ChargeFlow: A Unified Transport Framework

• ChargeFlow is a concept that unifies various charge-mediated transport phenomena by coupling a conserved charge field with a corresponding flow field.
• It underpins methods for refining electron densities in materials, driving electrokinetic flow in nanochannels, and analyzing anisotropic flows in heavy-ion collisions.
• By linking charge allocation to transport observables, ChargeFlow offers actionable insights for design improvements in nanofluidics, ion channels, and networked circuits.

ChargeFlow is a research term used in several technical literatures to denote charge-mediated transport, charge-conditioned flow, or charge/flux-conservation formalisms. In current usage it spans at least six distinct but structurally related settings: flow-matching refinement of electron densities in periodic materials, self-generated and pressure-driven electrokinetic transport in charged micro- and nanochannels, charge-dependent anisotropic flow in heavy-ion collisions, nonlinear charge transport in ion channels and correlated quantum systems, triboelectric charging in particulate transport, and charge-allocation or charge-flow descriptions in engineered networks and circuits (Nguyen et al., 25 Mar 2026, Shrestha et al., 2024, Sousa et al., 21 May 2025, Wang, 2024). This suggests a family of concepts rather than a single canonical definition: in each case, a charge field couples to a transport field, but the transported object may be electron density, electrolyte momentum, conserved current, particle charge, electrical power, or magnetic flux.

1. Domain structure and common mathematical pattern

Across the cited literatures, ChargeFlow denotes either a directly physical charge-transport process or a formalism in which charge is the state variable controlling transport. The dominant equations differ by domain—CNF ODEs in density refinement, PNPS or PNP in electrokinetics and ion channels, hydrodynamics plus conserved-current response in heavy-ion collisions, DQMOM transport in powders, and conservation laws on circuit containers and pumps in electromagnetic circuit theory—but each formulation couples charge storage or redistribution to a flow field or a flow observable (Nguyen et al., 25 Mar 2026, Petersen et al., 25 Oct 2025, Voloshin et al., 2014, Zeybek et al., 2021, Carvalho et al., 2015, Wang, 2024).

Domain	ChargeFlow meaning	Representative papers
Electronic-structure ML	flow-matching refinement of charge-conditioned electron densities	(Nguyen et al., 25 Mar 2026)
Electrokinetics and nanofluidics	charge-pattern-driven electrolyte flow, ionic current, selectivity, and energy conversion	(Shrestha et al., 2024, Petersen et al., 25 Oct 2025, Kateb et al., 2019)
Heavy-ion collisions	charge-dependent anisotropic flow and net-charge flow observables	(Sousa et al., 21 May 2025, Collaboration, 2015, Gürsoy et al., 2018, Voloshin et al., 2014)
Complex media transport	permanent-charge effects, porous electrohydromechanics, triboelectric charging, anomalous multi-charge flow	(Huang et al., 2020, Yang, 2024, Zeybek et al., 2021, Ozler et al., 4 Dec 2025, Guan et al., 6 Jun 2025)
Infrastructure and circuit formalisms	EV charging congestion control; electric-charge-flow and magnetic-flux-flow diagrams	(Carvalho et al., 2015, Wang, 2024)

The unifying pattern is not phenomenological identity but structural analogy. Charge enters as a conserved, conditioned, or externally patterned quantity; transport responds through drift, advection, diffusion, hydrodynamic body force, or optimization. Where the term becomes potentially misleading is that “flow” may mean fluid velocity, anisotropic flow coefficients, current allocation, or a learned probability-flow field. The literature therefore uses ChargeFlow as a domain-specific technical label rather than as a universally standardized concept.

