Generic Model of Resonant Interaction Matter-Wave

I analyze in Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation a mathematical model of radiative heat transfer of the form:

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

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

1. $U_{tt} - U_{xx}$: material force from vibrating string with U displacement
2. $- \gamma U_{ttt}$: radiation pressure from outgoing radiation = emission
3. $- \delta^2U_{xxt}$: viscous force from internal heating = absorption
4. $f$ is exterior forcing,

and $\gamma$ and $\delta^2$ are (small) positive constants connected to dissipative losses as outgoing radiation = emission and internal heating = absorption. Let me here clarify the physics behind the dissipative terms:

• $- \gamma U_{ttt}$ represents the Abraham-Lorentz (radiation reaction) force
• $- \delta^2U_{xxt}$ represents a viscous force from internal friction in the string.

The corresponding dissipation energies are obtained by multiplication of the forces with $U_t$ followed by integration (by parts) in space-time over a period, assuming periodicity, to give

• $R=\int \gamma U_{tt}^2dxdt$ Larmor's formula for radiance $R$
• $IE =\int \delta^2 U_{xt}^2dxdt$ internal energy IE stored as heat energy.

Recall that damping in the wave equation $U_{tt} - U_{xx} = f$ can take the from

$\nu U_{t}$ with $\nu$ a viscosity coefficient, or as above $-\gamma U_{ttt}$ and $- \delta^2U_{xxt}$, thus with the following derivatives:

• $U_{t}$
• $U_{ttt}$
• $U_{xxt}$

where $U_t$ appears in acoustic viscous damping, and can be seen as a simplified version of both $U_{ttt}$ and $U_{xxt}$.

The above model thus can be viewed as the generic model of wave motion with damping describing of interaction matter (string) and electromagnetic waves (radiation term and forcing) with a very rich area of application including:

• radiative heat transfer
• acoustic resonance in string instruments.

The mathematical analysis exhibits a fundamental phenomenon of near resonance in wave motion with small damping, as explained in How to Push a Swing: In near resonance the forcing is in phase with the displacement, while in perfect resonance (with large damping) the forcing is in phase with the velocity. The effect is that in near resonance the exterior forcing is balanced by both the string force and the Abraham-Lorentz radiation force, while in perfect resonance only by the radiation force.

Near resonance appears when the radiative damping is small ($\gamma$ is small), which is the real case of radiation with full interaction between matter (string) and radiation.