# Thermal diffusion of Gaussian

Thermal diffusion of Gaussian refers to the specific solution

$$\Psi=\Psi(x,t)=\alpha(t) \exp(-\beta(t) x^2)=\alpha \epsilon$$

of the equation of thermal conductivity

$$\dot \Psi=\Psi''$$

where dot differentiates with respect to time $$t$$ and prime differentiates with respect the spatial variables $$x$$. Parameters $$\alpha$$ and $$\beta$$ are supposed to be functions of time. Evolution of these parameters can be determined from the thermal diffusion equation.

## Calculation of $$\alpha$$ and $$\beta$$

Derivative of $$\Psi$$ with respect to $$j$$th coordinates appears as follows:

$$\Psi'_{,j}=-2 \alpha \beta x_j \epsilon$$

and the second derivative

$$\Psi''_{,j,j}=-2 \alpha \beta x_{j,j} \epsilon+ 4\alpha\beta^2 x_j x_j \epsilon =-6 \alpha \beta \epsilon+ 4\alpha\beta^2 x^2 \epsilon$$

where summation with respect to repeating index $$j$$ is assumed. For 3-dimensional case, $$j$$ takes values 1,2,3; so, $$x_{j,j}=3$$.

The time derivative of solution is expressed as follows:

$$\dot \Psi = \dot \alpha\epsilon - \alpha \dot \beta \epsilon x^2$$

Then, the thermal diffusion equation gives

$$\dot\alpha=-6\alpha \beta~$$, and $$~ \dot \beta=-4\beta^2$$

The last equation has solution $$~\displaystyle \beta=\frac{1}{4t}$$

Then, for $$\alpha$$,

$$\displaystyle \frac{\dot\alpha}{\alpha}=-6 \frac{1}{4t}= -\frac{3}{2} \frac{1}{t}$$

$$\displaystyle (\ln(\alpha))^{\bullet} = -\frac{3}{2} \frac{1}{t}$$

$$\displaystyle \ln(\alpha) = -\frac{3}{2} \ln(t)=\ln\big( t^{-3/2} \big)$$

$$\alpha=t^{-3/2}$$

## Constants of integration

In the deduction above, the constants of integration are omitted. They can be recovered using the translational invariance of the diffusion equation. For example, after such a recovery, the temperature at coordinate zero can be expressed as follows:

$$y=y(t)=a + b\, (t\!-\!c)^{-3/2}$$

where $$a$$ refers to the background constant temperature, $$c$$ indicate the place of singularity, and $$b$$ refers to the thermal capacity and thermal conductivity, that can easy included into the initial equation by scaling of the coordinate $$x$$ and/or time $$t$$.

Such a representation of temperature can be used to fit experimental data of evolution of temperature at the center of localised (punctual) warming or cooling.

## Recovery of dimensions

The scaling of coordinate and/or scaling of time allows to write the temperature distribution as follows:

$$\displaystyle \Phi=\Phi(X,T)= \frac{W r^{1/2}T^{-3/2} }{\pi^{3/2} s^{3/2} } \exp\left( - \frac{r/s}{4T} X^2 \right)$$

where
$$X$$ are dimensional coordinates, proportional to $$x$$
$$T$$ is dimensional time, proportional to $$t$$
$$r$$ is thermal capacity of the material
$$s$$ is its termal conductivity
$$W$$ is total energy of pulse used to hit the material. In the case, if the half–space is occupied by the material, this $$W$$ is doubled energy of the initial heating pulse.

This $$\Phi$$ satisfies the physical, dimensional equation of thermal conductivity

$$r\, \dot \Phi = k\, \Phi ''$$

Measurements of temperature at $$X=0$$ as function of time and approximating it with power function $$\displaystyle f(T)= \frac{W }{\pi^{3/2}} q^{1/2} T^{-3/2}$$ gives the fitting parameter $$q=r/s^3$$.

In such way, with the single pulse, ratio of the thermal capacity of the material to the cube of the thermal conductivity can be measured.

## References

http://web.csulb.edu/~kmerry/FourierAnalysis/Fourier_09.pdf Kent G. Merryfield. 9. The Gaussian and the Heat Equation on the Line.