chapter 9.pdf - Chapter 9 Critical Phenomena and...

Chapter 9
Critical Phenomena and
Renormalization Theory
9.1 WHAT IS A PHASE TRANSITION?
This chapter will be devoted to the study of second-order phase transitions, but, before considering the
particular case of second-order phase transitions, it is best to look in more detail at the question of how
one defines a phase transition in general. The most familiar phase transitions, such as those between water,
ice, and steam, are almost misleadingly clear, in that the properties of the phases involved are so strikingly
different. However, even in this seemingly obvious case, the meaning of the word phase begins to become
more subtle when we realize that we can convert a sample of pure liquid to pure gas with no phase transition,
simply by moving the system along a trajectory around the critical point in the p - T plane (see Fig. 9.1). The
phase transition lines do not necessarily separate the thermodynamic space into disjoint regions of different
phases. Although it is possible to define precisely a phase transition point in the p - T plane, and we will do
so presently, it is not possible to separate clearly the liquid phase points from the gas phase points.
•
•
•
critical
point
solid
gas
liquid
p
T
Fig. 9.1
By varying the values of p and T to move the system along the curve shown,
it is possible to transform a liquid to a gas without encountering a phase transition line.
The phase transition encountered in the ideal Bose gas is typical of many, more subtle, phase transitions.
Across the phase transition line, the density is continuous, the energy density is continuous, and even the
specific heat is continuous. One must go to the derivative of C
V
with respect to T in order to find a property
that changes discontinuously. If phase transitions can be so delicate, what is the defining characteristic of a
phase transition point?

CRITICAL PHENOMENA AND RENORM. THEORY 9.1 WHAT IS A PHASE TRANSITION? 2
“The ordinary gaseous and liquid states are, in short, only widely separated forms of the same condition
of matter and can be made to pass into one another by a series of gradations so gentle that the passage shall
nowhere present any interuption or break of continuity.”
—— Thomas Andrews Bakerian Lecture to the Royal Society. (1869)
Andrews was the first person to demonstrate clearly the existence of a critical point in the liquid–gas
transition.
The unambiguous definition of a phase transition involves the mathematical concept of analyticity. A
mathematical function of many variables (we will consider the case of two variables) is called analytic at a
point ( x, y ) if its Taylor expansion converges to the value of the function at every point within some finite
circle about ( x, y ). That is, if
f ( x + ∆ x, y + ∆ y ) = f ( x, y ) +
∞ ∑
m,n =0
f
mn
(∆ x )
m
(∆ y )
n
m ! n !