Minimum Electromagnetic Entropy Production (MEMEP)

• MEMEP is a variational principle that determines superconducting current distributions by minimizing electromagnetic dissipated energy under Maxwell equations and material constraints.
• It effectively handles non-linear, hysteretic behavior in high-field REBCO magnets, accurately modeling screening currents, AC loss, and quench propagation using 2D/3D numerical methods.
• MEMEP benchmarks against alternative methods and extends to stochastic and quantum formulations, demonstrating versatility in both superconducting engineering and thermodynamic applications.

Searching arXiv for recent and foundational papers on MEMEP and related minimum entropy production formulations. Minimum Electromagnetic Entropy Production (MEMEP), also written in some sources as Minimum Electro-Magnetic Entropy Production, is a variational principle used most explicitly in superconducting-electromagnetic modeling to determine current distributions by minimizing electromagnetic entropy production, i.e. dissipated energy, under Maxwellian and constitutive constraints. In the recent REBCO-magnet literature, MEMEP is presented as a numerical framework for non-linear, hysteretic superconducting behavior, for screening currents, and for coupled electrothermal quench analysis in full-scale high-field magnets (Pardo et al., 8 Sep 2025). In a broader thermodynamic literature, minimum entropy production ideas are also formulated for electrical circuits, network thermodynamics, continuous-time Markov processes, and quantum transport or quantum-state evolution, although these works differ in assumptions, scope, and the meaning assigned to “minimum entropy production” (Miranda et al., 2012, Polettini, 2011, Dechant, 2021, Márkus et al., 2021).

1. Definition and scope

In superconducting magnet modeling, MEMEP is a variational approach specifically suited to model non-linear, hysteretic superconducting behavior, as found in high-field REBCO magnets. It seeks the current density distribution that minimizes the total electromagnetic entropy production, identified in the cited literature with dissipated energy, and this is presented as critically relevant for superconductors operating near their critical current and undergoing flux motion, including screening currents (Pardo et al., 8 Sep 2025).

The method is used in several closely related contexts. One line of work develops a fast and accurate two-dimensional cross-sectional model for the electromagnetic response of non-insulated (NI), metal-insulated (MI), and soldered REBCO coils, with attention to current density, screening current induced field (SCIF), and AC loss (Pardo et al., 2023). Another line couples MEMEP to finite-difference thermal solvers for quench analysis in axi-symmetric full-scale magnets of more than 32 T field strength, including screening currents and temperature-dependent material properties (Dadhich et al., 2024). A further application adapts MEMEP to partially coupled REBCO stacks with end connections, where the objective is accurate and computationally tractable calculation of coupling loss in configurations that would otherwise require full 3D modeling (Li et al., 2020).

Beyond superconducting engineering, the phrase minimum entropy production appears in works on simple electrical circuits, network thermodynamics, stochastic thermodynamics, and information geometry. In those settings, MEMEP is not always a numerical electromagnetic solver. Instead, it may denote a thermodynamic principle, a variational statement about steady states, or an optimal-path criterion on a probability manifold (Miranda et al., 2012, Polettini, 2011, Gassner et al., 2020). This suggests that “MEMEP” names a family of related minimization ideas rather than a single universally identical formalism.

2. Variational formulation

The central electromagnetic relation used in the REBCO-quench formulation is derived from the Maxwell-Faraday equation in Coulomb’s gauge,

$$E(J) = -\frac{\partial \mathbf{A}[J]}{\partial t} - \frac{\partial \mathbf{A}_a}{\partial t} - \nabla \phi .$$

After time discretization,

$$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$

with $\Delta J = J(t)-J(t-\Delta t)$. The quantity minimized is the functional

$$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$

where $U(J)=\int_0^J J' \cdot E(J')$, and $J_p$ is the current at the previous time step (Pardo et al., 8 Sep 2025).

