Diffusion in a bistable potential. General inverse friction expansion

The derivation of the characteristic times and of the density probability distribution for the motion of a Brownian particle in a bistable potential at intermediate friction was, until now, essentially limited to low orders in the inverse frictionγ ^?1. On the other hand, at least for temperatures low with respect to the barrier height, the Kramers time, which is the lowest nonzero eigenvalue in the bistable potential problem, is known exactly. This paper presents a systematic approach for the determination of the solution of the Fokker-Planck equation in an arbitrary potential in the overdamped regime. This calculation includes anharmonicity corrections up to orderγ ^?5. One feature of this paper is to show that the problem is equivalent to replacing the original potentialφ(x) by a free energy which, for a velocity distribution at equilibrium, simply is \(\widetilde\phi \) =φ(x) ?k _BT ln[g(x)], where $$g(x) = \left\{ {{{m\gamma } \mathord{\left/ {\vphantom {{m\gamma } {[2\phi ''(x)]}}} \right. \kern-\nulldelimiterspace} {[2\phi ''(x)]}}} \right\}\left\{ {1 - [{{1 - 4\phi ''(x)} \mathord{\left/ {\vphantom {{1 - 4\phi ''(x)} {m\gamma ^2 }}} \right. \kern-\nulldelimiterspace} {m\gamma ^2 }}]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\}$$ For out-of-equilibrium velocity distribution an effective potential is also explicitly given. In every case the function g(x) plays a crucial role. This approach is then applied to the exact determination, in the low-temperature limit, of all the characteristic times and of the probability distribution in bistable potentials. Moreover, from the knowledge of the characteristic times and probability density distribution, it would be easy to determine the general and exact Suzuki scaling law for the relaxation from the instability point at intermediate friction.     

Power Series Representations for Bosonic Effective Actions

We develop a power series representation and estimates for an effective action of the form $$\ln\frac{\int e^{f(\phi,\psi)}d\mu(\phi)}{\int e^{f(\phi,0)}d\mu(\phi)}$$ Here, f(φ,ψ) is an analytic function of the real fields φ(x),ψ(x) indexed by x in a finite set X, and d μ(φ) is a compactly supported product measure. Such effective actions occur in the small field region for a renormalization group analysis. The customary way to analyze them is a cluster expansion, possibly preceded by a decoupling expansion. Using methods similar to a polymer expansion, we estimate the power series of the effective action without introducing an artificial decomposition of the underlying space into boxes.     

Search for rare radiative decays ofΦ-meson at VEPP-2M

Results of the search for rare radiative decay modes of the ?-meson performed with the Neutral Detector at the VEPP-2M collider are presented. For the first time upper limits for the branching ratios of the following decay modes have been placed at 90% confidence level: $$\begin{gathered} B(\phi \to \eta '\gamma )< 4 \cdot 10^{ - 4} , \hfill \\ B(\phi \to \pi ^0 \pi ^0 \gamma )< 10^{ - 3} , \hfill \\ B(\phi \to f_0 (975)\gamma )< 2 \cdot 10^{ - 3} , \hfill \\ B(\phi \to H\gamma )< 3 \cdot 10^{ - 4} , \hfill \\ \end{gathered} $$ whereH is a scalar (Higgs) boson with a mass 600 MeV<m _H <1000 MeV, the real measurement isB(φ→H γ)·B(H→2π⁰)<0.8·10^-4, the quoted result is model dependent, as explained in the text, $$\begin{gathered} B(\phi \to a\gamma ) \cdot B(a \to e^ + e^ - )< 5 \cdot 10^{ - 5} , \hfill \\ B(\phi \to a\gamma ) \cdot B(a \to \gamma \gamma )< 2 \cdot 10^{ - 3} , \hfill \\ \end{gathered} $$ wherea is a particle with a low mass and a short lifetime, $$B(\phi \to a\gamma )< 0.7 \cdot 10^{ - 5} ,$$ wherea is a particle with a low mass not observed in the detector.     

