Introduction to Thermodynamics and Statistical Physics phần 7 pptx

... integral, a nd to a good approximation the upper limit N/2canbe substituted by infinity, thus Eyal Buks Thermodynamics and Statistical Physics 106 3.2. Phonons in Solids x D = Θ τ , (3 .76 ) and U =9Nτ ³ τ Θ ´ 3 x D Z 0 dx x 3 exp ... spacing, k n = 2πn aN , (3. 57) and n is in teger ranging from −N/2toN/2. 0 0.2 0.4 0.6 0.8 -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 x The function...

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... (4.53) ThelastresultistheNyquist’snoiseformula. Eyal Buks Thermodynamics and Statistical Physics 135 4.1. Classical Hamiltonian H = m ˙q 2 − m ˙q 2 2 + V (q) = p 2 2m + V (q) . (4.9) Hamilton-Ja cobi equations (4 .7) and (4.8) read ˙q ... energy stored in the inductor, and the potential energy V = q 2 /2C is t he energy stored in the capacitor. The canonical conjugate momentum is...

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... (−βJ)[exp(−2βµ 0 H)+exp(2βµ 0 H)] + 2 exp (βJ) , (6. 37) Eyal Buks Thermodynamics and Statistical Physics 159 Chapter6. ExamWinter2010B 2. The k vector is restricted due to boundary conditions to the values k = πn L , (6.21) where n ... e −η V π 2 ~ 3 c 3 β 3 . (6. 27) a) The a v erage energy U and average number of particle N are calcu- lated using Eqs. (1.80) and (1.81) re...

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... Buks Thermodynamics and Statistical Physics 2 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 ... 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...

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... (1.69) Z c =exp µ βε 2 ¶ +exp µ − βε 2 ¶ =2cosh µ βε 2 ¶ , (1 .75 ) thus using Eqs. (1 .70 ) and (1 .71 ) one finds hUi = − ε 2 tanh µ βε 2 ¶ , (1 .76 ) and D (∆U) 2 E = ³ ε 2 ´ 2 1 cosh 2 βε 2 , (1 .77 ) where β =1/τ. -1 -0.8 -0.6 -0.4 -0.2 0 -tanh(1/x) 12345 x 1.3.3 ... systems denoted as S 1 and S 2 .Letσ 1 = σ 1 (U 1 ,N 1 )and σ 2 = σ 2 (U 2 ,N 2 )betheentropyofthefirst and second...

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... probability for for taking n 1 steps to the right and n 2 = N − n 1 steps to the left, where m = n 1 − n 2 ,andN = n 1 + n 2 . Using Eyal Buks Thermodynamics and Statist ical Physics 29 1.8. Solutions Set ... Buks Thermodynamics and Statist ical Physics 37 Chapter 1. The Principle of Largest Uncertainty 24. In general, g (N,m)={# o s ways to distribute m identical balls i...

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... λZ int exp (−βε n ) . (2 .77 ) The total grandcanonical partition function is given by Z gc = Y n ζ n , (2 .78 ) thus Eyal Buks Thermodynamics and Statistical Physics 58 2.3. Fermions and Bosons occupation ... Fermio ns becomes ζ FD,n = Y l (1 + λ exp (−βε n )exp(−βE l )) , (2 .71 ) Eyal Buks Thermodynamics and Statistical Physics 57 Chapter 2. Ideal Gas 2.3.2 Bos e-Einstei...

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... Buks Thermodynamics and Statistical Physics 72 2.8. Problems Set 2 Corollary 2 .7. 1. All reversible heat engines operating bet ween a hotter heat reservoir and a colder one at temperatures τ h and ... (2.134) Eyal Buks Thermodynamics and Statistical Physics 71 2.8. Problems Set 2 8. A lattice contains N sites, each is occupied by a single atom. The set of eigenstates...

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... Z int ¶ . (2. 175 ) b) Using Eqs. (2. 172 ) and (2. 173 ) U = N µ 3τ 2 − ∂ log Z int ∂β ¶ . (2. 176 ) c) Using the relations F = U −τσ , (2. 177 ) σ =logZ gc + βU + ηN , (2. 178 ) one finds F = U −τσ (2. 179 ) = Nτ ... Z int − 1 ¶ . (2.182) (2.183) Eyal Buks Thermodynam ics and Statistical Physics 80 Chapter 2. Ideal Gas 4. Using Eqs. (1. 87) , (1 .70 ) and (1 .71 ), and noting tha...

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... hni) 2 E . (3.1 47) Eyal Buks Thermodynamics and Statistical Physics 120 Chapter 4. Classical Limit of Statistical Mechanics ¯p =(p 1 ,p 2 , , p d ) , (4.4) namely H = H (¯q, ¯p) , (4.5) and it is defined ... Solutions Set 3 thus µ = τ log · exp µ nπ~ 2 m e τ ¶ − 1 ¸ . (3. 179 ) Eyal Buks Thermodynamics and Statistical Physics 125 Chapter 3. Boso nic and Fermionic System...

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