Fractional Diffusion Equation for a Power-Law-Truncated Levy Process

Truncated Levy flights are stochastic processes which display a crossover from a heavy-tailed Levy behavior to a faster decaying probability distribution function (pdf). Putting less weight on long flights overcomes the divergence of the Levy distribution second moment. We introduce a fractional generalization of the diffusion equation, whose solution defines a process in which a Levy flight of exponent alpha is truncated by a power-law of exponent 5 - alpha. A closed form for the characteristic function of the process is derived. The pdf of the displacement slowly converges to a Gaussian in its central part showing however a power law far tail. Possible applications are discussed

Fluctuations, Higher Order Anharmonicities, and Landau Expansion for Barium Titanate

Correct phenomenological description of ferroelectric phase transitions in barium titanate requires accounting for eighth-order terms in the free energy expansion, in addition to the conventional sixth-order contributions. Another unusual feature of BaTiO_3 crystal is that the coefficients B_1 and B_2 of the terms P_x^4 and P_x^2*P_y^2 in the Landau expansion depend on the temperature. It is shown that the temperature dependence of B_1 and B_2 may be caused by thermal fluctuations of the polarization, provided the fourth-order anharmonicity is anomalously small, i. e. the nonlinearity of P^4 type and higher-order ones play comparable roles. Non-singular (non-critical) fluctuation contributions to B_1 and B_2 are calculated in the first approximation in sixth-order and eighth-order anharmonic constants. Both contributions increase with the temperature, which is in agreement with available experimental data. Moreover, the theory makes it possible to estimate, without any additional assumptions, the ratio of fluctuation (temperature dependent) contributions to coefficients B_1 and B_2. Theoretical value of B_1/B_2 appears to be close to that given by experiments.Comment: 5 pages, 1 figur

The 30-GHz monolithic receive module

Key requirements for a 30 GHz GaAs monolithic receive module for spaceborne communication antenna feed array applications include an overall receive module noise figure of 5 dB, a 30 dB RF to IF gain with six levels of intermediate gain control, a five-bit phase shifter, and a maximum power consumption of 250 mW. The RF designs for each of the four submodules (low noise amplifier, some gain control, phase shifter, and RF to IF sub-module) are presented. Except for the phase shifter, high frequency, low noise FETs with sub-half micron gate lengths are employed in the submodules. For the gain control, a two stage dual gate FET amplifier is used. The phase shifter is of the passive switched line type and consists of 5-bits. It uses relatively large gate width FETs (with zero drain to source bias) as the switching elements. A 20 GHz local oscillator buffer amplifier, a FET compatible balanced mixer, and a 5-8 GHz IF amplifier constitute the RF/IF sub-module. Phase shifter fabrication using ion implantation and a self-aligned gate technique is described. Preliminary RF results obtained on such phase shifters are included

From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit to anomalous diffusion. We consider here the case of subdiffusive processes, which are semi-Markovian and correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to know fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power-law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time

Towards deterministic equations for Levy walks: the fractional material derivative

Levy walks are random processes with an underlying spatiotemporal coupling. This coupling penalizes long jumps, and therefore Levy walks give a proper stochastic description for a particle's motion with broad jump length distribution. We derive a generalized dynamical formulation for Levy walks in which the fractional equivalent of the material derivative occurs. Our approach will be useful for the dynamical formulation of Levy walks in an external force field or in phase space for which the description in terms of the continuous time random walk or its corresponding generalized master equation are less well suited