Demonstration of the origins of cooperativity within SU2×L 6 mapping over\left\{ {I\tilde H_\upsilon } \right\} Liouvillian carrier subspaces characterising the MQ-NMR cluster [A]6(L 6) and its\{ |kq\upsilon :[\tilde \lambda ] > > \} tensorial bases

The fundamental mappings over carrier subspace and substructures associated with \(\{ |kq\upsilon > > \} \) augmented spin algebras of Liouville space, and their mapping onto a subduced symmetry, are derived for [A]₆(L ₆) spin clusters within the combinatorial context of Rota-Cayley algebra over a field. Use of suitable lexical sets of combinatorialp-tuples (number partitions) over {|IM(M ₁?M _n )>}M, followed by the subsequent use ofL _n inner tensor product (ITP) algebra, allows the substructure of Liouville space to be derived. For SU2×L ₆ mapping over the simply-reducible \(\left\{ {I\tilde H_\upsilon } \right\}\) carrier subspaces, the \(D^k \left( {\tilde U} \right) \times \tilde \Gamma ^{\left[ {\tilde \lambda } \right]} \left( \upsilon \right)\) (L ₆) dual irreps, also arise as a consequence of the Liouville space recoupling termsv≡{k ₁?k _n } being distinct labels for \(\left\{ {I\tilde H_\upsilon } \right\}\) which are themselves amenible to combinatorial analysis within the concept of Rota-Cayley algebra. Hence, theL _n -induced symmetry aspects of multiquantum NMR density matrix formalisms and their dual \(\{ |kq\upsilon :[\tilde \lambda ] > > \} \) tensorial bases of spin cluster problems are derived and the nature of the cooperative, aspect between the individual symmetries comprising the duality is demonstrated, i.e. in the context of the operator bases of Liouville space. These practical arguments correlate, well with those based on an augmented boson pattern algebra derived from a Heisenburg algebra for superoperators, ?_±,?₀. An earlier, treatment of conventional Hilbert space SU2×L ₆ dualitycould only be realised in terms of standard SU2 boson algebra. Since the recoupling Rota-‘field’v for Liouville space is an explicit aspect of the dual mapping, a direct demonstration of cooperativity exists.     

Limit behavior of saturated approximations of nonlinear Schrödinger equation

We consider the solutionu _?(t) of the saturated nonlinear Schrödinger equation (1) $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} u + \varepsilon \left| u \right|^{q - 1} uandu(0,.) = \varphi (.)$$ where \(N \geqslant 2,\varepsilon > 0,1 + 4/N< q< (N + 2)/(N - 2),u:\mathbb{R} \times \mathbb{R}^N \to \mathbb{C},\varphi \) , ? is a radially symmetric function inH ¹(R ^N ). We assume that the solution of the limit equation is not globally defined in time. There is aT>0 such that \(\mathop {\lim }\limits_{t \to T} \left\| {u(t)} \right\|_{H^1 } = + \infty \) , whereu(t) is solution of (1) $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} uandu(0,.) = \varphi (.)$$ For ?>0 fixed,u _?(t) is defined for all time. We are interested in the limit behavior as ?→0 ofu _?(t) fort≥T. In the case where there is no loss of mass inu _? at infinity in a sense to be made precise, we describe the behavior ofu _? as ? goes to zero and we derive an existence result for a solution of (1) after the blow-up timeT in a certain sense. Nonlinear Schrödinger equation with supercritical exponents are also considered.     

On the Lieb-Thirring constantsL γ,1 for γ≧1/2

LetE _i(H) denote the negative eigenvalues of the one-dimensional Schrödinger operatorHu??u″?Vu,V≧0, onL ₂(∝). We prove the inequality (1) $$\mathop \sum \limits_i |E_i (H)|^{ \gamma } \leqq L_{\gamma ,1} \mathop \smallint \limits_\mathbb{R} V^{\gamma + 1/2} (x)dx,$$ for the “limit” case γ=1/2. This will imply improved estimates for the best constantsL _γ,1 in (1) as 1/2<γ<3/2.     

