Mesoscopic Non-Equilibrium Thermodynamics Part 2

Tham khảo tài liệu 'mesoscopic non-equilibrium thermodynamics part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | New Microscopic Connections of Thermodynamics 21 12. Appendix I Here we prove that Eqs. 24 are obtained via the MaxEnt variational problem 27 . Assume now that you wish to extremize S subject to the constraints of fixed valued for i U and ii the M values Av. This is achieved via Lagrange multipliers 1 fi and 2 M Yv. We need also a normalization Lagrange multiplier ỉ. Recall that Av Rv E pit 76 i with aV ẠRv ộ the matrix elements in the chosen basis i of Rv. The MaxEnt variational problem becomes now U Ei piei 5 Pi M S - fiU -E YvAv - ỉE Pi 0 77 leading with Yv ỘẢy to the vanishing of M . E Pif Pi - fi Pi E Kv a i Ỉ Pi H so that the 2 quantities below vanish f Pi Pifl Pi - fi EM 1 R av i if ỉ fiK f Pi Pif Pi - p Pi Ev 1 R a K 0 i T 2 . 78 79 Clearly 26 and the last equality of 79 are one and the same equation Our equivalence is thus proven. 13. Acknowledgments This work is founded by the Spain Ministry of Science and Innovation Project FIS2008-00781 and by FEDER funds EU . 14. References 1 R. B. Lindsay and H. Margenau Foundations of Physics NY Dover 1957. 2 J. Willard Gibbs Elementary PrinciPles in Statistical Mechanics New Haven Yale University Press 1902. 3 . Jaynes Probability Theory The Logic of Science Cambridge University Press Cambridge 2005. 4 . Grandy Jr. and P. W. Milonni Editors Physics and Probability. Essays in Honor of Edwin T. Jaynes NY Cambridge University Press 1993. 5 E. T. Jaynes PaPers on Probability statistics and statistical Physics edited by R. D. Rosenkrantz Dordrecht Reidel 1987. 6 E. A. Desloge Thermal Physics NY Holt Rhinehart and Winston 1968. 7 E. Curado A. Plastino Phys. Rev. E 72 2005 047103. 8 A. Plastino E. Curado Physica A 365 2006 24 22 Thermodynamics 9 A. Plastino E. Curado International Journal of Modern Physics B 21 2007 2557 10 A. Plastino E. Curado Physica A 386 2007 155 11 A. Plastino E. Curado M. Casas Entropy A 10 2008 124 12 International Journal of Modern Physics B 22 2008 4589 13 E. Curado F Nobre A. Plastino .