1. 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

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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
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Lattice
The assemblage is compose of the elements of a cooperative systems spatially in a regular manner.
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4. 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
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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>
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6. 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
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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
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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.
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9. 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.
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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
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