--- title: 'Li-Vary Models: Variability Across Domains' url: https://www.emergentmind.com/topics/li-vary-model type: topic --- # Li-Vary Models: Variability Across Domains Searching arXiv for the term and the cited papers to ground the article. arxiv_search(query="\"Li-Vary Model\" OR \"Li-vary\" OR \"Li Vary\"", max_results=10) arxiv_search(query="2006.02287 OR 2210.13152 OR 2103.04179 OR 1901.04260", max_results=10) “Li-Vary Model” is not a single standardized designation in the arXiv literature. In the available corpus, the label functions as an *Editor’s term* for several distinct variability-aware models involving lithium or Li-based systems: a multi-phase astrophysical framework for primordial, interstellar, and Galactic lithium evolution [2006.02287]; a diffusion-enhanced enrichment scenario for Li-rich and super Li-rich giants [2210.13152]; the BELIEVER behavioral ReRAM model for leakage and inter-cycle variability [2103.04179]; and a non-ideal linear Li-ion battery operation model with state-dependent efficiencies and power limits [1901.04260]. This suggests that the term denotes a modeling style—explicitly representing variation with state, phase, or cycle—rather than a single formalism. ## 1. Terminological scope and defining features Across the cited works, “Li-Vary” denotes different objects. In stellar and Galactic astrophysics, it refers to lithium-abundance variability across primordial nucleosynthesis, pre-main-sequence processing, interstellar chemistry, and disc chemical evolution. In stellar-evolution modeling of red giants, it denotes lithium enrichment driven by element diffusion and flash-linked mixing. In nanoelectronic compact modeling, it is effectively synonymous with BELIEVER, a model for leakage and inter-cycle variability in ReRAM. In power-systems optimization, it refers to a linear Li-ion battery model that captures variation of efficiencies and power limits with state of charge and requested power [2006.02287; 2210.13152; 2103.04179; 1901.04260]. | Domain | Principal state variables | Variation represented | |---|---|---| | Galactic/stellar lithium evolution | $A(\mathrm{Li})$, $[\mathrm{Fe}/\mathrm{H}]$ | PMS destruction, residual accretion, ISM LiH, disc enrichment | | Li-rich giant stars | $A(\mathrm{Li})$, $\nabla_\mu$, $D_{\rm mix}$ | diffusion-enhanced thermohaline transport during the He flash | | ReRAM compact modeling | $w(t)$, $\Theta(t)$ | leakage/drift and device or cycle variability | | Li-ion battery optimization | $E_t$, $SOC_t$, $P_t^{\mathrm{cha/dis}}$ | SOC- and power-dependent efficiencies and limits | A common structural feature is rejection of constant-parameter idealizations. Each model introduces explicit state dependence, memory, transport, or statistical dispersion, but the governing equations, observables, and intended applications differ fundamentally by domain. ## 2. Primordial-to-Galactic lithium evolution In the astrophysical usage, the Li-Vary framework is a synthesis of the proceedings paper “From the cosmological Li problem to the Galactic Li evolution” [2006.02287]. The starting point is the cosmological lithium problem: standard BBN predicts a primordial Li abundance about three times higher than the values observed in old, metal-poor halo stars, typically expressed through the Spite plateau. The abundance scale is $$ A(\mathrm{Li})=\log_{10}\!\Big(\frac{n(\mathrm{Li})}{n(\mathrm{H})}\Big)+12, $$ and the metallicity convention is $$ [\mathrm{Fe}/\mathrm{H}] = \log_{10}\!\Big((\mathrm{Fe}/\mathrm{H})_\star\Big)-\log_{10}\!\Big((\mathrm{Fe}/\mathrm{H})_\odot\Big). $$ The stellar solution proposed in Fu et al. (2015), implemented with the PARSEC stellar evolution code, combines pre-main-sequence convective overshooting and residual mass accretion with main-sequence nuclear burning and microscopic diffusion. The key sequence is explicitly temporal. First, lithium is “significantly depleted by convective OV in the PMS phase.” The relevant destruction channel is $$ ^{7}\mathrm{Li}(p,\gamma)\,{}^{4}\mathrm{He}+{}^{4}\mathrm{He}, $$ active at temperatures of several $10^{6}\,\mathrm{K}$. Second, lithium is “partially restored in the stellar atmosphere by a tail of matter accretion,” with residual accretion self-consistently regulated by stellar EUV photoevaporation. Third, conventional nuclear burning and