Radiative decays and quark content of f 0(980) and φ(1020)

The quark structure of φ(1020) and f ₀(980) is studied on the basis of data on the radiative decays φ(1020) → γπ ⁰, γη, γη′, γ a ₀(980), γ f ₀(980) and f ₀(980) → γγ. The partial widths are calculated under the assumption that all the mesons under consideration are $\bar qq$ states: φ(1020) is a dominantly $s\bar s$ state ( $\eta ' = n\bar nsin\theta + s\bar scos\theta $ component contributes not more than 1%); η, η′, and π ⁰ are standard $q\bar q$ states, $\eta = n\bar ncos\theta - s\bar ssin\theta $ and $\eta ' = n\bar nsin\theta + s\bar scos\theta $ with θ?37°; and f ₀(980) is a e5 meson with the flavor wave function $n\bar ncos\varphi + s\bar ssin\varphi $ . The transition φ → γπ ⁰ specifies the admixture of the $n\bar n$ component in the φ meson: it is on the order of 0.5%. We argue that this order of $n\bar n$ value does not contradict data on the decay φ(1020) → γ a ₀(980). The partial widths calculated for the decays φ → γη, γη? are in reasonable agreement with experimental data. The measured branching-ratio value Br(φ→γf ₀(980))=(3.4±0.4 _?0.5 ^+1.5 ×10^?4) requires 25°≤|?|≤90°. For the decay f ₀(980) → γγ, the agreement with data, Γ(f ₀(980) → γγ)=0.28 _?0.13 ^+0.09 keV, is attained at either ?=85°±8° or ?=?46°±8°. A simultaneous analysis of the decays φ(1020) → γ f ₀(980) and f ₀(980) → γγ favors the solution with the negative mixing angle of ?=?48°±6°, setting f ₀(980) very close to the flavor octet (? _octet=±54.7°).     

Krein formula with compensated singularities for the ND-mapping and the generalized Kirchhoff condition at the Neumann Schrödinger junction

The Neumann Schrödinger operator \(\mathcal{L}\) is considered on a thin 2D star-shaped junction, composed of a vertex domain Ω_int and a few semi-infinite straight leads ω _m , m = 1, 2, ..., M, of width δ, δ ? diam Ω_int, attached to Ω_int at Γ ? ?Ω_int. The potential of the Schrödinger operator l ^ω on the leads vanishes, hence there are only a finite number of eigenvalues of the Neumann Schrödinger operator L _int on Ω_int embedded into the open spectral branches of l ^ω with oscillating solutions χ _±(x, p) = \(e^{ \pm iK_ + x} e_m \) of l ^ω χ _± = p ² χ _±. The exponent of the open channels in the wires is

$K_ + (\lambda ) = p\sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = \sqrt \lambda P_ + $

, with constant e _m , on a relatively small essential spectral interval Δ ? [0, π ² δ ^?2). The scattering matrix of the junction is represented on Δ in terms of the ND mapping

$\mathcal{N} = \frac{{\partial P_ + \Psi }}{{\partial x}}(0,\lambda )\left| {_\Gamma \to P_ + \Psi _ + (0,\lambda )} \right|_\Gamma $

as

$S(\lambda ) = (ip\mathcal{N} + I_ + )^{ - 1} (ip\mathcal{N} - I_ + ), I_ + = \sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = P_ + $

. We derive an approximate formula for \(\mathcal{N}\) in terms of the Neumann-to-Dirichlet mapping \(\mathcal{N}_{\operatorname{int} } \) of L _int and the exponent K _? of the closed channels of l ^ω . If there is only one simple eigenvalue λ ₀ ∈ Δ, L _intφ0 = λ _0φ0 then, for a thin junction, \(\mathcal{N} \approx |\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} \) with

