熵大偏差和統(tǒng)計(jì)力學(xué)

內(nèi)容概要

　　《熵、大偏差和統(tǒng)計(jì)力學(xué)》是一部教程，內(nèi)容上相對(duì)獨(dú)立，自成體系。書(shū)中大偏差的講述除了為這科目做出了巨大貢獻(xiàn)，也將統(tǒng)計(jì)力學(xué)的好多方面完美結(jié)合，并且很具有數(shù)學(xué)吸引力。而且作者在沒(méi)有假設(shè)讀者具有豐富的物理知識(shí)背景下講述，使得本書(shū)能夠讓更多的讀者學(xué)習(xí)理解。每章末都附有一節(jié)注解和一節(jié)問(wèn)題，這100來(lái)道練習(xí)題，附有許多提示，使得本書(shū)更加易于學(xué)習(xí)理解。目次：（第一部分）大偏差和統(tǒng)計(jì)力學(xué)：大偏差導(dǎo)論；大偏差性質(zhì)和積分漸近；大偏差和離散理想氣體；z上的鐵磁模型；zd和圓周上的磁模型
；（第二部分）大偏差定理上的復(fù)雜度和證明：復(fù)函數(shù)和legendre-fenchel變換；大偏差的隨機(jī)向量；i. i. d.
隨機(jī)變量的2級(jí)大偏差；i. i. d.
隨機(jī)變量的3級(jí)大偏差；附錄：概率論；ii.7中兩個(gè)定理的證明；自旋系統(tǒng)中無(wú)限體積測(cè)度的等價(jià)觀點(diǎn)；特殊gibbs自由能量的存在性。
　　讀者對(duì)象：數(shù)學(xué)專業(yè)的研究生，教師和相關(guān)專業(yè)的科研人員。

作者簡(jiǎn)介

作者：(美國(guó))艾里斯 (Richard S.Ellis)艾里斯，Richard S.Ellis，received his B.A. degree in mathematics and German literature from Harvard University in 1969 and his Ph.D. degree in mathematics from New York University in 1972. After spending three years at Northwestern University, he moved to the University of Massachusetts, Amherst, where he is a Professor in the Department of Mathematics and Statistics and Adjunct Professor in the Depart-ment of Judaic and Near Eastern Studies. His research interests in mathematics focus on the theory of large deviations and on applica-tions to statistical mechanics and other areas.

書(shū)籍目錄

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 a
finite state space
 i.3. levels-1 and 2 for coin tossing
 i.4. levels-1 and 2 for i.i.d. random variables with a
finite 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 of
integrals
 ii.1. introduction
 ii.2. levels-l, 2, and 3 large deviations for i.i.d. random
vectors
 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 and
exponential convergence
 ii.7. varadhan's theorem on the asymptotics of
integrals
 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 microcanonical
ensemble
 iii.4. thermodynamic limit, exponential convergence, and
equilibrium values
 iii.5. the maxweli-boltzmann distribution and
temperature
 iii.6. the canonical ensemble and its equivalence with the
microcanonical 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-weiss
model
 iv.5. spontaneous magnetization for general ferromagnets
on
 iv.6. infinite-volume gibbs states and phase
transitions
 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 free
energy
 v.5. spontaneous magnetization on z, d ≥ 2, via the peierls
argument
 v.6. infinite-volume gibbs states and phase
transitions
 v.7. infinite-volume gibbs states and the central limit
theorem
 v.8. critical phenomena and the breakdown of the central
limit 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-fenchel
transform
 vii.1. introduction
 vi.2. basic definitions
 vi.3. properties of convex functions
 vi.4. a one-dimensional example pf the legendre-fenchel
transform
 vi.5. the legendre-fenchel transform for convex functions on
ra
 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. random
vectors
 vii.6. exponential convergence and proof of theorem
ii.6.3
 vii.7. notes
 vii.8. problems
chapter viii. level-2 large deviations for i.i.d. random
vectors
 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. random
vectors
 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, and
regular conditional distribution
 a.7. the koimogorov existence theorem
 a.8. weak convergence of probability measures on a metric
space
 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 spin
systems
 c.i. introduction
 c.2. two-body interactions and infinite-volume gibbs
states
 c.3. many-body interactions and infinite-volume gibbs
states
 c.4. dlr states
 c.5. the gibbs variational formula and principle
 c.6. solution of the gibbs variational formula for
finite-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

章節(jié)摘錄

版權(quán)頁(yè)：插圖：In the next three chapters we apply the theory of large deviations to analyze some basic models in equilibrium statistical mechanics.' This branch of physics applies probability theory to study equilibrium properties of systems consisting of a large number of particles. The systems fall into two groups:continuous systems, which include the solids, liquids, and gases common to everyday experience; and lattice systems, of which ferromagnets are the main example. This chapter introduces the continuous theory by treating a simple model called a discrete ideal gas. This model, which has no interactions, is a physical analog of i.i.d, random variables.The macroscopic description of a physical system such as an ideal gas isgiven by thermodynamics. Thermodynamics summarizes the properties ofthe gas in terms of macroscopic variables such as pressure, volume, tempera- ture, and internal energy. But this theory takes no account of the fact that the gas is composed ofrnolecules. The main aim of statistical mechanics is to derive properties of the gas from a probability distribution which describes its microscopic （i.e., molecular） behavior. This distribution is called an ensemble.

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