# Materials Science and Engineering/Equations/Kinetics

## Mathematical Foundation

### Time-Dependent Field

 ${\displaystyle {\frac {dc}{dt}}=\nabla c\cdot {\overrightarrow {v}}+{\frac {\partial c}{\partial t}}}$

${\displaystyle {\overrightarrow {v}}({\overrightarrow {r}})}$: Velocity
${\displaystyle c({\overrightarrow {r}},t)}$: Time-Dependent Field

### Accumulation

Rate of accumulation is the negative of the divergence of the flux of the quantity plus the rate of production

 ${\displaystyle {\frac {\partial c_{i}}{\partial t}}=-\nabla \cdot {\overrightarrow {J_{i}}}+{\dot {\rho _{i}}}}$

${\displaystyle {\dot {\rho _{i}}}({\overrightarrow {r}})}$: Rate of production of the density of ${\displaystyle i}$ in ${\displaystyle \Delta V}$
${\displaystyle \nabla \cdot {\overrightarrow {J_{i}}}}$: The divergence of ${\displaystyle {\overrightarrow {J_{i}}}}$
 ${\displaystyle {\dot {M_{i}}}=\int _{\Delta V}-\nabla \cdot {\overrightarrow {J_{i}}}+{\dot {\rho _{i}}}\,dx}$

${\displaystyle {\dot {M_{i}}}}$: Rate at which ${\displaystyle i}$ flows through area ${\displaystyle \Delta {\overrightarrow {A}}}$

## Divergence Theorem

 ${\displaystyle \int _{\mathrm {B} (\Delta V)}{\overrightarrow {J}}\cdot {\dot {n}}\,dA=\int _{\Delta V}\nabla \cdot {\overrightarrow {J}}\,dV}$

${\displaystyle \mathrm {B} (\Delta V)}$: Oriented surface around a volume

## General Set of Linear Equations

 {\displaystyle {\begin{alignedat}{7}M_{11}x_{1}&&\;+\;&&M_{12}x_{2}&&\;+\cdots +\;&&M_{1n}x_{n}&&\;=\;&&&y_{1}\\M_{21}x_{1}&&\;+\;&&M_{22}x_{2}&&\;+\cdots +\;&&M_{2n}x_{n}&&\;=\;&&&y_{2}\\\vdots \;\;\;&&&&\vdots \;\;\;&&&&\vdots \;\;\;&&&&&\;\vdots \\M_{m1}x_{1}&&\;+\;&&M_{m2}x_{2}&&\;+\cdots +\;&&M_{mn}x_{n}&&\;=\;&&&y_{m}\\\end{alignedat}}}


The vector equation is equivalent to a matrix equation of the form

 ${\displaystyle M\mathbf {x} =\mathbf {y} }$


where M is an m×n matrix, x is a column vector with n entries, and y is a column vector with m entries.

${\displaystyle M={\begin{bmatrix}M_{11}&M_{12}&\cdots &M_{1n}\\M_{21}&M_{22}&\cdots &M_{2n}\\\vdots &\vdots &\ddots &\vdots \\M_{m1}&M_{m2}&\cdots &M_{mn}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {y} ={\begin{bmatrix}y_{1}\\y_{2}\\\vdots \\y_{m}\end{bmatrix}}}$

### Eigenvalue Equation

 ${\displaystyle {\underline {M}}{\overrightarrow {e}}=\lambda {\overrightarrow {e}}}$

${\displaystyle {\underline {M}}}$: ${\displaystyle nxn}$ square matrix or tensor
${\displaystyle {\overrightarrow {e}}}$: eigenvector (special vector)
${\displaystyle \lambda }$: eigenvalue (special scalar multiplier)

### Transformation of Rank-Two Tensor

 ${\displaystyle \left[{\mbox{diagonalized matrix}}\right]=\left[{\mbox{eigenvector column matrix}}\right]^{-1}\left[{\mbox{square matrix}}\right]\left[{\mbox{eigenvector column matrix}}\right]}$


