10: Electrokinetics in a Tube

9.2.1 Setup and definitions

Consider a classical system with phase-space coordinates \(\Gamma=(\mathbf{q}, \mathbf{p})\) and unperturbed Hamiltonian \(H_{0}\). Let \(A(\Gamma)\) be an observable of interest. The equilibrium auto-correlation function is

\[ C(t)=\langle A(t) A(0)\rangle_{0}, \tag{9.1} \label{9.1} \]

where \(\langle\cdots\rangle_{0}\) denotes the canonical ensemble average with respect to \(\rho_{0}=e^{-\beta H_{0}} / Z_{\text {part }}\).

When a small external force \(h(t)\) is applied through the perturbation \(\Delta H=-A(\Gamma) h(t)\), the response of \(A\) is linear for small \(h\) :

\[ \langle\Delta A(t)\rangle=\int_{-\infty}^{t} \chi\left(t-t^{\prime}\right) h\left(t^{\prime}\right) d t^{\prime}, \tag{9.2} \label{9.2} \]

where causality is encoded by the upper limit \(t\).

Rate of average decay. Differentiating Equation \ref{9.2} with respect to \(t\) gives

\[ \frac{d}{d t}\langle\Delta A(t)\rangle=\chi(0) h(t)+\int_{-\infty}^{t} \dot{\chi}\left(t-t^{\prime}\right) h\left(t^{\prime}\right) d t^{\prime} . \tag{9.3} \label{9.3} \]

The first term is the instantaneous kick from the current force; the second encodes the memory of all past forcing weighted by the decay rate of \(\chi\). For a spontaneous fluctuation there is no external force, so the relaxation rate is governed entirely by \(\dot{\chi}(t)\). Since the FDT will identify \(\chi(t)=-\beta \dot{C}(t) \theta(t)\), equation Equation \ref{9.3} tells us directly that the rate of average decay of a fluctuation is proportional to \(\ddot{C}(t)\)-the curvature of the correlation function.

The goal of this section is to prove:

\[ \chi(t)=-\beta \dot{C}(t) \theta(t) . \tag{9.4} \label{9.4} \]

9.2.2 Poisson brackets and the Liouville equation

The Poisson bracket of two phase-space functions \(f\) and \(g\) is

\[ \{f, g\}=\sum_{i}\left(\frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}}-\frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}}\right) . \tag{9.5} \label{9.5} \]

The Liouville equation governing \(\rho(\Gamma, t)\) is

\[ \frac{\partial \rho}{\partial t}=\{H, \rho\} \equiv-\mathcal{L} \rho, \tag{9.6} \label{9.6} \]

where \(\mathcal{L} \equiv-\{H, \cdot\}\) is the Liouville operator. Hamilton's equations give \(\dot{A}=\left\{A, H_{0}\right\}\) for any observable.

9.2.3 A key thermal identity

Theorem 9.1 (Thermal identity). For the canonical equilibrium density \(\rho_{0}=e^{-\beta H_{0}} / Z_{\text {part }}\),

\[ \left\{A, \rho_{0}\right\}=-\beta \rho_{0} \dot{A} . \tag{9.7} \label{9.7} \]

Proof. Using the chain rule \(\{A, f(g)\}=f^{\prime}(g)\{A, g\}\) with \(f=e^{-\beta H_{0}} / Z_{\text {part }}\) and \(g=H_{0}\) :

\[ \left\{A, \rho_{0}\right\}=\frac{1}{Z_{\text {part }}} \frac{d\left(e^{-\beta H_{0}}\right)}{d H_{0}}\left\{A, H_{0}\right\}=-\beta \rho_{0}\left\{A, H_{0}\right\}=-\beta \rho_{0} \dot{A} . \tag{9.8} \label{9.8} \]\(\square\)

Physical meaning. Equation \ref{9.7} is the classical analogue of the quantum identity \(\left[A, \rho_{0}\right]=\) \(-i \beta \rho_{0} \dot{A}\) (with \(\{,\} \rightarrow-i[,] / \hbar\) as \(\hbar \rightarrow 0\) ). It encodes how the equilibrium distribution "knows about" dynamical quantities through \(\beta\).

