Finite Phase Space Research Articles

Ultrasonic attenuation in clean d-wave superconductors

We calculate the low temperature longitudinal ultrasonic attenuation α S in clean d-wave superconductors. We consider the contribution of previously ignored processes involving the excitation of a pair of quasi-holes or quasi-particles. These processes, which are forbidden by energy conservation in conventional s-wave superconductors, have a finite phase space in d-wave superconductors due to the presence of nodes in the gap which give rise to soft low-energy electronic excitations. We find the contribution to α S from these processes to be proportional to T in the regime k B T≪ Qv Δ ≪ Δ 0 (ultra-low temperature regime) and to be proportional to 1/ T in the region Qv F≪ k B T≪ Δ 0 (low temperature regime), where Q is the ultrasound wave vector and Δ 0 is the maximum gap amplitude. We explicitly evaluate these terms, for parameters appropriate to the cuprates, for Q along the nodal and the antinodal directions and compare it with the contribution from processes considered earlier [Phys. Rev. B 59 (1999) 7123]. In the ultra-low temperature regime, the processes considered by us make a contribution which is smaller by about a factor of 10 for Q along the nodal direction, while along the antinodal direction it is larger by a factor of 100 or so. In the low temperature regime on the other hand the contribution made by these terms is small. However taken together with the original terms we describe a possible way to evaluate the parameter v F/ v Δ .

A coordinate independent formulation of the Weyl–Wigner transform theory

We present the Weyl–Wigner (WW) transform theory in a much more compact way than usual, by introducing the Δ basis in an intrinsic form. This permits the derivation of new identities and also leads to generalizations, like the inclusion of finite-dimensional systems in the WW theory, which is also discussed. We show, in this case, some striking differences in the structure of finite phase space depending on the underlying dimension of quantum space being an even or odd integer.

Computation of the viscosity of a liquid from time averages of stress fluctuations.

The shear viscosity can be calculated from the standard deviation of an equilibrium ensemble of time averages of the shear stress computed along finite duration phase space trajectory segments. The mean square of the segment averages of the shear stress is proportional to the shear viscosity and inversely proportional to the duration of the trajectory segments and the number of particles. We test the fluctuation relation for the shear viscosity and show that it provides a simple but viable means of computing the zero strain rate shear viscosity. We decompose the shear viscosity computed using this fluctuation method, into its "kinetic" and "configurational" components. We also calculate the relevant relaxation times. We compare the computed results with standard nonequilibrium molecular dynamics simulations. Finally we compute the bulk viscosity using an analogous fluctuation method.

On chemical equilibrium in high-energy heavy-ion collisions

The thermal fitting parameters depend on the phase space coverage. It isquestionable to interpret the fitting results extracted from theexperimental measured particle yields within finite phase space. We testthis idea with the transport models RQMD and NEXUS for heavy-ioncollisions at both SPS and RHIC energies.

The action-angle Wigner function: a discrete, finite and algebraic phase space formalism

The action-angle representation in quantum mechanics is conceptually quite different from its classical counterpart and motivates a canonical discretization of the phase space. In this work, a discrete and finite-dimensional phase space formalism, in which the phase space variables are discrete and the time is continuous, is developed and the fundamental properties of the discrete Weyl-Wigner-Moyal quantization are derived. The action-angle Wigner function is shown to exist in the semi-discrete limit of this quantization scheme. A comparison with other formalisms which are not explicitly based on canonical discretization is made. Fundamental properties that an action-angle phase space distribution respects are derived. The dynamical properties of the action-angle Wigner function are analysed for discrete and finite-dimensional model Hamiltonians. The limit of the discrete and finite-dimensional formalism including a discrete analogue of the Gaussian wavefunction spread, viz. the binomial wavepacket, is examined and shown by examples that standard (continuum) quantum mechanical results can be obtained as the dimension of the discrete phase space is extended to infinity.