2. ChargeFlow in electronic-structure theory and machine learning

In electronic-structure theory, ChargeFlow is the name of a generative refinement model that maps a charge-conditioned superposition of isolated atomic densities to the corresponding self-consistent DFT density on the native periodic VASP real-space grid (Nguyen et al., 25 Mar 2026). The source density is

$$\rho_{\mathrm{sup}}(r)=\sum_i \rho^{\mathrm{atom}}_{Z_i,q}(r-R_i),$$

the target is $\rho_{\mathrm{DFT}}(r\mid q)$, and the refinement signal is the deformation density $$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$. The model uses a continuous normalizing flow with state evolution

$\frac{dz_t}{dt}=v_\theta(z_t,t,c),$

trained by flow matching on the linear interpolation path $x_t=t x_1+(1-t)x_0$ with target velocity $u_t=x_1-x_0$. The loss is

$L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$

with $\alpha=2.0$, and inference uses Heun’s method with 50 function evaluations and EMA weights with decay $0.999$. The velocity field is parameterized by a 3D U-Net with four resolution levels, base channels $C=16$, multipliers $\rho_{\mathrm{DFT}}(r\mid q)$0, circular padding for periodic boundary conditions, residual blocks with dropout $\rho_{\mathrm{DFT}}(r\mid q)$1, and self-attention only at the coarsest level. No explicit charge-state embedding is used; the charge state is implicit in the SAD input.

The training corpus contains 11,878 charged-density calculations from 4,145 Materials Project-derived parent structures, with 9,502 training examples and 2,376 development examples after filtering invalid grid-size pairs. External evaluation uses 1,671 periodic structures spanning perovskites, charged defects, multisite diamond defects, special diamond defects, MOFs, extreme MOFs, organic crystals, and extreme organic crystals. Performance is deliberately nonuniform across classes: regression baselines remain better on some in-distribution regimes such as perovskites and charged defects, whereas ChargeFlow is strongest when nonlocal redistribution and charge-state extrapolation dominate. On the external benchmark it reduces deformation-density $\rho_{\mathrm{DFT}}(r\mid q)$2 from 3.62% to 3.21% relative to a ResNet baseline and improves charge-response cosine similarity from 0.571 to 0.655. It yields successful Bader partitioning on all 1,671 benchmark structures, with atom-level $\rho_{\mathrm{DFT}}(r\mid q)$3 and MAE $\rho_{\mathrm{DFT}}(r\mid q)$4, and it provides an approximate three-order-of-magnitude speed-up over conventional DFT for inference.

The significance of this usage is methodological. ChargeFlow does not predict energies or forces directly; it predicts charge-state-dependent densities and evaluates them through density fidelity, deformation-density fidelity, charge-response similarity, Bader partitioning, and Hartree potentials. Its limitations are explicit: the internal 80/20 split is not parent-structure disjoint, spin density and magnetization are not modeled, and it is not uniformly best across all benchmark classes. Even so, it establishes a technical interpretation of ChargeFlow as deterministic transport in density space rather than as fluid or current flow.

3. Electrokinetic ChargeFlow in capillaries and nanochannels

In electrokinetics, ChargeFlow refers to several mechanisms by which heterogeneous charge distributions generate or control electrolyte motion. A central formulation considers a capillary whose wall simultaneously supports a modulated surface charge

$\rho_{\mathrm{DFT}}(r\mid q)$5

and active ionic fluxes

$\rho_{\mathrm{DFT}}(r\mid q)$6

with the coupled electrochemical-hydrodynamic response governed by steady Poisson, Nernst–Planck, and Stokes equations (Shrestha et al., 2024). The key symmetry-breaking quantity is the zero-mode selector

$\rho_{\mathrm{DFT}}(r\mid q)$7

which vanishes unless both charge modulation and active flux are present with phase mismatch. When $\rho_{\mathrm{DFT}}(r\mid q)$8, the system develops a steady unidirectional through-flow; the $\rho_{\mathrm{DFT}}(r\mid q)$9 harmonics generate counter-rotating toroidal vortices. The wavelength-averaged speed obeys

$$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$0

and vanishes asymptotically as $$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$1. The same analysis shows that diffusivity mismatch $$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$2 can prevent cancellation when cationic and anionic fluxes are co-generated at the same site. The paper proposes a microfluidic generator based on enzyme-coated patches and static wall charge heterogeneity, operating without externally imposed bulk fields or gradients.

