energy, in the case of a general translation-invariant spin interaction (not necessarily only between nearest neighbors). We estimate the partition function from above and below in the plane of the complex variable $\beta$ $(\beta$ is the inverse temperature) and consider the conditions under which these estimates are asymptotically equivalent as $N\to\infty$.
Keywords:
Ising model, statistical model, specific free energy, asymptotics, Toeplitz matrix.
Received: 29.11.2019 Revised: 06.02.2020
Citation:
V. M. Kaplitskii, “Thermodynamical averages for the Ising model and spectral invariants of Toeplitz matrices”, TMF, 203:3 (2020), 401–416; Theoret. and Math. Phys., 203:3 (2020), 780–793
Linking options:
https://www.mathnet.ru/eng/tmf9855https://doi.org/10.4213/tmf9855 https://www.mathnet.ru/eng/tmf/v203/i3/p401
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Abstract page: | 543 | Full-text PDF : | 70 | References: | 45 | First page: | 17 |
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