Approximation Of Small Fluctuations Research Articles

In Ukrainian

Brownian motors belong to the class of nanoscale devices that use the thermal noise of the environment as one of the necessary components in the mechanism of their operation. Today, there are a lot of practical implementations of such nanomachines, both inorganic, fairly simple mechanisms produced artificially, and more complex ones created from separate biological components available at the cellular level. One of the options for implementing the mechanism of straightening the chaotic thermal noise of the environment into unidirectional motion is the presence of a motor particle in the field of action of an asymmetric periodic stationary potential, which undergoes certain small disturbances (fluctuations) periodically over time. To describe such asymmetric one-dimensional structures (for example, dipole chains or fibers of the cytoskeleton) in the theory of Brownian motors, two model potentials are most often used: piecewise linear sawtooth and double sinusoidal. In this work, within the framework of the approximation of small fluctuations, a model of a pulsating Brownian motor with a stationary double sinusoidal potential and a disturbing small harmonic signal is considered. A new method of parametrization of such a problem is proposed, which allows to separate the contributions from various factors affecting the operation of the ratchet, and the numerical procedure for calculating the average speed of the directional movement of nanoparticles for the selected type of model potentials is specified. A number of numerical dependences of the average speed on the main parameters of the system were obtained. Peculiarities of the behavior of the motor as dependent on the parameter responsible for asymmetry and the number of potential wells on the spatial period of the stationary potential have been investigated. It is shown that the direction of the generated flux of nanoparticles depends not only on the phase shift between the stationary and fluctuating components of the potential, but also on the temperature of the system and the frequency of fluctuations, i.e., a possibility of temperature-frequency control of the direction of movement in the considered model has been found. Diagrams have been constructed that allow you to choose the ratio between the parameters of the nanomotor to create a flux of particles in the desired direction.

The influence of atomic collisions on the spectrum of light scattered from an f-deformed Bose–Einstein condensate

In this paper, we investigate the spectrum of light scattered from a Bose–Einstein condensate (BEC) in the framework of an f-deformed boson model. We use an f-deformed quantum model in which Gardiner's phonon operators for the BEC are deformed by an operator-valued function, , of the particle-number operator . We also consider the collisions between the atoms as a special kind of f-deformation where the collision rate κ is regarded as the corresponding deformation parameter. By applying the small fluctuation approximation, we obtain the spectrum of light scattered from the f-deformed BEC. By analysing the scattering spectrum we find that by increasing the values of the deformation parameters κ and (N is the total number of condensate atoms) the spectrum shows deviation from the spectrum associated with the non-deformed Bose–Einstein condensate.

Squeezing spectra of the output field for a high-density exciton laser

The effect of the non-linear interaction between high density excitons is investigated. We use the Fokker–Planck equation and the corresponding stochastic differential equation to study the composite system of cavity field and exciton. The small-fluctuation approximation is made to obtain the quadrature squeezing spectra of the output light field. The squeezing conditions for each quadrature component of the output field are given.

Tunneling splittings in model 2D potentials. II. V(X, V) = λ(X2 − X02)2 − CX2(Y − Y0) + ½Ω2(Y − Y0 + CX02/Ω2)2 − C2X04/2Ω2

Abstract We propose a method for calculating semiclassical wave functions of the vibrationally excited states in model 2D potentials with two degenerate minima. The procedure includes two steps. First, the approximate solution of the Schrodinger equation is found variationally in the vicinity of one of the minima. This solution is made separable along effective vibrations, other than the actual anharmonic ones. The parameters of the effective vibrations depend on quantum numbers of a given stationary state. Second, the Hamilton—Jacobi and transport equations are solved in the classically forbidden region in the approximation of small fluctuations around the extreme tunneling path, and the “tunneling” WKB wave function is constructed. Due to the freedom in decomposition of the total energy between the WKB equations, this wave function is extended to the potential minimum, where its form coincides with the variational one. This enables one to correctly normalize the WKB wave function avoiding matching at the caustics. Application of the proposed method to calculating tunneling splittings of the vibrationally excited states in the “gated” and “squeezed” potentials with different sets of parameters (corresponding, in particular, to hydrogen transfer in malonaldehyde molecule and puckering of cyclopentanone molecule) gives results that agree well with the quantum mechanical results obtained by diagonalization of Hamiltonian matrices.

Time dependent photon transport in a three dimensional interstellar cloud with stochastic clumps

We consider time dependent photon transport in a three dimensional interstellar cloud which occupies a three dimensional regionV. One or more clumps of given shapes are present withinV and their positions are determined by a suitable set of stochastic variables. Iff is the photon number density in the cloud or in the clumps, then our mathematical model leads to two coupled initial value problems for the average photon density over the stochastic variables 〈f〉 and forf* =f -〈f〉. By using the theory of semigroups, we prove existence and uniqueness of a strongly continuous solution and examine the small fluctuation approximation of such a solution.

