Designing Directional Traffic Flow With Edge Mode Combination in 2-D Topological Structures

We demonstrate directional vehicular traffic using a two-dimensional honeycomb-shaped topological structure. We consider a hexagonal street network modeled with vertices and edges, and numerically simulate vehicular transport as a symmetric random walk between vertices. We show that two topologically protected modes lead to traffic flows in orthogonal directions. Additionally, we introduce a synthesized mode that combines topological edge modes. This synthesized mode enables traffic to flow in specific direction by adjusting the combined weights. Our investigation offers an approach for optimizing urban traffic management, enhancing traffic efficiency, and reducing congestion in urban environments.

SECTION I. Introduction

The design and control of vehicular traffic is crucial in urban development as it leads to multifaceted benefits [1], [2], including smoother traffic, ensured safety, and reduced environmental impacts. These benefits are directly linked to improvements in the sustainability and quality of life in cities. To effectively design and control vehicular traffic, researchers have developed traffic models [3], [4], [5], [6], [7]. Traffic models provide a mathematical representation of traffic dynamics and enable simulations to develop optimal traffic management strategies. Another approach to traffic design and control is transport management using intelligent transportation systems (ITS) [8], [9], [10], [11], [12]. An ITS utilizes information and communication technologies to enhance the efficiency and safety of transportation systems [13], [14].

Topological properties of materials have attracted considerable attention in condensed matter physics [15], [16]. This is because topological materials exhibit unique physical properties and functions not found in classical materials. Triggered by the discovery of topological insulators, research on topological materials has progressed rapidly. In addition, there has been significant interest in exploiting topological properties in various physical systems, such as electric [17], [18], [19], [20] and photonic devices [21], [22], [23], [24]. Recent studies have extended the topological properties to thermal diffusion systems [25], [26], [27], [28], [29], [30], [31], [32], [33]. Heat localization has been demonstrated numerically and experimentally in one- and two-dimensional structures. Subsequent studies have shown that topological states enable directional diffusion [34], [35].

Beyond these traditional diffusion systems, recent research has explored the potential of topological concepts in vehicular traffic design in urban environments [36]. In [36], it has been demonstrated that unidirectional flow of vehicular traffic emerges in a hexagonal street network with a two-dimensional topological structure. Such unidirectional property deteriorates due to the collapse of the topological edge mode. To mitigate this decay and maintain traffic unidirectionality, a strategy has been proposed for controlling the entry-exit timing of vehicles in a street network. We note that the direction of traffic flow was not tunable in [36], indicating that subsequent studies are required for flexible designs of unidirectional traffic flow.

In this study, we extend the concept of topologically oriented design of directional traffic flow. We show that two topologically protected edge modes can direct vehicular traffic flow in specific direction within a honeycomb-shaped street network. First, we provide a brief overview of a topological diffusion system where directional diffusion occurs. Analogous to diffusion systems, we apply the concept of directional diffusion to vehicular transport. We model the network structure using vertices and edges and describe vehicular transport as a symmetric random walk between vertices. We numerically demonstrate that the two topological states result in traffic flows in orthogonal directions. Furthermore, we introduce a synthesized mode that combines the two topological edge modes. This synthesized mode permits traffic to flow in specific direction by adjusting the combined weights of the edge modes.

We note again that the direction of traffic flow was not tunable in the previous study [36]. In contrast, the synthesized mode represented in this study allows for the adjustment of the direction of vehicular traffic flow, thereby offering a crucial advantage for flexible design of vehicular traffic. This capability provides a promising new tool for optimizing urban transportation networks and contributes significantly to the development of smart cities.

SECTION II. Design of Diffusion Direction With Topologically Protected Edge Modes

