By Professor Dr. Franz Schwabl (auth.)

The thoroughly revised re-creation of the classical booklet on Statistical Mechanics covers the elemental strategies of equilibrium and non-equilibrium statistical physics. as well as a deductive method of equilibrium statistics and thermodynamics in keeping with a unmarried speculation - the shape of the microcanonical density matrix - this ebook treats an important components of non-equilibrium phenomena. Intermediate calculations are awarded in entire aspect. difficulties on the finish of every bankruptcy aid scholars to consolidate their figuring out of the fabric. past the basics, this article demonstrates the breadth of the sphere and its nice number of functions. smooth components equivalent to renormalization workforce concept, percolation, stochastic equations of movement and their functions to serious dynamics, kinetic theories, in addition to primary issues of irreversibility, are mentioned. The textual content could be beneficial for complex scholars of physics and different traditional sciences; a simple wisdom of quantum mechanics is presumed.

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It holds also for a time-dependent H. It should not be confused with the equation of motion of Heisenberg operators, which has a positive sign on the right-hand side. 8). The time dependence of the expectation value is referred to by the index t. We shall meet up with the von Neumann equation in the next chapter where we set up the equilibrium density matrices, and it is naturally of fundamental importance for all time-dependent processes. We now treat the transformation to the Heisenberg representation.

The Hamiltonian of the three-dimensional ideal gas is N H= i=1 p2i + Vwall . 10) Here, the pi are the cartesian momenta of the particles and Vwall is the potential representing the wall of the container. The surface area of the energy shell is in this case Ω (E) = N 1 d3 x1 . . h3N N ! V d3 xN V d3 p1 . . 2 Microcanonical Ensembles 31 where the integrations over x are restricted to the spatial volume V deﬁned by the walls. It would be straightforward to calculate Ω (E) directly. We shall ¯ carry out this calculation here via Ω(E), the volume inside the energy shell, which in this case is a hypersphere, in order to have both quantities available: ¯ Ω(E) = × 1 h3N N !

5) i This mean value can also be represented in terms of the density matrix deﬁned by pi |ψi ρ= ψi | . 7c) ρ† = ρ . 7d) The derivations of these relations and further remarks about the density matrices of mixed ensembles will be given in Sect. 2. 2 The Von Neumann Equation From the Schr¨ odinger equation and its adjoint i ∂ |ψ, t = H |ψ, t , ∂t −i ∂ ψ, t| = ψ, t| H , ∂t it follows that i ∂ ρ=i ∂t pi |ψ˙i ψi | + |ψi ψ˙ i | i pi (H |ψi = i ψi | − |ψi ψi | H) . 16 1. 8) it is the quantum-mechanical equivalent of the Liouville equation.