Biointelligence Laboratory, Seoul National University Ch 3. Cooperativity in Lattice Systems 3.1 ~ 3.2 Adaptive Cooperative Systems, Martin Beckerman, 1997. Summarized by M.-O. Heo Biointelligence Laboratory, Seoul National University http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Contents 3.1 Introduction 3.1.1 Configuration and the Counting Problem 3.1.2 Objectives of the Chapter 3.2 Key Concepts 3.2.1 Entropic and Energetic Potentials 3.2.2 Phase Transitions 3.2.3 Order Parameters 3.2.4 Critical Exponents and Scaling 3.2.5 Correlation Lengths (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Lattice The assemblage is compose of the elements of a cooperative systems spatially in a regular manner. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

Configurations and the Counting Problem Interaction Energy of the model system This system has 2 types of elements which are distributed among lattice sites in such a way that there is one and only one element at each site. The coordination number ν the # of nearest neighbors of each site Some formulae can be derived from the structure. N1 : # of elements of type 1 N2 : # of elements of type 2 N11 : # of 1-1 nearest-neighbor pairs N12 : # of 1-2 nearest-neighbor pairs N22 : # of 2-2 nearest-neighbor pairs ε11 : 1-1 pairs interaction energies ε12 : 1-2 pairs interaction energies ε22 : 2-2 pairs interaction energies (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ We can derive new interaction energies using the above. The partition function Representing a sum over all possible configurations of the lattice system 4 lines of attack to evaluate the partition function Analytic: transfer matrix and combinatorial methods Mean-Field Theory Renormalization Group Monte Carlo methods The density of states  It’s not easy to evaluate <Counting Problem> (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

Entropic and Energetic Potentials Order The most remarkable feature of the stable equilibrium state. Correspond to values to a maximum in the entropy Correspond to values to a minimum in the (free) energy As T is lowered, the short-range interactions become more important, the system seeks a configuration in which the potential energy is a minimum. Helmholtz free energy A Internal energy Entropic potential (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Phase Transitions Ordinary (1st-order) phase transition Discontinuities in the molar parameters of state Ex) the molar entropy, energy, and volume Latent heat the entropic change TΔS, comes from the increase in configuration order that occurs in the more condensed phase. The heat absorbed in transforming from one phase to another. Continuous (2nd-order) phase transition No latent heat, but singularities in the heat capacity. With strongly cooperative interactions. Ex) paramagnetic-ferromagnetic transitions (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Order Parameters Order parameters A fluctuating variable whose average provides a measure of the amount of order present in the system. Examples The spontaneous magnetization for the ferromagnet The pairing gap for a superconductor Vanishes above the TC Reaches its maximum at T=0 The spontaneous breaking of spatial symmetry Different possibilities exist under identical physical conditions. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

Critical Exponents and Scaling The free energy may be expanded in power series in the order parameter near the TC As T approaches to the TC, the leading term will dominate the power series. A ferromagnet case near the TC Heat Capacity Magnetization Magnetic susceptibility (differential increase in magnetization with increasing field strength) Universality Striking similarities in the critical behavior observed in different systems. The intra- and extrasystem findings support the notion that the essential properties of many phase transition are independent of the detailed interactions, except for the dependence on the dimensionality of the system. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Correlation Lengths Cooperative interactions Give rise to correlations between the fluctuations a different points in space. Correlation function r, r’ : the positions of two lattice sites. An estimate of the spatial extent of the correlated fluctuations Correlation length A measure of the size of the region, or the range, over which correlated fluctuations in the thermodynamic variables occur. Near the TC, the cooperative interactions promote a rapid onset of large-scale, correlated fluctuations in the system. These fluctuations in turn produce divergences in the range of correlations at TC that lead to singularities in the thermodynamic derivatives. Correlation Length (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/