Introduction to Thermodynamics and Statistical Physics phần 9 pptx

... (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 ... heorem for the e ne rg y stored in the inductor LI 2 /2. 5. Calculate for example Eyal Buks Thermodynamics and Statistical Physics 1 39...

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... Buks Thermodynamics and Statistical Physics 104 Chapter 3. Bosonic and Fermionic Systems Similarly to th e EM case, the average total energy is given by U = X n }ω n exp (β}ω n ) − 1 (3. 69) = 3π 2 n D Z 0 dnn 2 }ω n exp ... 1, thus U =9Nτ ³ τ Θ ´ 3 x D Z 0 dx x 3 exp x −1 ' 9Nτ ³ τ Θ ´ 3 ∞ Z 0 dx x 3 exp x − 1 | {z } π 4 /15 = 3π 4 5 Nτ ³ τ Θ ´ 3 , (3.80) Eyal Buks Thermod...

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... + 2 exp (βJ) , (6.37) Eyal Buks Thermodynamics and Statistical Physics 1 59 Chapter6. ExamWinter2010B 2. The k vector is restricted due to boundary conditions to the values k = πn L , (6.21) where n ... N B )log   µ V 2 V 1 ¶µ τ 2 τ 1 ¶ c V,A +c V,B N A +N B   , (5.54) and the requirement ∆σ =0leadsto Eyal Buks Thermodynamics and Statistical Physics 153 5.2. Solutio...

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... (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 ... system. Eyal 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 Inde...

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... (1.120) Use this result to evaluate the expectation value and the variance of M. 9. Consider a one dimensional random walk. The p robabilities of transiting to the right and left are p and q =1−p respectively. ... as µ = −τη . (1. 89) In the definition (1.2) the entropy σ is dimensionless. Historically, the entropy was defined as S = k B σ, (1 .90 ) where Eyal Buks Thermodynamics...

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... 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 1 In ... Buks Thermodynamics and Statist ical Physics 43 Chapter 1. The Principle of Largest Uncertainty is determined by the Boltzmann factor. The energy change due to flipping of o...

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... (2.88) c p = c V + N. (2. 89) Proof. See problem 3 of set 2 . Eyal Buks Thermodynamics and Statistical Physics 59 Chapter 2. Ideal Gas where {E l } are the eigenenergies of a particle due to internal degrees ... (−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. Fer...

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... (V 2 − V 1 ) . (2 .98 ) The change in energy ∆U can be found by in tegrating Eq. (2 .96 ) ∆U = τ 2 Z τ 1 c V dτ. (2 .99 ) The heat added to the system Q can be found using Eq. (2 .94 ) Q = W + ∆U = p ... as afunctionofτ and N only). Th us w e conclude that for processes for which dN = 0 the change in energy dU can be expressed as dU = c V dτ. (2 .96 ) Eyal Buks Thermodynamics and Sta...

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... n)! p n (1 − p) N−n . (2. 194 ) Using the solution of problem 4 of set 1 p n = λ n n! e −λ , where λ = Nv/V. Eyal Buks Thermodynam ics and Statistical Physics 82 2 .9. Solutions Set 2 19. The efficiency is ... H/τ,therefore τ 2 = τ 1 H 2 H 1 . (2.200) 9. The partition function of a single atom is given b y Eyal Buks Thermodynam ics and Statistical Physics 83 2 .9. Solutions Set 2...

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... ~cπ µ 3N πV ¶ 1/3 Eyal Buks Thermodynamics and Statistical Physics 123 Chapter 3. Boso nic and Fermionic Systems thus T planet = r R Sun 2D T Sun , and for Mars T Mars = r 6 .96 × 10 8 m 2 × 2.28 × ... Solutions Set 3 thus µ = τ log · exp µ nπ~ 2 m e τ ¶ − 1 ¸ . (3.1 79) Eyal Buks Thermodynamics and Statistical Physics 125 Chapter 3. Boso nic and Fermionic Systems I = ∞...

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