A. Diffusion System Consisting of Honeycomb-Shaped Periodic Structure

We provide a brief overview of a topological diffusion system with a honeycomb-shaped periodic structure [34], [36], as shown in Fig. 1. The unit cell consists of six reservoirs containing the diffusing material. Each reservoir is connected to the intra- and inter-channels with diffusivities $D_{1}$ and $D_{2}$ , respectively. According to the diffusion equation and Bloch’s theorem, the effective Hamiltonian of the system is expressed as follows: $$\begin{align*} \mathcal {H}=& \begin{bmatrix} \mathcal {Q} & ~ \mathcal {P}_{1} \\ \mathcal {P}_{2} & ~ \mathcal {Q}\end{bmatrix}, \tag {1a}\\ \mathcal {Q}\equiv & \begin{bmatrix} -2D_{1}-D_{2} & 0 & 0 \\ 0 & -2D_{1}-D_{2} & 0 \\ 0 & 0 & -2D_{1}-D_{2}\end{bmatrix}, \tag {1b}\\ \mathcal {P}_{1(2)}\equiv & \begin{bmatrix} D_{2} e^{-i\boldsymbol {k}\cdot \left ({{\boldsymbol {a}_{1(2)}-\boldsymbol {a}_{2(1)}}}\right) } & D_{1} & D_{1} \\ D_{1} & D_{1} & D_{2} e^{i\boldsymbol {k}\cdot \boldsymbol {a}_{1(2)}} \\ D_{1} & D_{2} e^{-i\boldsymbol {k}\cdot \boldsymbol {a}_{2(1)}} & D_{1}\end{bmatrix}, \quad \tag {1c}\end{align*}$$ where $\boldsymbol {k}$ is the wave number vector, and $\boldsymbol {a}_{1\,(\mathrm {or}\,2)}$ is the reciprocal lattice vector shown in Fig. 1. By adjusting the ratio of the diffusivities $r=D_{2}/D_{1}$ , we can design nontrivial topological and ordinary states for an infinite periodic structure. Specifically, the structures with $r\lt 1$ and $r\gt 1$ correspond to ordinary and topological states, respectively [34].

FIGURE 1.

Schematic of honeycomb-shaped unit cell. Each of the nearest neighboring reservoirs is connected by the channel (black) with effective diffusivity $D_{1}$ . Neighboring unit cells are connected by channels (orange) with effective diffusivity $D_{2}$ . a₁ and a₂ are reciprocal lattice vectors.

B. Diffusion Direction for Edge Modes With Distinct Polarization

We considered a supercell with a boundary between the lower topological and upper ordinary unit cells, as shown in Fig. 2(a). The unit cells are indexed as $m = 1,2,3,4$ along the direction of $\boldsymbol {a}_{1}$ . Figure 2(b) shows the eigenvalue spectra of the supercell. We observed two independent topologically protected edge modes at $k_{\mathrm {x}}=0$ , indicated by the lower (orange) and upper (green) arrows, referred to as Modes A and B, respectively. These edge modes macroscopically provide orthogonal and directional flows for diffusing materials [34].

FIGURE 2.

(a) Schematic of supercell with boundary between topological ($m=1$ and 2) and ordinary ($m=3$ and 4) unit cells. Effective diffusivities are $D_{\mathrm {t}1} = 0.765$ , $D_{\mathrm {t}2} = 0.518$ , $D_{\mathrm {o}1} =0.600$ , and $D_{\mathrm {o}2} = 0.850$ . (b) Eigenvalue spectra for each reservoir. Lower (orange) and upper (green) arrows depict topological edge modes with distinct polarization; Modes A and B.

To capture the flow direction over unit cells, we calculated the diffusion flux $\boldsymbol {J}_{s}^{(m)}$ at the point $\mathrm {p}_{s}~(s=1,2,\ldots, 6)$ on the outer surface $\Omega$ encircling the mth unit cell, as shown in Fig. 3(a). By summing up the diffusion fluxes over the surface $\Omega$ , the flux through the mth unit cell can be expressed as follows: $$\begin{align*} \boldsymbol {J}_{\mathrm {cell}}^{(m)}\equiv & \begin{bmatrix} J_{\mathrm {x}}^{(m)} \\ J_{\mathrm {y}}^{(m)}\end{bmatrix} \\=& \sum _{s}^{6}\boldsymbol {J}_{s}^{(m)}. \tag {2}\end{align*}$$ Details of the calculations in (2) are presented in the Appendix.

FIGURE 3.

(a) Reservoir index. Flux was calculated at the point ${}\mathrm {p}_{s}$ ($s=1,2,\ldots, 6$ ) and summed over surface $\Omega$ in (2) and (4). Diffusion fluxes for (b) Mode A and (c) Mode B. The right panels in (b) and (c) depict fluxes with red arrows. Their lengths represent the quantity of fluxes normalized by their maximum value, i.e., $\max _{m} |\boldsymbol {J}_{\mathrm {cell}}^{\text {(}m\text {)}}|$ . Reservoir surfaces are shaded according to the amplitudes of the eigenfunction of each mode.