Quantum Statistics of the Optical Parametric Process for Arbitrary Initial States

Abstract Photon statistics and non-classical behaviour of light produced in the optical parametric process are studied in the approximation of small fluctuations around a stationary point. The depletion of the pump mode is included. An initial generalized superposition of a coherent state and noise as well as the influence of reservoir noise and internal losses are considered.

Statistical analysis of a circular plate on a random winkler foundation

Statistical analysis of the problem of a circular plate on a random Winkler support is presented using the small fluctuation approximation technique. In particular, we consider the specific case of an infinite circular plate loaded by a uniform circumferential load. The explicit statistical solutions are obtained for deflections, moments and shear forces at every point along the circular plate. The statistical information about the relative deflection is given. All these statistical functions are required for constructing the probability density function of deflections, relative deflections, moments and shear forces. The structural reliability analysis of the circular plate can be performed based on these probability density functions. In general, the power spectral density of the stochastic field for modulus of subgrade reaction is not usually available and hence the upper bounds of the variance and standard deviation of the deflection at the specified points of the circular plate for engineering use are proposed.

Forward and Backward Four-wave Mixing of Non-classical Light with Pump Depletion

Abstract Photon statistics and non-classical behaviour of forward and backward four-wave mixing are examined including the coupling of modes in the approximation of small fluctuations around a stationary point, which is analytically determined and conditions for its stability are obtained. The depletion of pump modes is involved and effects of nonlinear dynamics, non-classical behaviour of input radiation, external noise and losses are included.

Quantum Statistics of Stimulated Raman and Hyper-Raman Scattering of Squeezed Light with Pump Depletion

Abstract Using the approximation of small fluctuations around a stationary solution, photon statistics and quadrature variances are derived in stimulated Raman and hyper-Raman scattering including the coupling of pumping, Stokes, anti-Stokes and phonon modes. The depletion of pump light is included. Compared to Raman scattering, additional regimes to generate anti-bunched, sub-Poisson and squeezed light were found caused by self-interaction of the pump mode. It was possible to include effects of nonlinear dynamics, initial squeezing of radiation and phonon system, external noise and losses.

Stimulated Raman Scattering of Squeezed Light with Pump Depletion

Abstract Using the approximation of small fluctuations around a stationary point, photon statistics and squeezing of vacuum fluctuations are derived in stimulated Raman scattering including the coupling of photon-photon and photon-phonon modes. The depletion of pump light is described and it is shown that a regime of inversion of the process is related to an increase in quantum fluctuations, which are finally reduced again, more in the anti-Stokes mode than in the Stokes mode. If the anti-Stokes coupling is stronger than the Stokes coupling, then an oscillating behaviour of the statistical characteristics occurs and substantial squeezing can be found in photon-phonon modes.

Small fluctuation solution of infinite beam on random Winkler foundation

An analysis of an infinite beam on random Winkler foundation is presented. The solution is based on the small fluctuation approximation which is similar to a first order perturbation procedure. The displacement of the beam is found to be a nonhomogeneous random function with a space dependent spectrum. Explicit expression is presented for the spatial distribution of the standard deviation of the displacement along the beam. Statistical characterization of the small fluctuation approximation makes it possible to derive the spatial distribution of the error introduced by this approximation. It is shown that this error increases exponentially near the zeros of the deterministic solution, but the range of x values where the error exceeds an acceptable value is quite limited. If it is assumed that such local violation of the small fluctuation approximation do not have a significant effect on the solution in the vicinity of the loading point, then it is possible to establish the zone of validity (in the x space) of this approximation. Based on this argument, it is shown that the small fluctuation approximation is valid in the zone of major engineering interest, i.e. from the loading point, down to the point where the deterministic displacement reduces to approximately 20% of its maximum value. In the random variable limit when the autocorrelation distance is very large in comparison with the characteristic length of the beam support system, the present procedure yields results that are identical with the first order Cornell approximation. In general the error introduced by the small fluctuation assumption is less than or equal to the error associated with the first order Cornell approximation; this error decreases with the autocorrelation distance of the coefficient of subgrade reaction. In the zero correlation limit the displacement become deterministic and the error introduced by the small fluctuation approximation vanishes.

Spatial statistics of pulsed partly coherent radiation in a medium with a thermal nonlinearity

The approximation of small fluctuations of the optical field is used to solve the transient problem of conversion of the spatial statistics of partly coherent radiation in a randomly inhomogeneous weakly absorbing medium. Pulsed radiation is considered and it is assumed that the pulse duration does not exceed the lifetime of inhomogeneities. Equations derived for the coherence functions of field perturbations are used to study in detail the dynamics of the statistical characteristics of radiation, such as the correlation radius, variance of the fluctuations, correlation functions of the amplitudes, and phase of the field perturbations.