Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

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Let interaction events occur at times {ti}\{t_i\}{ti} with energy quanta ΔEi=hνi\Delta E_i = h\nu_iΔEi=hνi. Define the instantaneous energy budget (including kinetic/bound EM contributions) as

E(t)=γ(t)M0c2+∑iΔEi Θ(t−ti)−Lrad(t),E(t)=\gamma(t)M_0c^2+\sum_i \Delta E_i\,\Theta(t-t_i)-\mathcal{L}_{\rm rad}(t),E(t)=γ(t)M0c2+i∑ΔEiΘ(t−ti)−Lrad(t),
where Lrad\mathcal{L}_{\rm rad}Lrad is the outgoing radiative loss (Poynting flux integrated over a sphere enclosing the system) and Θ\ThetaΘ is the Heaviside step.

Stationary regime (ergodic/time-averaged limit):

(γM0c2)∞:=lim⁡T→∞1T∫0T[E(t)+Lrad(t)−∑i:ti≤tΔEi] dt.(\gamma M_0 c^2)_\infty
:= \lim_{T\to\infty}\frac{1}{T}\int_0^T \Big[E(t) + \mathcal{L}_{\rm rad}(t) - \sum_{i:t_i\le t}\Delta E_i\Big]\,dt.(γM0c2)∞:=T→∞limT1∫0T[E(t)+Lrad(t)−i:ti≤t∑ΔEi]dt.
Interpretation: the asymptotic, time-averaged non-radiative energy equals the “effective” γM0c2\gamma M_0 c^2γM0c2. This moves your subscript “∞\infty∞” from an informal marker to a precise long-time average of a jump–diffusion process driven by photon events.

==== If events form a stationary process with rate Γ\GammaΓ and mean quantum ⟨ΔE⟩\langle\Delta E\rangle⟨ΔE⟩, write ====

ddt⟨E⟩=Γ ⟨ΔE⟩−⟨Lrad⟩.\frac{d}{dt}\langle E\rangle = \Gamma\,\langle\Delta E\rangle - \langle\mathcal{L}_{\rm rad}\rangle.dtd⟨E⟩=Γ⟨ΔE⟩−⟨Lrad⟩.
A steady state gives ⟨E⟩∞=const\langle E\rangle_\infty = \text{const}⟨E⟩∞=const, so

(γM0c2)∞≡⟨E⟩∞.(\gamma M_0c^2)_\infty \equiv \langle E\rangle_\infty.(γM0c2)∞≡⟨E⟩∞.
This makes your notation mathematically clean: “∞_\infty∞” = stationary time-average under continuous photon exchange.