1. Ch 2. THERMODYNAMICS, STATISTICAL MECHANICS, AND METROPOLIS ALGORITHMS 2.6 ~ 2.8 Adaptive Cooperative Systems, Martin Beckerman, 1997. Summarized by J.-W. Ha Biointelligence Laboratory, Seoul National University http://bi.snu.ac.kr/

2. 2(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Contents 2.6 Statistical Mechanics  2.6.1 Formal Structure of the Theory  2.6.2 External Parameters 2.7 Thermodynamics  2.7.1 Equilibrium States  2.7.2 The Correspondence between Statistical Mechanics and Thermodynamics 2.8 The Ensembles of Statistical Mechanics  2.8.1 Microcanonical Ensemble  2.8.2 The Canonical Ensemble  2.8.3 Helmholts Free Energy  2.8.4 Energy Fluctations  2.8.5 Grand Canonical Ensemble

3. 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, http://bi.snu.ac.kr/http://bi.snu.ac.kr/ Normalized term = Partition function

4. External Parameters The Effects of External Parameters 4(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/http://bi.snu.ac.kr/

5. 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, http://bi.snu.ac.kr/http://bi.snu.ac.kr/

6. 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, http://bi.snu.ac.kr/http://bi.snu.ac.kr/

7. 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, http://bi.snu.ac.kr/http://bi.snu.ac.kr/

8. Microcanonical Ensembles  No information available expect the normalization condition 8(C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/http://bi.snu.ac.kr/ 0 elsewhere

9. 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, http://bi.snu.ac.kr/http://bi.snu.ac.kr/