2,757 research outputs found
On the existence of quantum representations for two dichotomic measurements
Under which conditions do outcome probabilities of measurements possess a
quantum-mechanical model? This kind of problem is solved here for the case of
two dichotomic von Neumann measurements which can be applied repeatedly to a
quantum system with trivial dynamics. The solution uses methods from the theory
of operator algebras and the theory of moment problems. The ensuing conditions
reveal surprisingly simple relations between certain quantum-mechanical
probabilities. It also shown that generally, none of these relations holds in
general probabilistic models. This result might facilitate further experimental
discrimination between quantum mechanics and other general probabilistic
theories.Comment: 16+7 pages, presentation improved and minor errors correcte
Deterministic Walks in Quenched Random Environments of Chaotic Maps
This paper concerns the propagation of particles through a quenched random
medium. In the one- and two-dimensional models considered, the local dynamics
is given by expanding circle maps and hyperbolic toral automorphisms,
respectively. The particle motion in both models is chaotic and found to
fluctuate about a linear drift. In the proper scaling limit, the cumulative
distribution function of the fluctuations converges to a Gaussian one with
system dependent variance while the density function shows no convergence to
any function. We have verified our analytical results using extreme precision
numerical computations.Comment: 18 pages, 9 figure
Persistent random walk on a one-dimensional lattice with random asymmetric transmittances
We study the persistent random walk of photons on a one-dimensional lattice
of random asymmetric transmittances. Each site is characterized by its
intensity transmittance t (t') for photons moving to the right (left)
direction. Transmittances at different sites are assumed independent,
distributed according to a given probability density Distribution. We use the
effective medium approximation and identify two classes of probability density
distribution of transmittances which lead to the normal diffusion of photons.
Monte Carlo simulations confirm our predictions.Comment: 7 pages, submitted to Phys. Rev.
Zur allgemeinen Theorie der halbgeordneten Räume
Foreword by K. Kopotun11Correspondence to: K. Kopotun, Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada ^^IR3T 2N2. Email: [email protected] paper “On the general theory of semi-ordered spaces” (“Zur allgemeinen Theorie der halbgeordneten Räume”) was written by L.V. Kantorovich and G.R. Lorentz22Until 1946, G.G. (Georg Gunter) Lorentz was using the name Geogrij Rudolfovich (G.R.) Lorentz. sometime in 1937–1939, and this is the first time it appears in print.The following is a short history of this manuscript.In his letter to I.P. Natanson written on October 11, 1937, G.G. Lorentz mentioned a talk on joint work with L.V. Kantorovich that he gave at a Session on Functional Analysis in Moscow earlier that year. The records of the Academy of Sciences of USSR indicate that a Session on Functional Analysis took place in Moscow during September 27–29, 1937, and that G.R. Lorentz gave a talk “Topological theory of semi-ordered spaces” there, and that L.V. Kantorovich was speaking on “Theory of linear operations in semi-ordered spaces”.The manuscript “On the general theory of semi-ordered spaces” was found in the archives of L.V. Kantorovich. According to Vsevolod Leonidovich Kantorovich, L.V. Kantorovich’s son, it was submitted to Trudy Tomskogo Gosudarstvennogo Universiteta imeni V. V. Kuibysheva (Proceedings of Tomsk State University). The typed version33See www.math.ohio-state.edu/~nevai/LORENTZ/KANTOROVICH_LORENTZ_typed.pdf/. of the manuscript has a handwritten note by N. Romanov44N.P. Romanov (1907–1972) was a Professor at Tomsk University from 1935 until 1944. After 1944 he worked in Uzbekistan. His main area of research was Number Theory and Theory of Functions of Complex Variables. For more information see “Nikolaĭ Pavlovich Romanov (on the eightieth anniversary of his birth)”, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 1987, no. 3, 92–93, MR0914654 (89b:01069). dated by August 31, 1939 stating that the manuscript is accepted for publication. The manuscript was never published (probably because of the World War II) and around 1945 was returned to L.V. Kantorovich.It has been decided to publish this manuscript in its original language (German), and translate the extended abstract accompanying this manuscript from Russian to English. The manuscript appears here in its original form with only minor editorial corrections.Publication of this historical document would not have been possible without the assistance and