9.2.4 First-order perturbation of the density

Write \(\rho(t)=\rho_{0}+\Delta \rho(t)\) and insert into the Liouville equation with \(H=H_{0}-A h(t)\). Keeping only terms linear in \(h(t)\) :

\[ \frac{\partial \Delta \rho}{\partial t}=\left\{H_{0}, \Delta \rho\right\}-h(t)\left\{A, \rho_{0}\right\} . \tag{9.9} \label{9.9} \]

Applying the thermal identity Equation \ref{9.7} converts the second term: \(-h(t)\left\{A, \rho_{0}\right\}=\beta h(t) \rho_{0} \dot{A}\). The formal solution via the unperturbed propagator \(e^{\mathcal{L}_{0}\left(t-t^{\prime}\right)}\) is:

\[ \Delta \rho(t)=\beta \int_{-\infty}^{t} e^{\mathcal{L}_{0}\left(t-t^{\prime}\right)}\left[\rho_{0} \dot{A}(\Gamma)\right] h\left(t^{\prime}\right) d t^{\prime} . \tag{9.10} \label{9.10} \]

9.2.5 Computing the response and identifying \(\chi(t)\)

The observable response is \(\langle\Delta A(t)\rangle=\int A(\Gamma) \Delta \rho(t) d \Gamma\). Shifting the time evolution from the density to the observable and using stationarity of the equilibrium ensemble:

\[ \langle\Delta A(t)\rangle=\beta \int_{-\infty}^{t} h\left(t^{\prime}\right)\left\langle\dot{A}(0) A\left(t-t^{\prime}\right)\right\rangle_{0} d t^{\prime} \tag{9.11} \label{9.11} \]

By time-translation invariance, \(\left\langle\dot{A}(0) A\left(t-t^{\prime}\right)\right\rangle_{0}=\left\langle\dot{A}\left(t^{\prime}-t\right) A(0)\right\rangle_{0}\). Differentiating Equation \ref{9.1} gives \(\dot{C}(s)=\langle A(0) \dot{A}(s)\rangle_{0}\). Under time-reversal symmetry for an even-parity observable \({ }^{1}\) \(\langle A(0) \dot{A}(s)\rangle_{0}=-\langle\dot{A}(0) A(s)\rangle_{0}\), so \(\left\langle\dot{A}(0) A\left(t-t^{\prime}\right)\right\rangle_{0}=-\dot{C}\left(t-t^{\prime}\right)\). Comparing with Equation \ref{9.2} and invoking causality:

\[ \chi(t)=-\beta \dot{C}(t) \theta(t) . \tag{9.12} \label{9.12} \]

This is the classical FDT. The response function is completely determined by the equilibrium correlation function; no separate knowledge of the out-of-equilibrium dynamics is needed.

9.3.1 Protocol: abrupt removal of a static force

Consider a static force \(f_{0}\) applied for all \(t \leq 0\) and then suddenly removed:

\[ f(t)=f_{0} \theta(-t)= \begin{cases}f_{0} & t \leq 0 \\[4pt] 0 & t>0\end{cases} \tag{9.13} \label{9.13} \]

At \(t=0^{-}\)the system is in a shifted equilibrium with mean \(\langle\delta A(0)\rangle=\beta f_{0} C(0)\), obtained by integrating Equation \ref{9.12} against a constant force.

9.3.2 Relaxation for \(t>0\)

For \(t>0\) there is no applied force, but the system was perturbed for all \(t^{\prime} \leq 0\). Inserting \(f\left(t^{\prime}\right)=f_{0} \theta\left(-t^{\prime}\right)\) into Equation \ref{9.2} and using Equation \ref{9.12}:

\[ \begin{align*} \langle\delta A(t)\rangle & =\int_{-\infty}^{0} \chi\left(t-t^{\prime}\right) f_{0} d t^{\prime}=-\beta f_{0} \int_{-\infty}^{0} \dot{C}\left(t-t^{\prime}\right) d t^{\prime} \\[4pt] & =-\beta f_{0}\left[C\left(t-t^{\prime}\right)\right]_{t^{\prime}=-\infty}^{t^{\prime}=0}=-\beta f_{0}[C(t)-C(\infty)] \tag{9.14} \label{9.14} \end{align*} \]

