Symmetry and Topology in Quantum Logic

A test space is a collection of non-empty sets, usually construed as the catalogue of (discrete) outcome sets associated with a family of experiments. Subject to a simple combinatorial condition called algebraicity, a test space gives rise to a “quantum logic”—that is, an orthoalgebra. Conversely, all orthoalgebras arise naturally from algebraic test spaces. In non-relativistic quantum mechanics, the relevant test space is the set ℱ F(H) of frames (unordered orthonormal bases) of a Hilbert space H. The corresponding logic is the usual one, i.e., the projection lattice L(H) of H. The test space ℱ F(H) has a strong symmetry property with respect to the unitary group of H, namely, that any bijection between two frames lifts to a unitary operator. In this paper, we consider test spaces enjoying the same symmetry property relative to an action by a compact topological group. We show that such a test space, if algebraic, gives rise to a compact, atomistic topological orthoalgebra. We also present a construction that generates such a test space from purely group-theoretic data, and obtain a simple criterion for this test space to be algebraic. PACS: 02.10.Ab; 02.20.Bb; 03.65.Ta.     

A New Method for the Solution of Models of Biological Evolution: Derivation of Exact Steady-State Distributions

We investigate well-known models of biological evolution and address the open problem of how construct a correct continuous analog of mutations in discrete sequence space. We deal with models where the fitness is a function of a Hamming distance from the reference sequence. The mutation-selection master equation in the discrete sequence space is replaced by a Hamilton-Jacobi equation for the logarithm of relative frequencies of different sequences. The steady-state distribution, mean fitness and the variance of fitness are derived. All our results are asymptotic in the large genome limit. A variety of important biological and biochemical models can be solved by this new approach. PACS numbers: 87.10.+e, 87.15.Aa, 87.23.Kg, 02.50.-r     

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.     

Discrete, Amorphous Physical Models

Discrete models of physical phenomena are an attractive alternative to continuous models such as partial differential equations. In discrete models, such as cellular automata, space is treated as having finitely many locations per unit volume, and physical processes are modelled by rules that depend on a small number of nearby locations. Such models depend critically on a regular (crystalline) lattice, as well as the global synchronization of all sites. We should ask, on the grounds of minimalism, whether the global synchronization and crystalline lattice are inherent in any discrete formulation. Is it possible to do without these conditions and still have a useful physical model? Or are they somehow fundamental? We will answer this question by presenting a class of models that are “extremely local” in the sense that the update rule does not depend on synchronization with the other sites, or on knowledge of the lattice geometry. All interactions involve only a single pair of sites. The models have the further advantage that they exactly conserved the analog of quantities such as momentum and energy which are conserved in physics. An example model of waves is given, and evidence is given that it agrees well qualitatively and quantitatively with continuous differential equations.     

An Adaptive Wavelet–Vaguelette Algorithm for the Solution of PDEs

The paper first describes a fast algorithm for the discrete orthonormal wavelet transform and its inverse without using the scaling function. This approach permits to compute the decomposition of a function into a lacunary wavelet basis, i.e., a basis constituted of a subset of all basis functions up to a certain scale, without modification. The construction is then extended to operator-adapted biorthogonal wavelets. This is relevant for the solution of certain nonlinear evolutionary PDEs where a priori information about the significant coefficients is available. We pursue the approach described in (J. Fröhlich and K. Schneider,Europ. J. Mech. B/Fluids13,439, 1994) which is based on the explicit computation of the scalewise contributions of the approximated function to the values at points of hierarchical grids. Here, we present an improved construction employing the cardinal function of the multiresolution. The new method is applied to the Helmholtz equation and illustrated by comparative numerical results. It is then extended for the solution of a nonlinear parabolic PDE with semi-implicit discretization in time and self-adaptive wavelet discretization in space. Results with full adaptivity of the spatial wavelet discretization are presented for a one-dimensional flame front as well as for a two-dimensional problem.     