Figures 3(b) and (c) show the diffusion flux as a function of the cell index, with the fluxes shown in the right panel. When edge modes are present, the system exhibits a directional diffusion flux in the positive x (or y) direction. It is noteworthy that the reservoir surfaces in Figs. 3(b) and (c) are shaded, with the amplitudes of the eigenfunction corresponding to the density distribution of the diffusing material for each mode.

SECTION III. Design of Traffic Direction With Topologically Protected Edge Modes

This diffusion phenomenon is equivalent to the random walk of particles [37]. Therefore, we can apply the design method for directional flow in diffusion systems to vehicular transportation when vehicles behave as diffusive particles. Here, we numerically show that vehicular traffic flow is macroscopically directed in a topological structure where vehicles are modeled as Brownian particles. Furthermore, we demonstrate that combining the edge modes enables traffic flow in specific direction.

A. The Model

The diffusion phenomenon is equivalent to the random walk of particles because diffusion corresponds to the macroscopic picture of a sufficiently large number of particles moving with distinct velocities in different random directions. This interpretation indicates that topological edge modes allow directional diffusion in particle transportation. In this study, analogous to the particles, we designed the flow direction in vehicle traffic using a topological structure in which vehicles are considered Brownian particles.

Figure 4(a) shows the outline sketch of the scenario. We considered an urban area where streets are arranged along hexagonal sites. The vehicles navigate through a grid street network. For simplicity, we represent the street network as a lattice consisting of $4\times 2$ unit cells with vertices and edges, as shown in Fig. 4(b). We modeled vehicle transport in a hopping-like manner, specifically as a symmetric multiparticle random walk over the vertices.

FIGURE 4.

(a) Outline sketch of scenario. Streets are arranged in mesh-like hexagonal pattern throughout the urban area. (b) Simulation model of grid street network. Indices (m, n) are written inside the unit cell. Edges with corresponding symbols (e.g., $\boxminus$ ) are connected to each other. We determined the average speed of vehicles at each vertex using $D_{\mathrm {t}1} = 0.600$ , $D_{\mathrm {t}2} = 0.850$ , $D_{\mathrm {o}1} =0.765$ , $D_{\mathrm {o}2} = 0.518$ , and $l\gamma =1$ .

In our numerical simulations, the vehicles were initially allocated to each vertex based on the density distribution associated with the eigenfunctions of Modes A and B, $\boldsymbol {\phi }^{\mathrm {A}}$ and $\boldsymbol {\phi }^{\mathrm {B}}$ . At vertex s and cell m for all n, the vehicle density is expressed as follows: $$\begin{equation*} d_{s,m} = \alpha \phi ^{\mathrm {A} \,\left ({{\mathrm {or}\,\mathrm {B}}}\right)}_{s,m} + \beta, \tag {3}\end{equation*}$$ where $\phi ^{\mathrm {A} \,(\mathrm {or}\,\mathrm {B})}_{s,m}$ is the eigenfunction at vertex s and cell m for Mode A (or B), and $\alpha$ and $\beta$ are the conversion coefficient and bias, respectively. In the subsequent steps, the vehicles immediately jump to the neighboring vertex. The vehicle transition was determined using equal probabilities $p=1/b$ , where b denotes the number of edges connected to the targeted vertex. We assumed that the vehicles pass through the edges at a regulated speed proportional to the diffusivities $D_{\mathrm {t}1}$ , $D_{\mathrm {t}2}$ , $D_{\mathrm {o}1}$ , and $D_{\mathrm {o}2}$ . Thus, the time required for the jump between vertices is expressed as $\gamma l/D_{\mathrm {t}1}$ , $\gamma l/D_{\mathrm {t}2}$ , $\gamma l/D_{\mathrm {o}1}$ , and $\gamma l/ D_{\mathrm {o}2}$ , depending on the diffusivity of the edges. Here, l is the length of the vertex and $\gamma$ is the scaling coefficient.

