How Does Boltzmann’s Transformation Reduce the Heat Equation?

Boltzmann’s transformation is a similarity change of variables used to reduce certain diffusion equations from partial differential equations (PDEs) to ordinary differential equations (ODEs). In the common heat-conduction setting, it combines position and time into a single variable proportional to $x/\sqrt{t}$.

Here is the chain-rule derivation for the one-dimensional heat equation, followed by a worked example. The term “Boltzmann transformation” here refers to a diffusion similarity variable, not to the Boltzmann factor $e^{-E/(k_BT)}$ from statistical mechanics.

The similarity variable

Consider heat flow in a uniform material with constant thermal diffusivity $\alpha$:

$$\frac{\partial T}{\partial t}=\alpha\frac{\partial^2 T}{\partial x^2}.$$

The diffusivity has units of length squared per time. Consequently, a natural length scale for diffusion over time $t$ is proportional to $\sqrt{\alpha t}$. Define the dimensionless similarity variable

$$\eta=\frac{x}{2\sqrt{\alpha t}},$$

and assume the temperature profile can be written as $T(x,t)=\Theta(\eta)$. The factor of 2 is a convenient convention; choosing $x/\sqrt{\alpha t}$ instead is also valid, but changes the coefficients in the resulting ODE.

This assumption is appropriate when the boundary and initial conditions allow a self-similar profile. It does not mean every solution of the heat equation depends only on this one variable.

Apply the chain rule

When differentiating with respect to time, hold $x$ fixed. When differentiating with respect to position, hold $t$ fixed. From the definition of $\eta$,

$$\frac{\partial\eta}{\partial x}=\frac{1}{2\sqrt{\alpha t}}, \qquad \frac{\partial\eta}{\partial t}=-\frac{\eta}{2t}.$$

Therefore,

$$\frac{\partial T}{\partial t} =\Theta'(\eta)\frac{\partial\eta}{\partial t} =-\frac{\eta}{2t}\Theta'(\eta),$$

and

$$\frac{\partial T}{\partial x} =\frac{1}{2\sqrt{\alpha t}}\Theta'(\eta), \qquad \frac{\partial^2 T}{\partial x^2} =\frac{1}{4\alpha t}\Theta''(\eta).$$

The second spatial derivative includes no extra term from differentiating its prefactor, because that prefactor depends on $t$, not $x$.

Substitute into the heat equation

Putting the derivatives into $T_t=\alpha T_{xx}$ gives

$$-\frac{\eta}{2t}\Theta' =\frac{1}{4t}\Theta''.$$

Multiplying by $4t$ yields the ODE

$$\Theta''+2\eta\Theta'=0.$$

To solve it, set $p=\Theta'$. Then $p'+2\eta p=0$, so $p=C e^{-\eta^2}$. Integrating once more gives

$$\Theta(\eta)=A+B\,\operatorname{erf}(\eta),$$

where $A$ and $B$ are constants and

$$\operatorname{erf}(\eta)=\frac{2}{\sqrt{\pi}}\int_0^\eta e^{-s^2}\,ds$$

is the error function. The transformation reduces the differential equation; the physical conditions are still needed to fix the constants.

Worked example: a suddenly heated half-space

Suppose a solid occupies $x\geq 0$. Initially its temperature is $T_i$, and from time $t=0$ onward its surface at $x=0$ is held at temperature $T_s$. Far from the surface, the temperature remains $T_i$:

$$T(0,t)=T_s, \qquad T(x,t)\longrightarrow T_i\quad\text{as }x\longrightarrow\infty.$$

Since $\operatorname{erf}(0)=0$ and $\operatorname{erf}(\eta)\to 1$ as $\eta\to\infty$, applying these two conditions to $A+B\operatorname{erf}(\eta)$ gives $A=T_s$ and $B=T_i-T_s$. Thus

$$T(x,t)=T_s+(T_i-T_s)\operatorname{erf}\!\left(\frac{x}{2\sqrt{\alpha t}}\right).$$

At the surface, the argument is zero and the solution gives $T_s$. For fixed $t>0$, as $x$ tends to infinity the error function tends to one, so the solution approaches $T_i$. For fixed $x>0$, as $t$ tends to zero the argument tends to infinity, recovering the initial temperature away from the surface. The idealized surface temperature changes abruptly at $t=0$, so the point $x=0,t=0$ is a discontinuity in the prescribed conditions.

More general diffusion equations

For a concentration or temperature-dependent diffusivity, a common model is

$$\frac{\partial C}{\partial t} =\frac{\partial}{\partial x}\left(D(C)\frac{\partial C}{\partial x}\right).$$

Using $\eta=x/\sqrt{t}$ and $C=C(\eta)$ gives

$$-\frac{\eta}{2}C' =\frac{d}{d\eta}\left(D(C)C'\right).$$

This is still an ODE, although it may be nonlinear. The exact similarity variable and any constant scale factors depend on the equation and its conditions. In diffusion experiments, this same position-over-square-root-of-time scaling is used in Boltzmann–Matano analysis to infer a concentration-dependent diffusivity.

Common mistakes

• Differentiating as if $\eta$ were independent of $t$. It is a function of both $x$ and $t$; use the chain rule for each partial derivative.
• Forgetting what is held fixed. In $T_t$, hold $x$ fixed; in $T_x$, hold $t$ fixed.
• Mixing conventions for $\eta$. With $\eta=x/\sqrt{\alpha t}$, the ODE is $\Theta''+(\eta/2)\Theta'=0$. With the convention used above, $\eta=x/(2\sqrt{\alpha t})$, it is $\Theta''+2\eta\Theta'=0$. Re-derive the coefficients whenever you change the definition.
• Treating the similarity form as universal. It is an ansatz justified by the equation and conditions, not a replacement for all possible solutions.
• Using the wrong physical conditions. The ODE has two constants; specify two independent conditions to select the desired profile.