Representations of Uniformly Hyperfinite Algebras and Their Associated von Neumann Rings

Information is physical, so its manipulation through devices is subject to its own mechanics: the science and engineering of behavioral description, which is intermingled with classical, quantum and statistical mechanics principles. This chapter is a unification of these principles and physical laws with their implications for nanoscale. Ideas of state machines, Church-Turing thesis and its embodiment in various state machines, probabilities, Bayesian principles and entropy in its various forms (Shannon, Boltzmann, von Neumann, algorithmic) with an eye on the principle of maximum entropy as an information manipulation tool. Notions of conservation and non-conservation are applied to example circuit forms folding in adiabatic, isothermal, reversible and irreversible processes. This brings out implications of fluctuation and transitions, the interplay of errors and stability and the energy cost of determinism. It concludes discussing networks as tools to understand information flow and decision making and with an introduction to entanglement in quantum computing.

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Unbounded Linear Operators

10.1093/oso/9780198812050.003.0003 ◽

2018 ◽

Author(s):

D. E. Edmunds ◽

W. D. Evans

Keyword(s):

Hilbert Spaces ◽

Linear Operators ◽

Von Neumann ◽

Limit Circle ◽

Sectorial Operators ◽

Closed Operators ◽

The Stability ◽

Symmetric Operators ◽

Unbounded Linear Operators ◽

Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

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Hamilton-Jacobi Theory

10.1093/oso/9780198822370.003.0019 ◽

2018 ◽

Author(s):

Peter Mann

Keyword(s):

Phase Space ◽

Bbgky Hierarchy ◽

Distribution Functions ◽

Symplectic Integrators ◽

Partition Functions ◽

Von Neumann ◽

Equipartition Theorem ◽

Recurrence Theorem ◽

Phase Amplitude ◽

Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

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