The constitutive law in the same formulation separates angular and radial transport channels. For the engineering current density,

$$J_\varphi = \frac{E_\varphi}{\rho_n} + J_{ce} \left(\frac{|E_\varphi|}{E_c} \right)^{1/n} \mathrm{sign}(E_\varphi),$$

where $\rho_n$ is the homogenized resistivity of all normal conductor layers, $J_{ce}$ is the engineering critical current density, and $E_c,n$ are power-law parameters of the superconductor. Radial current between turns is represented by

$$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$0

with $$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$1 the contact resistance and $$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$2 the tape thickness (Pardo et al., 8 Sep 2025).

In the NI/MI-coil literature, the corresponding MEMEP functional is stated in 3D as

$$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$3

with

$$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$4

For the axisymmetric coil model, the functional reduces to a 2D cross-sectional expression over $$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$5, in terms of $$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$6, $$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$7, and $$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$8 (Pardo et al., 2023).

For partially coupled REBCO stacks, the 2D long-stack reduction takes the form

$$E(J) = -\frac{\mathbf{A}[\Delta J]}{\Delta t} - \frac{\Delta \mathbf{A}_a}{\Delta t} - \nabla \phi ,$$9

and, when finite end resistances are included,

$\Delta J = J(t)-J(t-\Delta t)$0

In that paper, $\Delta J = J(t)-J(t-\Delta t)$1 is given piecewise: for the superconductor,

$\Delta J = J(t)-J(t-\Delta t)$2

and for a normal conductor,

$\Delta J = J(t)-J(t-\Delta t)$3

These expressions make explicit that the minimization includes both non-linear superconducting dissipation and Ohmic loss (Li et al., 2020).

3. Numerical realizations

The principal coupled implementation is MEMEP-FD, in which the MEMEP electromagnetic solution is combined with a finite-difference solution of the heat diffusion equation,

$\Delta J = J(t)-J(t-\Delta t)$4

with $\Delta J = J(t)-J(t-\Delta t)$5 the local Joule heat source. The cited workflow is iterative at each time step: solve for $\Delta J = J(t)-J(t-\Delta t)$6 and $\Delta J = J(t)-J(t-\Delta t)$7 using MEMEP, calculate $\Delta J = J(t)-J(t-\Delta t)$8, update temperatures with the finite-difference method, update material properties such as $\Delta J = J(t)-J(t-\Delta t)$9 and $$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$0 as functions of temperature and magnetic field, and repeat until convergence. Adaptive time stepping is used to resolve time scales from microseconds to seconds (Pardo et al., 8 Sep 2025).

A closely related C++ software implementation is described for coupled electro-magnetic and electro-thermal analysis of high-field superconducting magnets. In that work, the electromagnetic and electrothermal solvers are coupled iteratively, screening currents are included, and the MEMEP solver is run on longer timesteps than the thermal finite-difference solver; intermediate power values are provided to the thermal solver by linear interpolation. The same work also states that a homogenized model can be used in which each cross-sectional element contains several turns, and it cites a parallel computing iterative algorithm for large windings (Dadhich et al., 2024).

For NI and MI coils, the cross-section is discretized into finite elements with current density assumed uniform in each, the superconducting layer is modeled with a non-linear power-law $$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$1-$$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$2 relation, the normal metal or insulator is modeled as Ohmic, and homogenization combines the response of superconductor, stabilizer, and turn-to-turn interface resistance. The method enforces current conservation in each turn or pancake during minimization and updates both angular and radial current densities, thereby capturing screening currents, radial bypass currents, and inductive effects in dynamic simulations (Pardo et al., 2023).

For REBCO stacks with end connections, the numerical model exploits the long-tape geometry so that current flow is almost entirely along the tapes and coupling currents close through resistive end connections. The algorithm discretizes the cross-section into $$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$3 elements and tries pairwise changes of current in elements, including cross-tape loop changes, that minimize the functional while respecting loop or Kirchhoff constraints. The paper states that this 2D MEMEP method handles realistic end resistances, agrees with measurements, and enables parameter studies that would be impractical in full 3D finite-element modeling (Li et al., 2020).

