  We show that in highly multimoded nonlinear photonic systems, the optical
thermodynamic pressures emerging from different species of the optical field
obey Dalton's law of partial pressures. In multimode settings, the optical
thermodynamic pressure is defined as the conjugate to the extensive variable
associated with the system's total number of modes and is directly related to
the actual electrodynamic radiation forces exerted at the physical boundaries
of the system. Here, we extend this notion to photonic configuration supporting
different species of the optical field. Under thermal equilibrium conditions,
we formally derive an equation that establishes a direct link between the
partial thermodynamic pressures and the electrodynamic radiation pressures
exerted by each polarization species. Our theoretical framework provides a
straightforward approach for quantifying the total radiation pressures through
the system's thermodynamics variables without invoking the Maxwell stress
tensor formalism. In essence, we show that the total electrodynamic pressure in
such arrangements can be obtained in an effortless manner from initial
excitation conditions, thus avoiding time-consuming simulations of the utterly
complex multimode dynamics. To illustrate the validity of our results, we carry
out numerical simulations in multimoded nonlinear optical structures supporting
two polarization species and demonstrate excellent agreement with the Maxwell
stress tensor method.