$\vec \phi _0 = P_ + \phi _0 = (\delta ^{ - 1} \int_{\Gamma _1 } {\phi _0 (\gamma )} d\gamma ,\delta ^{ - 1} \int_{\Gamma _2 } {\phi _0 (\gamma )} d\gamma , \ldots \delta ^{ - 1} \int_{\Gamma _M } {\phi _0 (\gamma )} d\gamma )$

and \(P_0 = \vec \phi _0 \rangle |\vec \phi _0 |^{ - 2} \langle \vec \phi _0 \),

$S(\lambda ) \approx \frac{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} - I_ + }}{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} + I_ + }} = :S_{appr} (\lambda )$

. The related boundary condition for the components P ₊Ψ(0) and P ₊Ψ′(0) of the scattering Ansatz in the open channel \(P_ + \Psi (0) = (\bar \Psi _1 ,\bar \Psi _2 , \ldots ,\bar \Psi _M ), P_ + \Psi '(0) = (\bar \Psi '_1 , \bar \Psi '_2 , \ldots , \bar \Psi '_M )\) includes the weighted continuity (1) of the scattering Ansatz Ψ at the vertex and the weighted balance of the currents (2), where

$\frac{{\bar \Psi _m }}{{\bar \phi _0^m }} = \frac{{\delta \sum\nolimits_{t = 1}^M { \bar \Psi _t \bar \phi _0^t } }}{{|\vec \phi _0 |^2 }} = \frac{{\bar \Psi _r }}{{\bar \phi _0^r }} = :\bar \Psi (0)/\bar \phi (0), 1 \leqslant m,r \leqslant M$

(1)

,

$\sum\limits_{m = 1}^M {\bar \Psi '_m } \bar \phi _0^m + \delta ^{ - 1} (\lambda - \lambda _0 )\bar \Psi /\bar \phi (0) = 0$

(1)

. Conditions (1) and (2) constitute the generalized Kirchhoff boundary condition at the vertex for the Schrödinger operator on a thin junction and remain valid for the corresponding 1D model. We compare this with the previous result by Kuchment and Zeng obtained by the variational technique for the Neumann Laplacian on a shrinking quantum network.

     

Commensurate-incommensurate phase transitions in one-dimensional chains

We consider one-dimensional systems of classical particles whose potential energy has the form: $$W_{\alpha ,\gamma } = \sum {[\alpha V(x_n )} + F(x_n - x_{n - 1} C\gamma )]$$ The limit of the Gibbs state as T→0 is described in terms of invariant measures of two-dimensional mappings which are constructed with the help ofW _{α, γ}. The dependence of these measures on parametersα, γ is investigated.     

Localization in the ground state of the ising model with a random transverse field

We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by $$H = - J\sum\limits_{\left\langle {x,y} \right\rangle } {\sigma _3 (x)\sigma _3 (y) - \sum\limits_x {h(x)\sigma _1 (x)} } $$ whereJ>0,x,y∈Z ^d, σ₁, σ₃ are the usual Pauli spin 1/2 matrices, andh={h(x),x∈Z ^d} are independent identically distributed random variables. We consider the ground state correlation function 〈σ₃(x)σ₃(y)〉 and prove:

1. Letd be arbitrary. For anym>0 andJ sufficiently small we have, for almost every choice of the random transverse fieldh and everyx∈Z ^d, that $$\left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle \leqq C_{x,h} e^{ - m\left| {x - y} \right|} $$ for ally∈Z ^d withC _{x h} <∞.
2. Letd≧2. IfJ is sufficiently large, then, for almost every choice of the random transverse fieldh, the model exhibits long range order, i.e., $$\mathop {\overline {\lim } }\limits_{\left| y \right| \to \infty } \left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle > 0$$ for anyx∈Z ^d.

