Source-Sink-Reservoir Diagrams

• Source-Sink-Reservoir diagrams are systematic models that map material, energy, or information flows from origins (sources) through storage intermediaries (reservoirs) to endpoints (sinks).
• They employ information-theoretic measures like Shannon entropy, Kullback–Leibler divergence, and mutual information to quantify throughput, asymmetry, and dynamic exchange rates.
• Widely applied in fields such as statistical physics, chemical dynamics, and network theory, SSR diagrams enable rigorous analysis of self-organizing and dissipative systems.

A source–sink–reservoir (SSR) diagram represents the flow and transformation of material, energy, or information within systems comprising three main components: a source (S), a reservoir (R), and a sink (K). These diagrams are foundational for modeling dissipative structures, signaling systems, self-organizing processes, and network flows, facilitating rigorous analysis via probability, entropy, and gradient-flow frameworks. SSR diagrams appear in fields spanning statistical physics, network theory, chemical physics, biological signaling, and mathematical graph theory. Their utility is enhanced by information-theoretic measures and variational principles, enabling quantification of throughput, storage, manipulation, and asymmetry of exchanges.

1. Core Definitions and Structural Roles

The SSR model divides system functionality among three distinct elements, each assigned a formal operational role:

• Source (S): The origin of raw material, energy, or information. In chemical applications, this may be a high-energy region where reaction precursors are created. In information theory, it is analogous to Shannon’s transmitter.
• Reservoir (R): An intermediary repository capable of storing, manipulating, enhancing, or degrading its received input. The reservoir features memory of arbitrary duration, variable storage capacity, and may itself be decomposed into a multi-stage subsystem. In networked systems, the reservoir executes the functions of a channel but with active, selective influence.
• Sink (K): The final recipient that accumulates output from the reservoir. The sink encompasses all forms of output, including transformed material, system state signals, and both usable and waste energy. This is akin to Shannon’s receiver.

In network contexts, vertices can act as sources or sinks depending on the direction and sign of mass or information flux associated with incident edges. Reservoir mass at vertices accumulates or releases content as dictated by edge-reservoir exchange (Ahmad, 2018).

2. Information-Theoretic Quantification

SSR diagrams support a detailed analysis using the following measures:

• Kolmogorov “Surprise”: For a stage (pulse) $n$ with probability $p(n)$, the self-information is $i(n) = \ln (1/p(n))$, diverging as events grow less probable.
• Shannon Entropy: For a component’s distribution over stages $n$, the cumulative entropy is $H = -\sum_n p(n)\ln p(n) = \sum_n p(n) i(n)$, capturing total information processed.
• Mutual Information: Quantifies the shared information between two components, such as source and sink, via $$I(X;Y) = \sum_{x,y} p_{XY}(x,y) \ln [p_{XY}(x,y)/(p_X(x)p_Y(y))]$$.
• Kullback–Leibler Divergence: Measures the difference between two distributions, modeling SSR asymmetry by evaluating both $D(p_S \Vert p_K)$ and $D(p_K \Vert p_S)$.
• Information (Fractal) Dimension: Defined as $d_I = H/\ln(1/m)$, reveals the extent of a component’s probabilistic “spread” over effective stages.
• Entropic Cost: The normalized total information required for output, computed by $\text{entropic-cost} = (H_S + H_R + H_K)/H_K$.

Asymmetry in exchange is established whenever $D(p_S \Vert p_K) \neq D(p_K \Vert p_S)$, with the asymmetry parameter $A = D(p_S \Vert p_K) / D(p_K \Vert p_S)$ providing a scalar diagnostic for manipulative capacity and exchange selectivity (Ahmad, 2018).

3. Box-Model Representation and Rate Coefficients

SSR diagrams are effectively visualized by a linear sequence of labeled boxes:

[Source] → [Reservoir] → [Sink]

Each directed edge is annotated with three concurrent flows:

1. Material (e.g., particle count, mass)
2. Process (chemical or physical transformation)
3. Information (probabilities, entropies)

Pairwise transfer rates govern the magnitude and direction of flow:

• $R_{S\to R}$: Content transfer from source to reservoir
• $R_{R\to K}$: Content transfer from reservoir to sink

The transport process is typically discretized into sequential pulses ($n=0,1,2,\dots$), tracking the evolution of normalized distributions $p_S(n)$, $p_R(n)$, and $p_K(n)$. Manipulability is controlled by modulating rate coefficients, including temporal variability and the introduction of reservoir sub-stages (Ahmad, 2018).

