**15**, 115-126 (2000)

https://doi.org/10.1007/s100510051105

## Phase transitions in "small"systems

Hahn-Meitner-Institut Berlin, Bereich Theoretische Physik, Glienickerstrasse 100,
14109 Berlin, Germany and Freie Universität Berlin, Fachbereich Physik, Arnimallee 14,
14159 Berlin, Germany

Received:
18
October
1999

Revised:
17
November
1999

Published online: 15 May 2000

Traditionally, phase transitions are defined in the thermodynamic
limit only. We discuss how phase transitions of first order (with
phase separation and surface tension), continuous transitions and
(multi)-critical points can be seen and classified for small
systems. "Small"systems are systems where the linear dimension is
of the characteristic range of the interaction between the
particles; *i.e.* also astrophysical systems are "small"in this
sense. Boltzmann defines the entropy as the logarithm of the area
of the surface in the mechanical *N*-body phase
space at total energy *E*. The topology of *S(E,N)* or more
precisely, of the curvature determinant
allows the classification of phase transitions without taking
the thermodynamic limit. Micro-canonical thermo-statistics and
phase transitions will be discussed here for a system coupled by
short range forces in another situation where entropy is not
extensive. The first calculation of the entire entropy surface
*S(E,N)* for the diluted Potts model (ordinary (*q=3*)-Potts
model plus vacancies) on a square lattice is shown. The
regions in {*E,N*}where *D> 0* correspond to pure phases, ordered
resp. disordered, and *D< 0* represent transitions of first order
with phase separation and "surface tension". These regions are
bordered by a line with *D=0*. A line of continuous transitions
starts at the critical point of the ordinary (*q=3*)-Potts model and
runs down to a branching point *P*_{m}. Along this line
vanishes in the direction of the
eigenvector of *D* with the largest
eigen-value . It characterizes a maximum of the
largest eigenvalue . This corresponds to a critical line
where the transition is continuous and the surface tension
disappears. Here the neighboring phases are indistinguishable. The
region where two or more lines with *D=0* cross is the region of the
(multi)-critical point. The micro-canonical ensemble allows to put
these phenomena entirely on the level of mechanics.

PACS: 05.20.Gg – Classical ensemble theory / 05.50.+q – Lattice theory and statistics (Ising, Potts, etc.) / 05.70.Fh – Phase transitions: general studies

*© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000*