     

Percival variational principle for invariant measures and commensurate-incommensurate phase transitions in one-dimensional chains

We prove for a one-dimensional system of classical particles with potential energy, $$U_{\alpha ,\gamma } = \sum\limits_n {\left[ {\alpha V(x_n ) + F(x_{n + 1} - x_n - \gamma )} \right]} $$ , the existence of such a smooth function γ(α), 0≦α≦α₀(ω) that the system with potential energyU _αγ(α) has the equilibrium state at the temperatureT=0. This is the incommensurate phase with the ratio of periods equal to the prescribed irrational number ω, badly approximated by rational ones. A simple geometric condition for the invariant curve of the corresponding dynamical system is established under which it is the support of the invariant measure minimizing Percival's energy functional.     

Some implications of inclusive electro- and neutrino production data

We systematically exploit the reported data on \(F_2^{\gamma p} ,F_2^{\gamma n} ,\sigma ^{vN} ,\sigma ^{\bar vN} ,\left\langle {xy} \right\rangle _{vN} ,\left\langle {xy} \right\rangle _{\bar vN} ,\left\langle {1 - y} \right\rangle _{vN} \) and \(\left\langle {1 - y} \right\rangle _{\bar vN} \) in order to test various versions of the quark parton model and to obtain further predictions.     

On Lévy (or stable) distributions and the Williams-Watts model of dielectric relaxation

This paper is concerned with the Lévy, or stable distribution function defined by the Fourier transform $$Q_\alpha \left( z \right) = \frac{1}{{2\pi }}\int {_{ - \infty }^\infty \exp \left( { - izu - \left| u \right|^\alpha } \right)du} with 0< \alpha \leqslant 2$$ Whenα=2 it becomes the Gauss distribution function and whenα=1, the Cauchy distribution. Whenα≠2 the distribution has a long inverse power tail $$Q_\alpha \left( z \right) \sim \frac{{\Gamma \left( {1 + \alpha } \right)\sin \tfrac{1}{2}\pi \alpha }}{{\pi \left| z \right|^{1 + \alpha } }}$$ In the regime of smallα, ifα¦logz¦?1, the distribution is mimicked by a log normal distribution. We have derived rapidly converging algorithms for the numerical calculation ofQ _α (z) for variousα in the range 0<α<1. The functionQ _α (z) appears naturally in the Williams-Watts model of dielectric relaxation. In that model one expresses the normalized dielectric parameter as $$ \in _n \left( \omega \right) \equiv \in '_n \left( \omega \right) - i \in ''_n \left( \omega \right) = - \int {_0^\infty e^{ - i\omega t} \left[ {{{d\phi \left( t \right)} \mathord{\left/ {\vphantom {{d\phi \left( t \right)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}} \right]} dt$$ with $$\phi \left( t \right) = \exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha $$ It has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form ofφ(t).ε″ _n (ω) is shown to be directly related toQ _α (z). It is also shown that if the Williams-Watts exponential is expressed as a weighted average of exponential relaxation functions $$\exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha = \int {_0^\infty } g\left( {\lambda , \alpha } \right)e^{ - \lambda t} dt$$ the weight functiong(λ, α) is expressible as a stable distribution. Some suggestions are made about physical models that might lead to the Williams-Watts form ofφ(t).     

Space tensors in general relativity III: The structural equations

A self-consistent theory of spatial differential forms over a pair (M,Γ)is proposed. The operators d(spatial exterior differentiation), d_T (temporal Lie derivative) andL (spatial Lie derivative) are defined, and their properties are discussed. These results are then applied to the study of the torsion and curvature tensor fields determined by an arbitrary spatial tensor analysis \((\tilde \nabla ,\tilde \nabla T)\) (M,Γ). The structural equations of \((\tilde \nabla ,\tilde \nabla T)\) and the corresponding spatial Bianchi identities are discussed. The special case \((\tilde \nabla ,\tilde \nabla T) = (\tilde \nabla *,\tilde \nabla T*)\) is examined in detail. The spatial resolution of the Riemann tensor of the manifold M is finally analysed; the resultingstructure of Eintein's equations over a pair (ν₄,Γ)is established. An application to the study of the problem of motion in terms of co-moving atlases is proposed.     