Homoclinic Tubes in Nonlinear Schr&ödinger Equation under Hamiltonian Perturbations

The concept of a homoclinic tube was introduced by Silnikov 14) in a study on the structure of the neighborhood of a homoclinic tube asymptotic to an invariant torus σ under a diffeomorphism F in a finite dimensional phase space. The asymptotic torus is of the saddle type. The homoclinic tube consists of a doubly infinite sequence of tori {σj , j = 0,±1,±2, · · ·} in the transversal intersection of the stable and unstable manifolds of σ, such that σj+1 = F ◦ σj for any j. This is a generalization of the concept of a transversal homoclinic orbit in which the points are replaced by tori. Silnikov obtained a theorem on the symbolic dynamics structures in the neighborhood of the homoclinic tube that is similar to Smale’s theorem for a transversal homoclinic orbit. 13), 12) We are interested in homoclinic tubes for several reasons: 1. Especially in high dimensions, it is often the case that the dynamics inside each invariant tube in the neighborhood of a homoclinic tube is also chaotic. We refer to such chaotic dynamics as “chaos in the small” and the symbolic dynamics of the invariant tubes as “chaos in the large”. Such cascade structures are more important than the structures in a neighborhood of a homoclinic orbit when high or infinite dimensional dynamical systems are studied. 2. Symbolic dynamics structures in the neighborhoods of homoclinic tubes are more observable than in the neighborhoods of homoclinic orbits in numerical and physical experiments. 3. When studying a high or infinite dimensional Hamiltonian system (for example, the cubic nonlinear Schrodinger equation under Hamiltonian perturbations), each invariant tube contains both KAM tori and stochastic layers (chaos in the small). Thus, not only is it the case that the dynamics inside each stochastic layer is chaotic, all these stochastic layers also move chaotically under Poincare maps. In this paper, we are interested in studying Hamiltonian partial differential equations. Denote by W (c) a normally hyperbolic center manifold, by W (cu) and W (cs)

Percolation and lack of self-averaging in a frustrated evolutionary model

We present a stochastic evolutionary model obtained through a perturbation of Kauffman's maximally rugged model, which is recovered as a special case. Our main results are: (i) existence of a percolation-like phase transition in the finite phase space case; (ii) existence of non-self-averaging effects in the thermodynamic limit. Lack of self-averaging emerges from a fragmentation of the space of all possible evolutions, analogous to that of a geometrically broken object. Thus the model turns out to be exactly solvable in the thermodynamic limit.

Discrete Wigner function for finite-dimensional systems

A phase-space approach to finite-dimensional systems is developed from basic principles. For a system describable by a Hilbert space of dimension d we define a one-to-one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d. The properties fulfilled by this correspondence and its uniqueness are examined. This formalism is applied to the number difference and phase difference of a two-mode field. This case is compared with the marginal distribution for these variables arising from a two-mode Wigner function for number and phase.

Factorization of complex canonical transformations

We investigate the Lie series representation of the canonical transformations in a finite dimensional complex phase space. It is shown that any transformation of this type can be factorized into a product of three factors associated with a pure imaginary generating function, a holomorphic function, and an element of the cyclic group C4. The imaginary function can be considered as an observable in the sense of classical mechanics. Some hints are given which suggest that the holomorphic function can be connected with the notion of the state of a physical system. Moreover, a special kind of mappings is studied which provides a link between entropy, action, and state functions. The occurrence of these important physical quantities shows that the mathematical structure goes beyond a formal analogy to quantum physics at least in the finite dimensional case.