Since \(C(\infty)=0\) for any ergodic system (decorrelation at long times):

\[ \langle\delta A(t)\rangle=\beta f_{0} C(t), \quad t>0 . \tag{9.15} \label{9.15} \]

9.3.3 The hypothesis as a ratio

Dividing the relaxation at time \(t\) by the initial displacement:

\[ \frac{\langle\delta A(t)\rangle}{\langle\delta A(0)\rangle}=\frac{\beta f_{0} C(t)}{\beta f_{0} C(0)}=\frac{C(t)}{C(0)} . \tag{9.16} \label{9.16} \]

Regression Hypothesis (proved). The normalised macroscopic relaxation (left-hand side) is identical to the normalised equilibrium auto-correlation function (right-hand side). The decay of a forced deviation from equilibrium follows the same temporal law as the spontaneous regression of a thermal fluctuation.

Physical interpretation. Equation \ref{9.16} says the system cannot "tell the difference" between relaxing from a forced displacement and regressing from a rare spontaneous fluctuation. This is a consequence of the linearity of the response and the time-reversal symmetry of the microscopic equations of motion [7].

9.4.1 Derivation

\[ \begin{align*} \chi(\omega) & =-\beta \int_{0}^{\infty} \dot{C}(t) e^{i \omega t} d t \\[4pt] & =-\beta\left[C(t) e^{i \omega t}\right]_{0}^{\infty}+i \omega \beta \int_{0}^{\infty} C(t) e^{i \omega t} d t \\[4pt] & =\beta C(0)+i \omega \beta \int_{0}^{\infty} C(t) e^{i \omega t} d t, \tag{9.17} \label{9.17} \end{align*} \]

where \(C(t) e^{i \omega t} \rightarrow 0\) as \(t \rightarrow \infty\) by decorrelation. Since \(C(t)\) is real and even \((C(t)=C(-t)\) from time-reversal symmetry of equilibrium dynamics), the integral in Equation \ref{9.17} contributes purely real and imaginary parts. Taking the imaginary part:

\[ \chi^{\prime \prime}(\omega) \equiv \operatorname{Im} \chi(\omega)=\omega \beta \int_{0}^{\infty} C(t) \cos (\omega t) d t \tag{9.18} \label{9.18} \]

Defining the two-sided power spectral density via the Wiener-Khinchin theorem,

\[ S(\omega)=\int_{-\infty}^{\infty} C(t) e^{i \omega t} d t=2 \int_{0}^{\infty} C(t) \cos (\omega t) d t \tag{9.19} \label{9.19} \]

we obtain the classical FDT in the frequency domain:

\[ \chi^{\prime \prime}(\omega)=\frac{\beta \omega}{2} S(\omega) . \tag{9.20} \label{9.20} \]

Remark 9.1. The factor of \(\omega\) reflects that \(\chi^{\prime \prime}\) describes dissipation (energy absorbed per cycle scales as frequency), while \(S(\omega)\) is the fluctuation spectrum. At low frequencies both vanish, but their ratio approaches a finite constant controlled by the zero-frequency diffusion coefficient (Einstein relation).

9.4.2 Generalisation to cross-correlations

When the driving force \(f(t)\) couples to observable \(B\) while we monitor observable \(A\), the same Liouville calculation yields the cross-susceptibility

\[ \chi_{A B}(t)=-\beta \dot{C}_{A B}(t) \theta(t), \quad C_{A B}(t) \equiv\langle A(t) B(0)\rangle_{0}, \tag{9.21} \label{9.21} \]

which is the natural generalisation of Equation \ref{9.12}. In the frequency domain:

\[ \chi_{A B}^{\prime \prime}(\omega)=\frac{\beta \omega}{2} S_{A B}(\omega), \quad S_{A B}(\omega)=\int_{-\infty}^{\infty} C_{A B}(t) e^{i \omega t} d t . \tag{9.22} \label{9.22} \]

Note that \(C_{A B}\) is generically not symmetric, i.e. \(C_{A B}(t) \neq C_{B A}(t)\) in general. This asymmetry is the classical seed of the Onsager reciprocal relations derived in the next section.