A second electrokinetic usage studies corrugated slit nanochannels with periodic aperture

$$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$3

and patterned wall charge

$$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$4

solved with the fully coupled PNPS equations (Petersen et al., 25 Oct 2025). Three regimes are identified. In Regime I, at low or zero pressure gradient, inhomogeneous wall charge generates a nonlinear torque

$$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$5

that drives recirculating flow, and asymmetric charge placement produces net axial flow. In Regime II, throughput is proportional to the applied pressure gradient but significantly diminished relative to channels without charge inhomogeneity, because electrostatic forces oppose displacement of ions from the diffuse layer. In Regime III, a transition occurs between electrostatically and mechanically controlled flow regimes, in which a marginal increase in pressure gradient can trigger an abrupt, orders-of-magnitude increase in mean velocity. The same study uses random-walk particle tracking to show selective control of ionic flux and ion dispersion, with selectivity $$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$6 approaching $$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$7 near the transition for some parameter choices.

A closely related device-level implementation uses electrostatic gating in a converging–diverging solid-state nanochannel to create heterogeneous wall charge and charge inversion (Kateb et al., 2019). The periodic geometry converts a uniform gate bias into a nonuniform radial field, stronger around the diverging section and weaker near the converging part. The reported local point-of-zero-charge thresholds are $$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$8, $$\Delta \rho(r)=\rho_{\mathrm{DFT}}(r)-\rho_{\mathrm{sup}}(r)$$9, and $\frac{dz_t}{dt}=v_\theta(z_t,t,c),$0. Low gate bias preserves the intrinsic wall-charge polarity and modulates forward EOF, whereas moderate gate bias produces charge inversion in diverging sections, reverse local flow, and circulation. The study further demonstrates particle trapping and rejection controlled by this recirculation.

Other nanofluidic realizations emphasize performance metrics. Introducing charged cylindrical obstacles into a narrow slit channel increases electrokinetic energy conversion efficiency from approximately 4% in a planar slit channel to approximately 18% with a regular obstacle pattern, with strong dependence on obstacle radius, pitch, wall hydrodynamic slip, finite ionic size, local permittivity variation, and obstacle-to-wall charge density ratio (Kahali et al., 2017). At the molecular scale, a ternary-mixture Enskog–BGK theory for nanochannels recovers Nernst–Planck and Navier–Stokes behavior without imposing constitutive flux laws, while also predicting layering, nonlocal drag, and strong departures from electroneutrality under extreme confinement (Marconi et al., 2012).

Taken together, these studies define electrokinetic ChargeFlow as boundary-programmed transport. The control variables are patterned $\frac{dz_t}{dt}=v_\theta(z_t,t,c),$1, active ionic fluxes, geometry, pressure gradient, and gate potential; the observables are net flow, vortices, ionic current, selectivity, and conversion efficiency.

4. ChargeFlow in heavy-ion collisions

In heavy-ion phenomenology, ChargeFlow denotes charge-dependent anisotropic flow or, more specifically, charge-conjugation-odd flow observables built from conserved-current transport (Sousa et al., 21 May 2025). Standard flow vectors are extended to net-charge flow vectors such as

$\frac{dz_t}{dt}=v_\theta(z_t,t,c),$2

and the initial-state estimator program is generalized by introducing charge-density cumulants through

$\frac{dz_t}{dt}=v_\theta(z_t,t,c),$3

The C-odd eccentricity estimator

$\frac{dz_t}{dt}=v_\theta(z_t,t,c),$4

predicts final net-charge flow through linear response $\frac{dz_t}{dt}=v_\theta(z_t,t,c),$5. In Au+Au collisions at $\frac{dz_t}{dt}=v_\theta(z_t,t,c),$6, simulations with superMC, MUSIC, NEoS-BQS, and UrQMD show that $\frac{dz_t}{dt}=v_\theta(z_t,t,c),$7 exceeds 0.9 in central and mid-central collisions, while $\frac{dz_t}{dt}=v_\theta(z_t,t,c),$8 is very small. The inferred $\frac{dz_t}{dt}=v_\theta(z_t,t,c),$9 is $x_t=t x_1+(1-t)x_0$0, with increasing relative importance toward peripheral collisions.