## Irreversible Thermodynamics

### Differential Change in Entropy

 ${\displaystyle TdS=du-\sum _{j}\Psi _{j}d\zeta _{j}\;}$

${\displaystyle \sum _{j}\Psi _{j}d\zeta _{j}=-Pdv+\phi dq+\gamma dA+\mu _{1}dc_{1}+\cdots \;}$

### Entropy Production

 ${\displaystyle T{\dot {\sigma }}=-{\frac {\overrightarrow {J_{Q}}}{T}}\cdot \nabla T-\sum _{j}{\overrightarrow {J_{i}}}\cdot \nabla \Psi _{j}}$

${\displaystyle {\dot {\sigma }}}$: Rate of entropy-density creation
${\displaystyle {\overrightarrow {J_{Q}}}}$: Flux of heat
${\displaystyle {\overrightarrow {J_{i}}}}$: Conjugate force
${\displaystyle \nabla \Psi _{j}}$: Conjugate flux

### Empirical Force-Flux Law

#### Fourier's

 ${\displaystyle {\overrightarrow {J_{Q}}}=-K\nabla T}$


#### Modified Fick's

 ${\displaystyle {\overrightarrow {J_{i}}}=-M_{i}c_{i}\nabla \mu _{i}}$


#### Ohm's

 ${\displaystyle {\overrightarrow {J_{q}}}=-\rho \nabla \phi }$


### Basic Postulate of Irreversible Thermodynamics

The local generation of entropy, ${\displaystyle {\dot {\sigma }}}$ is nonnegative

 ${\displaystyle {\dot {\sigma }}={\frac {\partial s}{\partial t}}+\nabla \cdot {\overrightarrow {J_{Q}}}\geq 0}$


### Coupling Between Forces and Fluxes

 {\displaystyle {\begin{alignedat}{7}{\frac {\partial J_{Q}}{\partial F_{Q}}}F_{Q}&&\;+\;&&{\frac {\partial J_{Q}}{\partial F_{q}}}F_{q}&&\;+\cdots +\;&&{\frac {\partial J_{Q}}{\partial F_{N_{c}}}}F_{N_{c}}&&\;=\;&&&J_{Q}(F_{Q},F_{q},F_{1},F_{2},...,F_{N_{c}})\\{\frac {\partial J_{q}}{\partial F_{Q}}}F_{Q}&&\;+\;&&{\frac {\partial J_{q}}{\partial F_{q}}}F_{q}&&\;+\cdots +\;&&{\frac {\partial J_{q}}{\partial F_{N_{c}}}}F_{N_{c}}&&\;=\;&&&J_{q}(F_{Q},F_{q},F_{1},F_{2},...,F_{N_{c}})\\\vdots \;\;\;&&&&\vdots \;\;\;&&&&\vdots \;\;\;&&&&&\;\vdots \\{\frac {\partial J_{N_{c}}}{\partial F_{Q}}}F_{Q}&&\;+\;&&{\frac {\partial J_{N_{c}}}{\partial F_{q}}}F_{q}&&\;+\cdots +\;&&{\frac {\partial J_{N_{c}}}{\partial F_{N_{c}}}}F_{N_{c}}&&\;=\;&&&J_{N_{c}}(F_{Q},F_{q},F_{1},F_{2},...,F_{N_{c}})\\\end{alignedat}}}


Abbreviated form:

 ${\displaystyle J_{\alpha }=\sum _{\beta }L_{\alpha \beta }F_{\beta }\;}$

${\displaystyle L_{\alpha \beta }={\frac {\partial J_{\alpha }}{\partial F_{\beta }}}}$

### Force-Flux Relations with Constrained Extensive Quantities

 ${\displaystyle TdS=du+dw-\sum _{i=1}^{N_{c}-1}\left(\mu _{i}-\mu _{N_{c}}\right)dc_{i}}$

${\displaystyle {\overrightarrow {F_{i}}}=-\nabla \left(\mu _{i}-\mu _{N_{c}}\right)}$