effort of many people. In particular, the significant help of C. de Boor, Ya.I. Fet, V.L. Kantorovich, V.N. Konovalov, and S.S. Kutateladze is acknowledged and greatly appreciated.Extended abstract55Translated from Russian by K. Kopotun.The current manuscript is devoted to the investigation of general semi-ordered spaces that are not necessarily linear. Hence, it may be considered a generalization of the work of L.V. Kantorovich [Linear semi-ordered spaces, Mat. Sbornik, 2 (1) 1937, 121–168].We say that a set Y={y} is a semi-ordered space if its elements are partially ordered using a relation “<” so that I.If y1<y2, y2<y3, then y1<y3.II.For any pair y1, y2, there exist elements y3,y4 such that y3⩽y1, y3⩽y2, y1⩽y4, and y2⩽y4.III.Every set E⊂Y bounded above has a least upper bound (supE).IV.For every set E⊂Y, there exists a countable subset E′ that has the same least upper and greatest lower bound as E. The above assumptions allow us to introduce notions of a limit superior, limit inferior, and of a convergent sequence in Y. For example, define lim¯yn=infn(sup(yn,yn+1,…)). It is possible to introduce, e.g., the limit superior differently, for example, by defining lim¯∗yn to be the least element y having the property that, for any subsequence {ynk}, there exists a subsequence {ynki} such that y⩾lim¯i→∞ynki. This type of convergence, ∗-convergence, turns out to be identical with the topological convergence that we arrive at if we turn Y into a topological space using the convergence defined initially. Relationships among various limits which we can define using the above approaches as well as some properties of these limits are studied in § 1 and § 2. In § 3, we study semi-ordered spaces equipped with a nonnegative metric function ρ(y1,y2) defined for all pairs y1, y2 such that y1⩽y2, and satisfying 1∘.ρ(y1,y2)=0 is equivalent to y1=y2.2∘.ρ(y1,y3)⩽ρ(y1,y2)+ρ(y2,y3) (y1⩽y2⩽y3).3∘.ρ(sup(y,y1),sup(y,y2))⩽ρ(y1,y2) (an analogous inequality holds with inf).4∘.If yn→y monotonically, then ρ(yn,y)→0 (or ρ(y,yn)→0).5∘.If yn monotonically tends to infinity, then the condition limn,m→∞ρ(yn,ym)=0 should not hold.Let ρ(y1,y2,…,yn)=ρ(inf(y1,…,yn),sup(y1,…,yn)). Then yn→y turns out to be equivalent to ρ(y,yn,…,yn+p)→0 when n→∞, and yn→y(∗) is equivalent to ρ(y,yn)→0. In addition, Cauchy’s convergence principle holds. Moreover, if Y is distributive, i.e., inf(y,sup(y1,y2))=sup(inf(y,y1),inf(y,y2)), then it is also strongly distributive: inf(y,supnyn)=supn(inf(y,yn)). In § 4, we study similar spaces under weaker assumptions. Particular examples of such spaces are the Hausdorff space of closed sets (see Hausdorff “Set theory”, p. 165) and the space of semicontinuous functions. § 5 is devoted to applications of the general theorems to the theory of semicontinuous functions y=f(x) that map a metric space {x}=X into a semi-ordered space {y}=Y. Under some additional assumptions (Y is regular, distributive, and between any two elements y1 and y2 such that y1<y2 there is a third element y3, y1<y3<y2) it is possible to develop a complete theory of semicontinuous functions including a theorem that every semicontinuous function is a limit of a monotone sequence of continuous functions as well a theorem on separation by a continuous function
Transport Properties of the Diluted Lorentz Slab
We study the behavior of a point particle incident from the left on a slab of
a randomly diluted triangular array of circular scatterers. Various scattering
properties, such as the reflection and transmission probabilities and the
scattering time are studied as a function of thickness and dilution. We show
that a diffusion model satisfactorily describes the mentioned scattering
properties. We also show how some of these quantities can be evaluated exactly
and their agreement with numerical experiments. Our results exhibit the
dependence of these scattering data on the mean free path. This dependence
again shows excellent agreement with the predictions of a Brownian motion
model.Comment: 14 pages of text in LaTeX, 7 figures in Postscrip
A Paradox of State-Dependent Diffusion and How to Resolve It
Consider a particle diffusing in a confined volume which is divided into two
equal regions. In one region the diffusion coefficient is twice the value of
the diffusion coefficient in the other region. Will the particle spend equal
proportions of time in the two regions in the long term? Statistical mechanics
would suggest yes, since the number of accessible states in each region is
presumably the same. However, another line of reasoning suggests that the
particle should spend less time in the region with faster diffusion, since it
will exit that region more quickly. We demonstrate with a simple microscopic
model system that both predictions are consistent with the information given.
