Detection Time Distribution for Several Quantum Particles

We address the question of how to compute the probability distribution of the time at which a detector clicks, in the situation of $n$ non-relativistic quantum particles in a volume $\Omega\subset \mathbb{R}^3$ in physical space and detectors placed along the boundary $\partial \Omega$ of $\Omega$. We have recently [http://arxiv.org/abs/1601.03715] argued in favor of a rule for the 1-particle case that involves a Schr\"odinger equation with an absorbing boundary condition on $\partial \Omega$ introduced by Werner; we call this rule the "absorbing boundary rule." Here, we describe the natural extension of the absorbing boundary rule to the $n$-particle case. A key element of this extension is that, upon a detection event, the wave function gets collapsed by inserting the detected position, at the time of detection, into the wave function, thus yielding a wave function of $n-1$ particles. We also describe an extension of the absorbing boundary rule to the case of moving detectors.Comment: 15 pages LaTeX, no figure

A non-compact deduction rule for the logic of provability and its algebraic models

In this paper, we introduce a proof system with a non-compact deduction rule, that is, a deduction rule with countably many premises, to axiomatize the logic $\mathbf{GL}$ of provability, and show its Kripke completeness in an algebraic manner. As $\mathbf{GL}$ is not canonical, a standard proof of Kripke completeness for $\mathbf{GL}$ is given by a Kripke model which is obtained by changing the binary relation of the canonical model, while our proof is given by a submodel of the canonical model of $\mathbf{GL}$ which is obtained by making use of an infinitary extension of the J\'{o}nsson-Tarski representation. We also show the three classes of algebras defined by $\Box x\leq\Box\Box x$ and one of the following three conditions, $\bigwedge_{n\in\omega}\Diamond^{n}1=0$, the non-compact deduction rule and the L\"{o}b formula, are mutually different, while all of them define $\mathbf{GL}$

Predicative proof theory of PDL and basic applications

Propositional dynamic logic (PDL) is presented in Schütte-style mode as one-sided semiformal tree-like sequent calculus Seq pdl ω with standard cut rule and the omega-rule with principal formulas [P * ]A. The omega-rule-free derivations in Seq pdl ω are finite (trees) and sequents deducible by these finite derivations are valid in PDL. Moreover the cut-elimination theorem for Seq pdl ω is provable in Peano Arithmetic (PA) extended by transfinite induction up to Veblen's ordinal ϕ ω (0). Hence (by the cutfree subformula property) such predicative extension of PA proves that any given [P * ]-free sequent is valid in PDL iff it is deducible in Seq pdl ω by a finite cut-and omega-rule-free derivation, while PDL-validity of arbitrary star-free sequents is decidable in polynomial space. The former also implies a Herbrand-style conclusion that e.g. a given formula S = P * A ∨ Z for star-free A and Z is valid in PDL iff there is a k ≥ 0 and a cut-and omega-rule-free derivation of sequent A, P 1 A, · · · , P k A, B where P i A is an abbreviation for P · · · P i times A. This eventually leads to PSPACE-decidability of PDL-validity of S, provided that P is atomic and A is in a suitable basic conjunctive normal form. Furthermore we consider star-free formulas A in dual basic disjunctive normal form, and corresponding expansions S = P * A ∨ Z whose PDL-validity problem is known to be EXPTIME-complete. We show that cutfree-derivability in Seq pdl ω (hence PDL-validity) of such S is equivalent to plain validity of a suitable "transparent" quantified boolean formula S. Hence EXPTIME = PSPACE holds true iff the validity problem for any S involved is solvable by a polynomial-space deterministic TM. This may reduce the former problem to a more transparent complexity problem in quantified boolean logic. The whole proof can be formalized in PA extended by transfinite induction along ϕ ω (0)-actually in the corresponding primitive recursive weakening, PRA ϕ ω (0)

Quantum to classical crossover in generalized spin systems -- the temperature-dependent spin dynamics of FeI2_2

Simulating quantum spin systems at finite temperatures is an open challenge in many-body physics. This work studies the temperature-dependent spin dynamics of a pivotal compound, FeI$_2$, to determine if universal quantum effects can be accounted for by a phenomenological renormalization of the dynamical spin structure factor $S(\mathbf{q}, \omega)$ measured by inelastic neutron scattering. Renormalization schemes based on the quantum-to-classical correspondence principle are commonly applied at low temperatures to the harmonic oscillators describing normal modes. However, it is not clear how to extend this renormalization to arbitrarily high temperatures. Here we introduce a temperature-dependent normalization of the classical moments, whose magnitude is determined by imposing the quantum sum rule, i.e. $\int d\omega d\mathbf{q} S(\mathbf{q}, \omega) = N_S S (S+1)$ for $N_S$ dipolar magnetic moments. We show that this simple renormalization scheme significantly improves the agreement between the calculated and measured $S(\mathbf{q}, \omega)$ for FeI$_{2}$ at all temperatures. Due to the coupled dynamics of dipolar and quadrupolar moments in that material, this renormalization procedure is extended to classical theories based on SU(3) coherent states, and by extension, to any SU(N) coherent state representation of local multipolar moments.Comment: Associated source code for reproducing calculations available at: https://github.com/SunnySuite/SunnyContribute

LoopW Technical Reference v0.3

This document describes the implementation in SML of the LoopW language, an imperative language with higher-order procedural variables and non-local jumps equiped with a program logic. It includes the user manual along with some implementation notes and many examples of certified imperative programs. As a concluding example, we show the certification of an imperative program encoding shift/reset using callcc/throw and a global meta-continuation

A Symbolic Transformation Language and its Application to a Multiscale Method

The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale methods approximate it by another partial differential equation which can be numerically simulated in a reasonable time. The challenge consists in taking into account a wide range of geometries combining thin and periodic structures with the possibility of multiple nested scales. In this paper we present a transformation language that will make the development of MEMSALab more feasible. It is proposed as a Maple package for rule-based programming, rewriting strategies and their combination with standard Maple code. We illustrate the practical interest of this language by using it to encode two examples of multiscale derivations, namely the two-scale limit of the derivative operator and the two-scale model of the stationary heat equation.Comment: 36 page

Stereo-Aware Extension of HOSE Codes

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Descriptions of molecular environments have many applications in chemoinformatics, including chemical shift prediction. Hierarchically ordered spherical environment (HOSE) codes are the most popular such descriptions. We developed a method to extend these with stereochemistry information. It enables distinguishing atoms which would be considered identical in traditional HOSE codes. The use of our method is demonstrated by chemical shift predictions for molecules in the nmrshiftdb2 database. We give a full specification and an implementation