Linear and quadratic static response functions and structure functions in Yukawa liquids

We compute linear and quadratic static density response functions of three-dimensional Yukawa liquids by applying an external perturbation potential in molecular dynamics simulations. The response functions are also obtained from the equilibrium fluctuations (static structure factors) in the system via the fluctuation-dissipation theorems. The good agreement of the quadratic response functions, obtained in the two different ways, confirms the quadratic fluctuation-dissipation theorem. We also find that the three-point structure function may be factorizable into two-point structure functions, leading to a cluster representation of the equilibrium triplet correlation function.

Figure 1

The normalized density distribution of the system in case of a harmonic external potential with ${\mathbf{k}}_{1}a=(35,0,0)k_{\text{min}}a$. (a) Total density response; (b) the nonlinear part of the response. The plots show only a part of the simulation box.

Figure 2

Normalized linear response functions $-{\widehat{\chi }}^{\left(1\right)}\left(k_1\right)/\beta n_0$ obtained from the PMD simulations (symbols), in comparison with the static structure factors $S\left(k_1\right)$ resulting from UMD simulations (lines) for the $\Gamma$ values indicated, $\kappa =1$.

Figure 3

(a) Normalized quadratic response function $2{\widehat{\chi }}^{\left(2\right)}(k_1,k_1)/{\beta }^2{n}_0$ obtained from the PMD simulations (symbols), in comparison with the static structure factor $S(k_1,k_1)$ resulting from an UMD simulation (line), for $\Gamma =50$, $\kappa =1$. The error bars indicate $\pm 2$ standard deviations of the UMD data obtained via averaging 60 independent simulation runs. (b) Dependence of $2{\widehat{\chi }}^{\left(2\right)}(k_1,k_1)/{\beta }^2{n}_0$ on the coupling parameter $\Gamma$, at $\kappa =1$. (c) Amplitudes $B_2$ of $2{\widehat{\chi }}^{\left(2\right)}(k_1,k_1)/{\beta }^2{n}_0$ and $B_1$ of $S\left(k_1\right)$ [equivalent to $-{\widehat{\chi }}^{\left(1\right)}\left(k_1\right)/\beta n_0$], as well as its square, $B_1^2$, as a function of $\Gamma$, at ${\left(k_{1}a\right)}^{*}=28k_{\text{min}}a=4.34$ (position of the first peak of the functions).

Figure 4

(a) Perturbed density distribution of the system in case of a biharmonic external potential with $k_{1}a=25k_{\text{min}}a$ and $k_{2}a=2k_{1}a$. (b) The nonlinear part of the perturbed density. The plots show only a part of the simulation box.

Figure 5

Normalized quadratic response function $2{\widehat{\chi }}^{\left(2\right)}(k_1,2k_1)/{\beta }^2{n}_0$ obtained from the PMD simulations (symbols), in comparison with the static structure factor $S(k_1,2k_1)$ resulting from an UMD simulation. The error bars indicate $\pm 2$ standard deviations of the UMD data obtained via averaging 120 independent simulation runs. $\Gamma =50$, $\kappa =1$.

Figure 6

The perturbed density distribution of the system in case of a biharmonic external potential, at ${\mathbf{k}}_{1}a=(35,10,0)k_{\text{min}}a$ and ${\mathbf{k}}_{2}a=(25,0,0)k_{\text{min}}a$. $\Gamma =50$, $\kappa =1$.

Figure 7

Normalized quadratic response function $2{\widehat{\chi }}^{\left(2\right)}({\mathbf{k}}_1,{\mathbf{k}}_2)/{\beta }^2{n}_0$ obtained from the PMD simulations, in comparison with the static structure factor $S({\mathbf{k}}_1,{\mathbf{k}}_2)$ resulting from an UMD simulation. The error bars indicate $\pm 2$ standard deviations of the UMD data obtained via averaging 125 independent simulation runs. ${\mathbf{k}}_{1}a=(m+5,10,0)k_{\text{min}}a$ and ${\mathbf{k}}_{2}a=(m-5,0,0)k_{\text{min}}a$, $\Gamma =50$, $\kappa =1$.

Figure 8

Factorization ratio $R_S$ as a function of wave number, for $\Gamma =50$, $\kappa =1$. In the case of $S(k_1,k_1)$ and $S(k_1,2k_1)$ the data are plotted as a function of $k_{1}a$, while for the more general case $S({\mathbf{k}}_1,{\mathbf{k}}_2)$ the data are plotted as a function of $k_{2}a$.