熵、大偏差和统计力学

　　《熵、大偏差和统计力学》是一部教程，内容上相对独立，自成体系。书中大偏差的讲述除了为这科目做出了巨大贡献，也将统计力学的好多方面完美结合，并且很具有数学吸引力。而且作者在没有假设读者具有丰富的物理知识背景下讲述，使得本书能够让更多的读者学习理解。每章末都附有一节注解和一节问题，这100来道练习题，附有许多提示，使得本书更加易于学习理解。目次：（第一部分）大偏差和统计力学：大偏差导论；大偏差性质和积分渐近；大偏差和离散理想气体；z上的铁磁模型；zd和圆周上的磁模型；（第二部分）大偏差定理上的复杂度和证明：复函数和legendre-fenchel变换；大偏差的随机向量；i. i. d.随机变量的2级大偏差；i. i. d.随机变量的3级大偏差；附录：概率论；ii.7中两个定理的证明；自旋系统中无限体积测度的等价观点；特殊gibbs自由能量的存在性。读者对象：数学专业的研究生，教师和相关专业的科研人员。

暂缺《熵、大偏差和统计力学》作者简介

preface
comments on the use of this book
part i: large deviations and statistical mechanics
chapter i. introduction to large deviations
  i.1. overview
  i.2. large deviations for 1.i.d. random variables with afinite state space
  i.3. levels-1 and 2 for coin tossing
  i.4. levels-1 and 2 for i.i.d. random variables with afinite state space
  i.5. level-3: empirical pair measure
  i.6. level-3: empirical process
  i.7. notes
  i.8. problems
chapter ii. large deviation property and asymptotics ofintegrals
  ii.1. introduction
  ii.2. levels-l, 2, and 3 large deviations for i.i.d. randomvectors
  ii.3. the definition of large deviation property
  ii.4. statement of large deviation properties for levels-l,2, and 3
  ii.5. contraction principles
  ii.6. large deviation property for random vectors andexponential convergence
  ii.7. varadhan's theorem on the asymptotics ofintegrals
  ii.8. notes
  ii.9. problems
chapter iii. large deviations and the discrete ideal gas
  iii.1. introduction
  iii.2. physics prelude: thermodynamics
  iii.3. the discrete ideal gas and the microcanonicalensemble
  iii.4. thermodynamic limit, exponential convergence, andequilibrium values
  iii.5. the maxweli-boltzmann distribution andtemperature
  iii.6. the canonical ensemble and its equivalence with themicrocanonical ensemble
  iii.7. a derivation of a thermodynamic equation
  ill.8. the gibbs variational formula and principle
  iii.9. notes
  iii. 10. problems
chapter iv. ferromagnetic models on z
  iv.1. introduction
  iv.2. an overview of ferromagnetic models
  iv.3. finite-volume gibbs states on 77
  iv.4. spontaneous magnetization for the curie-weissmodel
  iv.5. spontaneous magnetization for general ferromagnetson
  iv.6. infinite-volume gibbs states and phasetransitions
  iv.7. the gibbs variational formula and principle
  iv.8. notes
  iv.9. problems
chapter v. magnetic models on 7/d and on the circle
  v.1. introduction
  v.2. finite-volume gibbs states on zd, d ≥ 1
  v.3. moment inequalities
  v.4. properties of the magnetization and the gibbs freeenergy
  v.5. spontaneous magnetization on z, d ≥ 2, via the peierlsargument
  v.6. infinite-volume gibbs states and phasetransitions
  v.7. infinite-volume gibbs states and the central limittheorem
  v.8. critical phenomena and the breakdown of the centrallimit theorem
  v.9. three faces of the curie-weiss model
  v. 10. the circle model and random waves
  v.11. a postscript on magnetic models
  v.12. notes
  v.13. problems
part ii: convexity and proofs of large deviation theorems
chapter vi. convex functions and the legendre-fencheltransform
  vii.1. introduction
  vi.2. basic definitions
  vi.3. properties of convex functions
  vi.4. a one-dimensional example pf the legendre-fencheltransform
  vi.5. the legendre-fenchel transform for convex functions onra
  vi.6. notes
  vi.7. problems
chapter vii. large deviations for random vectors
  vii. i. statement of results
  vii.2. properties of i
  vii.3. proof of the large deviation bounds for d = 1
  vii.4. proof of the large deviation bounds for d≥ 1
  vii.5. level-i large deviations for i.i.d. randomvectors
  vii.6. exponential convergence and proof of theoremii.6.3
  vii.7. notes
  vii.8. problems
chapter viii. level-2 large deviations for i.i.d. randomvectors
  viii. 1. introduction
  viii.2. the level-2 large deviation theorem
  viii.3. the contraction principle relating levels-i and 2 (d= 1)
  viii.4. the contraction principle relating levels-1 and 2 (d≥ 2)
  viii.5. notes
  viii.6. problems
chapter ix. level-3 large deviations for i.i.d. randomvectors
  ix. 1. statement of results
  ix.2. properties of the level-3 entropy function
  ix.3. contraction principles
  ix.4. proof of the level-3 large deviation bounds
  ix.5. notes
  ix.6. problems
appendices
appendix a: probability
  a.1. introduction
  a.2. measurability
  a.3. product spaces
  a.4. probability measures and expectation
  a.5. convergence of random vectors
  a.6. conditional expectation, conditional probability, andregular conditional distribution
  a.7. the koimogorov existence theorem
  a.8. weak convergence of probability measures on a metricspace
  a.9. the space ms((rd)z) and the ergodic theorem
  a.10. n-dependent markov chains
  a.11. probability measures on the space { 1, - 1}zd
appendix b: proofs of two theorems in section ii.7
  b.i. proof of theorem ii.7.1
  b.2. proof of theorem ii.7.2
appendix c: equivalent notions of infinite-volume measures for spinsystems
  c.i. introduction
  c.2. two-body interactions and infinite-volume gibbsstates
  c.3. many-body interactions and infinite-volume gibbsstates
  c.4. dlr states
  c.5. the gibbs variational formula and principle
  c.6. solution of the gibbs variational formula forfinite-range interactions on z
appendix d: existence of the specific gibbs free energy
  d.1. existence along hypercubes
  d.2. an extension
  list of frequently used symbols
  references
  author index
  subject index