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By Roger Balian, Dirk Haar, John F. Gregg

This textual content not just presents an intensive advent to statistical physics and thermodynamics but in addition indicates the universality of the chain of principles that leads from the legislation of microphysics to the macroscopic behaviour of topic. a variety of functions teaches scholars the best way to utilize the strategies, and lots of routines may also help to deepen their figuring out. Drawing on either quantum mechanics and classical physics, the e-book follows smooth learn in statistical physics.

Volume I discusses intimately the probabilistic description of quantum or classical platforms, the Boltzmann-Gibbs distributions, the conservation legislation, and the translation of entropy as lacking details. Thermodynamics and electromagnetism in topic are handled, in addition to purposes to gases, either dilute and condensed, and to part transitions.

This softcover version accomodates the numerous requests to make this generic and infrequently pointed out classical textual content to be had again.

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Example text

L’idea pu`o essere spiegata con un semplice calcolo combinatorio. Consideriamo una sequenza di lanci di una moneta truccata i cui possibili risultati sono testa (+1) o croce (−1) ed indichiamo il risultato dell’n-mo lancio con xn , ove testa ha probabilit`a p e croce 1 − p. Sia yN = (x1 + ... + xN )/N ovviamente yN = 2p − 1 e σy2N = 4p(1 − p)/N. ] quindi dalla distribuzione binomiale abbiamo N! 2k −1 = pk (1 − p)N−k . (N − k)! √ Usando l’approssimazione di Stirling n! nn e−n 2πn e scrivendo k = f N e N − k = (1 − f )N ove f = k/N e` la frequenza dell’evento testa in N lanci, si ha P yN = P(y = 2 f − 1) ∼ e−NI(p, f ) , dove I(p, f ) = f ln 1− f f + (1 − f ) ln .

Sia Nn−1 il numero di individui all’(n − 1)-ma generazione, indichiamo con x1 il numero di individui generati dal primo individuo, x2 il numero di individui generati dal secondo e cos`ı via. All’n-ma generazione, avremo: Nn = x1 + x2 + ... + xNn−1 . 12) ove G2 (s) = G(G(s)), G3 (s) = G2 (G(s)) = G(G(G(s))), ... In genere il calcolo della Gn ( ), pur matematicamente banale, pu`o portare ad espressioni scomode da maneggiare. Non e` sempre cos`ı. Citiamo un esempio interessante: intorno al 1920 A.

2 Funzioni generatrici: come contare senza sbagliare 23 Utilizzando la formula di Bayes si ha: P(A2 |C1 ) = P(C1 |A2 )P(A2 ) 1 × 13 2 = 1 = P(C1 ) 3 2 quindi conviene cambiare porta. 2 Funzioni generatrici: come contare senza sbagliare Molti dei problemi di probabilit`a con variabili intere sono riconducibili a calcoli combinatori. ,t6 (ovviamente con i vincoli ∑i pi = ∑i qi = ∑i ti = 1) e ci si chiede la probabilit`a che la somma faccia 12 oppure 8. Un calcolo esplicito basato solo sulle definizioni elementari e` chiaramente possibile, ovviamente le cose si complicano al crescere del numero dei dadi.

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