13: Mori-Zwanzig Formulation

13.3.1 Explicit Form and Evolution

The Liouville operator \(\mathcal{L}\) generates time evolution via the Poisson bracket:

\[ \mathcal{L} A=\{A, H\}=\sum_{\alpha}\left(\frac{\partial A}{\partial q_{\alpha}} \frac{\partial H}{\partial p_{\alpha}}-\frac{\partial A}{\partial p_{\alpha}} \frac{\partial H}{\partial q_{\alpha}}\right) . \tag{13.1} \label{13.1} \]

The equation of motion for \(A(t)\) is given formally by:

\[ \frac{d}{d t} A(t)=\mathcal{L} A(t), \quad A(t)=e^{\mathcal{L} t} A(0) . \tag{13.2} \label{13.2} \]

13.3.2 Projection Operators

The projector \(\mathcal{P}\) projects an arbitrary vector \(B\) onto the relevant variable \(A\) :

\[ \mathcal{P} B=\langle B, A\rangle \mathbb{C}^{-1} A(0), \tag{13.3} \label{13.3} \]

where \(\mathbb{C}=\langle A, A\rangle\) is the correlation matrix. The complementary projector \(\mathcal{Q}=1-\mathcal{P}\) projects onto the orthogonal subspace. These operators satisfy \(\mathcal{P}^{2}=\mathcal{P}, \mathcal{Q}^{2}=\mathcal{Q}\), and \(\mathcal{Q} \mathcal{P}=0\).

13.4.1 Alternate Derivation of the Mori-Zwanzig Identity

Show the identity

\[ e^{\mathcal{L} t}=e^{O_{1} \mathcal{L} t}+\int_{0}^{t} d s e^{\mathcal{L}(t-s)} O_{2} \mathcal{L} e^{O_{1} \mathcal{L} s} \tag{13.9} \label{13.9} \]

holds for any two operators which have properties \(O_{1}+O_{2}=1\). Hint: Use the operator identity

\[ \frac{1}{s-\mathcal{L}}=\frac{1}{s-O_{1} \mathcal{L}}+\frac{1}{s-\mathcal{L}}\left(1-O_{1}\right) \mathcal{L} \frac{1}{s-O_{1} \mathcal{L}} \tag{13.10} \label{13.10} \]

Now because \(O_{1}+O_{2}=1\), the Laplace inversion yields the Mori-Zwanzig Identity. Note that \(e^{\mathcal{Q} \mathcal{L} t} \mathcal{Q}=\mathcal{Q} e^{\mathcal{Q} \mathcal{L} t} \mathcal{Q}\). Hint: Expand the exponential and the left hand side and use \(\mathcal{Q}^{2}=\mathcal{Q}\).

13.5.1 The Systematic Term

The first term in Equation \ref{13.12} uses the definition of \(\mathcal{P}\) :

\[ e^{\mathcal{L} t} \mathcal{P} \mathcal{L} A(0)=e^{\mathcal{L} t}\left(\langle\mathcal{L} A, A\rangle \mathbb{C}^{-1} A(0)\right)=\Omega e^{\mathcal{L} t} A(0)=\Omega A(t), \tag{13.13} \label{13.13} \]

where \(\Omega=\langle\mathcal{L} A, A\rangle \mathbb{C}^{-1}\) is the frequency matrix.

13.5.2 The Memory and Noise Terms

For the second term in Equation \ref{13.12}, we substitute the identity derived in Equation \ref{13.8}:

\[ e^{\mathcal{L} t} \mathcal{Q} \mathcal{L} A(0)=e^{\mathcal{Q} \mathcal{L} t} \mathcal{Q} \mathcal{L} A(0)+\int_{0}^{t} d s e^{\mathcal{L}(t-s)} \mathcal{P} \mathcal{L} e^{\mathcal{Q} \mathcal{L} s} \mathcal{Q} \mathcal{L} A(0) \tag{13.14} \label{13.14} \]

We define the fluctuating force (noise) as:

\[ F(t)=e^{\mathcal{Q} \mathcal{L} t} \mathcal{Q} \mathcal{L} A(0) . \tag{13.15} \label{13.15} \]

The integral term then becomes:

\[ \int_{0}^{t} d s e^{\mathcal{L}(t-s)} \mathcal{P} \mathcal{L} F(s) \tag{13.16} \label{13.16} \]

Applying the projector \(\mathcal{P}\) and using the adjoint property \(\left\langle\mathcal{L} O_{1}, O_{2}\right\rangle=-\left\langle O_{1}, \mathcal{L} O_{2}\right\rangle\) :

\[ \mathcal{P} \mathcal{L} F(s)=\langle\mathcal{L} F(s), A\rangle \mathbb{C}^{-1} A(0)=-\langle F(s), \mathcal{L} A\rangle \mathbb{C}^{-1} A(0) . \tag{13.17} \label{13.17} \]

Since \(F(s)\) is in the \(\mathcal{Q}\)-subspace, we note \(\langle F(s), \mathcal{P} \mathcal{L} A\rangle=0\), thus:

\[ -\langle F(s), \mathcal{L} A\rangle=-\langle F(s), \mathcal{Q} \mathcal{L} A\rangle=-\langle F(s), F(0)\rangle . \tag{13.18} \label{13.18} \]

Substituting this into the integral and moving the propagator \(e^{\mathcal{L}(t-s)}\) onto \(A(0)\) to obtain \(A(t-s)\) :

\[ \int_{0}^{t} d s e^{\mathcal{L}(t-s)}\left(-\langle F(s), F(0)\rangle \mathbb{C}^{-1} A(0)\right)=-\int_{0}^{t} d s K(s) A(t-s) \tag{13.19} \label{13.19} \]

Combining these results yields the Generalized Langevin Equation:

\[ \frac{d}{d t} A(t)=\Omega A(t)-\int_{0}^{t} d s K(s) A(t-s)+F(t) \tag{13.20} \label{13.20} \]

Fluctuating force \(F(t)\) : This represents the effect of all the degrees of freedom orthogonal to \(A\). It captures the random, microscopic fluctuations.

