Introduction to Thermodynamics and Statistical Physics (114016) - Lecture Notes potx

Eyal Buks Introduction to Thermodynamics and Statistical Physics (114016) - Lecture Notes April 13, 2011 Technion Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Preface to be written Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Con tents 1. The Principle of Larg est Uncertain ty 1 1.1 EntropyinInformationTheory 1 1.1.1 Example- TwoStatesSystem 1 1.1.2 SmallestandLargest Entropy 2 1.1.3 Thecompositionproperty 5 1.1.4 Alternative Definitionof Entropy 8 1.2 LargestUncertaintyEstimator 9 1.2.1 Useful Relations 11 1.2.2 TheFreeEntropy 13 1.3 The Principle of Largest Uncertainty in Statistical Mechanics 14 1.3.1 Microcanonical Distribution 14 1.3.2 CanonicalDistribution 15 1.3.3 Grandcanonical Distribution 16 1.3.4 TemperatureandChemicalPotential 17 1.4 Time Evolutionof EntropyofanIsolatedSystem 18 1.5 Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.1 ExternallyAppliedPotential Energy 20 1.6 FreeEntropyandFreeEnergies 21 1.7 ProblemsSet1 21 1.8 SolutionsSet1 29 2. Ideal Gas 45 2.1 AParticleinaBox 45 2.2 GibbsParadox 48 2.3 FermionsandBosons 50 2.3.1 Fermi-DiracDistribution 51 2.3.2 Bose-EinsteinDistribution 52 2.3.3 ClassicalLimit 52 2.4 IdealGasintheClassicalLimit 53 2.4.1 Pressure 55 2.4.2 Useful Relations 56 2.4.3 HeatCapacity 57 2.4.4 InternalDegrees of Freedom 57 2.5 ProcessesinIdealGas 60 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Contents 2.5.1 IsothermalProcess 62 2.5.2 IsobaricProcess 62 2.5.3 IsochoricProcess 63 2.5.4 Isentropic Process 63 2.6 CarnotHeatEngine 64 2.7 Limits Imposed Upon the Efficiency 66 2.8 ProblemsSet2 71 2.9 SolutionsSet2 79 3. Bosonic and Fermionic Systems 97 3.1 ElectromagneticRadiation 97 3.1.1 ElectromagneticCavity 97 3.1.2 PartitionFunction 100 3.1.3 CubeCavity 100 3.1.4 AverageEnergy 102 3.1.5 Stefan-Boltzmann Radiation Law 103 3.2 PhononsinSolids 105 3.2.1 OneDimensionalExample 105 3.2.2 The3DCase 107 3.3 FermiGas 110 3.3.1 OrbitalPartitionFunction 110 3.3.2 PartitionFunctionof theGas 110 3.3.3 EnergyandNumberofParticles 112 3.3.4 Example: ElectronsinMetal 112 3.4 Semiconductor Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.5 ProblemsSet3 115 3.6 SolutionsSet3 117 4. Classical Limit of Statistical M echanics 127 4.1 ClassicalHamiltonian 127 4.1.1 Hamilton-Jacobi Equations 128 4.1.2 Example 128 4.1.3 Example 129 4.2 Density Function 130 4.2.1 EquipartitionTheorem 130 4.2.2 Example 131 4.3 NyquistNoise 132 4.4 ProblemsSet4 136 4.5 SolutionsSet4 138 5. Exam Wint er 2010 A 147 5.1 Problems 147 5.2 Solutions 148 Eyal Buks Thermodynamics and Statistical Physics 6 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Con tents 6. Exam Wint er 2010 B 155 6.1 Problems 155 6.2 Solutions 156 References 163 Index 165 Eyal Buks Thermodynamics and Statistical Physics 7 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 1. The Principle of Largest Uncertainty In this chapter w e discuss relations between information theory and statistical mechanics. We show that the canonical and grand canonical distributions can be obtained from Shannon’s principle of maximum uncertainty [1, 2, 3]. Moreover, the tim e evolution of the entropy of an isolated system and the H theorem are discussed. 1.1 Entropy in Information Theory The possible states of a given system are denoted as e m ,wherem =1, 2, 3, , and the prob ability that s tate e m is occupied is denoted by p m . The normal- ization condition reads X m p m =1. (1.1) For a giv en probability distribution {p m } the entropy is defined as σ = − X m p m log p m . (1.2) Below we show that this quantity characterizes the uncertaint y in the knowl- edge of the state of the syste m. 1.1.1 Example - Two States System Consider a system which can occupy either state e 1 with probability p,or state e 2 with probability 1 − p,where0≤ p ≤ 1. The entropy is given by σ = −p log p −(1 − p)log(1− p) . (1.3) Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Chapter 1. The Principle o f Largest Un certainty 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 x −p log p − (1 − p)log(1− p) As expected, the entropy vanishes at p =0andp = 1, since in bo th cases there is no uncertainty in wha t is the state which is occupied by the system. The largest uncertainty is obtained at p =0.5, for which σ =log2=0.69. 1.1.2 Smallest and Largest Entropy Smallest value. The term −p log p in the range 0 ≤ p ≤ 1 is plotted in the figure belo w. Note that the value of −p log p in the limit p → 0canbe calculated using L’H ospital’s rule lim p→0 (−p log p) = lim p→0 Ã − dlogp dp d dp 1 p ! =0. (1.4) From this figure , which shows that −p log p ≥ 0 in the range 0 ≤ p ≤ 1, it is easy to infer that the smallest possible value of the entropy is zero. Moreover, since −p log p =0iff p =0orp =1,itisevidentthatσ =0iff the system occupies one of the states with probability one and all the other states with probability zero. In this case ther e is no uncertainty in what is the state which is occupied by the system. Eyal Buks Thermodynamics and Statistical Physics 2 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com [...]