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We consider a Fokker–Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear. In this paper we exploit an interpretation of this equation as a Wasserstein gradient flow of a free energy F on a time-constrained manifold. First, we prove existence of solutions by passing to the limit in an implicit Euler scheme obtained by minimizing h F ( ϱ ) + W 2 2 ( ϱ 0 , ϱ ) among all ϱ satisfying the constraint for some ϱ 0 and time-step h > 0 . Second, we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant. The proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint. The rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic Sobolev inequalities. [ABSTRACT FROM AUTHOR]

This paper proposes a Monte Carlo (MC) based angular distribution estimation method of multiply scattered photons for underwater imaging. This method targets on turbid waters. Our method is based on applying typical Monte Carlo ideas to the present problem by combining all the points on a spherical surface. The proposed method is validated with the numerical solution of the radiative transfer equation (RTE). The simulation results based on typical optical parameters of turbid waters show that the proposed method is effective in terms of computational speed and sensitivity. [ABSTRACT FROM AUTHOR]

Analysis of the Stark–Einstein law as it applies to the retinal molecule, which is part of the rhodopsin molecule within the rod cells of the retina, reveals that it may provide the solution to the measurement problem from an animate perspective. That it represents a natural boundary where the Schrödinger equation or wave function automatically goes from linear to nonlinear while remaining in a deterministic state. It will be possible in the near future to subject this theory to empirical tests as has been previously proposed. This analysis provides a contrast to the many decades well studied and debated inanimate measurement problem and would represent an addition to the Stark–Einstein law involving information carried by the photon. [ABSTRACT FROM AUTHOR]