4. Benchmarking and relation to alternative methods

The recent quench-propagation study explicitly benchmarks MEMEP-FD against a Partial Element Equivalent Circuit (PEEC) model. PEEC treats the system as a network of resistances and inductances derived from Maxwell’s equations and is described there as valuable for certain geometries and for induced currents in normal metals. The same paper states that PEEC becomes computationally expensive or less accurate for the non-linear, hysteretic, and highly anisotropic superconducting screening current phenomena typical in REBCO, and that MEMEP-FD and PEEC show good agreement when screening currents are neglected, especially for global quantities such as maximum temperature and bore field (Pardo et al., 8 Sep 2025).

The NI/MI-coil paper benchmarks MEMEP against an $$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$4-$$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$5 formulation on a double pancake coil and reports excellent agreement in angular and radial current distributions under identical conditions. The variational approach is there presented as especially efficient in 2D axisymmetric layouts with thousands of turns and as suitable for fast parameter studies for optimization (Pardo et al., 2023).

The stack-loss study frames the comparison differently, against full 3D finite-element approaches such as the $$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$6-formulation and $$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$7-$$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$8 formulation. It states that 3D FEM requires full resolution of tape ends and current loops, leading to immense mesh sizes and long computation times, whereas the 2D MEMEP adaptation incorporates the essential 3D end-coupling effect through discrete resistance terms in a 2D scheme (Li et al., 2020).

These comparisons establish a consistent pattern. MEMEP is not introduced as a general replacement for all electromagnetic formulations. Rather, it is presented as a variational method that is particularly effective when the dominant physics includes non-linear superconducting response, hysteresis, screening currents, large turn counts, or coupled radial and angular current paths.

5. Screening currents, quench propagation, and device-level applications

A central result in the 2025 REBCO-magnet study is that screening currents greatly increase the speed of quench propagation. In that work, MEMEP’s variational formulation self-consistently produces the physically correct distribution of screening currents under changing external and transport conditions. The cited mechanisms are: redistribution of current spreads dissipation more rapidly in the cross-section; screening current collapse during local quench releases additional energy through AC loss; and inductive coupling between layers or turns accelerates normal-zone expansion. The same work states that neglecting screening currents artificially slows down quench propagation and overestimates peak local temperatures, yielding non-conservative and potentially unsafe quench predictions. Its quench criterion identifies local regions as quenched where $$F[\Delta J] = \int_{\Omega}\left\{ \frac{1}{2} \Delta J \cdot \frac{\mathbf{A}[\Delta J]}{\Delta t} + \Delta J \cdot \frac{\Delta \mathbf{A}_a}{\Delta t} + U(\Delta J + J_p) \right\} + \int_{\Omega}(J_p + \Delta J)\cdot \nabla \phi ,$$9 (Pardo et al., 8 Sep 2025).

The practical setting for that analysis is a 32 T all-superconducting magnet. The paper states that the findings will have an impact in the design of ultra-high-field magnets for NMR or user facilities, and possibly for other kinds of magnets, including those for fusion energy. It also states that for ultra-high-field $U(J)=\int_0^J J' \cdot E(J')$0–$U(J)=\int_0^J J' \cdot E(J')$1 T REBCO inserts, screening currents are non-negligible due to the tape geometry and are significant in dictating quench evolution, maximum temperature, and operation margin (Pardo et al., 8 Sep 2025).

In NI and MI REBCO magnets, the 2023 electromagnetic model analyzes current density, SCIF, and AC loss in a fully superconducting 32 T magnet with a REBCO insert and a low-temperature superconducting outsert. It states that metal-insulated coils enable transfer of angular current in the radial direction, and hence magnet protection, while keeping the same screening currents and AC loss of isolated coils, even at relatively high ramp rates of $U(J)=\int_0^J J' \cdot E(J')$2. The same paper reports that soldered coils with low resistance between turns present relatively low AC loss for over-current configuration, which might enable higher generated magnetic fields (Pardo et al., 2023).

