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Extra info for Introduction to Euler's Fluid mechanics
34) Note that we have not exhausted the list of possible thermodynamic potentials for the case in which the number of particles varies. Ω is the Legendre transform of F with respect to μ and N and is constant when T, V , μ are fixed. We may define similar Legendre transforms of E¯ and W . 42) In this form it is known as the Gibbs–Duhem relation. 39) also reduces to this same Gibbs– Duhem relation. 42). Some thermodynamic relations and techniques Here we review some thermodynamic relations and methods.
33) Thus, as the system gets bigger while the size of each region remains fixed, the calculated fluctuations in the energy get relatively smaller and smaller and the canonical 20 1 The classical distribution function distribution function becomes in this respect a better and better representation of the energy conserving behavior of the system. Since we used approximations requiring a large system in deriving the canonical distribution it is not surprising that it only gives a consistent description when the system is large.
5) where is the number of states associated with the equilibrium density matrix. 7) Some other perspectives on this definition will be illustrated in the problems. 8) where the subscript c has been added to ρ to specify the canonical density matrix. 12) If the quantity S is indeed the thermodynamic entropy, then E¯ − T S is the Helmholtz free energy, denoted F (or A in the chemical literature). 13) This establishes the needed relation between a thermodynamic quantity and the microscopic, quantum mechanical model.