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Basic Theoretical Physics: A Concise Overview P42

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Basic Theoretical Physics: A Concise Overview P42. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 428 55 Conclusion to Part IV For a grand canonical ensemble where the heat bath not only exchanges energy with the system but also particles such that the particle number in a volume element V fluctuates around the average N T a V in addition to 3 1 kBT one obtains for the distribution a second parameter a the so-called chemical potential for the analogous quantity to the Helmholtz free energy i.e. for the Gibbs grand canonical thermodynamic potential T p V . -kB T ln Z T a V . with the grand canonical partition function Ei V Nj -N Z T p V . e . i j The mathematical relation between the Helmholtz free energy and the Gibbs grand canonical potential L is a Legendre transform i.e. T a V . F T V N T a V . - a N T a V similarly to the way the internal energy U T V N . and the enthalpy I depend on each other I T p N . U T V T p N . N . p V T p N . with the pressure p as the conjugate Lagrange parameter regulating fluctuations in V. These Legendre transformations are mathematically analogous to the transition from the Lagrange function L v . . . 2 to the Hamilton function H p . in classical mechanics incidentally this may be used for mnemonic purposes the corresponding letters are similar i.e. V and v and p and p although the meaning is completely different. The relation between a and b can also be expressed as U T V N . fh V N . T . Where is the thermodynamic expectation with the suitable canonical and microcanonical and grand canonical Boltzmann-Gibbs distribution e.g. _ BPV N . e kBT V4 T 2 pi T i -4l i with pi T Z T y N----- 2 Actually by the Legendre transformation of -L. 55 Conclusion to Part IV 429 and the Hermitian operator A represents an observable i.e. a measurable quantity . The ffi represent the complete system of eigenfunctions of the Hamilton operator H the Ei are the corresponding eigenvalues. Concerning the entropy This is a particularly complex quantity whose complexity should not simply be glossed over by simplifications. In fact the entropy is a .