Overdamped motion of Brownian particles in tilted piecewise linear periodic potentials is considered. Explicit algebraic expressions for the diffusion coefficient, current, and coherence level of Brownian transport are derived. Their dependencies on temperature, tilting force, and the shape of the potential are analyzed. The necessary and sufficient conditions for the nonmonotonic behavior of the diffusion coefficient as a function of temperature are determined. The diffusion coefficient and coherence level are found to be extremely sensitive to the asymmetry of the potential. It is established that at the values of the external force, for which the enhancement of diffusion is most rapid, the level of coherence has a wide plateau at low temperatures with the value of the Péclet factor 2. An interpretation of the amplification of diffusion in comparison with free thermal diffusion in terms of probability distribution is proposed.
We present an alternative method for constructing the exact and approximate solutions of electromagnetic wave equations whose source terms are arbitrary order multipoles on a curved spacetime. The developed method is based on the higher-order Green's functions for wave equations which are defined as distributions that satisfy wave equations with the corresponding order covariant derivatives of the Dirac delta function as the source terms. The constructed solution is applied to the study of various geometric effects on the generation and propagation of electromagnetic wave tails to first order in the Riemann tensor. Generally the received radiation tail occurs after a time delay which represents geometrical backscattering by the central gravitational source. It is shown that for an arbitrary weak gravitational field it is valid that the truly nonlocal wave-propagation correction (the tail term) has a universal form which is independent of multipole structure of the gravitational source. In a particular case when the electromagnetic radiation pulse is generated by the wave source during a finite time interval, the structure of the wave tail at the time after the direct pulse has passed the gravitational source is in the first approximation independent of the higher multipole moments of the source of
Higher-order fundamental solutions are defined as the distributions that satisfy the wave equations with inhomogeneous terms which are point distributions of the corresponding order. Starting from the Hadamard fundamental solution, the construction of the local higherorder fundamental solutions of the covariant scalar wave equation on a causal domain is considered. A simple recurrent algorithm for calculating such solutions is found.
The paper studies the overdamped motion of Brownian particles in a tilted sawtooth potential. The dependencies of the diffusion coefficient and coherence level of Brownian transport on temperature, tilting force, and the shape of the potential are analyzed. It is demonstrated that at low temperatures the coherence level of Brownian transport stabilizes in the extensive domain of the tilting force where the value of the Péclet factor is P e = 2. This domain coincides with the one where the enhancement of the diffusion coefficient versus the tilting force is the most rapid. The necessary and sufficient conditions for the non-monotonic behaviour of the diffusion coefficient as a function of temperature are found. The effect of the acceleration of diffusion by bias and temperature is demonstrated to be very sensitive to the value of the asymmetry parameter of the potential.
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