The 2024 electrothermal-software paper applies MEMEP-based coupled analysis to full-scale magnets under the SuperEMFL project and states that magnets incorporating non-insulated coils are more reliable against quench than metal-insulated coils. It also states that realistic cooling conditions at boundaries are essential for such simulations (Dadhich et al., 2024).

In REBCO stacks with end connections, MEMEP is used to study coupling loss under parallel sinusoidal magnetic field excitation. The paper reports amplitude-dependence, frequency-dependence, resistance-dependence, and length-dependence of coupling loss, distinguishes superconductor dissipation from resistive dissipation, and presents analytical expressions for low-frequency and arbitrary-frequency regimes including inductance effects (Li et al., 2020). This places MEMEP not only in quench analysis, but also in AC-loss and interconnection-resistance design for multi-tape devices such as motor windings.

6. Broader theoretical context, debates, and limits

The broader literature cited alongside MEMEP shows that minimum entropy production is not a single uncontested universal law. In a network-based formulation, Schnakenberg’s observables are identified as the macroscopic constraints that prevent a system from relaxing to equilibrium, and in the linear regime the steady state minimizes entropy production subject to fixed cycle affinities. That work applies the result to master-equation systems and presents the principle as conforming to Prigogine’s original formulation while remaining distinct from maximum entropy production principles (Polettini, 2011).

In simple electrical circuits, a separate analysis of $U(J)=\int_0^J J' \cdot E(J')$3, $U(J)=\int_0^J J' \cdot E(J')$4, and $U(J)=\int_0^J J' \cdot E(J')$5 circuits states that entropy production is minimal in the stationary regime, in agreement with Prigogine’s theorem, and explicitly rejects the claim that entropy production is maximal in such circuits. The same paper emphasizes that the minimum applies to steady state, not necessarily to transients (Miranda et al., 2012).

For continuous-time Markov processes, one paper shows that a prescribed time evolution may be realized at arbitrarily small entropy production if activity is allowed to diverge, but that for fixed activity the entropy-production minimum is achieved by conservative forces and can be expressed through a graph-distance-based Wasserstein distance. It derives a speed limit relating dissipation, average number of transitions, and Wasserstein distance (Dechant, 2021). Another stochastic-thermodynamic study goes further and states that real nonequilibrium steady states generally violate both MINEP and MAXEP, yet for large interconnected continuous-time Markov chains the steady-state entropy production tends to converge toward the minimum as system size increases (Ray et al., 14 Jul 2025). This suggests that broad invocations of MEMEP outside the linear-regime or constrained-activity settings require care.

Quantum formulations push the principle in a different direction. One paper introduces quantized entropy conductance,

$U(J)=\int_0^J J' \cdot E(J')$6

with entropy current

$U(J)=\int_0^J J' \cdot E(J')$7

and, for a transferred quantum $U(J)=\int_0^J J' \cdot E(J')$8,

$U(J)=\int_0^J J' \cdot E(J')$9

It also writes the entropy production density as

$J_p$0

and proposes that quantized transport of energy and entropy obeys a minimal entropy production principle at the level of single quanta (Márkus et al., 2021).

Information-geometric analyses of quantum-state evolution formulate the problem in terms of geodesics on manifolds endowed with the Fisher information metric. In that setting, the path minimizing thermodynamic length or action is the geodesic connecting the initial and final states, and that path also minimizes total entropy production. The papers further state that faster transfer is associated with higher entropy production rates and lower entropic efficiency (Gassner et al., 2020, Cafaro et al., 2021).

Taken together, these works delimit the meaning of MEMEP. In superconducting engineering, MEMEP is an operational variational method for solving non-linear current distributions and coupled electrothermal evolution. In network, stochastic, and quantum thermodynamics, minimum entropy production appears as a principle whose validity depends on regime, constraints, and the definition of the admissible dynamics. A plausible implication is that the superconducting MEMEP literature derives much of its practical strength not from an unrestricted universal law, but from a carefully specified variational formulation aligned with Maxwell equations, constitutive laws, and device geometry.