13.5.3 Properties of the Fluctuating Force

13.6.1 Proof of Zero Mean: \(\langle F(t)\rangle=0\)

By definition, the fluctuating force at any time \(t\) is given by \(F(t)=e^{\mathcal{Q} \mathcal{L} t} F(0)\), where \(F(0)=\) \(\mathcal{Q} \mathcal{L} A(0)\). Taking the equilibrium average:

\[ \langle F(t)\rangle=\int d \Gamma \rho(\Gamma) e^{\mathcal{Q} \mathcal{L} t} \mathcal{Q} \mathcal{L} A(0) \tag{13.21} \label{13.21} \]

Using the property that the projector \(\mathcal{Q}\) ensures orthogonality to the relevant subspace, and assuming the identity vector 1 is part of the relevant subspace (or noting that \(\mathcal{Q} 1=0\) ), we observe that the projection \(\mathcal{Q}\) acting on any vector results in a vector with zero mean in the equilibrium ensemble:

\[ \langle\mathcal{Q} B\rangle=\langle B\rangle-\langle\mathcal{P} B\rangle=\langle B\rangle-\langle B, A\rangle \mathbb{C}^{-1}\langle A\rangle=0 . \tag{13.22} \label{13.22} \]

Since \(F(t)\) remains in the \(\mathcal{Q}\)-subspace for all \(t\), its equilibrium average vanishes.

13.6.2 Proof of Orthogonality: \(\langle F(t) A(0)\rangle=0\)

The fluctuating force is constructed to be orthogonal to the initial state of the relevant variables. To see this for all \(t\) :

\[ \langle F(t), A(0)\rangle=\left\langle e^{\mathcal{Q} \mathcal{L} t} \mathcal{Q} \mathcal{L} A(0), A(0)\right\rangle . \tag{13.23} \label{13.23} \]

Because the Liouville operator is anti-Hermitian \(\left(\left\langle\mathcal{L} O_{1}, O_{2}\right\rangle=-\left\langle O_{1}, \mathcal{L} O_{2}\right\rangle\right)\), and by extension its projection \(\mathcal{Q} \mathcal{L}\) preserves this orthogonality within the subspace:

\[ \left\langle e^{\mathcal{Q} \mathcal{L} t} \mathcal{Q} \mathcal{L} A(0), A(0)\right\rangle=\left\langle\mathcal{Q}\left(e^{\mathcal{Q} \mathcal{L} t} \mathcal{L} A(0)\right), A(0)\right\rangle . \tag{13.24} \label{13.24} \]

By the definition of the projection operator \(\mathcal{Q}=1-\mathcal{P}\), the vector \(\mathcal{Q} B\) is orthogonal to \(A(0)\) for any \(B\) :

\[ \langle\mathcal{Q} B, A(0)\rangle=\langle B, A(0)\rangle-\langle\mathcal{P} B, A(0)\rangle=\langle B, A(0)\rangle-\langle B, A(0)\rangle=0 . \tag{13.25} \label{13.25} \]

Thus, \(\langle F(t) A(0)\rangle=0\) for all \(t\).

13.6.3 The Fluctuation-Dissipation Relation: \(\langle F(t) F(0)\rangle=k_{B} T K(t)\)

The second fluctuation-dissipation theorem relates the correlation of the fluctuating forces to the memory kernel \(K(t)\). From the derivation of the GLE, we found the memory kernel to be:

\[ K(t)=\langle F(t), F(0)\rangle \mathbb{C}^{-1} . \tag{13.26} \label{13.26} \]

For a single variable \(A\) in a canonical ensemble, the correlation matrix \(\mathbb{C}=\langle A, A\rangle\) is related to the thermal energy. In many physical representations (such as velocity or momentum), equipartition implies \(\langle A, A\rangle=k_{B} T / m\). Substituting the definition of the inner product as an equilibrium correlation, we obtain:

\[ K(t)=\frac{\langle F(t) F(0)\rangle}{\left\langle A^{2}\right\rangle} . \tag{13.27} \label{13.27} \]

13.6.4 Summary of properties of the fluctuating forces

The fluctuating force has the following properties:

• 1.\(\langle F(t)\rangle=0\) (zero mean)
• 2.\(\langle F(t) A(0)\rangle=0\) (orthogonality)

13.6.5 Fluctuation-Dissipation Relation

The memory kernel and fluctuating force are related:

\[ \langle f(t) f(0)\rangle=k_{B} T K(t) \tag{13.28} \label{13.28} \]

This is a manifestation of the second fluctuation-dissipation theorem.[4]