### Diffusion Potential

 ${\displaystyle {\overrightarrow {F_{1}}}=-\nabla \Phi _{1}}$


### Onsager Symmetry Principle

 ${\displaystyle L_{\alpha \beta }=L_{\beta \alpha }\;}$

 ${\displaystyle {\frac {\partial J_{\alpha }}{\partial F_{\beta }}}={\frac {\partial J_{\beta }}{\partial F_{\alpha }}}}$


## Driving Forces and Fluxes

### Diffusion in Absence of Chemical Effects

• Components diffuse in chemically homogeneous material
• Diffusion measured with radioactive tracer
• Fick's law flux equation derived when self-diffusion occurs by the vacancy-exchange mechanism.
• The crystal is network-constrained
• There are three components:
• Inert atoms
• Vacancies
• C-frame: single reference frame
• Vacancies assumed to be in equilibrium throughout
• Raoultian behavior
 ${\displaystyle J_{^{*}1}^{C}=-kT\left[{\frac {L_{11}}{c_{1}}}-{\frac {L_{1^{*}1}}{c_{^{*}1}}}\right]{\frac {\partial c_{^{*}1}}{\partial x}}}$

 ${\displaystyle J_{^{*}1}^{C}=-^{*}D{\frac {\partial c_{^{*}1}}{\partial x}}}$


### Diffusion of i in Chemically Homogeneous Binary Solution

 ${\displaystyle J_{^{*}1}^{C}=-kT\left[{\frac {L_{11}}{c_{1}}}-{\frac {L_{1^{*}1}}{c_{^{*}1}}}\right]{\frac {\partial c_{^{*}1}}{\partial x}}}$
${\displaystyle J_{^{*}1}^{C}=-^{*}D_{1}{\frac {\partial c_{^{*}1}}{\partial x}}}$


### Diffusion of Substitutional Particles in Concentration Gradient

• Constraint associated with vacancy mechanism: ${\displaystyle {\overrightarrow {J_{1}^{c}}}+{\overrightarrow {J_{2}^{c}}}+{\overrightarrow {J_{v}^{c}}}=0}$
• Difference in fluxes of the two substitutional species requires net flux of vacancies.
• Gibbs-Duhem relation: ${\displaystyle c_{1}{\frac {\partial \mu _{1}}{\partial x}}+c_{2}{\frac {\partial \mu _{2}}{\partial x}}+c_{v}{\frac {\partial \mu _{v}}{\partial x}}}$
• Chemical potential gradients related to concentration gradients: ${\displaystyle \mu _{i}=\mu _{i}^{\circ }+kT\ln(\gamma _{i}<\Omega >c_{i})}$

Flux is proportional to the concentration gradient

 ${\displaystyle J_{1}^{c}=-kT\left[{\frac {L_{11}}{c_{1}}}-{\frac {L_{12}}{c_{2}}}\right]\left[1+{\frac {\partial \ln \gamma _{1}}{\partial \ln c_{1}}}+{\frac {\partial \ln <\Omega >}{\partial \ln c_{1}}}\right]{\frac {\partial c_{1}}{\partial x}}}$
${\displaystyle J_{1}^{c}=-D_{1}{\frac {\partial c_{1}}{\partial x}}}$


Assumptions that simplify ${\displaystyle D_{1}\;}$

• Concentration-independent average site volume ${\displaystyle <\Omega >\;}$
• The coupling (off-diagonal) terms, ${\displaystyle L_{12}/c_{2}\;}$ and ${\displaystyle L_{1^{*}1}/c_{^{*}1}\;}$, are small compared with the direct term ${\displaystyle L_{11}/c_{2}\;}$
 ${\displaystyle D_{1}\approx \left[1+{\frac {\partial \ln \gamma _{1}}{\partial \ln c_{1}}}\right]{}^{*}D_{1}}$


