Module 4 · Theory
The Nakajima–Zwanzig Memory Kernel
The Nakajima–Zwanzig (NZ) equation is a cornerstone of non-equilibrium statistical mechanics. It governs the exact, non-Markovian dynamics of a quantum system \(S\) coupled to a bath \(B\), after the bath degrees of freedom have been formally eliminated.
4.1 Setup: System–Bath Decomposition
The total Hamiltonian decomposes as:
\[ H = H_S \otimes \mathbb{1}_B + \mathbb{1}_S \otimes H_B + H_{SB} \tag{4.1}\]
In the protein context:
- System S: the reactive subsystem (proton-transfer coordinate, catalytic residue, transferring hydrogen).
- Bath B: the protein scaffold, distal residues, and solvent.
- Coupling \(H_{SB}\): electrostatic and steric interaction between reactive site and scaffold.
4.2 Zwanzig Projection Operators
Define projection superoperators on the Liouville space of total density matrices:
\[ \mathcal{P}\rho = \rho_B^{\text{eq}} \otimes \text{Tr}_B[\rho], \qquad \mathcal{Q} = 1 - \mathcal{P} \tag{4.2}\]
where \(\rho_B^{\text{eq}} = e^{-\beta H_B} / Z_B\) is the equilibrium bath state. \(\mathcal{P}\) projects onto the “relevant” part of the density matrix (system state tensored with equilibrium bath); \(\mathcal{Q}\) retains bath correlations.
4.3 The Generalised Master Equation
Applying \(\mathcal{P}\) to the von Neumann equation for the total density matrix and eliminating \(\mathcal{Q}\rho\) yields the Nakajima–Zwanzig equation for the reduced density matrix \(\rho_S(t) = \text{Tr}_B[\rho(t)]\):
\[ \frac{d}{dt}\rho_S(t) = -i\mathcal{L}_S\,\rho_S(t) - \int_0^t \mathcal{K}(t - t')\,\rho_S(t')\,dt' + \mathcal{I}(t) \tag{4.3}\]
The three terms are: (i) the coherent system evolution under \(\mathcal{L}_S = [H_S, \cdot]/\hbar\); (ii) the memory kernel \(\mathcal{K}(t-t')\) — a convolution integral encoding the retarded influence of all past system states through the bath; (iii) the inhomogeneous term \(\mathcal{I}(t)\) that vanishes if the initial state is factorised: \(\rho(0) = \rho_S(0) \otimes \rho_B^{\text{eq}}\).
4.4 The Memory Kernel in Detail
\[ \mathcal{K}(t - t') = \text{Tr}_B\!\left[\mathcal{P}\mathcal{L}_{SB}\, e^{-i\mathcal{Q}\mathcal{L}(t-t')}\, \mathcal{Q}\mathcal{L}_{SB}\,\rho_B^{\text{eq}}\right] \tag{4.4}\]
Key physical content of \(\mathcal{K}(t-t')\):
- Memory timescale \(\tau_{\text{mem}}\): the width of \(\mathcal{K}\) in time. If \(\tau_{\text{mem}} \ll \tau_{\text{sys}}\) (bath relaxes fast), the Markovian (Redfield) limit recovers. For protein scaffolds, \(\tau_{\text{mem}} \sim 1\)–100 ps due to slow conformational fluctuations.
- Fluctuation–dissipation: \(\mathcal{K}\) is related to the bath correlation function \(C(t) = \langle H_{SB}(t) H_{SB}(0)\rangle_B\) via the quantum regression theorem.
- Non-Markovian signatures: oscillatory \(\mathcal{K}(t)\) (underdamped bath modes), power-law tails (1/f noise in disordered proteins), and bistable bath states (conformational switches) all produce physically distinct memory effects.
Physical Intuition
The memory kernel is the protein’s answer to the question: “Given that the reactive site was in state \(\rho_S(t')\) a time \(t - t'\) ago, how does the scaffold’s response at the present time \(t\) depend on that past state?” Rigid scaffolds with fast-relaxing modes have short memory; flexible scaffolds with slow collective motions have long memory.