A parallel literature studies charge-dependent anisotropic flow as a possible signature of the Chiral Magnetic Wave. ALICE measured a three-particle correlator

$x_t=t x_1+(1-t)x_0$1

and a differential version

$x_t=t x_1+(1-t)x_0$2

in Pb–Pb collisions at $x_t=t x_1+(1-t)x_0$3 (Collaboration, 2015). The integral $x_t=t x_1+(1-t)x_0$4 signal shows opposite signs for positive and negative particles and grows toward peripheral collisions, qualitatively matching a CMW-like $x_t=t x_1+(1-t)x_0$5 splitting. However, a nonzero $x_t=t x_1+(1-t)x_0$6 signal, reduced by about a factor of 3 relative to $x_t=t x_1+(1-t)x_0$7, and a differential correlator peaked at small $x_t=t x_1+(1-t)x_0$8 indicate substantial background contributions, notably local charge conservation coupled with radial and anisotropic flow. The paper therefore treats the interpretation as non-unique rather than as a definitive observation of the CMW.

That interpretive problem motivated an efficiency-independent correlator proposed by Voloshin and Belmont,

$x_t=t x_1+(1-t)x_0$9

designed to avoid explicit dependence on the experimentally measured event charge asymmetry $u_t=x_1-x_0$0 and to support differential measurements in $u_t=x_1-x_0$1, $u_t=x_1-x_0$2, and species (Voloshin et al., 2014). In the Blast Wave estimates reported there, local charge conservation generates a positive peak at small $u_t=x_1-x_0$3, a zero crossing around $u_t=x_1-x_0$4–0.7, and a vanishing large-gap tail; this pattern is proposed as a discriminator against a long-range CMW-like signal.

Electromagnetic fields introduce an additional, non-anomalous charge-dependent flow sector. Perturbative simulations in iEBE-VISHNU with resistive Maxwell evolution predict rapidity-odd charge-odd $u_t=x_1-x_0$5 and $u_t=x_1-x_0$6, generated by Faraday and Lorentz effects plus spectator Coulomb fields, and rapidity-even $u_t=x_1-x_0$7 and $u_t=x_1-x_0$8, generated by Coulomb forces from the net plasma charge density (Gürsoy et al., 2018). The predicted magnitudes are of order few$u_t=x_1-x_0$9 for $L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$0 and $L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$1 at RHIC energies, with $L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$2 negative and $L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$3 of order $L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$4 for pions in one reported case.

In this literature, ChargeFlow is therefore both an observable class and a diagnostic framework. Its central technical issue is disentangling genuine conserved-current dynamics from kinematic and local-charge-conservation backgrounds.

5. ChargeFlow in ion channels, porous media, powders, and correlated quantum systems

In ion-channel theory, permanent charge acts as a structural control parameter for charge flow through a quasi-one-dimensional PNP model (Huang et al., 2020). The central diagnostic is the flux ratio

$L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$5

which compares the flux of species $L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$6 at permanent charge $L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$7 to the zero-charge case. For two species with $L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$8 and $L=L_{\mathrm{FM}}+\alpha L_{\mathrm{NormMAE}},$9, a rigorous universal ordering holds for $\alpha=2.0$0: $\alpha=2.0$1 Analytical asymptotics describe the $\alpha=2.0$2 and $\alpha=2.0$3 regimes, while adaptive moving-mesh finite elements reveal intermediate nonlinear behavior and saddle-node bifurcations of the $\alpha=2.0$4 contours. For sufficiently large positive permanent charge, cation flux is strongly suppressed, $\alpha=2.0$5, whereas anion enhancement occurs only in a finite voltage window.

In nonequilibrium quantum thermodynamics, ChargeFlow has been generalized to anomalous transport of any conserved charge in a correlated bipartite quantum system (Guan et al., 6 Jun 2025). For energy-only, initially uncorrelated, energy-conserving dynamics, the paper proves a no-go result for anomalous energy flow: $\alpha=2.0$6 and extends the no-go to catalytic protocols satisfying strict return conditions. By contrast, with multiple conserved charges the exact identity

$\alpha=2.0$7

permits anomalous flow in one channel if normal flows in other channels are sufficiently strong. The proposed mechanism is a multi-charge drag effect, not depletion of initial correlations.