9.5.1 Phenomenological setup

Consider a system near equilibrium characterised by thermodynamic variables \(\left\{A_{i}\right\}\) with thermodynamic forces

\[ X_{i}=\frac{\partial S}{\partial A_{i}}, \quad J_{i}=\frac{d A_{i}}{d t}, \tag{9.23} \label{9.23} \]

where \(S\) is the total entropy. The fluxes are assumed to depend linearly on the forces:

\[ J_{i}=\sum_{j} L_{i j} X_{j} . \tag{9.24} \label{9.24} \]

The entropy production rate is therefore

\[ \dot{S}=\sum_{i} J_{i} X_{i}=\sum_{i, j} L_{i j} X_{i} X_{j}, \tag{9.25} \label{9.25} \]

so the second law of thermodynamics \((\dot{S} \geq 0)\) requires only that the symmetric part of \(\mathbf{L}\) be positive semidefinite. The reciprocity \(L_{i j}=L_{j i}\) is an additional, stronger statement that follows from microscopic reversibility-not from thermodynamics alone.

The Onsager matrix L encodes all transport phenomena: thermal conductivity, electrical conductivity, thermoelectric coefficients, diffusion, and their cross-effects.

We now present two independent proofs of the reciprocal relations, each illuminating a different facet of the underlying physics.

9.5.2 Proof A: Green-Kubo formula and time-reversal parity

9.5.3 Proof B: Entropy-production and Gaussian fluctuations

This alternative proof works entirely at the level of fluctuation statistics, without invoking the Green-Kubo integral explicitly. It follows closely the original reasoning of Onsager [1, 3].

Equilibrium fluctuation measure

Near equilibrium, the joint distribution of the variables \(\left\{A_{i}\right\}\) (each with zero mean) is approximately Gaussian:

\[ P\left(\left\{A_{i}\right\}\right) \propto \exp \left(\frac{S}{k_{\mathrm{B}}}\right), \quad S \approx S_{\mathrm{eq}}-\frac{1}{2} \sum_{i, j} g_{i j} A_{i} A_{j}, \tag{9.35} \label{9.35} \]

where \(g_{i j}\) is a symmetric positive-definite matrix (the inverse of the fluctuation covariance matrix, \(g=\langle A A\rangle^{-1}\) ). The thermodynamic forces Equation \ref{9.23} are therefore linear in the \(A_{i}\) :

\[ X_{k}=\frac{\partial S}{\partial A_{k}}=-\sum_{j} g_{k j} A_{j} . \tag{9.36} \label{9.36} \]

Orthogonality relation

Lemma 9.2 (Gaussian orthogonality). Under the distribution Equation \ref{9.35},

\[ \left\langle A_{i} X_{j}\right\rangle=-k_{\mathrm{B}} \delta_{i j} . \tag{9.37} \label{9.37} \]

Proof. \(\left\langle A_{i} X_{j}\right\rangle=-\sum_{k} g_{j k}\left\langle A_{i} A_{k}\right\rangle=-\sum_{k} g_{j k}\left(g^{-1}\right)_{k i}=-\delta_{j i}\). \(\square\)

Connecting time derivatives of correlators to \(L_{i j}\)

By the Regression Hypothesis (Section 9.3), the decay of \(\left\langle A_{k}\right\rangle\) conditioned on an initial fluctuation \(\left\{A_{i}\right\}\) follows the same linear law as a macroscopic relaxation:

\[ \frac{d}{d t}\left\langle A_{k}(t) A_{i}(0)\right\rangle=\sum_{j} L_{k j}\left\langle X_{j}(t) A_{i}(0)\right\rangle . \tag{9.38} \label{9.38} \]

Setting \(t=0\) and using the orthogonality relation Equation \ref{9.37}:

\[ \left.\frac{d}{d t} C_{k i}(t)\right|_{t=0}=\sum_{j} L_{k j}\left\langle X_{j} A_{i}\right\rangle=-k_{\mathrm{B}} L_{k i} . \tag{9.39} \label{9.39} \]

Time-reversal symmetry of the correlator

The three-step argument exploiting microscopic reversibility:

• 1.Time-reversal invariance. Under microscopic reversibility, for even-parity variables:

  \[ \left\langle A_{i}(t) A_{k}(t+\tau)\right\rangle=\left\langle A_{i}(t) A_{k}(t-\tau)\right\rangle . \tag{9.40} \label{9.40} \]

