4,435 research outputs found
The Information Geometry of the Ising Model on Planar Random Graphs
It has been suggested that an information geometric view of statistical
mechanics in which a metric is introduced onto the space of parameters provides
an interesting alternative characterisation of the phase structure,
particularly in the case where there are two such parameters -- such as the
Ising model with inverse temperature and external field .
In various two parameter calculable models the scalar curvature of
the information metric has been found to diverge at the phase transition point
and a plausible scaling relation postulated: . For spin models the necessity of calculating in
non-zero field has limited analytic consideration to 1D, mean-field and Bethe
lattice Ising models. In this letter we use the solution in field of the Ising
model on an ensemble of planar random graphs (where ) to evaluate the scaling behaviour of the scalar curvature, and find
. The apparent discrepancy is traced
back to the effect of a negative .Comment: Version accepted for publication in PRE, revtex
The quantum brachistochrone problem for non-Hermitian Hamiltonians
Recently Bender, Brody, Jones and Meister found that in the quantum brachistochrone problem the passage time needed for the evolution of certain initial states into specified final states can be made arbitrarily small, when the time-evolution operator is taken to be non-Hermitian but PT-symmetric. Here we demonstrate that such phenomena can also be obtained for non-Hermitian Hamiltonians for which PT-symmetry is completely broken, i.e. dissipative systems. We observe that the effect of a tunable passage time can be achieved by projecting between orthogonal eigenstates by means of a time-evolution operator associated with a non-Hermitian Hamiltonian. It is not essential that this Hamiltonian is PT-symmetric
Martingale Models for Quantum State Reduction
Stochastic models for quantum state reduction give rise to statistical laws
that are in most respects in agreement with those of quantum measurement
theory. Here we examine the correspondence of the two theories in detail,
making a systematic use of the methods of martingale theory. An analysis is
carried out to determine the magnitude of the fluctuations experienced by the
expectation of the observable during the course of the reduction process and an
upper bound is established for the ensemble average of the greatest
fluctuations incurred. We consider the general projection postulate of L\"uders
applicable in the case of a possibly degenerate eigenvalue spectrum, and derive
this result rigorously from the underlying stochastic dynamics for state
reduction in the case of both a pure and a mixed initial state. We also analyse
the associated Lindblad equation for the evolution of the density matrix, and
obtain an exact time-dependent solution for the state reduction that explicitly
exhibits the transition from a general initial density matrix to the L\"uders
density matrix. Finally, we apply Girsanov's theorem to derive a set of simple
formulae for the dynamics of the state in terms of a family of geometric
Brownian motions, thereby constructing an explicit unravelling of the Lindblad
equation.Comment: 30 pages LaTeX. Submitted to Journal of Physics
Thermodynamic curvature and black holes
I give a relatively broad survey of thermodynamic curvature , one spanning
results in fluids and solids, spin systems, and black hole thermodynamics.
results from the thermodynamic information metric giving thermodynamic
fluctuations. has a unique status in thermodynamics as being a geometric
invariant, the same for any given thermodynamic state. In fluid and solid
systems, the sign of indicates the character of microscopic interactions,
repulsive or attractive. gives the average size of organized mesoscopic
fluctuating structures. The broad generality of thermodynamic principles might
lead one to believe the same for black hole thermodynamics. This paper explores
this issue with a systematic tabulation of results in a number of cases.Comment: 27 pages, 10 figures, 7 tables, 78 references. Talk presented at the
conference Breaking of Supersymmetry and Ultraviolet Divergences in extended
Supergravity, in Frascati, Italy, March 27, 2013. v2 corrects some small
problem
The Information Geometry of the Spherical Model
Motivated by previous observations that geometrizing statistical mechanics
offers an interesting alternative to more standard approaches,we have recently
calculated the curvature (the fundamental object in this approach) of the
information geometry metric for the Ising model on an ensemble of planar random
graphs. The standard critical exponents for this model are alpha=-1, beta=1/2,
gamma=2 and we found that the scalar curvature, R, behaves as
epsilon^(-2),where epsilon = beta_c - beta is the distance from criticality.
This contrasts with the naively expected R ~ epsilon^(-3) and the apparent
discrepancy was traced back to the effect of a negative alpha on the scaling of
R.
Oddly,the set of standard critical exponents is shared with the 3D spherical
model. In this paper we calculate the scaling behaviour of R for the 3D
spherical model, again finding that R ~ epsilon^(-2), coinciding with the
scaling behaviour of the Ising model on planar random graphs. We also discuss
briefly the scaling of R in higher dimensions, where mean-field behaviour sets
in.Comment: 7 pages, no figure
Quantum Chaos Versus Classical Chaos: Why is Quantum Chaos Weaker?
We discuss the questions: How to compare quantitatively classical chaos with
quantum chaos? Which one is stronger? What are the underlying physical reasons
Misleading signatures of quantum chaos
The main signature of chaos in a quantum system is provided by spectral
statistical analysis of the nearest neighbor spacing distribution and the
spectral rigidity given by . It is shown that some standard
unfolding procedures, like local unfolding and Gaussian broadening, lead to a
spurious increase of the spectral rigidity that spoils the
relationship with the regular or chaotic motion of the system. This effect can
also be misinterpreted as Berry's saturation.Comment: 4 pages, 5 figures, submitted to Physical Review
The Impact of Isospin Breaking on the Distribution of Transition Probabilities
In the present paper we investigate the effect of symmetry breaking in the
statistical distributions of reduced transition amplitudes and reduced
transition probabilities. These quantities are easier to access experimentally
than the components of the eigenvectors and were measured by Adams et al. for
the electromagnetic transitions in ^{26}Al. We focus on isospin symmetry
breaking described by a matrix model where both, the Hamiltonian and the
electromagnetic operator, break the symmetry. The results show that for partial
isospin conservation, the statistical distribution of the reduced transition
probability can considerably deviate from the Porter-Thomas distribution.Comment: 16 pages, 8 figures, submitted to PR
Geometrothermodynamics of black holes
The thermodynamics of black holes is reformulated within the context of the
recently developed formalism of geometrothermodynamics. This reformulation is
shown to be invariant with respect to Legendre transformations, and to allow
several equivalent representations. Legendre invariance allows us to explain a
series of contradictory results known in the literature from the use of
Weinhold's and Ruppeiner's thermodynamic metrics for black holes. For the
Reissner-Nordstr\"om black hole the geometry of the space of equilibrium states
is curved, showing a non trivial thermodynamic interaction, and the curvature
contains information about critical points and phase transitions. On the
contrary, for the Kerr black hole the geometry is flat and does not explain its
phase transition structure.Comment: Revised version, to be published in Gen.Rel.Grav.(Mashhoon's
Festschrift
Correlation functions of scattering matrix elements in microwave cavities with strong absorption
The scattering matrix was measured for microwave cavities with two antennas.
It was analyzed in the regime of overlapping resonances. The theoretical
description in terms of a statistical scattering matrix and the rescaled
Breit-Wigner approximation has been applied to this regime. The experimental
results for the auto-correlation function show that the absorption in the
cavity walls yields an exponential decay. This behavior can only be modeled
using a large number of weakly coupled channels. In comparison to the
auto-correlation functions, the cross-correlation functions of the diagonal
S-matrix elements display a more pronounced difference between regular and
chaotic systems
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