420 research outputs found
The energy partitioning of non-thermal particles in a plasma: or the Coulomb logarithm revisited
The charged particle stopping power in a highly ionized and weakly to
moderately coupled plasma has been calculated to leading and next-to-leading
order by Brown, Preston, and Singleton (BPS). After reviewing the main ideas
behind this calculation, we use a Fokker-Planck equation derived by BPS to
compute the electron-ion energy partitioning of a charged particle traversing a
plasma. The motivation for this application is ignition for inertial
confinement fusion -- more energy delivered to the ions means a better chance
of ignition, and conversely. It is therefore important to calculate the
fractional energy loss to electrons and ions as accurately as possible, as this
could have implications for the Laser Megajoule (LMJ) facility in France and
the National Ignition Facility (NIF) in the United States. The traditional
method by which one calculates the electron-ion energy splitting of a charged
particle traversing a plasma involves integrating the stopping power dE/dx.
However, as the charged particle slows down and becomes thermalized into the
background plasma, this method of calculating the electron-ion energy splitting
breaks down. As a result, the method suffers a systematic error of order T/E0,
where T is the plasma temperature and E0 is the initial energy of the charged
particle. In the case of DT fusion, for example, this can lead to uncertainties
as high as 10% or so. The formalism presented here is designed to account for
the thermalization process, and in contrast, it provides results that are
near-exact.Comment: 10 pages, 3 figures, invited talk at the 35th European Physical
Society meeting on plasma physic
Charged Particle Motion in a Highly Ionized Plasma
A recently introduced method utilizing dimensional continuation is employed
to compute the energy loss rate for a non-relativistic particle moving through
a highly ionized plasma. No restriction is made on the charge, mass, or speed
of this particle. It is, however, assumed that the plasma is not strongly
coupled in the sense that the dimensionless plasma coupling parameter
g=e^2\kappa_D/ 4\pi T is small, where \kappa_D is the Debye wave number of the
plasma. To leading and next-to-leading order in this coupling, dE/dx is of the
generic form g^2 \ln[C g^2]. The precise numerical coefficient out in front of
the logarithm is well known. We compute the constant C under the logarithm
exactly for arbitrary particle speeds. Our exact results differ from
approximations given in the literature. The differences are in the range of 20%
for cases relevant to inertial confinement fusion experiments. The same method
is also employed to compute the rate of momentum loss for a projectile moving
in a plasma, and the rate at which two plasmas at different temperatures come
into thermal equilibrium. Again these calculations are done precisely to the
order given above. The loss rates of energy and momentum uniquely define a
Fokker-Planck equation that describes particle motion in the plasma. The
coefficients determined in this way are thus well-defined, contain no arbitrary
parameters or cutoffs, and are accurate to the order described. This
Fokker-Planck equation describes the longitudinal straggling and the transverse
diffusion of a beam of particles. It should be emphasized that our work does
not involve a model, but rather it is a precisely defined evaluation of the
leading terms in a well-defined perturbation theory.Comment: Comments: Published in Phys. Rep. 410/4 (2005) 237; RevTeX, 111
Pages, 17 Figures; Transcription error corrected in temperature equilibration
rate (3.61) and (12.44) which replaces \gamma-2 by \gamma-
Temperature equilibration in a fully ionized plasma: electron-ion mass ratio effects
Brown, Preston, and Singleton (BPS) produced an analytic calculation for
energy exchange processes for a weakly to moderately coupled plasma: the
electron-ion temperature equilibration rate and the charged particle stopping
power. These precise calculations are accurate to leading and next-to-leading
order in the plasma coupling parameter, and to all orders for two-body quantum
scattering within the plasma. Classical molecular dynamics can provide another
approach that can be rigorously implemented. It is therefore useful to compare
the predictions from these two methods, particularly since the former is
theoretically based and the latter numerically. An agreement would provide both
confidence in our theoretical machinery and in the reliability of the computer
simulations. The comparisons can be made cleanly in the purely classical
regime, thereby avoiding the arbitrariness associated with constructing
effective potentials to mock up quantum effects. We present here the classical
limit of the general result for the temperature equilibration rate presented in
BPS. We examine the validity of the m_electron/m_ion --> 0 limit used in BPS to
obtain a very simple analytic evaluation of the long-distance, collective
effects in the background plasma.Comment: 14 pages, 4 figures, small change in titl
Shor's Factoring Algorithm and Modular Exponentiation Operators: A Pedagogical Presentation with Examples
These are pedagogical notes on Shor's factoring algorithm, which is a quantum
algorithm for factoring very large numbers (of order of hundreds to thousands
of bits) in polynomial time. In contrast, all known classical algorithms for
the factoring problem take an exponential time to factor large numbers. In
these notes, we assume no prior knowledge of Shor's algorithm beyond a basic
familiarity with the circuit model of quantum computing. The literature is
thick with derivations and expositions of Shor's algorithm, but most of them
seem to be lacking in essential details, and none of them provide a pedagogical
presentation. We develop the theory of modular exponentiation (ME) operators in
some detail, one of the fundamental components of Shor's algorithm, and the
place where most of the quantum resources are deployed. We also discuss the
post-quantum processing and the method of continued fractions, which is used to
extract the exact period of the modular exponential function from the
approximately measured phase angles of the ME operator. The manuscript then
moves on to a series of examples. We first verify the formalism by factoring
N=15, the smallest number accessible to Shor's algorithm. We then proceed to
factor larger numbers, developing a systematic procedure that will find the ME
operators for any semi-prime (where and~ are prime).
Finally, we factor the numbers N=21, 33, 35, 143, 247 using the Qiskit
simulator. It is observed that the ME operators are somewhat forgiving, and
truncated approximate forms are able to extract factors just as well as the
exact operators. This is because the method of continued fractions only
requires an approximate phase value for its input, which suggests that
implementing Shor's algorithm might not be as difficult as first suspected.Comment: 108 pages, 63 figures, typos fixe
Predicting the Aoki Phase using the Chiral Lagrangian
This work is concerned with the phase diagram of Wilson fermions in the mass
and coupling constant plane for two-flavor (unquenched) QCD. We show that as
the continuum limit is approached, one can study the lattice theory using the
continuum chiral Lagrangian, supplemented by additional terms proportional to
powers of the lattice spacing. We find two possible phase structures at
non-zero lattice spacing: (1) There is an Aoki phase of spontaneously broken
flavor and parity, with two massless Goldstone-pions, and a width ; (2) There is no spontaneous symmetry breaking, and all three pions
have equal mass of order . Present numerical simulations suggest that the
former option is realized.Comment: LATTICE98(spectrum), 3 pages, 2 figures, LaTex, uses espcrc2.st
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