1 (c) SNU CSE Biointelligence Lab, Chap 3.8 – 3.10 Joon Shik Kim BI study group

2 (c) SNU CSE Biointelligence Lab, Quiz The renormalization group method of ( ? ) produces a way of treating cooperative systems with many degrees of freedom distributed over many scales of lengths

3 (c) SNU CSE Biointelligence Lab, THE LATTICE GAS MODEL OF FLUIDS (1/2) Figure 3.10 illustrates the radial dependence of an intermolecular potential, such as a Lennard-Jones 6-12 potential.

4 (c) SNU CSE Biointelligence Lab, THE LATTICE GAS MODEL OF FLUIDS (2/2) Three characteristics of this potential  Strongly repulsive at small distances  Attractive at separation of one or two molecular diameters  The potential vanishes for large radial separations of two molecules Our goal is to derive an equation of state for a fluid system possession an interaction potential of this type We treat the problem using a lattice gas model and the model is equivalent to the Ising model We will then establish a correspondence between the van der Waals equation of state and Weiss molecular fluid

5 (c) SNU CSE Biointelligence Lab, The Lattice Gas We partition a fluid into cells that are arranged in a regular lattice Each cell of the lattice gas is approximately equal in size to that of one of the molecules Two molecules cannot occupy the same cell The two states of any cell i are described by the occupation number n i, whose value is 1 or 0 0 otherwise (3.85)

6 (c) SNU CSE Biointelligence Lab, We must introduce constraints on the mean values of both the energy and potential number

7 (c) SNU CSE Biointelligence Lab, n i is the occupation numbers and s i is the spin variables

8 (c) SNU CSE Biointelligence Lab, Densitiy ρof a lattice gas, ρ= N/Nce Density of up (+1) spin of an Ising ferromagnet,

9 (c) SNU CSE Biointelligence Lab, The van der Waals Equation Transformation of the mean-field formula for an Ising spin system into an equation describing a lattice gas Gibbs-Duhem equation, νdp – s dT = dμ ch, νis the molar volume and s is the molar entropy

10 (c) SNU CSE Biointelligence Lab,

11 (c) SNU CSE Biointelligence Lab, Van der Waals equation of state upon setting b=1, a=ε 0 /2 Van der Waals equation is analogous to the Weiss mean-field equation for an Ising model

12 (c) SNU CSE Biointelligence Lab, The Triple and Critical Points Two modification of van der Waals equation from ideal gas law The first is the addition of an internal pressure term, a/ ν 2 that takes into account the attractive intermolecular forces The second is the incorporation of an excluded volume term of the form ν-b, that reflects the short-range and repulsive interactions The behavior of the system at the critical point provides an interesting contrast to that at the triple point

13 (c) SNU CSE Biointelligence Lab, In the vicinity of the critical point, the fluid density is far lower than at the triple point and long range correlations (fluctuation) are important Near the triple point, the important correlations are associated with the short-range repulsive component Critical point phenomena can be treated using the full lattice gas model

14 (c) SNU CSE Biointelligence Lab, THE RENORMALIZATION GROUP

15 (c) SNU CSE Biointelligence Lab, Coupled Degree of Freedom Near critical points the behavior of a system depends mostly on the existence of cooperativity itself and on the nature of the degree of freedom, rather than on the detailed character of the Hamiltonian The renormalization group method of Wilson produces a way of treating cooperative systems with many degrees of freedom distributed over many scales of lengths

16 (c) SNU CSE Biointelligence Lab, Two principles consequences of the cascade picture  The first is scaling: fluctuations at intermediate wavelength regions tend to be identical  The second feature emerging from the coupling is amplification and deamplification Amplification: small temperature changes are amplified by the development of a cascade Deamplification: while two different materials has different atomic characteristics, these differences decrease in importance as the cascade develops  Universality: all critical phenomena possess a common set of characteristics

17 (c) SNU CSE Biointelligence Lab, The Kadanoff Block Spin Construction : total spin of the Jth blosck This block of spins can be treated as a single-spin element, which can take on the usual values of +1 or -1 by introducing the factor f

18 (c) SNU CSE Biointelligence Lab, The Renormalization Group and Triple Points Critical phenomena manifests itself through the appearance of fixed points These are values for the coupling strength where recursion no longer produces any change in the coupling strength, so the further removal of degrees of freedom has no effect

19 (c) SNU CSE Biointelligence Lab, S 1 =S 3 =1, Setting one of spin value equal to +1 and the other equal to -1,

20 (c) SNU CSE Biointelligence Lab, Kadanoff trasformations

21 (c) SNU CSE Biointelligence Lab, Recursion in the opposite direction K’ = 0.01, Z = 2 N, Φ (0.01) = ln 2. From Eqs. (3.118) and (3.119), K= , Φ=

22 (c) SNU CSE Biointelligence Lab, Table 3.1 Figure 3.12

23 (c) SNU CSE Biointelligence Lab, In one-dimension Ising chain, trivial fixed points are at zero and infinity The absence of nontrivial fixed points means that there is no phase transition In 2-dimension, the fixed point value Kc=0.506 found using the RG method compares favorably with the exact fixed point value Kc=0.441

24 (c) SNU CSE Biointelligence Lab, Widom-Kadanoff scaling ε= (T-Tc)/Tc : the system is said to exhibit scaling (3.120) h(x) is said to be homogeneous if it obeys A generalized homogeneous function

25 (c) SNU CSE Biointelligence Lab, Widom-Kadanoff scaling hypothesis asserts that Gibbs (or Helmholtz) free energy per spin g is a generalized homogeneous function Differentiate both the sides of the above equation with respect to the magnetic field,

26 (c) SNU CSE Biointelligence Lab, In the limit as H goes zero,,M=1

27 (c) SNU CSE Biointelligence Lab, Calculating the derivatives leading to the specific heat and susceptibility. Then we can find the scaling relationship in Eq. (3.120).

28 (c) SNU CSE Biointelligence Lab, SUMMARY When a system is brought into a condition where it loses its stability against fluctuations in its parameters, phase transitions occur Singularities or discontinuity in the specific heat and nonzero order parameters signify the presence of these consequences of cooperativity In the one-dimensional case, there is no temperature below which free energy minimization will stabilize an ordered state In a mean field theory, local fluctuations are neglected In the Bethe and Kikuchi approximations, some attempt is made to include local order by replacing isolated spins by small clusters of elements

29 (c) SNU CSE Biointelligence Lab, The van der Waals equation is approximate near the triple point while the full lattice gas model may be used to treat the fluid system in the vicinity of the critical point Designed to treat critical phenomena, in a renormalization group treatment, cooperativity generates a more complex set of interactions The detailed character of the cooperative interactions are not important, instead, we encounter scaling and universality