Construction of approximative and almost periodic solutions of perturbed linear schrödinger and wave equations

Consider 1D nonlinear Schrodinger equation (0.1) $$iu_t - u_{xx} + V(x)u + \varepsilon \frac{{\partial {\rm H}}}{{\partial \bar u}} = 0$$ and nonlinear wave equation (0.2) $$y_{tt} - y_{xx} + \rho y + \varepsilon F'(y) = 0$$ under Dirichlet boundary conditions. We assume hereH(u,ū) andF(y) polynomials. It is proved that for “typical” periodic potentialV in (0.1) and typical ρ∈R in (0.2) the following is true. Letu(0) (resp.y(0),y′(0)) be smooth initial data fort=0. Then the corresponding solutionu(t) of (0.1) (resp.y(t) of (0.2)) will be ɛ1/2-close to the unperturbed solution (with appropriate frequency adjustment), for times |t|<ɛ−M whereM may be any chosen number (letting ɛ → 0) (See Prop. 4.18 and Prop. 5.13). This result may be seen as a Nekhoroshev type result (cf. [N]) for Hamiltonian PDE, in the nonresonant regime (which is the easiest to study). In this spirit, results in finite dimensional phase space have been obtained by various authors but for different interactions, essentially of finite range, which does not cover natural PDE models. See for instance [BFG]. We started here to investigate this phenomenon in the PDE context. In the second part of the paper, we use the technique from [Bo] (see also relevant references in [Bo] on earlier work such as [CrW]) to construct almost periodic (in time) solutions of say a wave equation $$y_{tt} - y_{xx} + V(x)y + \varepsilon F'(y) = 0$$ under Dirichlet boundary conditions. HereV is a “typical” real analytic periodic potential. The frequencies of these solutions form a full set, i.e. $$\lambda '_j \approx \lambda _j = \sqrt {\mu _j } $$ where {μ j } is the Dirichlet spectrum of $$ - \frac{{d^2 }}{{dx^2 }} + V(x)$$ . However, they are obtained starting from an unperturbed solutionu 0(x, t)=Σ∞ j=1 a j cos λ j t.ϕ j (x), subject to a strong decay assumption |a j | → 0 on the initial amplitudes {a j }. The argument would need to be considerably refined to reach a more realistic decay. Again, the construction of invariant tori of infinite dimension (via usual KAM techniques) is achieved for certain models with finite range interaction (see [FSW]). There are also the results of [CP], but they require a very rapidly increasing frequency sequence {λ j }.

A first order approximation forthe convergence of distributionsof the cox processes with

A limit of a normalized difference between the distribution of the conditionally Poisson process with the intensity modulated by a “fast” Markov process with a finite phase space and the distribution of the Poisson process with the appropriately averaged intensity is calculated. The proof is based on the study of a singularly perturbed filtering equation

Remarkable phase oscillations appearing in the lattice dynamics of Einstein-Podolsky-Rosen states.

It is shown that the transformations of Einstein-Podolsky-Rosen states such as those used in communication and cryptography schemes can be described as a hopping motion on a finite phase space lattice associated with a finite Heisenberg group. Quantum mechanical Hamiltonians that generate the hopping are shown to cause phase oscillations characterized by the number-theoretic Legendre symbol.

On one controversy about the smoothed Wigner function

We give a simple proof that the smoothed Wigner function in the general case cannot be non-negative, whenever the smoothing function is zero outside a finite phase space region. Our proof is not based on whatever assumptions about the behavior of the smoothing function in the non-zero region and on its boundary. In this way, the long lasted conjecture claiming the opposite is refuted.

A TOPOLOGICAL QUANTUM THEORY INTERPRETATION OF INTEGRABLE MODELS

An integrable model can be interpreted as a constrained Hamiltonian system by treating constants of motion of the former as constraints of the latter. The new constrained Hamiltonian system, when we deal with a finite initial phase space, after quantization does not have local excitations if operator ordering does not cause anomalies. So that it is a topological quantum theory. As an example, operator quantization of the Toda lattice where the ordering is important, is studied.

