On the Thermodynamical‐Statistical Consistency of a Variational Approximation for the Free Energy

Abstract

An upper bound for the free energy 1sfound which is more general than the Bogolyubov inequality. It generates an approximation which can accurately preserve the consistency between thermodynamic and statistical mechanics definitions. E s wird eine obere Grenze fur die Freie Energie gefunden, die allgemeiner ist als die BogolyubovUngleichung. Sie liefert eine Naherung, die die Konsistenz zwischen thermodynamischen Definitionen und denen der statistischen Mechanik exakt bewahren kann.

A Weighted Linear Chain Approximation for the Ising Model Free Energy

“…The stationarity condition of the above expression (related to h,) defines a self-consistent equation for ho, 3 2 tanh (ph,) -p , sinh (ph,)/[sinh' (ph,) + exp (-4PJn)J"2 = 0 .…”

“…It has led to better qualitative and quantitative results than those obtained through the well-known Bogolyubov principle. It preserves short-range order effects above the approximate critical temperature and it also preserves the consistency between thermodynamic and statistical-mechanics calculations [3]. From a general upper bound for the exact free energy several approximations can be made.…”

“…The variational parameters are to be obtained by the minimum condition of a certain upper bound for the free energy. Several finite divisions have been made before [3] but in the present work each term will include a whole chain with infinite number of sites. Each chain may be weighted by its own exchange constant so that a chain with a strong coupling makes a major contribution to the approximate free energy, while one with a weak coupling makes a minor contribution and so on.…”