# 6.021/Notes/Equations

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## Contents |

## Diffusion

**Fick's 1st law**:

**Continuity**:

**Diffusion Equation**:

Solution of diffusion equation to impulse stimulus is Gaussian:

Time for half the solute to diffuse *x*_{1 / 2}:

**Fick's law for membranes**: ;

Membrane steady state time constant:

Solution for dissolve and diffuse: ;

## Osmosis

**Van't Hoff Law**: π(*x*,*t*) = *R**T**C*_{Σ}(*x*,*t*)

**Darcy's Law**:

**Continuity**:

Hydraulic conductivity:

Flux: Φ_{V} = *L*_{V}((*p*^{i} − π^{i}) − (*p*^{o} − π^{o}))

Cells: with solution

## Carrier Transport

Solution to simple symmetric 4-state carrier model:

;

## Electrodiffusion

**Nernst-Planck Equation**:

**Einstein's relation**: *D*_{n} = *u*_{n}*R**T*

**Continuity**:

**Poisson's Equation**:

### Membranes

*J*_{n} = *G*_{n}(*V*_{m} − *V*_{n})

(electrical conductivity)

**Nernst potential**:

### Cells

G_{m} = | ∑ | G_{n} |

n |

**Resting membrane potential**:

**Resting potential with active pumps**:

## Core conductor model

**THE core conductor equation**:

**wave equation**:

,

## Hodgkin-Huxley

,

, ,

## Cable model

**Cable Equation**:

Steady state solution of cable equation to impulse stimulus:

Dynamics: where (Diffusion equation with )

## Ion channels

*I* = γ(*V*_{m} − *V*_{n})

, ,

,