Impact of interaction forces in first order many-agent systems for swarm manufacturing

Figure 1.  Left: steady state distribution for the 1D problem with $ \delta = 0.5 $ and $ x_0 = 0 $. Right: steady state distribution for the 2D problem with $ \delta = \sqrt{\pi^{-1}} $ and $ \mathbf{x}_0 = (0, 0) $. In both cases we fixed several values of $ m_2>0 $ and $ \delta>0 $

Figure 2.  Test 1a. Evolution of the reconstructed distribution functions of the particles' systems (26)-(27) given by $ f^{N}_1(x) $, $ f^{N}_2(x) $ in the case $ P\equiv1 $ at different times $ t = 1, 5, 10, 20 $ and for $ \lambda = 0.2, 0.8 $. We considered an Euler-Maruyama scheme with $ N = 10^5 $ and $ \Delta t = 10^{-2} $, the histograms have been obtained in the interval $ [-5, 5] $ with $ N_x = 101 $ gridpoints and the target domain is $ D = \{x \in \mathbb R: |x-x_0|\le \frac{1}{2}\} $, $ x_0 = 0 $ and $ m_1, \sigma^2>0 $ solution to the system (5) in such a way $ m_2 = 0.8 $ and $ \delta = \frac{1}{2} $

Figure 3.  Test 1a. Comparison between the reconstructed distribution functions $ f_1^{N}(x, t) $ (top row) and $ f_2^{N}(x, t) $ (bottom row) for increasing $ N = 10^{4} $, $ N = 10^5 $ of the particles' systems (26)-(27) in the case $ P\equiv 1 $, with the numerical solution of Fokker-Planck models defined in (1)-(16) and denoted by $ f_1(x, t) $, $ f_2(x, t) $. We considered $ \lambda = 0.2 $, a discretization of the interval $ [-5, 5] $ obtained with $ N_x = 101 $ gridpoints. The target domain is $ D = \{x \in \mathbb R: |x-x_0|\le \frac{1}{2}\} $, $ x_0 = 0 $ and $ m_1, \sigma^2>0 $ solution to (5) with $ m_2 = 0.8 $ and $ \delta = \frac{1}{2} $. Initial condition defined in (28)

Figure 4.  Test 1b. Comparison between the reconstructed distribution function $ f^{N}(x, t) $ for increasing $ N = 10^{4} $, $ N = 10^5 $ of the particles' systems (26) in the case $ P(x, y) $ in (29), with the numerical solution of Fokker-Planck models defined in (1) and denoted by $ f(x, t) $. We considered $ \lambda = 0.2 $, a discretization of the interval $ [-5, 5] $ obtained with $ N_x = 101 $ gridpoints. The target domain is $ D = \{x \in \mathbb R: |x-x_0|\le \frac{1}{2}\} $, $ x_0 = 0 $ and $ \sigma^2 = 0.2 $ of Figure 3 and $ \delta = \frac{1}{2} $. Initial condition defined in (28)

Figure 5.  Test 1b. Evolution of the particles' mean position. We denote with $ \bar u_1(t) $ the mean position obtained from (26), with $ \bar u_2(t) $ the one obtained from (27) and with $ \bar u_3(t) $ the one obtained with space dependent interactions $ P(x, y) $ (29) in (26). In all the tests we considered $ N = 10^4 $ particles evolving over the time interval $ [0, 10] $ with $ \Delta t = 10^{-2} $. The target domain is $ D = \{x \in \mathbb R: |x-x_0|\le \frac{1}{2}\} $ and we fixed the relevant parameters of Figure 4. We considered the case $ \lambda = 0.2 $ (left) and $ \lambda = 0.8 $ (right). Initial distribution defined in (28)

Figure 6.  Test 1c. Top row: evolution of the particles' distribution $ f^N( \mathbf{x}, t) $ at times $ t = 0, 1, 10 $ obtained from (26) with $ P\equiv1 $. Second row: evolution o the numerical solution of the Fokker-Planck equation (16) over the same grid. Bottom row: evolution of the marginal densities of $ f^N( \mathbf{x}, t) $ and $ f( \mathbf{x}, t) $. We considered as target domain $ D = \{ \mathbf{x} \in \mathbb R^2: | \mathbf{x}- \mathbf{x}_0|\le 1\} $, $ \mathbf{x}_0 = (0, 0) $, $ \sigma^2 = 0.2 $ and $ \lambda = 0.2 $, the nonconstant diffusion function $ \kappa( \mathbf{x}, \tilde {\mathbf{x}}_0) $ has been defined in (15). We introduced a grid of $ N_x = 81 $ gridpoints in $ [-5, 5] $, time discretization of $ [0, 10] $ with $ \Delta t = 10^{-2} $. Initial condition given in (30)

Figure 7.  Test 1c. Top row: evolution of the particles' distribution $ f^N( \mathbf{x}, t) $ at times $ t = 0, 1, 10 $ obtained from (26) with $ P( \mathbf{x}, \mathbf y) $ in (29) and $ N = 10^4 $ particles. Second row: evolution o the numerical solution of the Fokker-Planck equation (1) over the same grid. Bottom row: evolution of the marginal densities of $ f^N( \mathbf{x}, t) $ and $ f( \mathbf{x}, t) $. We considered as target domain $ D = \{ \mathbf{x} \in \mathbb R^2: | \mathbf{x}- \mathbf{x}_0|\le 1\} $, $ \mathbf{x}_0 = (0, 0) $, $ \sigma^2 = 0.2 $ and $ \lambda = 0.2 $. We introduced a grid of $ N_x = 81 $ gridpoints in $ [-5, 5] $, time discretization of $ [0, 10] $ with $ \Delta t = 10^{-2} $. Initial condition given in (30)

Figure 8.  Test 2. Left: evolution of the relative entropy functional $ H(f|f^\infty)(t) $ obtained from (16), $ d = 1 $, and analytical equilibrium $ f^\infty( \mathbf{x}) $ defined in (4). Right: evolution of the entropy functional $ H(f|f^ \rm{ref}_\lambda)(t) $ for the Fokker-Planck equation (1), $ d = 1 $, with space-dependent $ P(x, y) $ defined in (29). The reference solution $ f^ \rm{ref}_\lambda(x, T) $ have been obtained for a discrertization of $ [-5, 5] $ with $ N_x = 801 $ gridpoints and $ T = 50 $. In both cases $ \Delta t = \frac{\Delta x^2}{10} $ and the initial distribution is (33)

Figure 9.  Test 2. Left: evolution of the relative entropy functional $ H(f|f^\infty)(t) $ obtained from (16), $ d = 2 $, and analytical equilibrium $ f^\infty( \mathbf{x}) $ defined in (4). Right: evolution of the entropy functional $ H(f|f^ \rm{ref}_\lambda)(t) $ for the Fokker-Planck equation (1), $ d = 1 $, with space-dependent $ P( \mathbf{x}, \mathbf{y}) $ defined in (29). The reference solution $ f^ \rm{ref}_\lambda( \mathbf{x}, T) $ have been obtained for a discrertization of $ [-5, 5] \times [-5, 5] $ with $ N_x = 81 $ gridpoints in each space direction and $ T = 50 $. In both cases $ \Delta t = \frac{\Delta x^2}{10} $ and the initial distribution is (33)