Communications in Mathematical Sciences

Volume 15 (2017)

Number 2

Generalized Shockley–Ramo theorem in electrolytes

Pages: 555 – 564

(Fast Communication)



Pei Liu (School of Mathematical Sciences and Institute of Nature Sciences, Shanghai Jiao Tong University, Shanghai, China)

Chun Liu (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Zhenli Xu (School of Mathematical Sciences, Institute of Nature Sciences, and MoE Key Lab of Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai, China)


The charge motion in vacuum and the induced currents on the electrodes can be related through the Shockley–Ramo (SR) theorem. In this paper, we develop a generalized Shockley–Ramo (GSR) theorem, which could be used to study the motion of macro charged particles in electrolytes. It could be widely applied to biological and physical environments, such as the voltage-gated ion channels. With the procedure of renormalizing of charge and dipole, the generalized theorem provides a direct relationship between the induced currents and the macro charge velocity. Compared with the original Shockley–Ramo theorem, the generalized Shockley–Ramo theorem avoids integrating all the ionic flux, which could reduce the computational cost significantly.


hockley–Ramo theorem, electrolytes, charge renormalization, inverse problem

2010 Mathematics Subject Classification

35Q70, 82B21, 82C05, 92C30

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