By Herbert Jr. Oertel, M. Böhle, J. Delfs, D. Hafermann, H. Holthoff

Dieses Buch wendet sich an Studenten der Ingenieurwissenschaften und Ingenieure der Raumfahrtindustrie und der Energieverfahrenstechnik. Es verkn?pft die klassischen Gebiete der Aerodynamik mit der Nichtgleichgewichts-Thermodynamik hei?er Gase. Am Beispiel des Wiedereintritts einer Raumkapsel in die Erdatmosph?re werden die aerothermodynamischen Grundlagen und numerischen Methoden zur Berechnung des Str?mungsfeldes der Raumkapsel im gaskinetischen und kontinuumsmechanischen Bereich der Wiedereintrittstrajektorie behandelt. Am Beispiel von Raumfahrtprojekten werden die Methoden entwickelt. Die Autoren sind anerkannte Spezialisten f?r dieses Fachgebiet.

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In fact, the ensuing history of the study of spin systems has an amusing twist. The Ising model is a classical model and as such was regarded unsatisfactory at a time when quantum mechanics was just being developed and celebrated its first great successes. Heisenberg proposed in 1928 a quantum spin model where the classical, twostate Ising spins are replaced by spin-(1/2) Pauli matrices (Heisenberg, 1928). The Hamiltonian of this model reads (k,l) k 34 G . M . ~ + tykYcriy + t~zt~lz and the physically immaterial constant 1 has been introduced for later convenience.

This behaviour is expressed in the Ising energy of a given spin configuration r / = {s1 . . . 2) where the first sum runs over all nearest neighbour pairs of sites and the second sum runs over all lattice sites. With this energy function one calculates the statistical properties of the model following the usual rules of classical statistical mechanics. 3) gives the equilibrium probability of finding a state r/with energy E at temperature T = 1/(k/3). From the partition function Z = Y~contigexp (-/~ E) one obtains the free energy and all other thermodynamic properties.

The spin F(I') = 1, F(,I,) = - 1, or the position x = F(k) -- ka of a particle at site k in a lattice with lattice constant a. In a series of measurements the system may be found in states 17of the system with probabilities Po(t). Hence the expression ( F ) is the average value of what one measures in a series of many identical experiments, using the same initial state. If the initial states are not always the same fixed state, but some collection of different states, given by an initial distribution P0 = Po (0), then the expression ( F ) involves not only averaging over many realizations of the same process, but also averaging over the initial states.