Conference, Proceedings

94
A statistical mechanical description of phases and first‐order
phase transitions
Igor Medved’
Department of Physics, Constantine the Philosopher University, Tr. A. Hlinku 1, 949 74
Nitra, Slovakia
Abstract: We briefly review the Pirogov‐Sinai theory that allows one to rigorously study phases and
first‐order phase transitions from a statistical mechanical point of view. We wish to demonstrate that it is
a powerful tool that may be used in various applications from materials physics to physical chemistry and
biology.
Keywords: Phase, phase transition of first order, Pirogov‐Sinai theory
1 Introduction
Phase transitions are intriguing phenomena that we often met in practice. The melting of solids,
dehydroxylation in kaolinite, α→β transition in quartz, condensation of gases, phenomena of
ferromagnetism and antiferromagnetism, order‐disorder transitions in alloys, or transition from
a normal to a superconducting material are all examples of phase transitions.
There are basically two ways to describe phase transitions. First, one says that a phase transition
occurs if thermodynamic functions have discontinuities or singularities. If it is a discontinuity in
the first derivative of a thermodynamic potential (such as specific energy e, magnetization m, or
density ρ), the phase transition is of first order; we will focus on this type of phase transition in the
following. Second, one says that a phase transition occurs if there is a macroscopic instability
with respect to boundary conditions: keeping, for instance, the temperature and pressure of a
system constant, a change in the external conditions leads to dramatic changes in macroscopic
quantities (like e, m, or ρ). Thus, changing forces acting near the boundary (that is small with
respect to the system volume) causes a change in the macroscopic state of the system. Under
such conditions, the free energy (and/or other boundary‐condition independent thermodynamic
functions) cannot be a smooth function.
However, the proof of existence of phase transitions in realistic systems at the microscopic level
is still an open problem, although the procedure is in principle clear. Indeed, it should basically
run as follows. Consider a system of N interacting particles with the Hamiltonian of the form,
H
N
N 2
r r r r pk
r r
( r1
, p1,
K,
rN
, p N ) = ∑ + ∑U
( | rk
− rj
| ),
2m
k = 1 k
1≤k
< j≤
N
where U is a realistic (e.g., the Lenard‐Jones) potential, and its grand‐canonical partition
function
Z
N
3N
/ 2
∞ N
N
z ⎛ 2π
⎞ ⎛
( | | )
3 / 2 ⎞ −β
∑U
rk
−r
j
1≤k
< j ≤ N
3N
( V , β , z)
= ∑ ⎜ ⎟ ⎜∏
mk
⎟ e d r,
N 0 N!
h ∫
(2)
= ⎝ β ⎠ ⎝ k = 1 ⎠ N
V
r
r
(1)