B. Flux

To analyze the flow direction of the transporting vehicles over the unit cell, we considered the vector flux during $t_{1} \leq t \lt t_{2}$ in the cell of (m, n) as follows: $$\begin{align*} \boldsymbol {F}^{(m,n)}\left ({{t_{1},t_{2}}}\right)\equiv & \begin{bmatrix} F_{\mathrm {x}}^{(m,n)}\left ({{t_{1},t_{2}}}\right) \\ F_{\mathrm {y}}^{(m,n)}\left ({{t_{1},t_{2}}}\right)\end{bmatrix} \\=& \sum _{s}^{6} \frac {\Delta \boldsymbol {Q}^{(m,n)}_{s}}{\Delta t}, \tag {4}\end{align*}$$ where $\Delta t=t_{2}-t_{1}$ is the time interval considered, and $\Delta \boldsymbol {Q}^{(m,n)}_{s}\equiv [\Delta Q^{(m,n)}_{\mathrm {x},s},\Delta Q^{(m,n)}_{\mathrm {y},s}]^{\mathrm {T}}$ is a vector representing the quantity that passes through point p_s on the surface $\Omega$ during $\Delta t$ .

C. Results

1) Direction of Traffic Flow in Modes A and B

Figures 5(a) and (b) depict the macroscopic flux of the vehicles in each cell. We observe the directional flow parallel to the boundary for Mode A [Fig. 5(a)], and perpendicular for Mode B [Fig. 5(b)]. This trend is similar to that of the diffusion flux depicted in Figs. 3(b) and (c). These results indicate that a topologically oriented network structure and vehicle allocation enable traffic to flow along the x- and y-directions. We calculated the traffic flow for time intervals of $0 \leq t \lt 2$ when the edge mode was maintained. This time interval was determined considering the decay rate of the edge mode, which was discussed in detail in [36].

FIGURE 5.

Flux $\boldsymbol {F}^{\text {(}m,n\text {)}}\text {(}0,2$ ) is illustrated using red arrows when vehicles are initially allocated using eigenfunctions (a) $\boldsymbol {\phi }^{\mathrm {A}}$ , (b) $\boldsymbol {\phi }^{\mathrm {B}}$ , (c) $-\boldsymbol {\phi }^{\mathrm {A}}$ , and (d) $-\boldsymbol {\phi }^{\mathrm {B}}$ . Arrow length represents the flux quantity. Amplitudes of flux are normalized by $\max _{m,n} |\boldsymbol {F}^{\text {(}m,n\text {)}}\text {(}0,2\text {)}|$ . Initial distribution of vehicles at $t=0$ is determined as rounded values of $d_{s,m}$ with $\alpha =7 \times 10^{4}$ and $\beta =10^{6}/28$ , resulting in whole number of vehicles in single cell array as $\sum _{s,m} d_{s,m} \sim 10^{6}$ .

It should be noted that $\boldsymbol {\phi }^{\mathrm {A}\,(\mathrm {or}\, \mathrm {B})}$ is the eigenfunction corresponding to Mode A (or B), and any scalar multiple of this vector, such as $-\boldsymbol {\phi }^{\mathrm {A}\,(\mathrm {or}\, \mathrm {B})}$ , is also an eigenfunction. These negatively weighted eigenfunctions provide an asymmetric distribution for the initial vehicle allocation. Consequently, negatively weighted eigenfunctions can produce traffic flow in the opposite direction relative to the original eigenfunctions. To confirm this hypothesis, we ran a simulation and calculated the flow direction when vehicles were initially allocated to the vertices using a negatively weighted eigenfunction: $d_{s,m} = - \alpha \phi ^{\mathrm {A} \,(\mathrm {or}\,\mathrm {B})}_{s,m} + \beta$ . Figures 5(c) and (d) show the flow directions for this vehicle allocation. Comparing Figs. 5(a) and (c) [and Figs. 5(b) and (d)], we can clearly observe the opposite direction of traffic flow. These results demonstrate that traffic flow can be selected from four directions (± x and ± y) using two distinct edge modes.