Propagation of a transient electromagnetic disturbance in an inhomogeneous, collisional, Magnetized plasma

Summary The time response of an anisotropic, inhomogeneous magnetized plasma to a time-dependent externally applied electric field is studied. The plasma is considered as a conducting medium embedded in a static magnetic fieldB _o, with finite and constant parallel conductivity σ_∥ and normal conductivity σ_⊥ exponentially varying; a constant ion polarization conductivity (σ_pol) normal toB ₀ is also included. The driving termE ₀ is assumed to be normal to the ambient magnetic field, with a space dependence only in the (E ₀, B₀)-plane (two-dimensional case). The problem is treated as an initial (in time) and boundary (in space) value problem, and an analytic solution is obtained for the time response of the system to a unit step of electric field; the solution for any other driving field is then given via a convolution integral. This model can be used in particular to study the propagation through the ionosphere of ULF waves generated by the motion of an artificially created ion cloud in the upperF region. Moreover, the mathematical technique is rather general, so it may be used also for different plasma models.

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Riassunto Si presenta lo studio della risposta temporale (transitorio) di un plasma magnetizzato, anisotropo e inomogeneo ad un campo elettrico esterno impresso. Il plasma è trattato come un mezzo conduttore, immerso in un campo magnetico staticoB ₀, la cui conducibilità elettrica parallela al campo magnetico σ_∥ è costante, mentre la conducibilità normale σ_⊥ varia esponenzialmente nello spazio; si considera inoltre una conducibilità dovuta alla polarizzazione degli ioni (σ_pol) costante e normale aB ₀. Il campo impressoE ₀ è assunto normale al campo magnetico ambiente con dipendenza spaziale solo nel piano (E ₀, B₀) (caso bidimensionale). In queste ipotesi si perviene ad un problema ai valori iniziali e al contorno, e si deriva una soluzione analitica per la risposta transitoria del sistema ad un gradino di campo elettrico impresso; la soluzione per una qualsiasi funzione di campo applicato è data come integrale di convoluzione con la risposta al gradino. Questo modello può essere usato, in particolare, per studiare la propagazione attraverso la ionosfera di un'onda ULF generata dal moto di una nuvola ionica creata artificialmente nella parte superiore della regioneF. Un ulteriore interesse deriva dal fatto che, essendo la tecnica matematica utilizzata molto generale, questa può essere applicata pure a differenti modelli di plasma.

     

Marginal Distributions of Wigner Function in a Mesoscopic L-C Circuit at Finite Temperature and Thermal Wigner Operator

Wigner function in phase space has its physical meaning as marginal probability distribution in coordinate space and momentum space respectively, here we endow the Wigner function with a new physical meaning, i.e., its marginal distributions’ statistical average for q ²/(2C) and p ²/(2L) are the energy stored in capacity and in inductance of a mesoscopic L-C circuit at finite temperature, respectively. PACS numbers: 03.65.-w, 73.21.-b     

Realizations of causal manifolds by quantum fields

Quantum mechanical operators and quantum fields are interpreted as realizations of timespace manifolds. Such causal manifolds are parametrized by the classes of the positive unitary operations in all complex operations, i.e., by the homogenous spacesD(n)=GL(C _R ⁿ )/U(n) withn=1 for mechanics andn=2 for relativistic fields. The rankn gives the number of both the discrete and continuous invariants used in the harmonic analysis, i.e., two characteristic masses in the relativistic case. ‘Canonical’ field theories with the familiar divergencies are inappropriate realizations of the real 4-dimensional causal manifoldD(2). Faithful timespace realizations do not lead to divergencies. In general they are reducible, but nondecomposable—in addition to representations with eigenvectors (states, particle), they incorporate principal vectors without a particle (eigenvector) basis as exemplified by the Coulomb field. In theorthogonal andunitary groupsO(N ₊,N ₋), respectively, thepositive orthogonal and unitary ones areO(N) andU(N), respectively.     

Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I: Mathematical Derivation of Adjoint Lattice Boltzmann Equations

The significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector. Moreover, this approach was not limited to flow fields and could be employed for steady as well as unsteady flows. Initially, the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time, respectively. Meanwhile, new adjoint concepts in lattice space were introduced. Finally, the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.     

