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Progress of Theoretical Physics

Approximate Method for the Free Energy

Akihide Oguchi

Published Nov 1, 1976• 28 citations
Publisher

Abstract

We propose an upper bound and a lower bound of the Helmholtz free energy in the statistical physics. A new approximate expression of the free energy is obtained. This approximate value of the free energy is proved to be greater than the lower bound and less than the upper bound. A systematic method which can be extended to improve the approximation is studied. The upper bound and the lower bound of the approximate free energy converge to the true free energy as the successive approximation proceeds. The method is first explained with the Ising ferromagnet and then applied to the Heisenberg ferromagnet. In the simplest approximation the result agrees with the Bethe-Peierls approximation for the Ising model and the constant coupling approximation for the Heisenberg model. A more accurate approximation is studied for the Ising ferromagnet.

DOI
10.1143/ptp.56.1442

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We propose an upper bound and a lower bound of the Helmholtz free energy in the statistical physics. A new approximate expression of the free energy is obtained. This approximate value of the free energy is proved to be greater than the lower bound and less than the upper bound. A systematic method which can be extended to improve the approximation is studied. The upper bound and the lower bound of the approximate free energy converge to the true free energy as the successive approximation proceeds. The method is first explained with the Ising ferromagnet and then applied to the Heisenberg ferromagnet. In the simplest approximation the result agrees with the Bethe-Peierls approximation for the Ising model and the constant coupling approximation for the Heisenberg model. A more accurate approximation is studied for the Ising ferromagnet.
Abstract. We consider the quantum ferromagnetic Heisenberg model in three di-mensions, for all spins S ≥ 1/2. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial con-tribution to the free energy at low temperatures. Our proof comes with explicit, constructive upper and lower bounds on the error term. It uses in an essential way the bosonic formulation of the model in terms of the Holstein–Primakoff representa-tion. In this language, the model describes interacting bosons with a hard-core on-site repulsion and a nearest-neighbor attraction. This attractive interaction makes the lower bound on the free energy particularly tricky: the key idea there is to prove a differential inequality for the two-particle density, which is thereby shown to be smaller than the probability density of a suitably weighted two-particle random process on the lattice. 1.
A variational approximation to the Ising model in a magnetic field (T< TcT<~{T}_{c}T< Tc​), based on the zero-field solution, yields an upper bound on the free energy, an approximate equation of state, and a lower bound on the magnetization, all having the correct critical indices. In two dimensions, these are numerically evaluated, and compared with the results of series expansions.
To calculate thermodynamic functions in statistical mechanics the authors propose a new variational approach which has as particular cases the approaches of Bogoliubov, Oguchi and Tsallis-da Silva (1982), but introduces some improvements, mainly in the short-range order effects above the critical temperature. It requires a choice of a trial Hamiltonian that defines an approximate ensemble through which the thermodynamic functions may be calculated. To check the efficiency of their approach they apply it to the ferromagnetic Ising model by using the simplest possible trial Hamiltonian.
We present and demonstrate an accurate, reliable, and computationally cheap method for the calculation of free energies in Monte Carlo simulations of lattice models. Even in the critical region it yields good results with comparatively short simulation runs. The method combines upper and lower bounds on the thermodynamic limit entropy density to yield not only an accurate estimate of the free energy but a bound on the possible error as well. The method is demonstrated on the two- and three-dimensional Ising models and the three-dimensional, three-states Potts model.
A class of quasiclassical adiabatic approximations is described that enables upper bounds to be determined for the partition function of a system and, hence, lower bounds for its Helmholtz function. Based upon a neglect of the lack of commutativity of selected observables with others, these adiabatic approximations become exact in classical statistical mechanics. The effects of carrying out sequences of such adiabatic approximations and of relaxing intrinsic exchange-symmetry restrictions is to produce additional upper-bound expressions for the partition function. The lower-bound property of the adiabatically approximated Helmholtz function provides an alternative proof of the inequality that relates the lowest-energy eigenvalue of a system and that given by the Born-Oppenheimer approximation, as well as improvements of it.
The use of variational approximations in the study and application of renormalization groups is discussed. In particular, a simple approximation is derived which yields an upper bound to the free energy of Ising models on d-dimensional lattices. The optimal transformation, which yields the least upper bound, is determined analytically. The criterion proposed by Kadanoff (1975) to determine the 'best' approximation to the fixed point is found to fail in this case. The reasons for this failure and several of the basic problems posed by variational approximations are discussed.