Introduction to Thermodynamics and Statistical Physics phần 3

1.7. Problems Set 1 a) Show that α is given by · µ ¶¸ α = 2 1+tanh 2τ . (1.135) b) Show that in the limit of high temperature the spring constant is given approximately by k ` 4τ2 . (1.136) Nα 19. A long elastic molecule can be modelled as a linear chain of N links. The state of each link is characterized by two quantum numbers l and n. The length of a link is either l = a or l = b. The vibrational state of a link is modelled as a harmonic oscillator whose angular frequency is ωa for a link of length a and ωb for a link of length b. Thus, the energy of a link is En,l = ½}ωa ¡n+ 1¢ for l = a , (1.137) 2 n = 0,1,2,... The chain is held under a tension F. Show that the mean length hLi of the chain in the limit of high temperature T is given by hLi = N aωb +bωa +N F (ωωa (a−b)2 β +O¡β2¢ , (1.138) where β = 1/τ. 20. The elasticity of a rubber band can be described in terms of a one-dimensional 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 τ. Show that the force f required to maintain a length L is given by f = d tanh−1 Nd. (1.139) 21. Consider a system which has two single particle states both of the same energy. When both states are unoccupied, the energy of the system is Eyal Buks Thermodynamics and Statistical Physics 27 Chapter 1. The Principle of Largest Uncertainty zero; when one state or the other is occupied by one particle, the energy is ε. We suppose that the energy of the system is much higher (infinitely higher) when both states are occupied. Show that in thermal equilibrium at temperature τ the average number of particles in the level is hNi = 2+exp[β(ε−µ)] , (1.140) where µ is the chemical potential and β = 1/τ. 22. Consider an array of N two-level particles. Each one can be in one of two states, having energy E1 and E2 respectively. The numbers of particles in states 1 and 2 are n1 and n2 respectively, where N = n1 +n2 (assume n1 À 1 and n2 À 1). Consider an energy exchange with a reservoir at temperatureτ leadingtopopulationchangesn2 → n2−1andn1 → n1+1. a) Calculate the entropy change of the two-level system, (∆σ) . b) Calculate the entropy change of the reservoir, (∆σ) . c) What can be said about the relation between (∆σ) and (∆σ) in thermal equilibrium? Use your answer to express the ration n2/n1 as a function of E1, E2 and τ. 23. Consider a lattice containing N sites of one type, which is denoted as A, and the same number of sites of another type, which is denoted as B. The lattice is occupied by N atoms. The number of atoms occupying sites of type A is denoted as NA, whereas the number of atoms occupying atoms of type B is denoted as NB, where NA + NB = N. Let ε be the energy necessary to remove an atom from a lattice site of type A to a lattice site of type B. The system is in thermal equilibrium at temperature τ. Assume that N,NA,NB À 1. a) Calculate the entropy σ. b) Calculate the average number hNBi of atoms occupying sites of type B. 24. Consider a microcanonical ensemble of N quantum harmonic oscillators in thermal equilibrium at temperature τ. The resonance frequency of all oscillators is ω. The quantum energy levels of each quantum oscillator is given by µ ¶ εn = }ω n+ 2 , (1.141) where n = 0,1,2,... is integer. The total energy E of the system is given by µ ¶ E = }ω m+ 2 , (1.142) where Eyal Buks Thermodynamics and Statistical Physics 28 1.8. Solutions Set 1 N m = nl , (1.143) l=1 and nl is state number of oscillator l. a) Calculate the number of states g(N,m) of the system with total energy }ω(m+N/2). b) Use this result to calculate the entropy σ of the system with total energy }ω(m+N/2). Approximate the result by assuming that N À 1 and m À 1. c) Use this result to calculate (in the same limit of N À 1 and m À 1) the average energy of the system U as a function of the temperature τ. 25. The energy of a donor level in a semiconductor is −ε when occupied by an electron (and the energy is zero otherwise). A donor level can be either occupied by a spin up electron or a spin down electron, however, it cannot be simultaneously occupied by two electrons. Express the average occupation of a donor state hNdi as a function of ε and the chemical potential µ. 1.8 Solutions Set 1 1. Final answers: N! 1 N (N )!(N )! 2 b) 0 2. Final answers: a) 5 N b) 5 N−1 1 c) In general ∞ ∞ N=0 NxN−1 = dx N=0 xN = dx 1−x = (1−x)2 , thus X µ ¶N−1 N = 6 N=0 N 6 = 6 ¡1− 5¢2 = 6 . 3. Let W (m) be the probability for for taking n1 steps to the right and n2 = N − n1 steps to the left, where m = n1 − n2, and N = n1 + n2. Using Eyal Buks Thermodynamics and Statistical Physics 29 Chapter 1. The Principle of Largest Uncertainty n1 = N +m , N −m 2 2 one finds N! N+m N−m N+m N−m 2 2 2 2 It is convenient to employ the moment generating function, defined as φ(t) = ­etm® . In general, the following holds X k φ(t) = m , k=0 thus from the kth derivative of φ(t) one can calculate the kth moment of m ­mk® = φ(k) (0) . Using W (m) one finds N φ(t) = W (m)etm m=−N N N! N+m N−m tm N+m N−m m=−N 2 2 or using the summation variable n1 = N +m , one has N N! n1 N−n1 t(2n1−N) n1=0 n1!(N −n1)! −tN N N! ¡ 2t¢n1 N−n1 n1=0 n1!(N −n1)! = e−tN pe2t +q N . Using p = q = 1/2 Eyal Buks Thermodynamics and Statistical Physics 30 1.8. Solutions Set 1 µ t −t ¶N φ(t) = 2 = (cosht) . Thus using the expansion (cosht)N = 1+ 2!Nt2 + 4! [N +3N (N −1)]t4 +O¡t5¢ , one finds hmi = 0 , m2 = N , m3 = 0 , m4 = N (3N −2) . 4. Using the binomial distribution W (n) = n!(N −n)!pn (1−p)N−n = N (N −1)(N −1)×...(N −n +1)pn (1−p)N−n = (Np)n exp(−Np) . 5. X X a) W (n) = e n! = 1 n=0 n=0 b) As in Ex. 1-6, it is convenient to use the moment generating function φ(t) = ­etn® = XetnW (n) = e−λ X λnetn = e−λ X (λet)n = exp£λ¡et −1¢¤ . n=0 n=0 n=0 Using the expansion exp£λ¡et −1¢¤ = 1+λt + 2λ(1+λ)t2 +O¡t3¢ , one finds hni = λ . c) Using the same expansion one finds ­n2® = λ(1+λ) , thus D(∆n)2E = ­n2®−hni2 = λ(1+λ)−λ2 = λ . Eyal Buks Thermodynamics and Statistical Physics 31 ... - tailieumienphi.vn