#### Diffusion in a Volume-Fixed (V-Frame)

• Velocity of a local C-frame with respect to the V-frame: velocity of any inert marker with respect to the V-frame
• Flux of 1 in the V-frame:
 ${\displaystyle J_{1}^{v}=-[c_{1}\Omega _{1}D_{2}+c_{2}+\Omega _{2}D_{1}]{\frac {\partial c_{1}}{\partial x}}}$

• The interdiffusivity, ${\displaystyle c_{1}\Omega _{1}D_{2}+c_{2}+\Omega _{2}D_{1}\;}$, can be simplified through ${\displaystyle \Omega _{1}=\Omega _{2}=<\Omega >\;}$
• The L-frame and the V-frame are the same

### Diffusion of Interstitial Particles in Concentration Gradient

• ${\displaystyle {\overrightarrow {J_{1}^{c}}}=L_{11}{\overrightarrow {F_{1}}}}$
• ${\displaystyle {\overrightarrow {J_{1}^{c}}}=-L_{11}\nabla \Phi _{1}}$
• ${\displaystyle {\overrightarrow {J_{1}^{c}}}=-L_{11}\nabla \mu _{1}}$
• ${\displaystyle \mu _{1}=\mu _{1}^{\circ }+kT\ln(K_{1}c_{1})}$
• ${\displaystyle \nabla \mu _{1}={\frac {kT}{c_{1}}}\nabla c_{1}}$
 ${\displaystyle {\overrightarrow {J_{1}^{c}}}=-L_{11}{\frac {kT}{c_{1}}}\nabla c_{1}}$

• Evaluate ${\displaystyle L_{11}}$ by substitution of interstitial mobility, ${\displaystyle M_{1}}$
• ${\displaystyle {\overrightarrow {v_{1}^{c}}}=-M_{1}\nabla \mu _{1}}$
• ${\displaystyle {\overrightarrow {v_{1}^{c}}}={\frac {-M_{1}kT}{c_{1}}}\nabla c_{1}}$
• ${\displaystyle {\overrightarrow {J_{1}^{c}}}={\overrightarrow {v_{1}^{c}}}c_{1}}$
 ${\displaystyle {\overrightarrow {J_{1}^{c}}}=-M_{1}kT\nabla c_{1}}$


### Diffusion of Charged Ions in Ionic Conductors

• ${\displaystyle {\overrightarrow {J_{1}}}=L_{11}{\overrightarrow {F_{1}}}}$
• ${\displaystyle {\overrightarrow {J_{1}}}=-L_{11}\nabla \Phi _{1}}$
• ${\displaystyle {\overrightarrow {J_{1}}}=-L_{11}\nabla (\mu _{1}+q_{1}\phi )}$
 ${\displaystyle {\overrightarrow {J_{1}}}=-D_{1}\nabla c_{1}-{\frac {D_{1}c_{1}q_{1}}{kT}}\nabla (\phi )}$

• ${\displaystyle {\overrightarrow {E}}=-\nabla \phi \;}$: Electric field
• ${\displaystyle {\overrightarrow {J_{q}}}=q_{1}{\overrightarrow {J_{1}}}}$
• ${\displaystyle {\overrightarrow {J_{q}}}=-{\frac {D_{1}c_{1}q_{1}^{2}}{kT}}\nabla (\phi )}$
• Electrical conductivity:
• ${\displaystyle \rho ={\frac {D_{1}c_{1}q_{1}^{2}}{kT}}}$

### Electromigration in Metals

• Two fluxes when electric field is applied to a dilute solution of interstitial atoms in metal
• ${\displaystyle J_{q}\;}$: Flux of conjuction electrons
• ${\displaystyle J_{1}\;}$: Flux of interstitials
• ${\displaystyle F_{q}=E\;}$
• ${\displaystyle F_{q}=-\nabla \phi }$
 ${\displaystyle {\overrightarrow {J_{1}}}=-L_{11}\nabla \mu _{1}+L_{1q}{\overrightarrow {E}}}$
${\displaystyle {\overrightarrow {J_{1}}}=-D_{1}\left(\nabla c_{1}-{\frac {c_{1}\beta }{kT}}{\overrightarrow {E}}\right)}$


