Download e-book for kindle: An introduction to statistical physics (1982) by William Geraint V. Rosser

By William Geraint V. Rosser

ISBN-10: 0853122725

ISBN-13: 9780853122722

Advent to Statistical Physics (Mathematics and Its purposes) [Paperback

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Extra resources for An introduction to statistical physics (1982)

Example text

Con probabilit`a Pk . , cio`e da G(s) si possono sempre determinare le probabilit`a {Pk }. 12. Siano X ed Y variabili Poissoniane indipendenti, con parametri rispettivamente λx e λy , calcolare P(X = k|X +Y = n). 13. Si consideri un dado non truccato, calcolare un numero medio di lanci per avere per la prima volta 6. 14. Per incentivare le vendite un supermercato inserisce nelle scatole di biscotti un bollino numerato da 1 a 5. Dopo aver collezionato tutti i numeri si ha diritto ad una scatola gratis.

3 e` evidente che, se y1 = f (x1 ) e y2 = f (x2 ) P(y ∈ [y1 , y2 ]) = P(x ∈ [x1 , x2 ]) , se x2 = x1 + Δ x con Δ x piccolo, allora y2 = y1 + Δ y con Δ y = f (x1 )Δ x, poich´e px (x)Δ x = py (y)| f (x)|Δ x , (il modulo e` stato introdotto per tener conto dei casi con f < 0) si ottiene py (y) = px (x∗ ) , con x∗ = f −1 (y) . 14) 28 2 Qualche risultato con un po’ di formalismo Fig. 14) possono essere scritte in forma compatta (e facile da ricordare): py (y) = px (x)δ (y − f (x))dx . ,yN (y) = ∑ |detA (x(k) )| x(k) :f(x(k) )=y ove A e` la matrice con elementi ∂ fi /∂ x j .

Con distribuzione gaussiana a media nulla e varianza unitaria, calcolare la densit`a di probabilit`a delle variabili r= x12 + x22 , z = x2 . 8. , con distribuzione gaussiana, mostrare che le variabili z = x1 + x2 , q = x1 − x2 sono gaussiane ed indipendenti. 9. d. uniformamente distribuite in [0, 1], si considerino le variabili z1 = r cos ψ , z2 = r sin ψ ove r= −2 ln x1 , ψ = 2πx2 . Si mostri che z1 e z2 sono variabili gaussiane indipendenti a media nulla e varianza unitaria. 10. Due amici si danno un appuntamento probabilistico: ci si vede al bar tra le 17 e le 18, si aspetta 5 minuti e poi si esce.

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An introduction to statistical physics (1982) by William Geraint V. Rosser

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