Ch 2. THERMODYNAMICS, STATISTICAL MECHANICS, AND METROPOLIS ALGORITHMS 2.6 ~ 2.8 Adaptive Cooperative Systems, Martin Beckerman, Summarized by J.-W. Ha Biointelligence Laboratory, Seoul National University

2(C) 2009, SNU Biointelligence Lab, Contents 2.6 Statistical Mechanics  Formal Structure of the Theory  External Parameters 2.7 Thermodynamics  Equilibrium States  The Correspondence between Statistical Mechanics and Thermodynamics 2.8 The Ensembles of Statistical Mechanics  Microcanonical Ensemble  The Canonical Ensemble  Helmholts Free Energy  Energy Fluctations  Grand Canonical Ensemble

Formal Structure of the Theory Maximal Entropy Principle  We should make inferences using probability distributions that maximize the entropy s. t. to the given constraints.  Expectation value  Variance 3(C) 2009, SNU Biointelligence Lab, Normalized term = Partition function

External Parameters The Effects of External Parameters 4(C) 2009, SNU Biointelligence Lab,

Equilibrium States Parameters of state  Equilibrium states are completely described by a small number of macroscopic variables called parameters of state.  If the integrated changes in a system produced by a small increment in that quantity Stable  Once formed, they do not easily change in time  States of maximum entropy for a given value of the total energy In cooperative systems  Tend to be disordered at high tempartures  Ordering is propagated throughout the system at transition temperature 5(C) 2009, SNU Biointelligence Lab,

The Correspondence between Statistical Mechanics and Thermodynamics Thermodynamics application  External variables : volume, magnetic fields, work(heat)  Inexact differentials : small differential changes in work done by a system, and heat added to a system 6(C) 2009, SNU Biointelligence Lab,

The second law of thermodynamics  For any equilibrium state of a thermodynamic closed system, it is possible to define a thermodynamic entropy 7(C) 2009, SNU Biointelligence Lab,

Microcanonical Ensembles  No information available expect the normalization condition 8(C) 2009, SNU Biointelligence Lab, 0 elsewhere

The Canonical and Grand Canonical Ensemble The Canonical Ensemble Helmholts Free Energy Energy Fluctuations Grand Canonical Ensemble  Composed of kinetic term(single particle) and potential term(coordinates) 9(C) 2009, SNU Biointelligence Lab,