### Mass Diffusion in Thermal Gradient

• Interstitial flux with thermal gradient where both heat flow and mass diffusion of interstitial component occurs:
 ${\displaystyle {\overrightarrow {J_{1}}}=-L_{11}\nabla \mu _{1}-{\frac {L_{1q}}{T}}\nabla T}$
${\displaystyle {\overrightarrow {J_{1}}}=-D_{1}\nabla c_{1}-{\frac {D_{1}c_{1}Q_{1}^{\mbox{trans}}}{kT^{2}}}\nabla T}$


### Mass Diffusion Driven by Capillarity

• The system consists of two network-constrained components:
• Host atoms
• Vacancies
• No mass flow within the crystal (the crystal C-frame is also the V-frame)
• Constant temperature and no electric field
• ${\displaystyle {\overrightarrow {J_{A}}}=L_{AA}{\overrightarrow {F_{A}}}}$
• ${\displaystyle {\overrightarrow {J_{A}}}=-L_{AA}\nabla \Phi _{A}}$
• ${\displaystyle {\overrightarrow {J_{A}}}=-L_{AA}\nabla (\mu _{A}-\mu _{v})}$
• ${\displaystyle {\overrightarrow {J_{V}}}=-{\overrightarrow {J_{A}}}}$

## Fick's Second Law

### Diffusion Equation in the General Form

 ${\displaystyle {\frac {\partial c}{\partial t}}={\dot {n}}-\nabla \cdot {\overrightarrow {J}}}$

${\displaystyle {\dot {n}}}$: source or sink term
${\displaystyle {\overrightarrow {J}}}$: any flux in a V-frame

### Fick's Second Law

 ${\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot {\overrightarrow {J}}}$
${\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot (D\nabla c)}$


### Linearization of Diffusion Equation

 ${\displaystyle {\frac {\partial c}{\partial t}}=D_{o}\nabla ^{2}c}$


### Heat Equation

 ${\displaystyle {\frac {\partial h}{\partial t}}=-\nabla \cdot {\overrightarrow {J_{Q}}}}$
${\displaystyle c_{P}{\frac {\partial T}{\partial t}}=-\nabla \cdot (-K\nabla T)}$
${\displaystyle {\frac {\partial T}{\partial t}}=\nabla \cdot \left({\frac {K}{c_{P}}}\nabla T\right)}$
${\displaystyle \nabla \cdot (\kappa \nabla T)}$

${\displaystyle h\;}$: enthalpy density
${\displaystyle c_{P}\;}$: heat capacity
${\displaystyle K/c_{p}=\kappa \;}$: thermal diffusivity

### Constant Diffusivity

${\displaystyle {\frac {\partial c}{\partial t}}=D\nabla ^{2}c}$

### One-Dimensional Diffusion Along x from an Initial Step Function

${\displaystyle c(x,t)={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4Dt}}}\right)}$

### Localized Source

 ${\displaystyle c(x,t)={\frac {c_{o}\Delta x}{\sqrt {4\pi Dt}}}e^{-x^{2}/(4Dt)}}$
${\displaystyle c(x,t)={\frac {n_{d}}{\sqrt {4\pi Dt}}}e^{-x^{2}/(4Dt)}}$

• Source strength, ${\displaystyle n_{d}=\int _{-\infty }^{\infty }c(x)dx}$

### Diffusivity as a Function of Concentration

 ${\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot [D(c)\nabla c]}$

• Interdiffusivity: ${\displaystyle D(c_{1})=-{\frac {1}{2\tau }}{\frac {dx}{dc_{1}}}\int _{c_{1}^{R}}^{c_{1}}x(c')dc'}$

### Diffusivity as a Function of Time

 ${\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot [D(t)\nabla c]}$
${\displaystyle {\frac {\partial c}{\partial t}}=D(t)\nabla ^{2}c]}$

