Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 20.2-8

POSTLUDE: SYMMETRY AND THE CONCEPTUAL FOUNDATIONS OF THERMOSTATISTICS

Physics — Statistical Mechanics · mean-field-alternating-ab-sites

Problem

Consider a sequence of

2\tilde{N}

alternating

A

sites and

B

sites, the system being arranged in a circle so that the

(2\tilde{N})^{\text{th}}

site is the nearest neighbor of the first site. Even numbered sites are occupied by two-valued Ising spins, with

\sigma_j = \pm 1

. Odd numbered sites are occupied by three-valued Ising spins, with

\sigma_j = -1, 0, +1

. The Hamiltonian is

\mathscr{H} = -2J\sum_j \sigma_j\sigma_{j+1} - B\sum_j \sigma_j

a

) Formulate a mean field theory by choosing as a soluble model system a collection of independent

A

sites and a collection of independent

B

sites, each acted upon by a different mean field.

b

) Formulate a mean field theory by choosing as a soluble model system a collection of

\tilde{N}

independent

A

B

pairs, with the Hamiltonian of each pair being

\mathscr{H}_{\text{pair}} = -2J\sigma_{\text{odd}}\sigma_{\text{even}} + B_{\text{odd}}\sigma_{\text{odd}} + B_{\text{even}}\sigma_{\text{even}}

c

) Are these two procedures identical? If so, why? If not, which procedure would you judge to be superior, and why?