11,729 research outputs found
Why are probabilistic laws governing quantum mechanics and neurobiology?
We address the question: Why are dynamical laws governing in quantum
mechanics and in neuroscience of probabilistic nature instead of being
deterministic? We discuss some ideas showing that the probabilistic option
offers advantages over the deterministic one.Comment: 40 pages, 8 fig
A practical, unitary simulator for non-Markovian complex processes
Stochastic processes are as ubiquitous throughout the quantitative sciences
as they are notorious for being difficult to simulate and predict. In this
letter we propose a unitary quantum simulator for discrete-time stochastic
processes which requires less internal memory than any classical analogue
throughout the simulation. The simulator's internal memory requirements equal
those of the best previous quantum models. However, in contrast to previous
models it only requires a (small) finite-dimensional Hilbert space. Moreover,
since the simulator operates unitarily throughout, it avoids any unnecessary
information loss. We provide a stepwise construction for simulators for a large
class of stochastic processes hence directly opening the possibility for
experimental implementations with current platforms for quantum computation.
The results are illustrated for an example process.Comment: 12 pages, 5 figure
A practical, unitary simulator for non-Markovian complex processes
Stochastic processes are as ubiquitous throughout the quantitative sciences
as they are notorious for being difficult to simulate and predict. In this
letter we propose a unitary quantum simulator for discrete-time stochastic
processes which requires less internal memory than any classical analogue
throughout the simulation. The simulator's internal memory requirements equal
those of the best previous quantum models. However, in contrast to previous
models it only requires a (small) finite-dimensional Hilbert space. Moreover,
since the simulator operates unitarily throughout, it avoids any unnecessary
information loss. We provide a stepwise construction for simulators for a large
class of stochastic processes hence directly opening the possibility for
experimental implementations with current platforms for quantum computation.
The results are illustrated for an example process.Comment: 12 pages, 5 figure
Modified Sch\"odinger dynamics with attractive densities
The linear Schr\"{o}dinger equation does not predict that macroscopic bodies
should be located at one place only. Quantum mechanics textbooks generally
solve the problem by introducing the projection postulate, which forces
definite values to emerge during position measurements; many other
interpretations have also been proposed.\ Here, in the same spirit as the GRW
and CSL theories, we modify the Schr\"{o}dinger equation in a way that
efficiently cancels macroscopic density fluctuations in space.\ Instead of
introducing the stochastic dynamics of GRW or CSL, we assume a deterministic
dynamics that includes an attraction term towards the density in space of the
de Broglie-Bohm position of particles. This automatically ensures macroscopic
uniqueness, so that the state vector can be considered as a direct
representation of physical reality.Comment: 14 pages, no figure. This is the version accepted by the European
Physical Journal, with a few additions in the text and a few more references.
A typo has been corrected in Eq (4
The non-Markovian stochastic Schrodinger equation for open systems
We present the non-Markovian generalization of the widely used stochastic
Schrodinger equation. Our result allows to describe open quantum systems in
terms of stochastic state vectors rather than density operators, without
approximation. Moreover, it unifies two recent independent attempts towards a
stochastic description of non-Markovian open systems, based on path integrals
on the one hand and coherent states on the other. The latter approach utilizes
the analytical properties of coherent states and enables a microscopic
interpretation of the stochastic states. The alternative first approach is
based on the general description of open systems using path integrals as
originated by Feynman and Vernon.Comment: 9 pages, RevTe
Review of Some Promising Fractional Physical Models
Fractional dynamics is a field of study in physics and mechanics
investigating the behavior of objects and systems that are characterized by
power-law non-locality, power-law long-term memory or fractal properties by
using integrations and differentiation of non-integer orders, i.e., by methods
of the fractional calculus. This paper is a review of physical models that look
very promising for future development of fractional dynamics. We suggest a
short introduction to fractional calculus as a theory of integration and
differentiation of non-integer order. Some applications of
integro-differentiations of fractional orders in physics are discussed. Models
of discrete systems with memory, lattice with long-range inter-particle
interaction, dynamics of fractal media are presented. Quantum analogs of
fractional derivatives and model of open nano-system systems with memory are
also discussed.Comment: 38 pages, LaTe
Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators
Nonlinearities in finite dimensions can be linearized by projecting them into
infinite dimensions. Unfortunately, often the linear operator techniques that
one would then use simply fail since the operators cannot be diagonalized. This
curse is well known. It also occurs for finite-dimensional linear operators. We
circumvent it by developing a meromorphic functional calculus that can
decompose arbitrary functions of nondiagonalizable linear operators in terms of
their eigenvalues and projection operators. It extends the spectral theorem of
normal operators to a much wider class, including circumstances in which poles
and zeros of the function coincide with the operator spectrum. By allowing the
direct manipulation of individual eigenspaces of nonnormal and
nondiagonalizable operators, the new theory avoids spurious divergences. As
such, it yields novel insights and closed-form expressions across several areas
of physics in which nondiagonalizable dynamics are relevant, including
memoryful stochastic processes, open non unitary quantum systems, and
far-from-equilibrium thermodynamics.
The technical contributions include the first full treatment of arbitrary
powers of an operator. In particular, we show that the Drazin inverse,
previously only defined axiomatically, can be derived as the negative-one power
of singular operators within the meromorphic functional calculus and we give a
general method to construct it. We provide new formulae for constructing
projection operators and delineate the relations between projection operators,
eigenvectors, and generalized eigenvectors.
By way of illustrating its application, we explore several, rather distinct
examples.Comment: 29 pages, 4 figures, expanded historical citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht
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