A Dynamical Classification of the Range of Pair Interactions

We formalize a classification of pair interactions based on the convergence properties of the forces acting on particles as a function of system size. We do so by considering the behavior of the probability distribution function (PDF) P(F) of the force field F in a particle distribution in the limit that the size of the system is taken to infinity at constant particle density, i.e., in the “usual” thermodynamic limit. For a pair interaction potential V(r) with V(r→∞)∼1/r ^γ defining a bounded pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the pair force is absolutely integrable, i.e., for γ>d−1, where d is the spatial dimension. We refer to this case as dynamically short-range, because the dominant contribution to the force on a typical particle in this limit arises from particles in a finite neighborhood around it. For the dynamically long-range case, i.e., γ≤d−1, on the other hand, the dominant contribution to the force comes from the mean field due to the bulk, which becomes undefined in this limit. We discuss also how, for γ≤d−1 (and notably, for the case of gravity, γ=d−2) P(F) may, in some cases, be defined in a weaker sense. This involves a regularization of the force summation which is generalization of the procedure employed to define gravitational forces in an infinite static homogeneous universe. We explain that the relevant classification in this context is, however, that which divides pair forces with γ>d−2 (or γ<d−2), for which the PDF of the difference in forces is defined (or not defined) in the infinite system limit, without any regularization. In the former case dynamics can, as for the (marginal) case of gravity, be defined consistently in an infinite uniform system.     

Limit Processes for TASEP with Shocks and Rarefaction Fans

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density ρ ₋ and right density ρ ₊. We study the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of ρ _±. We characterize the large time scaling limit of the multipoint fluctuations as a function of the densities ρ _± and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open.     

Tunneling through discrete levels in the continuum

We study the ballistic transport in quantum channels containing attractive impurities. We show that coherent interaction between asymptotic resonances may cause resonances to disappear and discrete levels to appear in the continuum at certain (critical) values of the parameters of the system. For the first time the tunneling of an electron through discrete levels is investigated. We find that the transmissivity changes dramatically when the scattered electrons at infinity have an energy coinciding with that of the discrete levels. It is found that a new type of degeneracy may arise in the system at critical values of the parameters, a degeneracy in which one state is described by a localized wave function and the other, by a propagating wave function. We calculate the critical values of the parameters of the structure and discuss ways of experimentally implementing this effect in two-dimensional channels. Zh. éksp. Teor. Fiz. 115, 211–230 (January 1999)     

Remark on the (non)convergence of ensemble densities in dynamical systems

We consider a dynamical system with state space M, a smooth, compact subset of some R(n), and evolution given by T(t), x(t)=T(t)x, x in M; T(t) is invertible and the time t may be discrete, t in Z, T(t)=T(t), or continuous, t in R. Here we show that starting with a continuous positive initial probability density rho(x,0)>0, with respect to dx, the smooth volume measure induced on M by Lebesgue measure on R(n), the expectation value of logrho(x,t), with respect to any stationary (i.e., time invariant) measure nu(dx), is linear in t, nu(logrho(x,t))=nu(logrho(x,0))+Kt. K depends only on nu and vanishes when nu is absolutely continuous with respect to dx.(c) 1998 American Institute of Physics.     

A prediction theory of random level distribution based on a generalized difference type of Fokker-Planck equation and its application to environmental noise and vibration,I: Theoretical study

For generalized discrete random signals, of arbitrary correlations among arbitrarily chosen samples, and also arbitrary distribution form, the short time prediction problem, in terms of the transition probability distribution, is theoretically considered, first for discrete time interval sampling. A general expression is derived from which any signal statistics, e.g., the average, the variance, the 90% range value, and so on, can be predicted. This general expression is equivalent to the well-known Fokker-Planck equation, with continuous time sampling, in the special case of a Markovian process. Explicit algorithms for estimating moment statistics of arbitrary order are derived, by introducing the generalized difference equation of Fokker-Planck type for the probability distribution function.     

