Derivation of the Singlet Yield Equation in the Radical Pair Mechanism
regarding the central observable in the radical pair mechanism (RPM) for avian magnetoreception and other quantum-biological processes
Derivation of the Singlet Yield Equation in the Radical Pair Mechanism
by Grok at my instruction
The singlet yield ΦS \Phi_S ΦS is the central observable in the radical pair mechanism (RPM) for avian magnetoreception and other quantum-biological processes. It quantifies the probability that the radical pair ends up in the singlet state and undergoes the downstream chemical reaction that produces the biological signal (e.g., the magnetic compass sense in birds).
Here is a complete, step-by-step derivation from first principles, using the density matrix formalism for open quantum systems.
Step 1: The Radical Pair State and Density Matrix
A radical pair is formed in a spin-correlated state, typically starting in the singlet state after photo-induced electron transfer:
Step 2: The Spin Hamiltonian
(For open systems we add decoherence later.)
EQUATION AVAILABLE
Step 3: Projectors for Singlet and Triplet Manifolds
The singlet projector is:
Step 4: Reaction Kinetics — The Singlet (or Triplet) Reaction Rate
Step 5: Integrating to Obtain the Singlet Yield
Step 6: Including Decoherence (Open Quantum System)
In real biological environments we must include relaxation and dephasing. The full dynamics follow the stochastic Liouville (or Lindblad) equation:
For exact non-Markovian dynamics, the hierarchical equations of motion (HEOM) are used, which is the state-of-the-art method in 2025–2026 simulations of cryptochrome and FMO complexes.
Step 7: Physical Interpretation and Connection to Magnetoreception
Final Singlet Yield Equation (Compact Form)
This equation is the quantitative heart of the radical pair mechanism. It directly links quantum spin coherence to the biological magnetic compass sense.
EQUATION AVAILABLE