• Change of variable: ${\displaystyle \tau _{D}=\int _{0}^{t}D(t')dt'}$
• Transformed equation: ${\displaystyle {\frac {\partial c}{\partial \tau _{D}}}=\nabla ^{2}c}$
• Solution:
${\displaystyle c(x,\tau _{D})={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4\tau _{D}}}}\right)}$
${\displaystyle c(x,t)={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4\int _{0}^{t}D(t')dt'}}}\right)}$

### Diffusivity as a Function of Direction

 ${\displaystyle {\overrightarrow {J}}=-\mathbf {D} \nabla c}$

• The diagonal elements of ${\displaystyle {\hat {\mathbf {D} }}}$ are the eigenvalues of ${\displaystyle \mathbf {D} }$, and the coordinate system of ${\displaystyle {\hat {\mathbf {D} }}}$ defines the principal axes.
• ${\displaystyle {\frac {\partial c}{\partial t}}=-\nabla \cdot {\overrightarrow {J}}}$
• ${\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot {\hat {\mathbf {D} }}\nabla c}$
• Relation of ${\displaystyle \mathbf {D} }$ and ${\displaystyle {\hat {\mathbf {D} }}}$:
 ${\displaystyle {\hat {\mathbf {D} }}={\underline {R}}\mathbf {D} {\underline {R}}^{-1}}$


### Harmonic Functions

 ${\displaystyle \nabla ^{2}c=0}$


### One Dimension

 ${\displaystyle J=-D{\frac {dc}{dx}}}$
${\displaystyle J=D{\frac {c^{0}-c^{L}}{L}}}$


### Cylindrical Shell

• Laplacian Operator: ${\displaystyle \nabla ^{2}c={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial c}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}c}{\partial \theta ^{2}}}+{\frac {\partial ^{2}c}{\partial x^{2}}}}$
• Integrate Twice and Apply the Boundary Conditions:
 ${\displaystyle c(r)=c^{\mbox{in}}-{\frac {c^{\mbox{in}}-c^{\mbox{out}}}{\ln(r^{\mbox{out}}/r^{\mbox{in}})}}\ln \left({\frac {r}{r^{\mbox{in}}}}\right)}$


### Spherical Shell

• Laplacian operator in spherical coordinates

${\displaystyle \nabla ^{2}c={\frac {1}{r^{2}}}\left[{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial c}{\partial r}}\right)+{\frac {1}{\sin {\theta }}}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial c}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}c}{\partial \phi ^{2}}}\right]}$

### Variable Diffusivity

• ${\displaystyle D}$ varies with position
 ${\displaystyle \partial \cdot (D\nabla c)=0}$

• Solution is obtained by integration:
 ${\displaystyle c(x)=c(x_{1})+a_{1}\int _{x_{1}}^{x}{\frac {d\zeta }{D(\zeta )}}}$


### Infinite Media with Instantaneous Localized Source

 ${\displaystyle c(x,y,z,t)={\frac {n_{d_{x}}}{\sqrt {4\pi Dt}}}e^{-x^{2}/(4Dt)}\times {\frac {n_{d_{y}}}{\sqrt {4\pi Dt}}}e^{-y^{2}/(4Dt)}\times {\frac {n_{d_{z}}}{\sqrt {4\pi Dt}}}e^{-z^{2}/(4Dt)}}$


### Solutions with the Error Function

• Uniform distribution of point, line, or plana source placed along ${\displaystyle x>0}$
• Contribution at a general position ${\displaystyle x}$ from the source:
${\displaystyle c_{\zeta }(x,t)={\frac {c_{o}d\zeta }{\sqrt {4\pi Dt}}}e^{-(\zeta -x)^{2}/(4Dt)}}$
• Integral over all sources:
${\displaystyle c(x,t)={\frac {c_{o}}{\sqrt {4\pi Dt}}}\int _{0}^{\infty }e^{-(\zeta -x)^{2}/(4Dt)}d\zeta }$
 ${\displaystyle c(x,t)={\frac {c_{o}}{2}}+{\frac {c_{o}}{2}}{\mbox{erf}}\left({\frac {x}{2{\sqrt {Dt}}}}\right)}$