2) Traffic Direction Provided by Combining Edge Modes

We reiterate that Modes A and B individually provide directional traffic flows along the x- and y-axes, respectively. Due to the orthogonality of Modes A and B, we treat $\boldsymbol {\phi }^{\mathrm {A}}$ and $\boldsymbol {\phi }^{\mathrm {B}}$ as orthonormal basis functions. We then represent Mode C as a linear combination of these edge modes. Subsequently, the vehicle density can be expressed as follows: $$\begin{equation*} d_{s,m}= \alpha \phi ^{\mathrm {C}}_{s,m} + \beta, \tag {5a}\end{equation*}$$ where $$\begin{equation*} \boldsymbol {\phi }^{\mathrm {C}}= w_{\mathrm {A}} \boldsymbol {\phi }^{\mathrm {A}} + w_{\mathrm {B}} \boldsymbol {\phi }^{\mathrm {B}}. \tag {5b}\end{equation*}$$ where $w_{\mathrm {A} \,(\mathrm {or}\,\mathrm {B})}\in \mathbb {R}$ is the combining weight, satisfying $|w_{\mathrm {A} \,(\mathrm {or}\,\mathrm {B})} | \leq 1$ and $|w_{\mathrm {A}}|^{2}+|w_{\mathrm {B}}|^{2}=1$ . The traffic direction produced by Mode C is determined in the polar coordinate system (refer to the coordinates in Fig. 6) as follows: $$\begin{equation*} \arg \left ({{ \frac {w_{\mathrm {B}}}{w_{\mathrm {A}}} }}\right), \tag {6}\end{equation*}$$ which is calculated using a two-argument arctangent function.

FIGURE 6.

Flux $\boldsymbol {F}^{\text {(}m,n\text {)}}\text {(}0,2$ ) is illustrated using red arrows; (a) ($w_{\mathrm {A}}, w_{\mathrm {B}}\text {)} = \text {(}\sqrt {0.5},\sqrt {0.5}$ ), (b) ($\sqrt {0.5},-\sqrt {0.5}$ ). Arrow length indicates the flux quantity. Amplitudes of flux are normalized by $\max _{m,n} |\boldsymbol {F}^{\text {(}m,n\text {)}}\text {(}0,2\text {)}|$ . With the exception of initial vehicle allocation, other simulation settings are the same as in Fig. 5. (c) Traffic direction as a function of combined weight. Solid lines and circle symbols represent designed directions, $\arg \text {(}w_{\mathrm {B}}/w_{\mathrm {A}}$ ), and numerically calculated directions, $\arg \text {(}F_{\mathrm {y}}^{\text {(}3,1\text {)}}/F_{\mathrm {x}}^{\text {(}3,1\text {)}}$ ), respectively.

Next, we investigated the tunability of the traffic flow direction provided by Mode C. We performed numerical simulations where vehicles were initially allocated to the vertices according to (5a), and we calculated the resulting traffic flow direction. Figures 6(a) and (b) show the traffic direction, $\theta$ , for two sets of combining weights: $(w_{\mathrm {A}}, w_{\mathrm {B}}) = (\sqrt {0.5},\sqrt {0.5})$ in (a) and $(\sqrt {0.5},-\sqrt {0.5})$ in (b). The direction of traffic flow for $(w_{\mathrm {A}}, w_{\mathrm {B}}) = (\sqrt {0.5},\sqrt {0.5})$ and $(\sqrt {0.5},-\sqrt {0.5})$ becomes $\arg (F_{\mathrm {y}}^{(m,n)}/F_{\mathrm {x}}^{(m,n)}) \sim \pi /4$ and $-\pi /4$ at all cells, respectively.

Moreover, Fig. 6(c) shows the traffic flow direction in cell $(m,n)=(3,1)$ as a function of the combined weight. The traffic direction in the numerical simulation $\arg (F_{\mathrm {y}}^{(3,1)}/F_{\mathrm {x}}^{(3,1)})$ (circle symbols) is in good agreement with the designed direction $\arg (w_{2}/w_{1})$ (solid line).

SECTION IV. Conclusion

In urban areas (Fig. 4(a)), the flow direction of vehicular traffic can be engineered in a multi-hexagonal street network using topologically protected edge modes. We identified that two distinct edge modes exist in the diffusion system comprising the supercell with a boundary between the ordinary and topological unit cells (Figs. 2(a) and (b)). The network structure was modeled using the topology of the vertices and edges (Fig. 4(b)), and vehicular transport was described as a symmetric random walk between vertices. Through a numerical simulation of vehicular dynamics, we showed that the topological structure provided directional traffic flow that was positive/negative and parallel/perpendicular to the boundary, depending on the edge modes (Fig. 5). Furthermore, we introduced a mode represented by a linear combination of edge modes (Fig. 6). This synthesized mode allowed traffic to flow in specific direction by adjusting the combined weights.

