794,571 research outputs found
Generalized quantum Fokker-Planck, diffusion and Smoluchowski equations with true probability distribution functions
Traditionally, the quantum Brownian motion is described by Fokker-Planck or
diffusion equations in terms of quasi-probability distribution functions, e.g.,
Wigner functions. These often become singular or negative in the full quantum
regime. In this paper a simple approach to non-Markovian theory of quantum
Brownian motion using {\it true probability distribution functions} is
presented. Based on an initial coherent state representation of the bath
oscillators and an equilibrium canonical distribution of the quantum mechanical
mean values of their co-ordinates and momenta we derive a generalized quantum
Langevin equation in -numbers and show that the latter is amenable to a
theoretical analysis in terms of the classical theory of non-Markovian
dynamics. The corresponding Fokker-Planck, diffusion and the Smoluchowski
equations are the {\it exact} quantum analogues of their classical
counterparts. The present work is {\it independent} of path integral
techniques. The theory as developed here is a natural extension of its
classical version and is valid for arbitrary temperature and friction
(Smoluchowski equation being considered in the overdamped limit).Comment: RevTex, 16 pages, 7 figures, To appear in Physical Review E (minor
revision
Approach to Quantum Kramers' Equation and Barrier Crossing Dynamics
We have presented a simple approach to quantum theory of Brownian motion and
barrier crossing dynamics. Based on an initial coherent state representation of
bath oscillators and an equilibrium canonical distribution of quantum
mechanical mean values of their co-ordinates and momenta we have derived a
-number generalized quantum Langevin equation. The approach allows us to
implement the method of classical non-Markovian Brownian motion to realize an
exact generalized non-Markovian quantum Kramers' equation. The equation is
valid for arbitrary temperature and friction. We have solved this equation in
the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier
crossing and analyze its variation as a function of temperature and friction.
While almost all the earlier theories rest on quasi-probability distribution
functions (like Wigner function) and path integral methods, the present work is
based on {\it true probability distribution functions} and is independent of
path integral techniques. The theory is a natural extension of the classical
theory to quantum domain and provides a unified description of thermal
activated processes and tunneling.Comment: RevTex, 18 pages, 2 figures; Minor corrections; To appear in Phys.
Rev.
Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
We show that the geometric lifting of the RSK correspondence introduced by
A.N. Kirillov (2001) is volume preserving with respect to a natural product
measure on its domain, and that the integrand in Givental's integral formula
for GL(n,R)-Whittaker functions arises naturally in this context. Apart from
providing further evidence that Whittaker functions are the natural analogue of
Schur polynomials in this setting, our results also provide a new
`combinatorial' framework for the study of random polymers. When the input
matrix consists of random inverse gamma distributed weights, the probability
distribution of a polymer partition function constructed from these weights can
be written down explicitly in terms of Whittaker functions. Next we restrict
the geometric RSK mapping to symmetric matrices and show that the volume
preserving property continues to hold. We determine the probability law of the
polymer partition function with inverse gamma weights that are constrained to
be symmetric about the main diagonal, with an additional factor on the main
diagonal. The third combinatorial mapping studied is a variant of the geometric
RSK mapping for triangular arrays, which is again showed to be volume
preserving. This leads to a formula for the probability distribution of a
polymer model whose paths are constrained to stay below the diagonal. We also
show that the analogues of the Cauchy-Littlewood identity in the setting of
this paper are equivalent to a collection of Whittaker integral identities
conjectured by Bump (1989) and Bump and Friedberg (1990) and proved by Stade
(2001, 2002). Our approach leads to new `combinatorial' proofs and
generalizations of these identities, with some restrictions on the parameters.Comment: v2: significantly extended versio
Interest Rates and Information Geometry
The space of probability distributions on a given sample space possesses
natural geometric properties. For example, in the case of a smooth parametric
family of probability distributions on the real line, the parameter space has a
Riemannian structure induced by the embedding of the family into the Hilbert
space of square-integrable functions, and is characterised by the Fisher-Rao
metric. In the nonparametric case the relevant geometry is determined by the
spherical distance function of Bhattacharyya. In the context of term structure
modelling, we show that minus the derivative of the discount function with
respect to the maturity date gives rise to a probability density. This follows
as a consequence of the positivity of interest rates. Therefore, by mapping the
density functions associated with a given family of term structures to Hilbert
space, the resulting metrical geometry can be used to analyse the relationship
of yield curves to one another. We show that the general arbitrage-free yield
curve dynamics can be represented as a process taking values in the convex
space of smooth density functions on the positive real line. It follows that
the theory of interest rate dynamics can be represented by a class of processes
in Hilbert space. We also derive the dynamics for the central moments
associated with the distribution determined by the yield curve.Comment: 20 pages, 3 figure
Quantum Turing Machines Computations and Measurements
Contrary to the classical case, the relation between quantum programming
languages and quantum Turing Machines (QTM) has not being fully investigated.
In particular, there are features of QTMs that have not been exploited, a
notable example being the intrinsic infinite nature of any quantum computation.
In this paper we propose a definition of QTM, which extends and unifies the
notions of Deutsch and Bernstein and Vazirani. In particular, we allow both
arbitrary quantum input, and meaningful superpositions of computations, where
some of them are "terminated" with an "output", while others are not. For some
infinite computations an "output" is obtained as a limit of finite portions of
the computation. We propose a natural and robust observation protocol for our
QTMs, that does not modify the probability of the possible outcomes of the
machines. Finally, we use QTMs to define a class of quantum computable
functions---any such function is a mapping from a general quantum state to a
probability distribution of natural numbers. We expect that our class of
functions, when restricted to classical input-output, will be not different
from the set of the recursive functions.Comment: arXiv admin note: substantial text overlap with arXiv:1504.02817 To
appear on MDPI Applied Sciences, 202
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