In porous electrohydromechanics, ChargeFlow denotes a three-continuum theory of an elastic solid, a viscous fluid, and a mobile charge continuum occupying the same spatial region (Yang, 2024). The framework imposes mass conservation for solid and fluid, charge conservation for the mobile charge continuum,

$\alpha=2.0$8

linear and angular momentum balances for the constituents and the mixture, and quasistatic Maxwell equations with $\alpha=2.0$9 and $0.999$0. The resulting theory is nonlinear, valid for large deformations and strong fields, and can be specialized to rigid-matrix electroosmosis, Biot-type poroelasticity, or purely conductive limits.

In triboelectric powder transport, ChargeFlow appears in two complementary forms. An Eulerian DQMOM formulation solves the joint distribution $0.999$1 for particle diameter, velocity, and charge, including wall uptake of charge, inter-particle exchange, electrostatic field feedback, and gas drag (Zeybek et al., 2021). Two-dimensional steady channel tests show charging from the walls toward the center and in the downstream direction, with strong sensitivity to particle size distribution, inlet velocity distribution, and a charge diffusion coefficient. A later DNS study at $0.999$2 shows that square ducts charge particles faster than channels because secondary flows intensify wall collisions and improve cross-sectional mixing (Ozler et al., 4 Dec 2025). For particles with $0.999$3, 85% of the equilibrium charge is reached approximately 1.5 times faster in duct flow than in channel flow, and once the powder reaches half of its equilibrium charge, particles increasingly accumulate at the wall, reducing central concentration and altering the carrier flow.

This cluster of usages broadens the concept beyond electrokinetics narrowly defined. ChargeFlow can describe charge-modulated selectivity in ionic channels, charge transport in deformable porous media, triboelectric charge build-up in dispersed particulate systems, or anomalous transport among conserved charges in closed quantum dynamics.

6. Networked and circuit-theoretic uses

One systems-level use of ChargeFlow concerns electric-vehicle charging networks. A rooted radial distribution feeder is modeled as a directed tree with convexified branch-flow constraints, and charging allocations are computed either by maximizing instantaneous aggregate power or by maximizing

$0.999$4

the proportional-fairness objective (Carvalho et al., 2015). As the Poisson arrival rate $0.999$5 increases, the system undergoes a continuous phase transition to congestion, characterized by the order parameter

$0.999$6

and susceptibility

$0.999$7

On the SCE 47-bus feeder, proportional fairness maintains free flow up to a higher $0.999$8 than max-flow and yields much lower charging-time inequality, with high-load Gini coefficients reaching about 0.45 under proportional fairness versus about 0.91 under max-flow. Here the term ChargeFlow is infrastructural rather than microscopic: it concerns allocation of electrical charging throughput under voltage constraints.

A more abstract but physically explicit use appears in electromagnetic-field-based circuit theory, which introduces Electric-Charge-Flow and Magnetic-Flux-Flow diagrams as a unified language for phase-independent and phase-dependent circuits (Wang, 2024). In the ECF description, electric-charge pumps transfer accumulated charges $0.999$9 between electric-charge containers with branch current

$C=16$0

node storage

$C=16$1

and charge conservation law

$C=16$2

In the MFF dual, magnetic-flux pumps transfer accumulated fluxes $C=16$3 with two-terminal voltage equal to flux-flow rate,

$C=16$4

loop storage

$C=16$5

and flux conservation law

$C=16$6

The formalism applies equally to resistor–inductor–capacitor and semiconductor circuits, Josephson junction circuits, and QPS junction circuits. The paper explicitly presents ECF and MFF as new languages for circuit design and analysis and states that they are promising for machine learning.

This last usage makes the family resemblance especially clear. ChargeFlow need not denote material transport of charged matter; it can also denote a conservation-based representation in which charge is the transported state variable organizing the system equations.