One-dimensional fluids with second nearest-neighbor interactions

As is well known, one-dimensional systems with interactions restricted to first nearest neighbors admit a full analytically exact statistical-mechanical solution. This is essentially due to the fact that the knowledge of the first nearest-neighbor probability distribution function, $p_1(r)$, is enough to determine the structural and thermodynamic properties of the system. On the other hand, if the interaction between second nearest-neighbor particles is turned on, the analytically exact solution is lost. Not only the knowledge of $p_1(r)$ is not sufficient anymore, but even its determination becomes a complex many-body problem. In this work we systematically explore different approximate solutions for one-dimensional second nearest-neighbor fluid models. We apply those approximations to the square-well and the attractive two-step pair potentials and compare them with Monte Carlo simulations, finding an excellent agreement.Comment: 26 pages, 12 figures; v2: more references adde

Lagrange Interpolation Learning Particle Swarm Optimization

<div><p>In recent years, comprehensive learning particle swarm optimization (CLPSO) has attracted the attention of many scholars for using in solving multimodal problems, as it is excellent in preserving the particles’ diversity and thus preventing premature convergence. However, CLPSO exhibits low solution accuracy. Aiming to address this issue, we proposed a novel algorithm called LILPSO. First, this algorithm introduced a Lagrange interpolation method to perform a local search for the global best point (gbest). Second, to gain a better exemplar, one gbest, another two particle’s historical best points (pbest) are chosen to perform Lagrange interpolation, then to gain a new exemplar, which replaces the CLPSO’s comparison method. The numerical experiments conducted on various functions demonstrate the superiority of this algorithm, and the two methods are proven to be efficient for accelerating the convergence without leading the particle to premature convergence.</p></div

results for D = 50, N = 100, FEs = 500,000.

<p>results for D = 50, N = 100, FEs = 500,000.</p

PID Results optimized by some algorithms.

<p>PID Results optimized by some algorithms.</p

Iterative forms of each algorithms.

<p>Iterative forms of each algorithms.</p

Details of benchmarks.

<p>Details of benchmarks.</p

Results for D = 10, N = 50, FEs = 100,000.

<p>Results for D = 10, N = 50, FEs = 100,000.</p

Selection of the exemplar dimensions for particle i.

<p>(a)CLPSO (b)CLPSO-LIL.</p

Results for D = 30, N = 40, FEs = 200,000.

<p>Results for D = 30, N = 40, FEs = 200,000.</p

The flowchart of LSLI.

<p>The flowchart of LSLI.</p