microscopic diffusion continue on the main sequence. The proceedings state that this model “successfully reproduce[s] the observed Li plateau and the declining branch at low temperatures, with an initial Li abundance of the BBN prediction value,” and also reproduces Li abundances in the globular clusters NGC 6397 and M4. The cluster comparisons involve low-mass stars in the range $0.57$–$0.82\,M_\odot$ and ages from $10$ to $13.8$ Gyr. The same synthesis extends beyond halo stars to the interstellar medium and Galactic discs. Molecular studies had suggested that “the majority of the interstellar Li is in the form of lithium hydride (LiH), which is well mixed in the molecular gas phase,” but the observational diagnostics are severe. NIR LiH lines around 8347 nm and 8357 nm are “ultra weak” and heavily blended, whereas the only possible line to measure LiH is the $J=1$–$0$ transition at 443.95 GHz (675 $\mu$m), for which ground transmission is very low. New ALMA observations toward two Milky Way clouds yielded only “marginal LiH detections,” with abundance lower than Galactic chemical evolution model predictions, indicating that “LiH might not be the main form of Li in the ISM.” The proceedings therefore favor a Galactic ISM lithium budget dominated by atomic or ionic Li rather than LiH. For Galactic chemical evolution, Gaia-ESO main-sequence stars observed with UVES/VLT are chemically separated into thin and thick discs using $[\alpha/\mathrm{Fe}]$. The stated trends are differential rather than fully parametrized: thin-disc stars have higher Li abundance than thick-disc stars at similar $[\mathrm{Fe}/\mathrm{H}]$, and the fraction of Li-enriched stars, defined as $A(\mathrm{Li})>2.2$ dex, is higher in the thin disc. An anti-correlation between $A(\mathrm{Li})$ and $[\alpha/\mathrm{Fe}]$ disfavors core-collapse supernovae as the dominant Li source for disc enrichment, while a positive correlation between $A(\mathrm{Li})$ and s-process elements suggests common sources in AGB stars. The implied contributors are BBN, cosmic-ray spallation, novae, AGB stars, Li-rich RGB stars, and possibly core-collapse supernovae, with novae highlighted among likely contributors. A standard chemical-evolution bookkeeping relation useful for synthesis is $$ \frac{d(M_g X_i)}{dt} = -X_i\,\psi(t)+E_i(t)+I_i(t)-O_i(t), $$ with gas mass $M_g$, abundance $X_i$, star formation rate $\psi$, stellar ejecta $E_i$, infall $I_i$, and outflow $O_i$. In this formulation, different thin- and thick-disc star-formation or infall histories naturally generate different Li trajectories. ## 3. Diffusion-enhanced lithium enrichment in red giants A second astrophysical Li-Vary usage appears in “Li-rich and super Li-rich giants produced by element diffusion” [2210.13152]. The model addresses the origin of Li-rich and super Li-rich giants by focusing on the compact He core of low-mass red giants and on the effect of element diffusion during the He-core flash. In MESA, diffusion is computed by solving Burgers’ equations with diffusion coefficients from Paquette et al. updated by Stanton and Murillo; radiative accelerations are set to zero. The physical claim is that strong gravity in the compact He core drives element diffusion, alters the local mean molecular weight $\mu$ and its gradient $\nabla_\mu$, and thereby enhances thermohaline mixing efficiency. The mechanism is explicitly tied to structural evolution. Diffusion reduces $\mu$ and steepens a negative $\mu$-gradient in the hydrogen-burning region, allowing the thermohaline mixing zone to expand inward and, during or after the He-core flash, connect to the inner convective region. This establishes a transport channel for freshly produced $^{7}\mathrm{Be}$, which is carried outward to cooler envelope layers and then converted into $^{7}\mathrm{Li}$ through the Cameron–Fowler chain, $$ ^{3}\mathrm{He}(\alpha,\gamma)^7\mathrm{Be}, \qquad ^{7}\mathrm{Be}(e^-,\nu_e)^7\mathrm{Li}. $$ The abundance definition is again $$ A(\mathrm{Li})=\log_{10}\left(\frac{N_{\mathrm{Li}}}{N_{\mathrm{H}}}\right)+12. $$ A generic 1D transport form consistent with the model is $$ \frac{\partial X_i}{\partial t} = \frac{1}{\rho r^2} \frac{\partial}{\partial r} \!