Notably, the vehicles were described as Brownian particles in our analysis. This simple modeling approach confirmed the proof of concept regarding the design methodology for unidirectional traffic flow provided by the topological structure. However, practical vehicle dynamics are generally more complex and should be investigated using more precise models in future studies.

ACKNOWLEDGMENT

The authors wish to acknowledge the fruitful discussions they had with Kenichi Yatsugi.

Appendix

According to Fick’s first law, the diffusion flux is expressed as $-D \nabla \phi$ , where D is the diffusivity at the cell edge and $\phi$ is the density of the diffusing material. In our case, the diffusion flux over the unit cell is expressed as follows: $$\begin{align*} J_{\mathrm {x}}^{(1)}=& \frac {D_{\mathrm {t}2}}{2l} \left ({{ \phi _{2}^{(1)} - 4\phi _{3}^{(1)} - \phi _{4}^{(1)} + 4\phi _{5}^{(1)} + \phi _{1}^{(2)} - \phi _{6}^{(2)} }}\right), \tag {7a}\\ J_{\mathrm {y}}^{(1)}=& \frac {\sqrt {3}D_{\mathrm {t}2}}{2l} \left ({{ \phi _{2}^{(1)} + \phi _{4}^{(1)} - \phi _{1}^{(2)} - \phi _{6}^{(2)} }}\right), \tag {7b}\\ J_{\mathrm {x}}^{(2)}=& \frac {D_{\mathrm {t}2}}{2l} \left ({{ \phi _{2}^{(1)} - \phi _{4}^{(1)} + \phi _{1}^{(2)} + \phi _{2}^{(2)} - 4\phi _{3}^{(2)} }}\right. \\& {}\left.{{ - \phi _{4}^{(2)} + 4\phi _{5}^{(2)} - \phi _{6}^{(2)} + \phi _{1}^{(3)} - \phi _{6}^{(3)} }}\right), \tag {7c}\\ J_{\mathrm {y}}^{(2)}=& \frac {\sqrt {3}D_{\mathrm {t}2}}{2l} \left ({{ \phi _{2}^{(1)} + \phi _{4}^{(1)} - \phi _{1}^{(2)} + \phi _{2}^{(2)} + \phi _{4}^{(2)} }}\right. \\& {}\left.{{ - \phi _{6}^{(2)} - \phi _{1}^{(3)} - \phi _{6}^{(3)} }}\right), \tag {7d}\\ J_{\mathrm {x}}^{(3)}=& \frac {D_{\mathrm {o}2}}{2l} \left ({{ \phi _{2}^{(2)} - \phi _{4}^{(2)} + \phi _{1}^{(3)} - \phi _{6}^{(3)} }}\right) \\& {}+ \frac {D_{\mathrm {t}2}}{2l} \left ({{ \phi _{2}^{(3)} - 4\phi _{3}^{(3)} - \phi _{4}^{(3)} + 4\phi _{5}^{(3)} + \phi _{1}^{(4)} - \phi _{6}^{(4)} }}\right), \tag {7e}\\ J_{\mathrm {y}}^{(3)}=& \frac {\sqrt {3}D_{\mathrm {o}2}}{2l} \left ({{ \phi _{2}^{(2)} + \phi _{4}^{(2)} - \phi _{1}^{(4)} - \phi _{6}^{(4)} }}\right) \\& {}+ \frac {\sqrt {3}D_{\mathrm {t}2}}{2l} \left ({{ \phi _{2}^{(3)} + \phi _{4}^{(3)} - \phi _{1}^{(3)} - \phi _{6}^{(3)} }}\right), \tag {7f}\\ J_{\mathrm {x}}^{(4)}=& \frac {D_{\mathrm {o}2}}{2l} \left ({{ \phi _{2}^{(3)} - \phi _{4}^{(3)} + \phi _{1}^{(4)} - 4\phi _{3}^{(4)} + 4\phi _{5}^{(4)} - \phi _{6}^{(4)} }}\right), \tag {7g}\\ J_{\mathrm {y}}^{(4)}=& \frac {\sqrt {3}D_{\mathrm {o}2}}{2l} \left ({{ \phi _{2}^{(3)} + \phi _{4}^{(3)} - \phi _{1}^{(4)} - \phi _{6}^{(4)} }}\right), \tag {7h}\end{align*}$$ where $\phi _{s}^{(m)}$ is the eigenfunction corresponding to the reservoir and cell indices, s and m.