Thus, specifying the diffusion rate as a function of position is not enough to
characterize the behaviour of a system, even assuming the absence of external
forces. We propose an alternative framework for modelling diffusive dynamics in
which both the diffusion rate and equilibrium probability density for the
position of the particle are specified by the modeller. We introduce a
numerical method for simulating dynamics in our framework that samples from the
equilibrium probability density exactly and is suitable for discontinuous
diffusion coefficients.Comment: 21 pages, 6 figures. Second round of revisions. This is the version
that will appear in Proc Roy So
Performance and selection of winter durum wheat genotypes in different European conventional and organic fields
Sustainability is a key factor for the future of agriculture. Productivity in agriculture has more than tripled in developed countries since the 1950s. Beyond the success of plant breeding, the increased use of inorganic fertilizers, application of pesticides, and spread of irrigation also contributed to this success. However, impressive yield increases started to decline in the 1980s because of the lack of sustainability. One of the most beneficial ways to increase sustainability is organic agriculture. In such agro-ecosystem-based holistic production systems the prerequisite of successful farming is the availability of crop genotypes that perform well. However, selection of winter durum wheat for sub-optimal growing conditions is still mainly neglected, and the organic seed market also lacks of information on credibly tested winter durum varieties suitable for organic agriculture
A real Lorentz-FitzGerald contraction
Many condensed matter systems are such that their collective excitations at
low energies can be described by fields satisfying equations of motion formally
indistinguishable from those of relativistic field theory. The finite speed of
propagation of the disturbances in the effective fields (in the simplest
models, the speed of sound) plays here the role of the speed of light in
fundamental physics. However, these apparently relativistic fields are immersed
in an external Newtonian world (the condensed matter system itself and the
laboratory can be considered Newtonian, since all the velocities involved are
much smaller than the velocity of light) which provides a privileged coordinate
system and therefore seems to destroy the possibility of having a perfectly
defined relativistic emergent world. In this essay we ask ourselves the
following question: In a homogeneous condensed matter medium, is there a way
for internal observers, dealing exclusively with the low-energy collective
phenomena, to detect their state of uniform motion with respect to the medium?
By proposing a thought experiment based on the construction of a
Michelson-Morley interferometer made of quasi-particles, we show that a real
Lorentz-FitzGerald contraction takes place, so that internal observers are
unable to find out anything about their `absolute ' state of motion. Therefore,
we also show that an effective but perfectly defined relativistic world can
emerge in a fishbowl world situated inside a Newtonian (laboratory) system.
This leads us to reflect on the various levels of description in physics, in
particular regarding the quest towards a theory of quantum gravity.Comment: 6 pages, no figures. Minor changes reflect published versio
Brownian motion and diffusion: from stochastic processes to chaos and beyond
One century after Einstein's work, Brownian Motion still remains both a
fundamental open issue and a continous source of inspiration for many areas of
natural sciences. We first present a discussion about stochastic and
deterministic approaches proposed in the literature to model the Brownian
Motion and more general diffusive behaviours. Then, we focus on the problems
concerning the determination of the microscopic nature of diffusion by means of
data analysis. Finally, we discuss the general conditions required for the
onset of large scale diffusive motion.Comment: RevTeX-4, 11 pages, 5 ps-figures. Chaos special issue "100 Years of
Brownian Motion
Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two
We consider the billiard dynamics in a non-compact set of R^d that is
constructed as a bi-infinite chain of translated copies of the same
d-dimensional polytope. A random configuration of semi-dispersing scatterers is
placed in each copy. The ensemble of dynamical systems thus defined, one for
each global realization of the scatterers, is called `quenched random Lorentz
tube'. Under some fairly general conditions, we prove that every system in the
ensemble is hyperbolic and almost every system is recurrent, ergodic, and
enjoys some higher chaotic properties.Comment: Final version for J. Stat. Phys., 18 pages, 4 figure
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