Abstract:
The diagram method is used to study phase transitions in systems with $R\to\infty$ Where $R$
is the range of the attractive potential between particles. If the thermodynamic functions
are to be calculated correctly in the neighborhood of a phase transition, it is necessary
to allow for diagrams with many vertices and lines. To allow for their contribution, a
recursion relation is obtained; it relates diagrams of different orders and structures.
The relation is used to estimate the contribution from all the many-vertex diagrams and
to obtain a differential equation for $p(\mu,T)$ that is valid as $R\to\infty$ ($p$ is the pressure, $T$
the temperature, and $\mu$ the chemical potential). The solution is investigated for the example
of the Ising model. In the two-phase region the $s(H)$ curve does not exhibit the unphysical
region with negative susceptibility found in the Curie–Weiss approximation ($s$ is
the polarization, $H$ the magnetic field). It follows from the solution that is found that the
point $R=\infty$ is an essential singularity, so that the thermodynamic functions cannot be expanded
in a Taylor series in powers of $1/R^3$ at points near the phase transition. It is
shown that allowing for many-vertex diagrams is equivalent to having an effective interaction
between the particles of the “all with all” type that is independent of the mutual
separations of the particles.