In networked settings, SSR diagrams map naturally onto metric graphs where mass density $\rho_e(t,x)$ evolves along edges, and vertex reservoirs $r_v(t)$ interact via incident fluxes $J_e(t,v)$. Mass conservation spans both edge and vertex compartments (Heinze et al., 2024).

4. Reservoir Dynamics and Multiscale Limits

The reservoir’s memory and storage capacity manifest in different operational regimes:

• Free-flow Mode ($R_{S\to R} = R_{R\to K}$): The reservoir serves as a passive channel with minimal memory.
• Source-friendly Mode ($R_{S\to R} > R_{R\to K}$): The reservoir aggregates content, maximizing memory and selectivity.
• Reservoir-friendly Mode ($R_{S\to R} < R_{R\to K}$): The reservoir rapidly emits its accumulated input, reducing storage.

In metric graph models, multiscale limits are explored via EDP-convergence:

• Kirchhoff-limit: Vanishing reservoirs ($r_v \to 0$) yield pure edge diffusion with Kirchhoff-type vertex boundary conditions.
• Fast-edge-diffusion Limit: Rapid edge transport reduces the system to coupled ODEs governing vertex and edge aggregate states.
• Combinatorial Graph Limit: Reservoirs and edge-nodes collapse to purely vertex ODEs describing direct vertex–vertex interactions (Heinze et al., 2024).

Transition between these limits enables investigation of emergent behaviors, dynamic exchange selectivity, and effective dimensional reduction (from PDE–ODE hybrid systems to purely discrete models).

5. Asymmetric Exchange and Diagnostic Metrics

SSR diagrams are characterized by fundamentally asymmetric interactions, as revealed by the dual evaluation of Kullback–Leibler divergences: $D(p_S \Vert p_K)$ and $D(p_K \Vert p_S)$. The observed inequality, maintained in all nontrivial SSR simulations, expresses the selective memory and manipulation inherent to the reservoir. The asymmetry parameter $A$ is tracked as a function of exchange rates and system architecture.

This diagnostic forms the basis for quantifying manipulative efficiency, irreversible transformation, and deviation from equilibrium transmission. Numerical schemes (finite difference, finite volume) can simulate birth–death–jump dynamics in networks, capturing swift and negligible exchange regimes as scaling parameters vary. Plots of reservoir mass versus edge fluxes reveal source/sink switching and validate predicted asymmetry patterns (Heinze et al., 2024).

6. Application to Self-Organizing Carbon Cages

A salient physical instantiation is the evolution of carbon fullerenes in hot vapor, mapped onto a multi-stage SSR box model (Ahmad, 2018):

• Source: Ensemble of large cages ($C_x$, $x>60$) formed by nucleation.
• Reservoir: Populations of intermediate cages experiencing fragmentation and reformation; process history encoded by cage lifetimes and fragmentation sequences.
• Sink: Stable, highly symmetric $C_{60}$ (“buckyball”) that accumulates the majority of surviving population.

Probabilities $p_x(n)$, self-information values $i_x(n)$, and entropies $H_x$ are computed per cage size per fragmentation pulse. The fractal dimension $d_x = H_x/\ln(1/N)$ specifies the dynamic spread over effective steps. Empirical findings reveal rapid disappearance of larger cages (low $H_x$), persistent accumulation of $C_{60}$ (dominant $H_{60}$), escalating entropic cost, and persistent asymmetry in fragmentation product distributions.

This example illustrates how the SSR framework supports empirical tracking, theoretical diagnostics, and mechanism elucidation in complex self-organizing systems.

7. Schematic Visualization and Network Generalization

In SSR diagrams mapped to metric graphs (Heinze et al., 2024):

• Edges: Represent transport intervals with mass-flux arrows ($J_e$)
• Vertices: Depicted with attached tanks ($r_v(t)$) denoting reservoirs
• Arrows: Directed flux indicates source or sink behavior depending on sign

Scaling analyses produce thin layers, quasi-stationarity, or jump-chain dynamics depending on system parameters, allowing flexible adaptation of SSR diagrams to represent evolving networks or engineered systems. Mass conservation and gradient-flow principles preserve the SSR identity in both microscopic (discrete) and macroscopic (continuum) representations.

SSR diagrams thus provide a principled, quantitative foundation for modeling and analyzing self-organizing, dissipative, and networked systems with rich manipulation, storage, and directional exchange properties (Ahmad, 2018, Heinze et al., 2024).