... denoted as S1 and S2 Let 1 = 1 (U1 , N1 ) and 2 = 2 (U2 , N2 ) be the entropy of the rst and second system respectively Eyal Buks Thermodynamics and Statistical Physics 19 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Chapter 1 The Principle of Largest Uncertainty and let = 1 + 2 be the total entropy The systems are brought to contact and now both energy and particles... contributes a length a to the total length of the chain, whereas in the other state the section has no contribution to the total length of the chain The total length of the chain in N , and the tension applied to the end points of the chain is F The system is in thermal equilibrium at temperature Eyal Buks Thermodynamics and Statistical Physics 26 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com... Ă Â Ă Â = k + , (1.9) (1.10) (1.11) p = ( p)k + ( p) , Ă Â Ă Â where k and ( p)k are parallel to g0 , and where and ( p) are orthogonal to g0 Using this notation Eq (1.8) can be expressed as Eyal Buks Thermodynamics and Statistical Physics 3 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Chapter 1 The Principle of Largest Uncertainty Ă Â Ă Â = k ã... tanh 2 2 (1.75) (1.76) and E 2 D 1 2 (U ) = 2 cosh2 2 , (1.77) where = 1/ 1 2 x 3 4 5 0 -0 .2 -0 .4 -tanh(1/x) -0 .6 -0 .8 -1 1.3.3 Grandcanonical Distribution Using Eq (1.47) one nds that the probability distribution is given by pm = 1 exp (U (m) N (m)) , Zgc (1.78) where and are the Lagrange multipliers associated with the given expectation values hUi and hN i respectively, and the partition function... Gibbs factor Moreover, Eq (1.48) yields Eyal Buks Thermodynamics and Statistical Physics 16 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 1.3 The Principle of Largest Uncertainty in Statistical Mechanics hU i = à à log Zgc log Zgc hN i = ả ả , (1.80) (1.81) Eq (1.52) yields E à 2 log Z ả D gc 2 , (U ) = 2 E à 2 log Z ả D gc (N )2 = , 2 (1.82) (1.83) and Eq (1.55)... Buks Thermodynamics and Statistical Physics (1.38a) (1.38b) 9 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Chapter 1 The Principle of Largest Uncertainty where l = 0, 1, 2, L A stationary point of occurs i for every small change p, which is orthogonal to all vectors g0 , g1 , g2 , , gL one has 0 = = ã p (1.39) This condition is fullled only when the vector ê... number of ips required Eyal Buks Thermodynamics and Statistical Physics 24 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 1.7 Problems Set 1 a) Find the entropy In this exercise use log in base 2 in the denition P of the entropy, namely = i pi log2 pi b) A random variable X is drawn according to this distribution Find an ecient sequence of yes-no questions of the form, Is... where C = lim x0 (1 + x) x (1.33) Integrating Eq (1.32) and using the initial condition (1.30) yields (K) = C log K Eyal Buks Thermodynamics and Statistical Physics (1.34) 8 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 1.2 Largest Uncertainty Estimator Moreover, the second property requires that C > 0 Choosing C = 1 and using Eq (1.28) yields (p1 , p2 , , pN ) = (M0 )... where = 1/ 20 The elasticity of a rubber band can be described in terms of a onedimensional model of N polymer molecules linked together end -to- end The angle between successive links is equally likely to be 0 or 180 The length of each polymer is d and the total length is L The system is in thermal equilibrium at temperature Show that the force f required to maintain a length L is given by f= L tanh1... (1.58) one can expressed the Lagrange multipliers and as à ả , (1.87) = U N à ả = (1.88) N U The chemical potential à is dened as à = (1.89) In the denition (1.2) the entropy is dimensionless Historically, the entropy was dened as S = kB , (1.90) where Eyal Buks Thermodynamics and Statistical Physics 17 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Chapter 1 The Principle . Eyal Buks Introduction to Thermodynamics and Statistical Physics (114016) - Lecture Notes April 13, 2011 Technion Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo. 163 Index 165 Eyal Buks Thermodynamics and Statistical Physics 7 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 1 (1.11) where ¡ ¯ ∇σ ¢ k and (δ¯p) k are parallel t o ¯ ∇g 0 ,andwhere ¡ ¯ ∇σ ¢ ⊥ and (δ¯p) ⊥ are orthogonal to ¯ ∇g 0 . Using this notation Eq. (1.8) can be expressed as Eyal Buks Thermodynamics and Statistical Physics