Numerical evaluation of the Casimir interaction between cylinders

We numerically evaluate the Casimir interaction energy for configurations involving two perfectly conducting eccentric cylinders and a cylinder in front of a plane. We consider in detail several special cases. For quasiconcentric cylinders, we analyze the convergence of a perturbative evaluation based on sparse matrices. For concentric cylinders, we obtain analytically the corrections to the proximity force approximation up to second order, and we present an improved numerical procedure to evaluate the interaction energy at very small distances. Finally, we consider the configuration of a cylinder in front of a plane. We first show numerically that, in the appropriate limit, the Casimir energy for this configuration can be obtained from that of two eccentric cylinders. Then we compute the interaction energy at small distances, and compare the numerical results with the analytic predictions for the first order corrections to the proximity force approximation.

Figure 1

Geometrical configuration for the eccentric cylinders. Two perfectly conducting cylinders of radii $a<b$, length $L\gg a,b$, and eccentricity $ϵ$ interact via the Casimir force.

Figure 2

Exact Casimir interaction energy difference $|\Delta E|$ between the eccentric and concentric configurations as a function of $\alpha =b/a$ for different values of $\delta =ϵ/a$. Here $\Delta E=E_{12}-E_{12}^{\mathrm{cc}}$. Energies are measured in units of $L/4\pi a^2$.

Figure 3

Exact Casimir interaction energy difference $\Delta E$ between the eccentric and concentric configurations as a function of $\delta =ϵ/a$ for different values of $\alpha =b/a$. Energies are measured in units of $L/4\pi a^2$. The maximum at $\delta =0$ shows the instability of the concentric equilibrium position.

Figure 4

Exact Casimir interaction energy difference $\Delta E$ between the eccentric and concentric configurations as a function of $\delta =ϵ/a$ for different values of $\alpha$. The solid line is the exact evaluation of the Casimir interaction energy, while the dashed line with triangles is the second order approximation and the dashed line with dots is the first order one. The analytic curve (dashed line with crosses) is the result of using Eq. (4). Energies are measured in units of $L/4\pi a^2$.

Figure 5

Exact Casimir interaction energy difference $\Delta E$ between the eccentric and concentric configurations as a function of $\delta =ϵ/a$ for different values of $\alpha$. The solid line is the exact evaluation of the Casimir interaction energy, while the dashed line with triangles is the second order approximation and the dashed line with dots is the first order one. The analytic curve is the result of using Eq. (4). Energies are measured in units of $L/4\pi a^2$.

Figure 6

Exact Casimir interaction energy difference $\Delta E$ between the eccentric and concentric configurations as a function of $\delta =ϵ/a$ for different values of $\alpha$ (a smaller range of $\delta$). The solid line is the exact evaluation of the Casimir interaction energy, while the dashed line with triangles is the second order approximation and the dashed line with dots is the first order one. The analytic curve (dashed line with crosses) is the result of using Eq. (4). Energies are measured in units of $L/4\pi a^2$.

Figure 7

Ratio between the exact Casimir energy for concentric cylinders $E_{12}^{\mathrm{cc}}$ and the Casimir energy estimated using the PFA up to the next to leading order $E_{\mathrm{PFA}}^{\mathrm{NTL}}$, as a function of the parameter $\alpha$. This is done for two different methods: the numerical (of slow convergence) and the numerical improved (subtraction method).

Figure 8

Cylinder-plane configuration. A perfectly conducting cylinder of radius $a$ is in front of a perfectly conducting plane at a distance $d$.

Figure 9

Comparison between the exact Casimir interaction energy for eccentric cylinders and the cylinder-plane configuration as a function of $d=H-a$ for different values of $\alpha$.

Figure 10

Leading term for the Casimir interaction energy of the cylinder-plane configuration. The leading term behaves proportional to $-0.0228d^{-5/2}$. A simple fit $f(x)=\gamma x^{\varsigma }$ of the numerical data for $d/a<1$ gives $\gamma =-0.021$ and $\varsigma =-2.53$.

Figure 11

Numerical result for the TM modes for the cylinder-plane configuration, and the corresponding fits presented in Table . A simple linear fit $f(x)=a+bx$ of the numerical data in the interval $0.04\le d/a\le 0.07$ gives $a=0.9999$ and $b=0.1900$. The theoretical values are $a=1$ and $b=0.1944$.

Figure 12

Numerical result for the TE modes for the cylinder-plane configuration, and the corresponding fits presented in Table . A simple linear fit $f(x)=a+bx$ of the numerical data in the interval $0.04\le d/a\le 0.07$ gives $a=0.9940$ and $b=-0.7808$. The theoretical values are $a=1$ and $b=-1.1565$.

Figure 13

A numerical fit of the results for the TE modes including cubic corrections $f(x)=1+bx+c{x}^2\log x+d{x}^3$. The coefficients are $b=-1.0478$, $c=-0.9485$, and $d=0.6708$.