\left(\rho r^2 D_{\mathrm{mix}} \frac{\partial X_i}{\partial r}\right) + S_i. $$ The model setup is specific. It uses MESA revision r12778, OPAL and Helmholtz equations of state, OPAL and low-temperature Ferguson opacities, Reimers mass loss, the pp_and_cno_extras network, JINA REACLIB rates, and the updated $^{7}\mathrm{Be}$ electron-capture rate from Simonucci et al. as tabulated by Vescovi et al. Thermohaline mixing is included with the Kippenhahn et al. prescription and $\alpha_{\rm th}=100$. Initial composition assumes the solar abundance pattern of Asplund et al. (2009), metallicity $Z=0.014$, and meteoritic lithium abundance $A(\mathrm{Li})=3.26$ dex. Simulated initial masses are $0.9$, $1.0$, $1.2$, $1.4$, $1.6$, and $1.8\,M_\odot$; for $M\gtrsim1.9\,M_\odot$, the envelope is hot enough that $^{7}\mathrm{Be}$ is destroyed and the mechanism is no longer effective. Quantitatively, diffusion alone raises the surface lithium abundance up to about $1.8$ dex, sufficient to explain Li-rich giants under the observational criterion $A(\mathrm{Li})>1.5$ dex. When the He-flash luminosity exceeds $L_{\rm He}>10^4\,L_\odot$, the model introduces constant diffusive mixing coefficients $D_{\rm mix}=10^{11}$–$10^{15}\,\mathrm{cm}^2\,\mathrm{s}^{-1}$. This extends the predicted abundance range from about $2.4$ dex to $4.5$ dex: approximately $2.4$ dex for $D_{\rm mix}\approx10^{11}\,\mathrm{cm}^2\,\mathrm{s}^{-1}$, about $3.0$–$3.4$ dex for $10^{12}$, about $3.5$–$4.0$ dex for $10^{13}$, and about $4.5$ dex for $10^{15}$. The enrichment becomes prominent about $0.2$ Myr after the first He flash and can remain elevated for several Myr if the enhanced mixing persists. The model therefore spans both Li-rich and super Li-rich regimes, including the abundance of TYC 429-2097-1 at approximately $4.5$ dex. The population-synthesis component uses a Kroupa et al. IMF and Monte Carlo generation of $10^6$ single stars. For $D_{\rm mix}=10^{11}$, $10^{12}$, $10^{13}$, and $10^{15}\,\mathrm{cm}^2\,\mathrm{s}^{-1}$, the predicted fraction of Li-rich giants among all giants is $0.5\%$, $1.2\%$, $1.1\%$, and $0.2\%$, respectively, matching the observed range of about $0.2$–$2\%$. The enrichment is triggered around the RGB tip during the first He flash and manifests in the subsequent core He-burning red-clump phase, consistent with the statement that most Li-rich giants identified asteroseismically are RC stars. ## 4. BELIEVER as a variability model for ReRAM In nanoelectronic modeling, the Li-Vary designation is attached to BELIEVER, the “BEhavioral Leakage and IntEr-cycle Variability Emulator model for ReRAMs” [2103.04179]. The paper explicitly states that it does not use the phrase “Li-Vary Model,” but BELIEVER was created to capture exactly two effects: leakage, meaning state drift in the absence of stimulus, and inter-cycle variability, meaning cycle-to-cycle parameter variation. The model augments the VTEAM voltage-controlled memristor with a leakage term and with SPICE-level statistical variability of key parameters. The underlying state variable is $w(t)$, bounded between $w_{\rm on}$ and $w_{\rm off}$, and the terminal behavior is a state-dependent resistance between $R_{\rm ON}$ and $R_{\rm OFF}$. BELIEVER introduces a leakage-capacity term $\Theta(t)$ that integrates recent switching activity and decays with a time constant $\tau_{\rm L}$. In the idle region, where $v_{\rm on}0$ and $\theta_{\rm on}<0$ weight how SET and RESET fill the leakage capacity. In the presented calibration, $\theta_{\rm on}=0$ and $\theta_{\rm off}>0$, consistent with measurements showing drift after SET but not after RESET. The variability machinery is statistical and implemented entirely in SPICE via `.gauss()` and `.flat()`. Gaussian dispersion is used for $R_{\rm OFF}$ and $R_{\rm ON}$; uniform distributions are used for thresholds $v_{\rm off}$ and $v_{\rm on}$ and for the dynamics parameters $k_{\rm off}$ and $k_{\rm on}$. The fitted parameter set for KNOWM Self-Directed Channel ReRAM includes $R_{\rm OFF}$ with $\mu=545.54\,\mathrm{k}\Omega$ and $\sigma=77.095\,\mathrm{k}\Omega$, $R_{\rm ON}$ with $\mu=4.92\,\mathrm{k}\Omega$ and $\sigma=858.8\,\Omega$, $v_{\rm off}=370.2\,\mathrm{mV}$, $v_{\rm on}=-373.8\,\mathrm{mV}$, $k_{\rm off}=780\,\mu\mathrm{m/s}$, $k_{\rm on}=-4.67\,\mu\mathrm{m/s}$, $\theta_{\rm off}=0.0173\,\mathrm{s}^{-1}$, $\theta_{\rm on}=0$, and $\tau_{\rm L}=10.3\,\mathrm{s}$. The average deviation