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Attraction of Like-Charged Walls with Counterions Only: Exact Results for the 2D Cylinder Geometry
Journal of Statistical Physics ( IF 1.6 ) Pub Date : 2020-09-29 , DOI: 10.1007/s10955-020-02642-9
Ladislav Šamaj

We study a 2D system of identical mobile particles on the surface of a cylinder of finite length d and circumference W , immersed in a medium of dielectric constant ε. The two end-circles of the cylinder are like-charged with the fixed uniform charge densities, the particles of opposite charge −e (e being the elementary charge) are coined as “counterions”; the system as a whole is electroneutral. Such a geometry is well defined also for finite numbers of counterions N . Our task is to derive an effective interaction between the end-circles mediated by the counterions in thermal equilibrium at the inverse temperature β. The exact solution of the system at the free-fermion coupling Γ ≡ βe/ε = 2 is used to test the convergence of the pressure as the (even) number of particles increases from N = 2 to ∞. The pressure as a function of distance d is always positive (effective repulsion between the likecharged circles), decaying monotonously; the numerical results for N = 8 counterions are very close to those in the thermodynamic limit N → ∞. For the couplings Γ = 2γ with γ = 1, 2, . . ., there exists a mapping of the continuous two-dimensional (2D) Coulomb system with N particles onto the one-dimensional (1D) lattice model of N sites with interacting sets of anticommuting variables. This allows one to treat exactly the density profile, two-body density and the pressure for the couplings Γ = 4 and 6, up to N = 8 particles. Our main finding is that the pressure becomes negative at large enough distances d if and only if both like-charged walls carry a nonzero charge density. This indicates a like-attraction in the thermodynamic limit N → ∞ as well, starting from a relatively weak coupling constant Γ in between 2 and 4. As a by-product of the formalism, we derive specific sum rules which have direct impact on characteristics of the long-range decay of 2D two-body densities along the two walls. Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, SK-84511 Bratislava, Slovakia E-mail: Ladislav.Samaj@savba.sk

中文翻译:

仅具有反离子的同电荷壁的吸引力:2D 圆柱几何的精确结果

我们研究了长度为 d 和周长为 W 的圆柱体表面上相同移动粒子的二维系统,浸入介电常数为 ε 的介质中。圆柱体的两个端圆带相同电荷,具有固定的均匀电荷密度,带相反电荷-e(e 为基本电荷)的粒子被称为“反离子”;整个系统是电中性的。对于有限数量的反离子 N 也很好地定义了这样的几何形状。我们的任务是在逆温度 β 下推导出由处于热平衡状态的反离子介导的端圆之间的有效相互作用。系统在自由费米子耦合 Γ ≡ βe/ε = 2 处的精确解用于测试当(偶数)粒子数从 N = 2 增加到 ∞ 时压力的收敛性。作为距离 d 的函数的压力始终为正值(相同带电圆之间的有效斥力),单调衰减;N = 8 个反离子的数值结果非常接近热力学极限 N → ∞ 中的数值结果。对于耦合 Γ = 2γ 且 γ = 1, 2, . . .,存在具有 N 个粒子的连续二维 (2D) 库仑系统到具有相互作用的反交换变量集的 N 个位点的一维 (1D) 晶格模型的映射。这允许我们精确地处理密度分布、二体密度和耦合 Γ = 4 和 6 的压力,最多 N = 8 个粒子。我们的主要发现是当且仅当两个带相同电荷的壁都带有非零电荷密度时,压力在足够大的距离 d 处变为负值。这也表明热力学极限 N → ∞ 中的相似吸引力,从介于 2 和 4 之间的相对较弱的耦合常数 Γ 开始。作为形式主义的副产品,我们推导出特定的求和规则,这些规则对沿两壁的二维双体密度的长程衰减特性有直接影响. 斯洛伐克科学院物理研究所,Dúbravská cesta 9, SK-84511 Bratislava, Slovakia 电子邮件:Ladislav.Samaj@savba.sk
更新日期:2020-09-29
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