arxiv: 2605.09367 · v1 · submitted 2026-05-10 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

A Stochastic Hybrid Automaton for Smartphone Battery Dynamics: Electro-Thermal Coupling and First-Passage Time-to-Empty Estimation

Runni Zhou, Xiaoyang Li

Pith reviewed 2026-05-12 04:32 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords smartphone batterytime-to-emptystochastic hybrid automatonelectro-thermal modelvoltage collapsefirst-passage timeuser activity model

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The pith

A stochastic model of smartphone battery dynamics shows shutdown can occur from voltage collapse during high-power use even when charge remains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a hybrid model that tracks battery charge, internal voltage, and temperature while user actions switch among idle, browsing, video, gaming, and poor-signal states. Shutdown is defined as the first moment terminal voltage falls below a cutoff threshold or power demand exceeds what the battery can supply at that instant. Simple charge counters miss this because they ignore how cold temperatures or battery wear raise resistance and amplify voltage sag under bursts. The approach therefore supplies an entire distribution of possible run times, letting the lower tail of early failures be measured directly.

Core claim

The authors formulate a stochastic hybrid automaton whose continuous states are state of charge, polarization voltage, and battery temperature, driven by a first-order Thevenin circuit coupled to a lumped thermal model; user behavior evolves as a piecewise deterministic Markov process across five modes, and battery exhaustion is the first-passage time at which terminal voltage crosses the shutdown threshold or instantaneous power becomes infeasible.

What carries the argument

The stochastic hybrid automaton that evolves continuous electro-thermal states between discrete switches in a five-mode user-activity Markov process and registers shutdown as the first time terminal voltage hits cutoff.

If this is right

Monte Carlo runs of the automaton produce full time-to-empty distributions that quantify premature-shutdown risk through lower-tail percentiles.
Sensitivity sweeps identify ambient temperature, internal resistance, weak-signal radio overhead, and screen brightness as dominant drivers of early cutoff.
The framework directly supports an operating-system policy that caps peak power once resistance limits the feasible envelope.
The same structure yields concrete guidance for users on avoiding high-load tasks when the phone is cold or aged.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Real-time resistance estimates from the phone could feed the model to trigger dynamic throttling before voltage sag becomes critical.
The same hybrid-automaton structure could be adapted to predict usable range in electric vehicles under cold-weather high-load conditions.
Field calibration against crowdsourced battery telemetry from many devices would allow the model to adjust its parameters automatically over time.

Load-bearing premise

The first-order Thevenin equivalent circuit, lumped thermal model, and five-mode user-activity process are accurate enough representations of actual smartphone behavior to produce reliable first-passage time-to-empty distributions.

What would settle it

Record actual shutdown times on multiple phones under controlled low ambient temperatures and repeated high-power bursts, then test whether the measured times fall inside the model's predicted first-passage distribution.

Figures

Figures reproduced from arXiv: 2605.09367 by Runni Zhou, Xiaoyang Li.

**Figure 1.** Figure 1: Motivating mechanisms for TTE variability. (a) Bursty usage creates highcurrent peaks; (b) low temperature increases internal resistance and amplifies voltage sag; (c) weak-signal radio states raise instantaneous power demand, increasing premature cutoff risk. 1.2 Restatement of the Problem We are tasked with developing a continuous-time mathematical model for smartphone battery discharge that predicts th… view at source ↗

**Figure 2.** Figure 2: Overview of Our Work. We integrate a stochastic usage process with a physicsbased electro-thermal battery model, define TTE via first-passage events, and derive sensitivity-ranked recommendations. 2 Model Preparation Before establishing the governing equations, we define the foundational assumptions, standardize the nomenclature, and detail the data processing pipeline used for parameter identification. 2… view at source ↗

**Figure 3.** Figure 3: Physical Parameter Identification Pipeline. (a) Raw pulse discharge profile showing the separation of instantaneous Ohmic drop (∆V0 ⇒ R0) and slow polarization relaxation. (b) Extracted OCV curve using PCHIP to capture the critical nonlinear knee near depletion. (c) Fitting the transient recovery to estimate the polarization time constant τp, validating the Thevenin model structure. 3 Model I: The Electro-… view at source ↗

