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

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Chapter 9Critical Phenomena andRenormalization Theory9.1WHAT IS A PHASE TRANSITION?This chapter will be devoted to the study of second-order phase transitions, but, before considering theparticular case of second-order phase transitions, it is best to look in more detail at the question of howone 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 strikinglydifferent. However, even in this seemingly obvious case, the meaning of the wordphasebegins to becomemore 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 thep-Tplane (see Fig. 9.1). Thephase transition lines do not necessarily separate the thermodynamic space into disjoint regions of differentphases. Although it is possible to define precisely a phase transition point in thep-Tplane, and we will doso presently, it is not possible to separate clearly the liquid phase points from the gas phase points.criticalpointsolidgasliquidpTFig. 9.1By varying the values ofpandTto 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 thespecific heat is continuous. One must go to the derivative ofCVwith respect toTin order to find a propertythat changes discontinuously. If phase transitions can be so delicate, what is the defining characteristic of aphase transition point?
CRITICAL PHENOMENA AND RENORM. THEORY9.1 WHAT IS A PHASE TRANSITION?2“The ordinary gaseous and liquid states are, in short, only widely separated forms of the same conditionof matter and can be made to pass into one another by a series of gradations so gentle that the passage shallnowhere present any interuption or break of continuity.”——Thomas AndrewsBakerian Lecture to the Royal Society.(1869)Andrews was the first person to demonstrate clearly the existence of a critical point in the liquid–gastransition.The unambiguous definition of a phase transition involves the mathematical concept of analyticity. Amathematical function of many variables (we will consider the case of two variables) is calledanalyticat apoint (x, y) if its Taylor expansion converges to the value of the function at every point within some finitecircle about (x, y). That is, iff(x+ ∆x, y+ ∆y) =f(x, y) +m,n=0fmn(∆x)m(∆y)nm!n!

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