Fluctuations of physical values in statistical mechanics

Walter Felipe Wreszinski

Jone-Zen Wang, Tzong‐Jer Yang

Based on the microcanonical ensemble theory in statistical mechanics, we devise a method that can be used to enhance the capability of numerical calculation on statistical models with large lattice sizes. In our method, we take the expectation value of the energy, as defined in quantum mechanics, instead of the eigenvalue as the energy of a physical system. We show mathematically that the relevant physical quantities obtained in this way are unchanged in the thermodynamical limit and we apply this method to numerical calculations. In this paper, we present our numerical results with the one-dimensional spinless fermionic model as a first test of our method. The numerical calculations are done to a 4096 lattice size using a computer with a speed of about 40 mflops. Our numerical data agree quite well with the exact values. Also, the fluctuation of data is small, in contrast to that obtained using the quantum Monte Carlo method. \textcopyright{} 1996 The American Physical Society.

Sergei Vladimirovich Tyablikov

Barry Loewer

Andrei Yu. Khrennikov

We show that quantum mechanics can be represented as an asymptotic projection of statistical mechanics of classical fields. Our approach does not contradict to a rather common opinion that quantum mechanics could not be reduced to statistical mechanics of classical particles. Notions of a system and causality can be reestablished on the prequantum level, but the price is sufficiently high — the infinite dimension of the phase space. In our approach quantum observables, symmetric operators in the Hilbert space, are obtained as the second order derivatives of functionals of classical fields: f(ψ) → Aquant = f″(0). Statistical states are given by Gaussian ensembles of classical fields with zero mean value (so these are vacuum fluctuations) and dispersion α which plays the role of a small parameter of the model (so these are small vacuum fluctuations). Our approach can be called Prequantum Classical Statistical Field Theory — PCSFT.

Gérard G. Emch