Variational Statement of the PB Eq.

Consider a system comprised of an arbitrary source charge density $\rho_s (\vec r)$, immersed in a solvent characterized by dielectric constant $\epsilon$.In the solvent are (``simple") molecular sized ions. For concreteness let us assume that all positive simple ions have charge +q and all negative simple ions have charge -q. (In the most common case the magnitude of q is one unit of electronic charge.) Let the bulk number densities of $\pm$ simple ions be $\bar n_{\pm}$.

Now we introduce the following dimensionless functional:

$$S = \int d \vec r [ \frac {-\epsilon}{8\pi \beta q^2} \psi ... ..._- e^{-\psi (\vec r) - v(\vec r)} + n_s (\vec r) \psi(\vec r)]$$ (37)

Here $\psi = -\beta q\phi$ is the dimensionless electric potential, v is the dimensionless excluded volume potential introduced above, and $n_s (\vec r) \equiv \rho_s (\vec r) / q$.(If the source charge consists of a distribution of elementary point charges with magnitude q, then $n_s (\vec r)$ is the number density of these elemental charges.) Assuming $\phi=0$ on the boundaries of the simulation box, it is simple to show that the field $\phi$ which extremizes this ``action" S is precisely the field that satisfies the PB Eq. (31) [37]. This is easily seen if the functional is discretized to read:

$$S = - \frac{\alpha}{2} \sum_{\vec n \vec m} \psi_{\vec n} \D... ...e^{-\psi_{\vec n} - v_{\vec n}} + \sum_{j} Q_j \psi_{\vec R_j}$$ (38)

where $\psi_{\vec n}$ is the dimensionless electric potential field at lattice point $\vec n$,$\alpha$ is the dimensionless constant $\alpha = \epsilon a/4\pi\beta q^2$ (we have replaced the grid size h with a here to be consistent with our published notation [38]), $v_{\vec n}$ is the dimensionless excluded volume potential at lattice site $\vec n$,$\vec R_j$ is the lattice location of the j'th permanent charge in the system and Q_j is its magnitude in units of the elemental charge q. Furthermore, $\gamma_{\pm}$ are dimensionless activity parameters related to the bulk density of simple ions according to $a^3 \bar n_{\pm} = \gamma_{\pm}$.Provided that the lattice discretization of the Laplacian operator, $\Delta_{\vec n \vec m}$,is symmetric (which it is for both pinned and wrap-around boundary conditions), then extremizing this multidimensional function of $\{ \psi_{\vec n} \}$ generates the conditions:

$$\alpha \sum_{\vec m} \Delta_{\vec n \vec m} \psi_{\vec m} = ... ...i_{\vec n} - v_{\vec n}} + \sum_j Q_j \delta_{\vec n, \vec R_j}$$ (39)

This is precisely the lattice discretized form of the PB Eq.

The variational version of the PB Eq. turns out to be useful for the same reasons as the variational version of Poisson's Eq. Namely, it is possible to show from the lattice action S in Eq. (38) that $\partial^2 S / \partial \psi_{\vec n} \partial \psi_{\vec m} \gt 0$for any field configuration [38]. Thus we know the action functional has no saddle points - if we find a solution of the field equations 39, this solution corresponds to a strict minimum in S and, moreover it is unique. Therefore, simple annealing strategies (e.g., the line minimization, conjugate gradient [40][a] and steepest descent techniques described above) are guaranteed to find the solution of the nonlinear PB Eq. for arbitrary macroion shapes, charge distributions and impurity ionic strengths.