Blackbody Dynamics

Authors

Model of One Blackbody

In our series on blackbody radiation we have mostly focussed on radiative equilibrium, and so let us now consider the more general question of the dynamics of radiative interaction of  two blackbodies of different temperature.

We recall our model for one blackbody subject to radiative forcing, explored in Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation, as a wave equation expressing force balance:

• $U_{tt} - U_{xx} - \gamma U_{ttt} - \delta^2U_{xxt} = f$      (1)

where the subindices indicate differentiation with respect to space $x$ and time $t$, and

1. $U_{tt} - U_{xx}$ is out-of-equilibrium force of a vibrating string with displacement $U$,
2. $- \gamma U_{ttt}$ is the Abraham-Lorentz (radiation reaction) force with $\gamma$ a small positive parameter,
3. $- \delta^2U_{xxt}$ is a friction force replacing the radiation reaction force for frequencies larger than a cut-off frequency $\frac{T}{h}$ and then contributing to the internal energy,
4. $\delta =\frac{h}{T}$ is a smallest coordination length with $h$ a measure of finite precision,
5. $T$ is the common energy/temperature of each frequency of the vibrating string,
6. $f$ is exterior forcing.

The model is specified by the parameters $\gamma$ and $h$. It is shown in Universality of Blackbody Radiation that all blackbodies can be assumed to have the same value of the radiation coefficient $\gamma$ and the cut-off (precision $h$),  given as the values of a chosen reference blackbody  with the property that $\gamma$ is maximal and $h$ minimal.

Stationary periodic solutions $U$ satisfy the energy balance

• $R + H = F$
• $R=\int\gamma U_{tt}^2dxdt$
• $H = \int \delta^2 U_{xt}^2dxdt$
• $F=\int f^2dxdt$,

which expresses that all incident radiation $F$ is absorbed and is either re-emitted as radiation $R$ or stored as internal energy from heating $H$ with a switch from $R$ to $H$ at the cut-off frequency. We here assume that all frequencies have the same energy $\int U_{\nu ,t}^2dxdt$, where $U_{\nu}$ is the amplitude of frequency $\nu$, and we refer to the common value $\int U_{\nu ,t}^2dxdt =T$ as the temperature.

With dynamics the wave equation (1) expressing force balance is complemented by an equation for the total energy $E$:

• $E(t)-E(0)+ R =\int_0^t\int fUdxds$        (2)

expressing that the change $E(t)-E(0)$ is balanced by the outgoing radiation $R$ and absorbed energy from the forcing $\int fUdxdt$, where $E=e+\epsilon$ with $e$ the string energy and  $\epsilon$ the internal energy as accumulated  dissipated energy $\int \delta^2 U_{xt}^2dx$. Equivalently, the change of the internal energy $\epsilon$ is given as

• $\epsilon (t)-\epsilon (0) = H$.

The temperature $T$ connects to $E$ by $T \sim \sqrt{h}E$, assuming all frequencies $U_\nu$ of $U$ have the same energy $U_{\nu ,t}^2=T$, because

• $e \sim\sum_{\nu\le\frac{T}{h}}$ $U_{\nu ,t}^2 =\frac{T^2}{h}$

assuming $\epsilon$ being dominated by $e$.

Our model thus consists of (1) + (2) combined with a mechanism for equidistribution of energy over all frequencies.

Let us now consider two blackbodies in radiative contact, one body $B$ with amplitude $U$ sharing a common forcing $F$ with another blackbody $\bar B$ with amplitude $\bar U$, modeled by the wave equation:

• $U_{tt} - U_{xx} - \gamma U_{ttt} - \delta^2U_{xxt} = F =$ $\bar U_{tt} - \bar U_{xx} - \gamma \bar U_{ttt} - \delta^2\bar U_{xxt}$
• $E_t + R =\int fUdxdt$ and $\bar E_t + \bar R =\int f\bar Udxdt$.

This system describes the dynamic interaction of $B$ and $\bar B$, with given initial values of $U$, $U_t$ and $E$ for $B$ and the same for $\bar B$.

In dynamic interaction different frequencies will have different times scales and thus to maintain that all frequencies have the same temperature some mechanism to this effect will have to be adjoined to the model.  We may think of this effect as a form of diffusion acting on frequencies.

We compare with the case of acoustic damping with an acoustic damping term of the form $\mu U_t$ in which case all frequencies will have the same damping and thus well tempered distributions will be preserved under dynamic interaction. Recall that a piano as a blackbody with acoustic damping is isotempered in the sense that the sustain of different tones is the same.

Let us now see what the model tells in different basic cases:

1. Both below cut-off.

Let us now consider the basic case of interaction with all frequencies below cut-off for both $B$ and $\bar B$. The difference $W = U -\bar U$ then satisfies the damped wave equation

• $W_{tt} - W_{xx} - \gamma W_{ttt} = 0$

which upon multiplication by $W_t$ and integration in space and time gives for $t>0$ (modulo two terms from integration by parts in time with small effect if the time scale is not short):

• $G(t) = G(0) -\int_0^t\int\gamma W_{tt}^2dxds$
• $G (t) =\int \frac{1}{2}(W_t^2+W_x^2)dx$.

It follows that $G(t)$ decays in time which effectively means equilibration in energy with a transfer of energy from the warmer to the colder body.

The effect of the dissipative radiation term ($-\gamma U_{ttt}$) is that the difference in energy (and thus temperature) between the two bodies decreases with time: Energy is transferred from the warmer to the colder body. We have thus proved a 2nd Law as an effect of dissipative radiation.

2. One below one above cut-off

For frequencies below cut-off for $B$ and above cut-off for $\bar B$, assuming $B$ is the warmer, the model shows a transfer from $R$ into $\bar H$ with a heating effect on $\bar B$ as a result of the energy balance

• $R + H = \bar R + \bar H$
with $H=0$ and $\bar R=0$.

3. Both above cut-off

This is analogous to case 1.