4,095 research outputs found
Gauss Sums and Quantum Mechanics
By adapting Feynman's sum over paths method to a quantum mechanical system
whose phase space is a torus, a new proof of the Landsberg-Schaar identity for
quadratic Gauss sums is given. In contrast to existing non-elementary proofs,
which use infinite sums and a limiting process or contour integration, only
finite sums are involved. The toroidal nature of the classical phase space
leads to discrete position and momentum, and hence discrete time. The
corresponding `path integrals' are finite sums whose normalisations are derived
and which are shown to intertwine cyclicity and discreteness to give a finite
version of Kelvin's method of images.Comment: 14 pages, LaTe
Static non-reciprocity in mechanical metamaterials
Reciprocity is a fundamental principle governing various physical systems,
which ensures that the transfer function between any two points in space is
identical, regardless of geometrical or material asymmetries. Breaking this
transmission symmetry offers enhanced control over signal transport, isolation
and source protection. So far, devices that break reciprocity have been mostly
considered in dynamic systems, for electromagnetic, acoustic and mechanical
wave propagation associated with spatio-temporal variations. Here we show that
it is possible to strongly break reciprocity in static systems, realizing
mechanical metamaterials that, by combining large nonlinearities with suitable
geometrical asymmetries, and possibly topological features, exhibit vastly
different output displacements under excitation from different sides, as well
as one-way displacement amplification. In addition to extending non-reciprocity
and isolation to statics, our work sheds new light on the understanding of
energy propagation in non-linear materials with asymmetric crystalline
structures and topological properties, opening avenues for energy absorption,
conversion and harvesting, soft robotics, prosthetics and optomechanics.Comment: 19 pages, 3 figures, Supplementary information (11 pages and 5
figures
Chaotic dynamics, fluctuations, nonequilibrium ensembles
A review of some recent results and ideas about the expected behaviour of
large chaotic systems and fluids.Comment: 10 pages: LaTeX-REVTe
Thermodynamics of Micro- and Nano-Systems Driven by Periodic Temperature Variations
We introduce a general framework for analyzing the thermodynamics of small
systems that are driven by both a periodic temperature variation and some
external parameter modulating their energy. This set-up covers, in particular,
periodic micro and nano-heat engines. In a first step, we show how to express
total entropy production by properly identified time-independent affinities and
currents without making a linear response assumption. In linear response,
kinetic coefficients akin to Onsager coefficients can be identified.
Specializing to a Fokker-Planck type dynamics, we show that these coefficients
can be expressed as a sum of an adiabatic contribution and one reminiscent of a
Green-Kubo expression that contains deviations from adiabaticity. Furthermore,
we show that the generalized kinetic coefficients fulfill an Onsager-Casimir
type symmetry tracing back to microscopic reversibility. This symmetry allows
for non-identical off-diagonal coefficients if the driving protocols are not
symmetric under time-reversal. We then derive a novel constraint on the kinetic
coefficients that is sharper than the second law and provides an
efficiency-dependent bound on power. As one consequence, we can prove that the
power vanishes at least linearly when approaching Carnot efficiency. We
illustrate our general framework by explicitly working out the paradigmatic
case of a Brownian heat engine realized by a colloidal particle in a
time-dependent harmonic trap subject to a periodic temperature profile. This
case study reveals inter alia that our new general bound on power is
asymptotically tight.Comment: 15 pages, 4 figure
Thermodynamic Bond Graphs and the Problem of Thermal Inertance
It is shown that an isolated thermal inertance does not obey the second law of thermodynamics. Consequently, such an element should not be used in physical systems theory. To eliminate the structural gap in the thermal domain of current physical systems theory, a new framework is introduced using Bond Graph concepts. These Thermodynamic Bond Graphs are the result of synthesis of methods used in thermodynamics and in mechanics
Constructing elliptic curves of prime order
We present a very efficient algorithm to construct an elliptic curve E and a
finite field F such that the order of the point group E(F) is a given prime
number N. Heuristically, this algorithm only takes polynomial time Otilde((\log
N)^3), and it is so fast that it may profitably be used to tackle the related
problem of finding elliptic curves with point groups of prime order of
prescribed size. We also discuss the impact of the use of high level modular
functions to reduce the run time by large constant factors and show that recent
gonality bounds for modular curves imply limits on the time reduction that can
be obtained.Comment: 13 page
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