# 泊松-玻尔兹曼方程

${\displaystyle \nabla ^{2}\phi ({\textbf {r}})=-{\frac {4\pi }{\epsilon }}\sum _{i}c_{i}^{0}z_{i}qe^{-\beta z_{i}q\phi ({\textbf {r}})}.}$

## 原理

${\displaystyle \nabla ^{2}\phi ({\textbf {r}})=-{\frac {4\pi }{\epsilon }}\sum _{i}z_{i}qc_{i}({\textbf {r}}),}$

${\displaystyle c_{i}({\textbf {r}})=c_{i}^{0}e^{-\beta w_{i}({\textbf {r}})},}$

${\displaystyle w_{i}({\textbf {r}})\simeq z_{i}q\phi ({\textbf {r}})}$ ,

## 求解

${\displaystyle e^{-\beta z_{i}q\phi ({\textbf {r}})}\simeq 1-\beta z_{i}q\phi ({\textbf {r}})}$

${\displaystyle \nabla ^{2}\phi ({\textbf {r}})+\kappa ^{2}\phi ({\textbf {r}})=0,}$

## 参考资料

1. ^ G.L. Gouy, j. de phys 9, 457 (1910)
2. ^ D.L. Chapman, Philos. Mag. 25, 475 (1913)
3. ^ Adaptive Poisson–Boltzmann Solver页面存档备份，存于互联网档案馆） - A free, open-source Poisson-Boltzmann electrostatics and biomolecular solvation software package.
4. ^ Zap - A Poisson–Boltzmann electrostatics solver.
5. ^ MIBPB Matched Interface & Boundary based Poisson–Boltzmann solver
6. ^ AFMPB Adaptive Fast Multipole Poisson–Boltzmann Solver, free and open-source.
7. ^ Donald Bashford and David A. Case GENERALIZED BORN MODELS OF MACROMOLECULAR SOLVATION EFFECTS Annu. Rev. Phys. Chem. 2000, 51, 129-152 doi:10.1146/annurev.physchem.51.1.129
8. ^ Y. Levin Electrostatic correlations: from plasma to biology. Rep. Prog. Phys. 65 1577 doi:10.1088/0034-4885/65/11/201
9. ^ A. Yu. Grosberg, T. T. Nguyen, and B. I. Shklovskii Colloquium: The physics of charge inversion in chemical and biological systems Rev. Mod. Phys. 74, 329 doi: 10.1103/RevModPhys.74.329