**Figure 4.** Figure 4: Schematic of the Coupled Electro-Thermal Model. Discharge current generates heat, heat changes temperature, and temperature modifies resistance via Arrhenius coupling, closing the feedback loop that governs voltage sag and power feasibility. 4 Model II: The Stochastic Hybrid Automaton While Model I provides deterministic battery physics, real-world usage is stochastic. Users switch unpredictably between h… view at source ↗

**Figure 5.** Figure 5: Topology of the Stochastic Hybrid Automaton. The usage mode m(t) follows a CTMC governed by Q; within each mode, the electro-thermal states evolve via coupled ODEs. Upon entering mode m, a constant session load Pload is sampled, producing a PDMP used for Monte Carlo estimation of the first-passage TTE. 5 Simulation and Performance Evaluation We conducted numerical simulations coupling the SHA stochastic dr… view at source ↗

**Figure 6.** Figure 6: Simulation Results demonstrating Electro-Thermal and Stochastic Effects. (a) Thermal Sensitivity: Voltage profiles across temperatures. Note premature shutdown at low temperature. (b) Stochastic Dynamics: A realization of the hybrid automaton vs. a mean-power baseline; arrows indicate voltage sags triggered by mode switches. (c) TTE Risk Distribution: PDF of TTE at 0 ◦C; shaded area indicates lower 5% risk… view at source ↗

**Figure 7.** Figure 7: Critical Validation: Model vs. Observed Data. Our SHA Model (Red) closely tracks the reference NASA discharge data (Gray Dots), accurately predicting the voltage collapse mechanism with a MAPE of < 2.1%. In contrast, the Linear Model (Blue) overestimates battery life by 1.5 hours. The shaded area represents the “False Positive Zone” where users face high shutdown risk. 6 Model III: Risk-Aware Optimal Cont… view at source ↗

**Figure 8.** Figure 8: Multi-Dimensional Analysis of the Risk-Aware Control Strategy. (a) Phase Plane: Controlled trajectory avoids collapse regions. (b) Robustness: The 5–95% voltage envelopes show the controller prevents lower-tail violations. (c) Risk Shift (CDF): The control strategy increases t0.05. (d) Pareto Frontier: Small throttling yields large reliability gains. 7 Sensitivity and Robustness Analysis To ensure reliabil… view at source ↗

**Figure 9.** Figure 9: Sensitivity Analysis Dashboard. (a) Tornado diagram of stressors on TTE. (b) Normalized importance across temperature regimes. (c) Response surface of t0.05 and risk frontier. 8 Recommendations and Practical Strategies Our results indicate that battery life is often power-limited near depletion (voltage sag / feasibility collapse) rather than purely capacity-limited. Therefore, the most effective strategie… view at source ↗

read the original abstract

Smartphone time-to-empty (TTE) is difficult to predict because shutdown is governed not only by remaining charge, but also by instantaneous power capability under temperature-, aging-, and load-dependent voltage sag. We develop a stochastic hybrid automaton for smartphone battery dynamics that couples a first-order Thevenin equivalent-circuit model with a lumped thermal model and a stochastic user-activity process. The continuous state includes state of charge, polarization voltage, and battery temperature; user behavior is represented as a piecewise deterministic Markov process switching among idle, social/web, video, gaming, and weak-signal modes. Shutdown is formulated as a first-passage event when terminal voltage crosses a cutoff threshold or when requested power exceeds the instantaneous feasibility envelope. The model captures a voltage-collapse mechanism that simple Coulomb-counting or linear discharge models miss: cold temperature or battery aging increases internal resistance, so high-power bursts can drive terminal voltage below cutoff even when substantial charge remains. Monte Carlo simulation yields a full TTE distribution rather than a single countdown, allowing lower-tail risk to be quantified by the 5th percentile. Sensitivity analysis identifies ambient temperature, internal resistance, weak-signal radio penalty, and screen brightness as major drivers of premature shutdown risk. These results motivate practical user guidance and an operating-system-level resistance-aware throttling policy that limits peak power in the power-limited regime. The framework provides a physically grounded, risk-aware approach for explaining and extending usable smartphone battery life under real-world uncertainty.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper assembles a stochastic hybrid automaton coupling Thevenin electro-thermal battery dynamics with a five-mode user Markov process to compute first-passage TTE distributions, but it contains no validation against real smartphone measurements.