On the rate of quantum ergodicity I: Upper bounds

One problem in quantum ergodicity is to estimate the rate of decay of the sums $$S_k (\lambda ;A) = \frac{1}{{N(\lambda )}}\sum\limits_{\sqrt {\lambda _j } \leqq \lambda } {\left| {(A\varphi _j ,\varphi _j ) - \bar \sigma _A } \right|^k } $$ on a compact Riemannian manifold (M, g) with ergodic geodesic flow. Here, {λ _j ,? _j } are the spectral data of the Δ of(M, g), A is a 0-th order ψDO, $\bar \sigma _A $ is the (Liouville) average of its principal symbol and $N(\lambda ) = \# \{ j:\sqrt {\lambda _j } \leqq \lambda \} $ . ThatS _k (λ;A)=o(1) is proved in [S, Z.1, CV.1]. Our purpose here is to show thatS _k (λ;A)=O((logλ) ^?k/2 ) on a manifold of (possibly variable) negative curvature. The main new ingredient is the central limit theorem for geodesic flows on such spaces ([R, Si]).     

On the free energy of the hopfield model

The general theory of inhomogeneous mean-field systems of Raggio and Werner provides a variational expression for the (almost sure) limiting free energy density of the Hopfield model $$H_{N,p}^{\{ \xi \} } (S) = - \frac{1}{{2N}}\sum\limits_{i,j = 1}^N {\sum\limits_{\mu = 1}^N {\xi _i^\mu \xi _j^\mu S_i S_j } } $$ for Ising spinsS _i andp random patterns ξ^μ=(ξ ₁ ^μ ,ξ ₂ ^μ ,...,ξ _N ^μ ) under the assumption that $$\mathop {\lim }\limits_{N \to \gamma } N^{ - 1} \sum\limits_{i = 1}^N {\delta _{\xi _i } = \lambda ,} \xi _i = (\xi _i^1 ,\xi _i^2 ,...,\xi _i^p )$$ exists (almost surely) in the space of probability measures overp copies of {?1, 1}. Including an “external field” term ?ξ _μ ^p h^μμξ _i=1 ^N ξ _i ^μ S_i, we give a number of general properties of the free-energy density and compute it for (a)p=2 in general and (b)p arbitrary when λ is uniform and at most the two componentsh ^μ1 andh ^μ2 are nonzero, obtaining the (almost sure) formula $$f(\beta ,h) = \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } + h^{\mu _2 } ) + \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } - h^{\mu _2 } )$$ for the free energy, wheref ^cw denotes the limiting free energy density of the Curie-Weiss model with unit interaction constant. In both cases, we obtain explicit formulas for the limiting (almost sure) values of the so-called overlap parameters $$m_N^\mu (\beta ,h) = N^{ - 1} \sum\limits_{i = 1}^N {\xi _i^\mu \left\langle {S_i } \right\rangle } $$ in terms of the Curie-Weiss magnetizations. For the general i.i.d. case with Prob {ξ _i ^μ =±1}=(1/2)±?, we obtain the lower bound 1+4?²(p?1) for the temperatureT _c separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero. This lower bound is exact forp=2, or ε=0, or ε=±1/2. Forp=2 we identify an intermediate temperature region between T_*=1?4?² and T_c=1+4?² where the overlap vector is homogeneous (i.e., all its components are equal) and nonzero.T _* marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct. We conjecture that the homogeneous nonzero regime exists forp≥3 and that T_*=max{1?4?²(p?1),0}.     

η , \eta{^\prime} mixing angle and \eta{^\prime} gluonium content extraction from the KLOE Rφ measurement⋆

In this report the extraction of the η , $ \eta{^\prime}$ mixing angle and of the $ \eta{^\prime}$ gluonium content from the R _φ = Br(φ(1020) → $ \eta{^\prime}$ γ)/Br(φ(1020) → ηγ) is updated. The $ \eta{^\prime}$ gluonium content is estimated by fitting R _φ , together, with other decay branching ratios. The extracted parameters are: Z ² _G = 0.12±0.04 and ?_P = (40.4±0.9)^° .