THERMODYNAMICS Edited by Tadashi Mizutani Thermodynamics Edited by Tadashi Mizutani Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Ana Nikolic Technical Editor Teodora Smiljanic Cover Designer Martina Sirotic Image Copyright Khotenko Volodymyr, 2010. Used under license from Shutterstock.com First published January, 2011 Printed in India A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Thermodynamics, Edited by Tadashi Mizutani p. cm. ISBN 978-953-307-544-0 free online editions of InTech Books and Journals can be found at www.intechopen.com [...]... = q pi (70) Eq (64) leads to 1 pi = ( )1/ (q 1) [1 − (1 − q) β i ]1/ (1 q) , q and, after normalization, one is led to the Curado-Tsallis distribution (35) pi = Zq = ( Zq ) 1 [1 − (1 − q) β i ]1/ (1 q) ∑ [1 − (1 − q) β i ]1/ (1 q) i ( 71) (72) 18 20 Thermodynamics Thermodynamics 10 .3 Exponential entropic form This measure is given in (36; 37) and also used in (38) One has 1 − exp (−bpi ) − S0 , pi f (... one (33) pi = − Zq 1 1 − (q − 1) β Zq = ∑ 1 − ( q − 1) β i 1/ (q 1) i 1/ (q 1) , (67) (68) i where β ≡ β/ (1 + (q − 1) βK ) 10 .2 Tsallis measure with non-linear constraints The information measure is still the one built up with the function f ( pi ) of (65), but we use now the so-called Curado-Tsallis constraints (35) that arise if one uses U= E = with q W ∑ i =1 g ( pi ) i , (69) q 1 g ( pi ) = pi ⇒.. .Part 1 Fundamentals of Thermodynamics 0 1 New Microscopic Connections of Thermodynamics 1 Facultad A Plastino1 and M Casas2 de C Exactas, Universidad Nacional de La Plata IFLP-CONICET, C.C 727, 19 00 La Plata 2 Physics Departament and IFISC-CSIC, University of Balearic Islands 0 712 2 Palma de Mallorca 1 Argentina 2 Spain 1 Introduction This is a work that discusses... ideas developed in (7; 8; 9; 10 ; 11 ; 12 ; 13 ) 2 4 Thermodynamics Thermodynamics 2 Thermodynamics axioms Thermodynamics can be thought of as a formal logical structure whose axioms are empirical facts, which gives it a unique status among the scientific disciplines (1) The four postulates we state below are entirely equivalent to the celebrated three laws of thermodynamics (6): 1 For every system there... β[ i g ( pi ) − K ] = 0, β ≡ 1/ kT (64) 10 .1 Tsallis measure with linear constraints We have, for any real number q the information measure (28) built up with (26; 33; 34) q 1 f ( pi ) = (1 − p i ) , q 1 (65) and, in the energy-constraint of Eq (30) g ( pi ) = pi , so that f ( pi ) = q −2 − pi (66) and Eq (64) becomes, with β = (1/ kT ), q 1 q pi = 1 + (q − 1) βK − (q − 1) β i , which after normalization... defining the Hilbert space of the system give also a contribution to the “heat part of the dU = TdS + δW relation 16 18 Thermodynamics Thermodynamics Example 2: In the Shannon instance discussed in Example 1 one has [Cf (48) and (49)] ∂pi = − βpi (1 − Z 1 ), ∂ i T (56) ∂S ∂pi = − β ( i − K ) pi , ∂pi ∂ i (57) ∂ i (1 − Z 1 ) ∂V QV = − ∑ β ( i − K ) pi i (58) Since the origin of the energy scale is... Variations in, respectively, pi , S, and U write W ∂pi ∂p ∂ j dT + ∑ i dV, ∂T ∂ j ∂V j =1 (32) W ∂S ∂pi ∂S ∂pi ∂ j dT + ∑ dV, ∂T ∂pi ∂ j ∂V i =1 i,j =1 (33) dpi = dS = W ∑ ∂pi and, last but not least, dU = W ∂g ∂pi ∂T i =1 ∑ ∂pi i dT + ∂g ∂pi ∂ j ∂pi ∂ j ∂V i,j =1 W ∑ W i dV + ∑ g ( pi ) i =1 ∂ i dV, ∂V (34) 12 14 Thermodynamics Thermodynamics where, for simplicity, we have considered non-degenerate levels Clearly,... (40) into two i= parts, i.e., W 1 ∑ i =1 ∂g ∂pi i −T ∂S ∂pi ∂pi dT + ∂T ∂g ∂pW W −T ∂S ∂pW ∂pW dT = 0 ∂T ( 41) Picking out level W for special attention is arbitrary Any other i −level could have been chosen as well, as the example given below will illustrate Taking into account now that, from Eq (39), W 1 ∂pW ∂pi =− ∑ , ∂T ∂T i =1 (42) we see that Eq ( 41) can be rewritten as W 1 ∑ i =1 ∂g ∂pi i −T... care above about such an expression, but we do now 14 16 Thermodynamics Thermodynamics Example 1 Consider the Shannon orthodox instance S = −k ∑ pi ln pi g ( pi ) = pi ∂S/∂pi = −k[ln pi + 1] = k[ β i + ln Z − 1] ] i (47) Here equation (46) yields the well known MaxEnt (and also Gibbs?) result ln pi pi ln Z = = = −[ β i + ln Z ]; i.e., Z 1 e− i /kT 1 − K/kT, and, finally, (48) ∂S/∂pi = kβ( i − K ), ∂... Clearly, p(i, j) = p1 (i ) Q( j|i ) (12 ) Then Kinchin’s fourth axiom states that I ( p ) = I ( p1 ) + ∑ p1 ( i ) I Q ( j | i ) (13 ) i An important consequence is that, out of the four Kinchin axioms one finds that Shannons’s measure N S = − ∑ p(i ) ln [ p(i )], i =1 is the one and only measure complying with them (14 ) New Microscopic Connections Thermodynamics New Microscopic Connections of of Thermodynamics . Olivier Baudouin Part 2 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Pref ac e Progress of thermodynamics has been stimulated. provide answers to the query, following ideas developed in (7; 8; 9; 10 ; 11 ; 12 ; 13 ). 1 2 Thermodynamics 2. Thermodynamics axioms Thermodynamics can be thought of as a formal logical structure whose. 8; 9; 10 ; 11 ; 12 ; 13 ) (remember that the Lagrange multipliers λ ν are identical to the generalized pressures P ν of Eq. (3)) C (1) i =[ ∑ M ν =1 λ ν a ν i + i ] C (2) i = −T ∂S ∂p i 10 Thermodynamics