PHY304 - Statistical Mechanics Spring Semester 2021 Dr. Anosh Joseph, IISER Mohali LECTURE 35 Monday, April 5, 2021 (Note: This is an online lecture due to COVID-19 interruption.) Contents 1 Classification of Phase Transitions 1 1.1 Order Parameter . .2 2 Critical Exponents 5 1 Classification of Phase Transitions The physics of phase transitions is a young research field of statistical physics. Let us summarize the knowledge we gained from thermodynamics regarding phases. The Gibbs’ phase rule is F = K + 2 − P; (1) with F denoting the number of intensive variables, K the number of particle species (chemical components), and P the number of phases. Consider a closed pot containing a vapor. With K = 1 we need 3 (= K + 2) extensive variables say, S; V; N for a complete description of the system. One of these say, V determines only size of the system. The intensive properties are completely described by F = 1 + 2 − 1 = 2 (2) intensive variables. For instance, by pressure and temperature. (We could also choose temperature and chemical potential.) The third intensive variable is given by the Gibbs’-Duhem relation X S dT − V dp + Ni dµi = 0: (3) i This relation tells us that the intensive variables PHY304 - Statistical Mechanics Spring Semester 2021 T; p; µ1; ··· ; µK , which are conjugate to the extensive variables S; V; N1; ··· ;NK are not at all independent of each other. In the above relation S; V; N1; ··· ;NK are now functions of the variables T; p; µ1; ··· ; µK , and the Gibbs’-Duhem relation provides the possibility to eliminate one of these variables. Due to the postulate of coexistence of two phases (solid-liquid, solid-gaseous or liquid-gaseous) 3 of the initial 6 intensive variables can be eliminated via Gibbs’ relations of phase equilibrium. Due to the Gibbs’- Duhem relations (one for each phase), 1 intensive variable remains, which can be independently fixed (e. g. T ). Then we can calculate, for instance, the pressure p as a function of T , which leads to the melting, sublimation and vapor pressure curves in the pT diagram. All three phases co-exist simultaneously at the crossing of the three curves, the triple point. Here F = 1 + 2 − 3 = 0; (4) and thus all intensive variables of the system (T , p, µ) are fixed. There are 9 intensive variables, but also 6 Gibbs’ phase co-existence relations, and additionally 3 Gibbs’-Duhem relations, so that the system of equations in the variables µi, Ti, and pi, i = 1; 2; 3 has a unique solution (which corresponds to the triple point). 1.1 Order Parameter It can be shown that most rearrangements of the structure in phase transitions can be described by a so-called order parameter (Landau, 1937). This order parameter should represent the main qualitative difference between the various phases. It is usually an extensive thermodynamic variable accessible to measurements. For the gas-liquid critical point, the order parameter is the volume difference (for a fixed particle number) of the co-existing phases, which tends to zero at the critical point. We could also use order parameter as density difference or entropy difference. Phase transitions which are connected with an entropy discontinuity are called discontinuous or first order phase transitions. Phase transitions where the entropy is continuous are called continuous or of second or higher order. To achieve a more general and unique classification of phase transitions, we start from the Gibbs’ free enthalpy G (we could also use the grand potential Φ). It is convenient to consider the free enthalpy as a function of the natural variables, say G ≡ G(N; T; p; H;~ E;~ ··· ): (5) Apart from the particle number and the temperature, further intensive field variables appear like pressure, magnetic field, electric field, etc., which represent the state variables which can be exter- nally controlled. 2 / 7 PHY304 - Statistical Mechanics Spring Semester 2021 Then the conjugated extensive quantities like entropy, volume, and magnetic field and electric dipole moments assume values according to @Ψ Ψ = ± ; (6) @h N;··· where we use the letter h for the relevant field variable and Ψ for the conjugated state quantity. For a first-order phase transition, one of the first derivatives of the free enthalpy with respect to the external fields is discontinuous. We have the entropy S, volume V , and magnetization M, respectively @G S = − ; (7) @T N;p;··· @G V = ; (8) @p N;T;··· @G M = − : (9) @H N;T;··· This discontinuity produces a divergence in the higher derivatives like the specific heat Cp, the compressibility κ, the expansion coefficient α, or the susceptibility χ. See Fig. 1. ∞ G S Cp p = const . p = const . p = const . Sg phase 1 phase 2 phase 1 phase 2 S1 phase 1 phase 2 Td T Td T Td T Figure 1: Free enthalpy, entropy, and specific heat as a function of temperature for a first-order phase transition. We have the specific heat Cp, compressibility coefficient κ, thermal expansion coefficient α, and 3 / 7 PHY304 - Statistical Mechanics Spring Semester 2021 magnetic susceptibility χ, respectively @S @2G Cp = T = −T ; (10) @T p @T 2 p 1 @V 1 @2G κ = − = − ; (11) V @p T V @p2 T 1 @V 1 @2G α = = ; (12) V @T p V @p@T and 1 @M @2G χ = = − : (13) V @H T @H2 T For instance, in the liquid-gas phase transition, Cp (as well as κ and α) diverges. For a phase transition of second or nth order, the first derivatives of the free enthalpy are continuous. However, second derivatives like the specific heat or the susceptibility, or nth order derivatives are discontinuous or divergent. See Fig. 2. In Fig. 2 the discontinuity of the specific heat at Td is due to a kink in the entropy. G S Cp phase 1 phase 2 phase 1 phase 2 Td T Td T Td T Figure 2: Free enthalpy, entropy, and specific heat as a function of temperature for a second-order phase transition. The transition to superconductivity without an external magnetic field is an example of phase transitions of this kind. Essentially, most kinds of second-order phase transitions are not due to kinks in the entropy, but are due to a vertical tangent at T = Td. Because of the characteristic shape of the specific heat, such transitions are called λ-transitions. See Fig. 3. Examples of λ-transitions are: transition to superfluidity in He-4, rearrangements in alloys 4 / 7 PHY304 - Statistical Mechanics Spring Semester 2021 ∞ S Cp Td T Td T Figure 3: Entropy, and specific heat as a function of temperature for a λ-transition. (order-disorder phase transitions), the transition to ferroelectricity in certain materials, and rear- rangements of the orientation in crystal lattices. 60 J CV [ mol ⋅ K ] 40 He II He I 20 0 0 1 2 3 4 T [K] Figure 4: The Lambda point is the temperature at which normal fluid helium (helium I) makes the transition to superfluid helium II (approximately 2.17 K at 1 atmosphere). 2 Critical Exponents A fundamental problem of the theory of phase transitions is the behavior of a system in the vicinity of the critical point. Several thermodynamic quantities begin to diverge at this point, and the order parameter vanishes. Let us make this statement more precise. Let us express the behavior of the most important thermodynamic quantities in the vicinity of the critical point as a function of temperature. To this end one uses power laws, the exponents of which are called critical exponents or critical indices. There are six commonly recognized critical points: α;β;γ;δ;η;ν. 5 / 7 PHY304 - Statistical Mechanics Spring Semester 2021 Denote the critical temperature by Tc, and introduce the quantity T − T t = c : (14) Tc We suppose that, in the limit t ! 0, any thermodynamic quantity can be decomposed into a “regular” part, which remains finite (but not necessarily continuous), plus a “singular” part that may be divergent, or have divergent derivatives. The singular part is assumed to be proportional to some power t, generally fractional. The first four critical exponents defined as follows Heat capacity : C ∼ jtj−α; (15) Order parameter : M ∼ jtjβ; (16) Susceptibility : χ ∼ jtj−γ; (17) Equation of state (t = 0) : M ∼ H1/δ: (18) In the above, ∼ means “ has a singular part proportional to.” Since the first three relations all refer to a phase transition, it is understood that H = 0. The last one, on the other hand, specifically refers to the case H 6= 0. The definitions above implicitly assume that the singularities are of the same type, whether we approach the critical point from above or from below. This has been borne out both theoretically and experimentally, except for M, which is identically zero above the critical point by definition. We have the correlation function (Correlation functions tell us how microscopic variables, such as spin and density, at different positions are related.) Γ(r) ! r−pe−r=ξ; when t ! 0: (19) Then ν and η are defined as Correlation length : ξ ∼ jtj−ν; (20) Power law decay at t = 0 : p = d − 2 + η: (21) The significance of critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Only two of the critical exponents mentioned above are independent, because of the following 6 / 7 PHY304 - Statistical Mechanics Spring Semester 2021 “scaling laws”: Fisher : γ = ν(2 − η); (22) Rushbrooke : α + 2β + γ = 2; (23) Widom : γ = β(δ − 1); (24) Josephson : νd = 2 − α; (25) with d denoting the dimensionality of space.