• 2.Stationarity. Shifting the time origin by \(\tau\) :

  \[ \left\langle A_{i}(t) A_{k}(t-\tau)\right\rangle=\left\langle A_{i}(t+\tau) A_{k}(t)\right\rangle . \tag{9.41} \label{9.41} \]

• 3.Commutativity. Classical phase-space functions commute:

  \[ \left\langle A_{i}(t+\tau) A_{k}(t)\right\rangle=\left\langle A_{k}(t) A_{i}(t+\tau)\right\rangle . \tag{9.42} \label{9.42} \]

Combining these three steps:

\[ C_{i k}(\tau) \equiv\left\langle A_{i}(t) A_{k}(t+\tau)\right\rangle=\left\langle A_{k}(t) A_{i}(t+\tau)\right\rangle \equiv C_{k i}(\tau) . \tag{9.43} \label{9.43} \]

Hence for even-parity variables the cross-correlator is symmetric: \(C_{i k}(\tau)=C_{k i}(\tau)\).

Concluding reciprocity

Differentiating both sides at \(\tau=0\) and applying Equation \ref{9.39}:

\[ -k_{\mathrm{B}} L_{i k}=-k_{\mathrm{B}} L_{k i}, \tag{9.44} \label{9.44} \]

and therefore

\[ L_{i j}=L_{j i} . \tag{9.45} \label{9.45} \]

The analogous argument for odd-parity variables (which pick up a sign under time reversal) yields the Casimir extension \(L_{i j}=-L_{j i}\) in the mixed case, consistent with Equation \ref{9.34}.

Remark 9.2. Proofs A and B are complementary, not redundant. Proof A makes the role of the current-current Green-Kubo correlator explicit and extends immediately to quantum systems via Equation \ref{9.26}. Proof B is closer to Onsager's original argument and makes the Gaussian fluctuation assumption and the orthogonality relation Equation \ref{9.37} explicit-ingredients that are sometimes hidden in the Green-Kubo approach.

Step-by-step proof outline:

1. Start from microscopic dynamics that are time-reversal invariant at \(\mathbf{B}=0\).

2. The thermal identity \(\left\{A, \rho_{0}\right\}=-\beta \rho_{0} \dot{A}\) (Theorem 15.2.1) converts the first-order Liouville perturbation into an equilibrium correlator.

3. Integration over the forcing history yields the classical FDT, \(\chi(t)=-\beta \dot{C}(t) \theta(t)\).

4. Applying the FDT to a suddenly removed static force proves Onsager's Regression Hypothesis: macroscopic relaxation and microscopic regression are both governed by \(C(t) / C(0)\).

5. Fourier-transforming the FDT yields \(\chi^{\prime \prime}(\omega)=\frac{1}{2} \beta \omega S(\omega)\), linking dissipation to the equilibrium fluctuation spectrum.

6. Route A: The Green-Kubo formula expresses \(L_{i j}\) as \(\int_{0}^{\infty}\left\langle J_{i}(t) J_{j}(0)\right\rangle_{0} d t\). Time-reversal parity gives \(C_{i j}(t)=\epsilon_{i} \epsilon_{j} C_{j i}(t)\), and hence \(L_{i j}=\epsilon_{i} \epsilon_{j} L_{j i}\).

7. Route B: The Regression Hypothesis applied to equilibrium fluctuation correlators, combined with the Gaussian orthogonality relation \(\left\langle A_{i} X_{j}\right\rangle=-k_{\mathrm{B}} \delta_{i j}\) and the symmetry \(C_{i j}(\tau)=C_{j i}(\tau)\) from microscopic reversibility, directly gives \(L_{i j}=L_{j i}\).

8. If a magnetic field B is present, time reversal maps \(\mathbf{B} \rightarrow-\mathbf{B}\), yielding \(L_{i j}(\mathbf{B})=\epsilon_{i} \epsilon_{j} L_{j i}(-\mathbf{B})\).

The entire argument uses only three inputs: (i) canonical equilibrium statistics, (ii) linearity of the response, (iii) time-reversal invariance of the Hamiltonian dynamics. No phenomenological assumptions are needed.