Overdamped dynamics of particles with repulsive power-law interactions

André A. Moreira, César M. Vieira, Humberto A. Carmona, José S. Andrade, Jr., and Constantino Tsallis

Phys. Rev. E 98, 032138 – Published 27 September 2018

Abstract

We investigate the dynamics of overdamped $D$ -dimensional systems of particles repulsively interacting through short-ranged power-law potentials, $V (r) \sim r^{- λ} (λ / D > 1)$ . We show that such systems obey a nonlinear diffusion equation, and that their stationary state extremizes a $q$ -generalized nonadditive entropy. Here we focus on the dynamical evolution of these systems. Our first-principle $D = 1, 2$ many-body numerical simulations (based on Newton's law) confirm the predictions obtained from the time-dependent solution of the nonlinear diffusion equation and show that the one-particle space distribution $P (x, t)$ appears to follow a compact-support $q$ -Gaussian form, with $q = 1 - λ / D$ . We also calculate the velocity distributions $P (v_{x}, t)$ , and, interestingly enough, they follow the same $q$ -Gaussian form (apparently precisely for $D = 1$ , and nearly so for $D = 2$ ). The satisfactory match between the continuum description and the molecular dynamics simulations in a more general, time-dependent framework neatly confirms the idea that the present dissipative systems indeed represent suitable applications of the $q$ -generalized thermostatistical theory.

Received 4 May 2018

DOI:https://doi.org/10.1103/PhysRevE.98.032138

Physics Subject Headings (PhySH)

Classical statistical mechanics

Continuum particle models Dynamical systems

Condensed Matter, Materials & Applied PhysicsNonlinear DynamicsStatistical Physics & ThermodynamicsGeneral PhysicsPolymers & Soft Matter

Authors & Affiliations

André A. Moreira^1,2, César M. Vieira^1,3, Humberto A. Carmona¹, José S. Andrade, Jr.^1,2, and Constantino Tsallis^4,2,5,6

¹Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Brazil
²National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
³Instituto Federal de Educação, Ciência e Tecnologia do Ceará, 62580-000 Acaraú, Brazil
⁴Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
⁵Santa Fe Institute, 1399 Hyde Park Road, New Mexico 87501, USA
⁶Complexity Science Hub Vienna, Josefstadter Strasse 39, 1080 Vienna, Austria

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Issue

Vol. 98, Iss. 3 — September 2018

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Images

Figure 1
Snapshots of the configuration of the system in three different moments of the dynamics: (a) $t = 1.0$ , (b) $t = 5.88$ , and (c) $t = 550$ . Due to the symmetry we show only the $x > 0$ half of the system. The particles start in a very narrow region and invade the system as time goes, eventually reaching an equilibrium position. This system comprises $N = 500$ particles, interacting through a potential $V_{i j} = ε {(r_{i j} / σ)}^{- λ}$ , with $λ = 4$ , in a cell of lateral length $L_{y} = 20 σ$ , and confined by an external force $- k x$ with $k = 1 \times 10^{- 6} ε / σ^{2}$ .
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Figure 2
Curves of $f (t)$ for $D = 1$ ( $λ = 2$ and 3) and $D = 2$ ( $λ = 4$ and 6) obtained from Eq. (11) and its derivative $\dot{f} (t)$ . In our numerical simulations, we start with all particles confined in a narrow stripe, leading us to use $t_{0} = 0$ in Eq. (11). The case $D = 1$ corresponds to $N = 3600$ particles with confining potential strength $k = 3.2 \times 10^{- 3} ε / σ^{2}$ . The case $D = 2$ corresponds to $N = 4000$ particles, with confining potential strength $k = 1 \times 10^{- 3} ε / σ^{2}$ , in a cell with transverse size $L_{y} = 20 σ$ .
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Figure 3
Distributions of scaled positions and velocities at different moments of the dynamics for the one-dimensional case ( $D = 1$ ). These results concern $N = 3600$ particles interacting through the power-law potential, $V = ε {(r / σ)}^{- λ}$ , for $λ = 2$ ( $q = - 1$ ) and $λ = 3$ ( $q = - 2$ ). The black curves are $q$ -Gaussians. Here we used the strength of the confining potential $k = 3.2 \times 10^{- 3} ε / σ^{2}$ .
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Figure 4
Distributions of scaled positions and velocities at different moments of the dynamics for the case of two dimensions ( $D = 2$ ). These results concern particles interacting through the power-law potential, $V = ε {(r / σ)}^{- λ}$ , for $λ = 4$ ( $q = - 1$ ) and $λ = 6$ ( $q = - 2$ ). One can compare the density profiles with $q$ -Gaussians (black curves). In the case of the velocity distributions the black curves show convolutions between $q$ -Gaussians and Laplacian distributions, as given by Eq. (15), with the parameter $h$ given by 0.045 and 0.051 for $λ = 4$ and 6, respectively. In these simulations we used $N = 4000$ particles in a cell of transverse size $L_{y} = 20 σ$ with periodic boundary conditions. In the longitudinal direction ( $x$ axis) we imposed a confining force $- k x$ with $k = 1 \times 10^{- 3} ε / σ^{2}$ .
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Figure 5
Distributions of the displacement of the average velocity, $ξ_{i} = {(v_{x})}_{i} / \dot{f} - x_{i} / f$ for some values of times $t$ considering $λ = 4$ and $λ = 6$ for $D = 2$ . The black curves represent Laplacian distributions, $P (ξ) \sim exp (- | ξ | / h)$ , where $h$ is an adjustment parameter, given by 0.045 and 0.051 for $λ = 4$ and 6, respectively.
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