On Fokker-Planck equations with In- and Outflow of Mass

Figure 1.  Sketch of a neuron (1) cell nucleus, 2) dendrite and 3) axon)

Figure 2.  Sketch of the geometry of boundary in- and outflux in 2D with a possible density

Figure 3.  Evolution over Time of the Particle Concentration: The time evolution of $ \rho $ solving (3) in comparison to the calculated stationary solution (21) for $ \alpha = 1, \beta = 0.9 $ and initial particle concentration $ \rho(x) = -0.1x +1.2 $. (a) The initial concentration at $ t = 0 $, (b) strong influence of the boundary terms at $ t = 0.05 $, (c) strong influence of the drift term and the diffusion at $ t = 1.5 $, (d) equilibrium state at $ t = 9 $

Figure 4.  Relative Entropy: The relative entropy (15) for the one dimensional version of (3) for the initial particle concentration $ \rho(x) = -0.1x +1.2 $. (a) Natural logarithm of the relative entropy for $ \alpha = 1, \; \beta = 1 $ and variable scaling factor $ \gamma $ of the potential term, (b) relation between the scaling factor $ \gamma $ and the corresponding slope of the logarithm of the relative entropy

Figure 5.  Mass Evolution: Two examples for non monotone mass evolution for $ \alpha = 1 $ and $ \beta = 0.9 $

Figure 6.  Sketch of the geometry of uniform spatial in- and outflux in 2D

Figure 7.  Evolution over Time of the Particle Concentration: Solution of (26) in comparison to the calculated stationary solution $ \frac{\alpha}{\beta}e^{V(x)} $ for $ \alpha = 1, \beta = 0.9 $, initial particle concentration $ \rho(x) = -0.1x+1.2 $. (a) The initial particle concentration at $ t = 0 $, (b) + (c) particle concentration at $ t = 0.05 $ and $ t = 1.5 $, (d) equilibrium state at $ t = 20 $

Figure 8.  Relative Entropy: Natural logarithm of the relative entropy(30) for the one dimensional version of (26) for $ \alpha = 1, \beta = 0.9 $, initial concentration $ \rho(x) = 0.1x +1 $

Figure 9.  Evolution over Time of the Particle Concentration: $ \rho(x,t) $ solving (35) in comparison to the calculated solution (44) for $ \alpha = 1, \beta = 0.9 $, initial particle concentration $ \rho_0(x) = -(x-0.5)^2+1 $. (a) The initial concentration in comparison with the calculated stationary solution at $ t = 0 $, (b) diffusion strongly visible $ t = 0.05 $, (c) transport term strongly visible $ t = 0.35 $, (d) equilibrium state at $ t = 3.7 $

Figure 10.  Relative Entropy: The logarithm of the relative entropy for the one dimensional version of (35) for $ \alpha = 1, \beta = 0.9 $ and $ \rho_0(x) = -(x-0.5)^2+1 $