metric used for validation is $$ \Delta=\frac{1}{N}\sum_{i=1}^N \frac{|\tilde{x}_i-x_i|}{x_i}. $$ Calibration is measurement-driven. Devices are formed using a 100 Hz sine with gradually increasing amplitude until hysteresis is observed, under current compliance of approximately $100\,\mu\mathrm{A}$. Resistance variation is extracted from repeated SET and RESET sequences; threshold voltages are obtained from forming hysteresis by detecting points with $\Delta I > 2\,\mathrm{A/s} \approx 80\,\mathrm{nA/sample}$; leakage is measured by low-voltage probing after RESET and after SET. The observed drift is directionally asymmetric: no significant drift after RESET, pronounced resistance increase after SET, initially strong and decaying over approximately $80$–$100$ s. BELIEVER reproduces this shape. The reported average absolute relative deviation for leakage over 20 tested cases is approximately $1.1\%$, with worst case at most $13.4\%$; for SET and RESET dynamics, the average deviation over 100 cases is approximately $4.6\%$, with worst case at most $8.2\%$. The practical significance lies in circuit-level consequences. BELIEVER is a pure-SPICE, LTSpice-compatible, drop-in replacement for VTEAM, with one extra first-order ODE for $\Theta(t)$ and minimal computational overhead. The case studies quantify correctness degradation in stateful logic. IMPLY NOR reaches overall correctness of approximately $96.6\%$; MAGIC NOR, under the stated technology and voltage constraints, about $48.8\%$; FELIX OR about $94.4\%$; and TMSL NOR about $93.2\%$. The model thereby recasts leakage and cycle dispersion as first-class compact-model variables rather than post hoc uncertainty. ## 5. Non-ideal linear operation model for Li-ion batteries In power-systems optimization, the Li-Vary designation refers to the non-ideal linear operation model for a Li-ion battery introduced in [1901.04260]. The model was developed because conventional operation and planning formulations assume constant charge and discharge efficiencies and constant power limits independent of state of charge, an assumption that can misestimate available storage flexibility. The proposed formulation instead treats efficiencies and power limits as functions of state of charge and requested power, while deriving a linear reformulation without binary variables. The state variables are the stored energy $E_t$ and the state of charge $$ SOC_t=\frac{E_t}{\overline{E}}, $$ with terminal charging and discharging powers $P_t^{\rm cha}$ and $P_t^{\rm dis}$, and internal electrochemical powers $P_t^{\rm in}$ and $P_t^{\rm out}$. The model uses an equivalent steady-state circuit with equilibrium voltage $v_{\rm eq}(SOC,T)$ and resistances $R_{\rm ohm}(SOC,T)$, $R_{\rm ct}(SOC,T)$, $R_{\rm dif,mem}(T)$, and $R_{\rm dif,elec}(T)$, together with a surface state of charge $SOC_{\rm sur}$ that captures diffusion limitations. The non-ideal behaviors are explicit: discharge efficiency decreases at low SOC and high current, charge efficiency decreases at high current and high SOC, discharge power capability falls at low SOC, and maximum charge current declines as SOC approaches unity. The original non-convex model derives SOC-dependent current and power limits from $SOC_{\rm sur}$ bounds and from steady-state KVL. The discharge and charge limits are $$ \overline{P}^{\rm dis}=v_{\rm eq}\,\overline{I}^{\rm dis}-(\overline{I}^{\rm dis})^2R_{\rm tot}, \qquad \overline{P}^{\rm cha}=v_{\rm eq}\,\overline{I}^{\rm cha}+(\overline{I}^{\rm cha})^2R_{\rm tot}, $$ where $R_{\rm tot}=R_{\rm ohm}+R_{\rm ct}+R_{\rm dif,mem}$. The variable efficiencies are $$ \eta^{\rm dis}=1-\frac{iR_{\rm tot}}{v_{\rm eq}}, \qquad \eta^{\rm cha}=\frac{v_{\rm eq}}{v_{\rm eq}+iR_{\rm tot}}, $$ and the energy balance is $$ E_t = E_{t-1} + P_{t-1}^{\rm cha}\eta_{t-1}^{\rm cha}\Delta - P_{t-1}^{\rm dis}\frac{1}{\eta_{t-1}^{\rm dis}}\Delta. $$ These terms make the original formulation nonlinear and non-convex. The central contribution is the linear reformulation. Auxiliary variables are introduced so that $$ P_t^{\rm out}=P_t^{\rm dis}\frac{1}{\eta_t^{\rm dis}}, \qquad P_t^{\rm in}=P_t^{\rm cha}\eta_t^{\rm cha}, $$ and the energy balance becomes affine: $$ E_t=E_{t-1}+\big(P_{t-1}^{\rm in}-P_{t-1}^{\rm out}\big)\Delta. $$ The nonlinear relations between terminal power, internal