### Method of Separation of Variables

• System : Three Dimensions, ${\displaystyle (x,y,z)}$
• Equation : ${\displaystyle {\frac {dc}{dt}}=D\nabla ^{2}c}$
• Solution : ${\displaystyle c(r,\theta ,z,t)=R(r)\Theta (\theta )Z(z)T(t)\;}$

### Method of Laplace Transforms

• Laplace transform of a function ${\displaystyle f(x,t)\;}$
 ${\displaystyle L\{f(x,t)\}={\hat {f}}(x,p)}$
${\displaystyle L\{f(x,t)\}=\int _{0}^{\infty }e^{-pt}f(x,t)dt}$


## Atomic Models of Diffusion

### Model of One-Particle with Step Potential-Energy Wells

 ${\displaystyle \mathrm {T} '={\sqrt {\frac {kT}{2\pi m}}}{\frac {1}{L^{\mbox{well}}}}e^{-(E^{A}-E^{\mbox{well}})/(kT)}}$
${\displaystyle \mathrm {T} '={\sqrt {\frac {kT}{2\pi m}}}{\frac {1}{L^{\mbox{well}}}}e^{-(E^{m})/(kT)}}$


### Model of One-Particle with Step Potential-Energy Wells

 ${\displaystyle \mathrm {T} '={\frac {1}{2\pi }}{\sqrt {\frac {\beta }{m}}}e^{-(E^{A}-E^{\mbox{well}})/(kT)}}$
${\displaystyle \mathrm {T} '=\nu e^{-E^{m}/(kT)}}$


### Many-Body Model

 ${\displaystyle \mathrm {T} '=\nu e^{-G^{m}/(kT)}}$


### Diffusion as Series of Discrete Jumps

 ${\displaystyle =N_{\tau }+2<(\sum _{j=1}^{N_{\tau }-1}\sum _{i=1}^{N_{\tau -j}}|{\overrightarrow {r_{i}}}||{\overrightarrow {r}}_{i+j}|\cos \theta _{i,i+j}>}$


### Diffusivity and Mean-Square Particle Displacement

 ${\displaystyle =6D\tau \;}$


### Relation of Macroscopic Diffusivity and Microscopic Jump Parameters

 ${\displaystyle D={\frac {\mathrm {T} }{2}}}$


### Diffusion and Correlated Jumps

• Correlation factor:
 ${\displaystyle \mathbf {f} =1+{\frac {2}{N_{\tau }}}<(\sum _{j=1}^{N_{\tau }-1}\sum _{i=1}^{N_{\tau -j}}|{\overrightarrow {r_{i}}}||{\overrightarrow {r}}_{i+j}|\cos \theta _{i,i+j}>}$

• Macroscopic Diffusivity and Microscopic Parameters:
 ${\displaystyle D={\frac {N_{\tau }}{6\tau }}\mathbf {f} }$
${\displaystyle D={\frac {\mathrm {T} \tau }{6\tau }}\mathbf {f} }$
${\displaystyle D={\frac {\mathrm {T} }{6}}\mathbf {f} }$


## Atomic Models of Diffusivity

### Metals

#### Correlation Factor

 ${\displaystyle \mathbf {f} ={\frac {1+<\cos \theta >}{1-<\cos \theta >}}}$
${\displaystyle \mathbf {f} \approx {\frac {z-1}{z+1}}}$


#### Isotope Effect

 ${\displaystyle {\frac {^{*}D(mass1)}{^{*}D(mass2)}}={\frac {\mathrm {T} _{1}}{\mathrm {T} _{2}}}={\frac {\mathrm {T} '_{1}}{\mathrm {T} '_{2}}}={\frac {\nu _{1}}{\nu _{2}}}={\frac {m_{1}}{m_{2}}}}$