read the letter

The core contribution is a hybrid model that treats battery state of charge, polarization voltage, and temperature as continuous variables while user activity switches among idle, social, video, gaming, and weak-signal modes via a piecewise deterministic Markov process. Shutdown is defined as the first time terminal voltage drops below cutoff or requested power exceeds the feasible envelope. Monte Carlo runs then produce full TTE distributions and sensitivity rankings on temperature, internal resistance, and load factors. This setup does let the model show how high-resistance bursts can trigger early voltage collapse even with charge remaining, which simpler Coulomb-counting approaches miss by construction. The integration itself is coherent and draws on standard domain pieces without obvious internal contradictions. The sensitivity results also flag concrete drivers like weak-signal radio penalty that could inform practical throttling policies. The clear gap is the total lack of empirical grounding. The manuscript reports only simulated outputs and parameter sweeps; there are no measured voltage traces, shutdown times, or comparisons under controlled temperature, aging, or load profiles from actual devices. As a result the claimed improvement over linear models remains an untested prediction of the modeling assumptions rather than a demonstrated result. Parameters such as mode switching rates and temperature-dependent resistance coefficients are left free without fitting or cross-validation. This is the kind of work that belongs in a battery-management or embedded-systems venue. It is worth sending to referees because the modeling is physically motivated and the problem is practical, but any positive review would hinge on the authors adding hardware validation and parameter estimation in revision.

Referee Report

2 major / 2 minor

Summary. The paper develops a stochastic hybrid automaton for smartphone battery dynamics that couples a first-order Thevenin equivalent-circuit model with a lumped thermal model and a five-mode piecewise deterministic Markov process (idle, social/web, video, gaming, weak-signal) for user activity. Continuous states track state of charge, polarization voltage, and temperature; shutdown is defined as a first-passage event when terminal voltage drops below cutoff or requested power exceeds the instantaneous feasibility envelope. Monte Carlo simulation produces full TTE distributions (including 5th-percentile risk) and sensitivity analysis identifies ambient temperature, internal resistance, weak-signal penalty, and screen brightness as key drivers. The central claim is that the coupled model captures a voltage-collapse mechanism (high-resistance bursts driving cutoff despite remaining charge) missed by Coulomb counting or linear discharge models, motivating resistance-aware throttling policies.

Significance. If experimentally validated, the framework would offer a physically grounded, risk-aware alternative to deterministic battery models, enabling quantification of premature-shutdown probability under temperature, aging, and load variability. The Monte Carlo first-passage formulation and sensitivity results provide concrete, actionable insights for OS-level power management that simpler models cannot supply. The hybrid automaton structure is a natural fit for electro-thermal-stochastic coupling and could be extended to other portable systems.

major comments (2)

[§5 (Numerical Results and Monte Carlo Simulations)] §5 (Numerical Results and Monte Carlo Simulations): The reported TTE distributions, 5th-percentile risk values, and sensitivity plots are generated entirely from the model assumptions with no comparison to measured voltage traces, shutdown times, or power-sag events from physical smartphone batteries under controlled temperature, aging, or load profiles. Without such grounding, the claim that the model captures a voltage-collapse mechanism missed by Coulomb counting remains an unverified output of the modeling choices rather than a demonstrated improvement.
[§3 (Stochastic User-Activity Process)] §3 (Stochastic User-Activity Process): The five-mode PDMP switching rates are treated as free parameters whose values are not calibrated or validated against real usage traces; because these rates directly shape the high-power burst statistics that trigger the voltage-collapse events, the resulting TTE distributions inherit unquantified uncertainty that undermines the sensitivity conclusions.

minor comments (2)

[Abstract] The definition of the 'feasibility envelope' for instantaneous power is introduced in the abstract but only formalized later; a brief forward reference or inline equation would improve readability.
[§2 (Electro-Thermal Model)] Notation for the temperature-dependent internal resistance parameters (R_0(T), R_p(T)) is used consistently but would benefit from an explicit table listing all free parameters and their physical units.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the potential of the stochastic hybrid automaton framework. We address each major comment below, providing clarifications on the modeling foundations while acknowledging the value of empirical grounding.

read point-by-point responses

Referee: §5 (Numerical Results and Monte Carlo Simulations): The reported TTE distributions, 5th-percentile risk values, and sensitivity plots are generated entirely from the model assumptions with no comparison to measured voltage traces, shutdown times, or power-sag events from physical smartphone batteries under controlled temperature, aging, or load profiles. Without such grounding, the claim that the model captures a voltage-collapse mechanism missed by Coulomb counting remains an unverified output of the modeling choices rather than a demonstrated improvement.