power, and SOC are then represented through convex combinations of sampled points. For discharge, $$ P_t^{\rm out}=\sum_j \widehat{P}^{\rm out}_j x_{j,t},\quad P_t^{\rm dis}=\sum_j \widehat{P}^{\rm dis}_j x_{j,t},\quad SOC_t=\sum_j \widehat{SOC}_j x_{j,t}, $$ with $\sum_j x_{j,t}=1$ and $x_{j,t}\ge0$; an analogous construction holds for charging with weights $y_{k,t}$. This removes bilinear efficiency products from the optimization and enforces feasible operating triplets without binary variables, SOS2 constructs, or McCormick envelopes. The model was tested in a network-constrained economic dispatch on the IEEE RTS 24-bus system over a 24 h horizon with 10-minute time steps. Objective values are nearly identical across models: $57{,}305$ for the full non-convex NLP, $57{,}312$ for the ideal model, $57{,}307$ for the MILP reformulation, and $57{,}308$ for the proposed piecewise-linear formulation, all within $0.01\%$ of the NLP. The computational differences are large: solver times are $231.3$ s for the NLP, $1.2$ s for the ideal LP, $143.3$ s for the MILP, and $3.4$ s for the proposed linear model. The reliability result is more consequential than the objective gap: schedules derived from the ideal model show an energy deviation of $12.2\%$ of scheduled stored energy once corrected to the feasible envelope, which is the detailed basis for the abstract’s statement of approximately $12\%$ mismatch. ## 6. Cross-domain interpretation, limits, and recurrent misconceptions The principal misconception is to treat “Li-Vary Model” as a single canonical formalism. The cited literature does not support that reading. In one case it is an editorial synthesis of primordial-to-Galactic lithium evolution [2006.02287]; in another, a diffusion-enhanced red-giant enrichment channel [2210.13152]; in another, BELIEVER for ReRAM leakage and inter-cycle variability [2103.04179]; and in another, a linearized non-ideal Li-ion battery model [1901.04260]. This suggests that the label is best understood as a cross-domain descriptor for models that make variation explicit, not as a unified theory. The uncertainties are likewise domain-specific. In the primordial-to-Galactic stellar framework, the main degeneracies concern PMS mixing physics, the extent and timing of residual accretion, EUV photoevaporation efficiency, and microscopic diffusion efficiency in low-metallicity regimes. In the Li-rich giant scenario, the dominant uncertainty is the magnitude and constancy of the added $D_{\rm mix}$ during the He-flash phase, together with sensitivity to diffusion prescriptions and thermohaline efficiency. In BELIEVER, leakage variability across runs was kept constant for simplicity, threshold extraction is difficult, and temperature and stochastic noise during a cycle are not parameterized. In the Li-ion battery formulation, temperature dynamics, auxiliary losses, aging, and electro-thermal coupling are neglected, and the model has no explicit binary exclusion of simultaneous charging and discharging. A second misconception is that variability-aware models are merely refinements of ideal baselines. The evidence summarized here is stronger. In halo-star lithium, PMS destruction plus residual accretion is used to reconcile the Spite plateau with standard BBN without altering BBN itself. In red giants, diffusion-enhanced transport is invoked to explain both Li-rich incidence and super Li-rich extremes. In ReRAM, leakage and cycle dispersion materially alter correctness probabilities and post-compute state stability. In Li-ion batteries, constant-efficiency idealizations generate infeasible schedules and about $12\%$ energy mismatch. In each case, the “variation” terms are not decorative; they change the feasible phenomenology. The cross-domain commonality is therefore methodological rather than ontological. Each Li-Vary usage introduces hidden-state dynamics, transport channels, or sampled feasible manifolds to describe behavior that constant-parameter models miss. The governing physics ranges from nuclear burning and diffusion to memristive state drift and electrochemical operating envelopes, but the formal motive is the same: to model lithium-related systems through their state dependence, time dependence, and non-ideal variability. Source: https://www.emergentmind.com/topics/li-vary-model