Towards a Model for Protein Production Rates

In the process of translation, ribosomes read the genetic code on an mRNA and assemble the corresponding polypeptide chain. The ribosomes perform discrete directed motion which is well modeled by a totally asymmetric simple exclusion process (TASEP) with open boundaries. Using Monte Carlo simulations and a simple mean-field theory, we discuss the effect of one or two “bottlenecks” (i.e., slow codons) on the production rate of the final protein. Confirming and extending previous work by Chou and Lakatos, we find that the location and spacing of the slow codons can affect the production rate quite dramatically. In particular, we observe a novel “edge” effect, i.e., an interaction of a single slow codon with the system boundary. We focus in detail on ribosome density profiles and provide a simple explanation for the length scale which controls the range of these interactions. PACS numbers: 05.70.Ln, 64.90.+b, 87.14.Gg     

Duality in Interacting Particle Systems and Boson Representation

In the context of Markov processes, we show a new scheme to derive dual processes and a duality function based on a boson representation. This scheme is applicable to a case in which a generator is expressed by boson creation and annihilation operators. For some stochastic processes, duality relations have been known, which connect continuous time Markov processes with discrete state space and those with continuous state space. We clarify that using a generating function approach and the Doi-Peliti method, a birth-death process (or discrete random walk model) is naturally connected to a differential equation with continuous variables, which would be interpreted as a dual Markov process. The key point in the derivation is to use bosonic coherent states as a bra state, instead of a conventional projection state. As examples, we apply the scheme to a simple birth-coagulation process and a Brownian momentum process. The generator of the Brownian momentum process is written by elements of the SU(1,1) algebra, and using a boson realization of SU(1,1) we show that the same scheme is available.     

Noncommutative Differential Calculus Approach to Discrete Symplectic Schemes on Regular Lattice

By means of a noncommutative differential calculus on function space of discrete Abelian groups and that of the regular lattice with equal spacing as well as the discrete symplectic geometry and a kind of classical mechanical systems with separable Hamiltonian of the type H(p, q) = T(p) + V(q) on regular lattice, we introduce the discrete symplectic algorithm, i.e., the phase-space discrete counterpart of the symplectic algorithm including original symplectic schemes and the jet-symplectic schemes in terms of the discrete time jet bundle formalism, on the regular lattice. We show some numerical calculation examples and compare the results of different schemes.     

A unified gas-kinetic scheme for continuum and rarefied flows

With discretized particle velocity space, a multiscale unified gas-kinetic scheme for entire Knudsen number flows is constructed based on the BGK model. The current scheme couples closely the update of macroscopic conservative variables with the update of microscopic gas distribution function within a time step. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate Navier–Stokes (NS) solutions in the continuum flow regime with a time step being much larger than the particle collision time. At the same time, the rarefied flow solution, even in the free molecule limit, can be captured accurately. The unified scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space. The success of the method is due to the un-splitting treatment of the particle transport and collision in the evaluation of local solution of the gas distribution function. For these methods which use operator splitting technique to solve the transport and collision separately, it is usually required that the time step is less than the particle collision time. This constraint basically makes these methods useless in the continuum flow regime, especially in the high Reynolds number flow simulations. Theoretically, once the physical process of particle transport and collision is modeled statistically by the kinetic Boltzmann equation, the transport and collision become continuous operators in space and time, and their numerical discretization should be done consistently. Due to its multiscale nature of the unified scheme, in the update of macroscopic flow variables, the corresponding heat flux can be modified according to any realistic Prandtl number. Subsequently, this modification effects the equilibrium state in the next time level and the update of microscopic distribution function. Therefore, instead of modifying the collision term of the BGK model, such as ES-BGK and BGK–Shakhov, the unified scheme can achieve the same goal on the numerical level directly. Many numerical tests will be used to validate the unified method.     

The Incompressible Navier–Stokes for the Nonlinear Discrete Velocity Models

Abstract

We establish the incompressible Navier–Stokes limit for the discrete velocity model of the Boltzmann equation in any dimension of the physical space, for densities which remain in a suitable small neighborhood of the global Maxwellian. Appropriately scaled families solutions of discrete Boltzmann equation are shown to have fluctuations that locally in time converge strongly to a limit governed by a solution of Incompressible Navier–Stokes provided that the initial fluctuation is smooth, and converges to appropriate initial data. As applications of our results, we study the Carleman model and the one-dimensional Broadwell model.