On the limiting distribution of pair-summable potential functions in many-particle systems

For systems with finite phase space volume, the density of states can be viewed as a multiple of the probability density of the energy, when the phase space variables are independent uniformly distributed random variables. We show that the distribution of a random variable proportional to the sum of pairwise interactions of independent identically distributed random variables converges to a limiting distribution as the number of variables goes to infinity, when the interaction satisfies certain homogeneity requirements. The moments of this distribution are simple combinations of cyclic integrals of the potential function. The existence of this limit gives information about the structure function of some systems in statistical mechanics having pair-summable interactions, even in the absence of a thermodynamic limit. The result is applied to several examples, including systems of two-dimensional point vortices.

Manin's quantum spaces and standard quantum mechanics

Manin's non-commutative coordinate algebra of quantum groups is shown to be identical, for unitary coordinates, with the conventional operator algebras of quantum mechanics. The deformation parameter q is a pure phase for unitary coordinates. When q is a root of unity, Manin's algebra becomes the matrix algebra of quantum mechanics for a discretized and finite phase space. Implications for quantum groups and the associated non-commutative differential calculus of Wess and Zumino are discussed.

Topological field theories

The outline of this review is as follows. In Section II a simple non-abelian C-S theory is described. It is the quantum mechanical theory of spin and it serves as an illustrative example for quantization of systems with finite phase space. In Section III general non-abelian Chern-Simons theories are discussed. It is shown how their canonical quantization on nontrivial Riemann surfaces leads to a system with finite number of degrees of freedom. The connection with conformal field theory is then explained using the abelian example. Implications to knot theory and quantum gravity are then discussed in Section IV. (orig.).

Von Neumann lattices and magnetic orbitals in finite phase space

It is proven that in the finite phase space of dimensionality M, the number of independent von Neumann states is M−r, where r is the number of distinct zeros of the kq function. When applied to magnetic orbitals, this leads to a linear dependence between them for each zero of the corresponding kq function. Different interesting identities follow from the linear dependence of von Neumann states.

Weyl-Heisenberg group and magnetic translations in finite phase space.

The symmetry of an electron in a magnetic field is given by the Weyl-Heisenberg group. When a periodic potential is added, the symmetry is lowered and is given by the magnetic-translation group. Irreducible representations and the group-subgroup relationship of these groups are investigated in finite phase space. Explicit formulas are derived for the splitting of magnetic states under periodic perturbations. A discussion is presented for extended and localized magnetic states. In particular, it is shown that despite the fact that the translation group and the magnetic translation group have the same number of elements in finite phase space, they nevertheless induce, in general, different numbers of localized orbitals (the latter fewer than the former).

Modeling adaptive biological systems

During the evolution of many systems found in nature, both the system composition and the interactions between components will vary. Equating the dimension with the number of different components, a system which adds or deletes components belongs to a class of dynamical systems with a finite dimensional phase space of variable dimension. We present two models of biochemical systems with a variable phase space, a model of autocatalytic reaction networks in the prebiotic soup and a model of the idiotypic network of the immune system. Each model contains characteristic meta-dynamical rules for constructing equations of motion from component properties. The simulation of each model occurs on two levels. On one level, the equations of motion are integrated to determine the state of each component. On a second level, algorithms which approximate physical processes in the real system are employed to change the equations of motion. Models with meta-dynamical rules possess several advantages for the study of evolving systems. First, there are no explicit fitness functions to determine how the components of the model rank in terms of survivability. The success of any component is a function of its relationship to the rest of the system. A second advantage is that since the phase space representation of the system is always finite but continually changing, we can explore a potentially infinite phase space which would otherwise be inaccessible with finite computer resources. Third, the enlarged capacity of systems with meta-dynamics for variation allows us to conduct true evolution experiments. The modeling methods presented here can be applied to many real biological systems. In the two studies we present, we are investigating two apparent properties of adaptive networks. With the simulation of the prebiotic soup, we are most interested in how a chemical reaction network might emerge from an initial state of relative disorder. With the study of the immune system, we study the self-regulation of the network including its ability to distinguish between species which are part of the network and those which are not.