16 Critical phenomena
16.1 Phase transitions and critical points
In Section 4.7, we studied the liquid–gas phase transition associatedwith the van der
Waals equation of state. We demonstrated the existence of a critical isotherm and
derived the thermodynamic state variables of the liquid–gas criticalpoint. We also
showed that the behavior of certain thermodynamic properties is determined by a
set of power laws characterized by exponents known ascritical exponents, and we
alluded to the phenomenon ofuniversality, which provides a rationalization for the
observation that large classes of physically very different systemspossess the same
critical exponents. In this chapter, we will explore the behavior ofsystems near their
critical points in greater detail.
To begin our discussion, consider a uniform gas of identical particlesin a container
of volumeV. The interactions between the particles are weak, and their motionis
primarily driven by the kinetic-energy (free particle) contribution to the Hamiltonian.
Collisions, which are overwhelmingly dominated by two-body interactions, are infre-
quent. If we assume that the collisions are approximately elastic, then each colliding
particle merely changes its direction of motion. The fact that an interaction event
determines when and where a particle’s next collision will occur is not particularly
important because the frequency of collision events is small. In this case, we speak of
a lack ofcorrelationbetween collision events.
If the gas is now compressed at a given temperature so that both the pressure
and the density are increased, the interactions between the particles become more
important, at least locally. Formation of small, short-lived clusters might occur, due
to cooperative interactions that have a greater influence on the system than merely
changing the direction of a particle’s motion. Collision events now exhibit short-range
correlations with each other, which leads to the formation of such local structures.
If the system is compressed even further, a change of phase orphase transition
occurs in which the gas becomes a liquid. Although phase transitions are an everyday
phenomenon, their underlying microscopic details are fascinating and merit further
comment. First, the macroscopic manifestation of a gas-to-liquid phase transition is
a discontinuous change in the volume. At the microscopic level, the interparticle in-
teractions give rise to long-range correlations—cooperative effects that cause the gas
particles to condense, forming well-defined solvation structures quantifiable through
spatial correlation functions such as those in Figs. 4.2 and 4.3.
Further compression leads to the formation of locally ordered structures that
resemble the solid. As the compression continues, long-range order sets in, and a liquid-