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#

The mean-field model

The models of Glasser, Privman [5], and Schulman avoid
some pathologies common to other infinite range models. They have a
``soft'' temperature (*T*) dependence which is an artifact of the infinite-range
model, and there are some other artificial features near *T*=0, but
the series is well defined and can be derived in closed form to any
fixed order given sufficiently powerful computational facilities.
The model has interacting scalar spins, ,
with energy

where *N* is the number of spins, and *J*>0.
The Gaussian-integral method [5] was used to show that in the
limit
the dimensionless free
energy, *f*, can be obtained as

where

The spins are weighed with measure
in the partition
function from whence the free energy is obtained, and the variable *x*
[5] is equal to
*y*(2*J*/*kTN*)^{1/2}, where *y* is the variable
introduced in the Gaussian integral.
If the minimum in (2.2) is at some *x*=*x*_{m}, then one can further
show that the magnetization *m* is

where the last equality follows from the fact that the
global minimum is obtained at one (or more) roots of

Thus, we note that *m*=*kTx*_{m}/*J*, i.e.,
for low
temperatures. This is one of those artificial infinite-range-model
features. It turns out convenient to work with *x*_{m} directly
rather than with *m*, as the order-parameter-like quantity for
series analysis. Of course, the actual
critical-tricritical-first-order behavior is at *T*>0 so the
difference only affects the form of analytic corrections to scaling.
In order to have a solvable model with tricritical behavior, *Q*(*x*)
was taken as an even, six-degree polynomial in *x*, the
*actual series* was conveniently generated [1] for

where *U* is a dimensionless coupling constant just like *T* is a dimensionless
temperature in this model.
Using *Mathematica* [6],
the order 50 double series
in *T* and *U* for this order-parameter quantity was derived in [1].
It is expressed as the first 2601 coefficients *c*_{ij},
for
,
in

Analysis of the exact solution [1] showed that
*U*_{c}=*T*_{c}=1,
and that near the tricritical point, one can write the low-*T*-side scaling
form in terms of the scaling variables

where the scaling form (2.9) applies for
and
the scaling function is

** Next:** Analytic Solution by PDA
** Up:** Series Analysis of Tricritical
** Previous:** Introduction

*1998-12-30*