Authors: We agree that direct experimental validation with physical smartphone measurements would provide stronger support for the claims. The model parameters are drawn from established literature on first-order Thevenin equivalent-circuit models and lumped thermal models for Li-ion batteries, and the voltage-collapse phenomenon under high internal resistance is a documented physical effect in power-limited discharge regimes. Our primary contribution is the stochastic hybrid formulation enabling full TTE distributions and sensitivity analysis. In the revised manuscript we will expand Section 5 with additional references to experimental observations of voltage sag in smartphone batteries, explicitly discuss the simulation-based nature of the results, and add a dedicated limitations subsection motivating future empirical validation. revision: partial
Referee: §3 (Stochastic User-Activity Process): The five-mode PDMP switching rates are treated as free parameters whose values are not calibrated or validated against real usage traces; because these rates directly shape the high-power burst statistics that trigger the voltage-collapse events, the resulting TTE distributions inherit unquantified uncertainty that undermines the sensitivity conclusions.

Authors: The switching rates are chosen to represent aggregate usage patterns drawn from publicly reported smartphone usage statistics (e.g., average session durations for gaming, video, and web browsing). In the revised Section 3 we will add explicit citations to these sources and provide a brief sensitivity study showing how moderate variations in the rates affect the lower tail of the TTE distribution. While full per-user calibration would require individual trace data outside the present scope, the existing Monte Carlo results already quantify the impact of load variability, and the identified key drivers (temperature, resistance, weak-signal penalty) remain robust under the chosen parameterization. revision: partial

Circularity Check

0 steps flagged

No circularity detected: model built from independent established components

full rationale

The derivation assembles a stochastic hybrid automaton by coupling a standard first-order Thevenin equivalent-circuit model, a lumped thermal model, and a piecewise deterministic Markov process for user modes. These components are introduced as external domain knowledge with no equations that define outputs in terms of themselves or rename fitted parameters as predictions. Monte Carlo first-passage simulations and sensitivity results follow directly from the forward dynamics without self-referential reductions or load-bearing self-citations. The voltage-collapse claim is a consequence of the coupled state evolution rather than an input presupposed by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard battery modeling assumptions plus several unspecified free parameters for user-mode transitions and component values that would require external data for calibration.

free parameters (3)

user activity mode switching rates
Transition probabilities or rates among the five user modes (idle, social/web, video, gaming, weak-signal) are required by the Markov process but not derived from first principles.
temperature-dependent internal resistance parameters
Values governing how resistance changes with temperature and aging in the Thevenin model are model inputs.
thermal model coefficients
Heat capacity and thermal resistance parameters in the lumped thermal model.

axioms (3)

domain assumption First-order Thevenin equivalent circuit sufficiently captures battery voltage and polarization dynamics
Used to define the continuous electrical state variables.
domain assumption Lumped-parameter thermal model adequately represents battery temperature evolution
Coupled directly to the electrical states.
domain assumption User behavior is adequately represented by a piecewise deterministic Markov process with the listed discrete modes
Defines the stochastic switching that drives load.

pith-pipeline@v0.9.0 · 5568 in / 1732 out tokens · 56469 ms · 2026-05-12T04:32:33.645629+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We employ a first-order Thevenin Equivalent Circuit Model (ECM)... coupled with a lumped thermal system... Arrhenius-type relationship R0(T) = Rref · exp(Ea/Rg (1/T − 1/Tref))

Reference graph

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AI Tools Used •Gemini 3 Pro(Google, Released Nov 2025): Used for high-level mathematical mod- eling, theoretical derivations (Electro-Thermal coupling), generating advanced La- TeX visualization code (TikZ/PGFPlots), and polishing the academic tone of the Abstract and Introduction. •ChatGPT 5.2(OpenAI, Released Dec 2025): Used for writing Python simulatio...

work page 2025
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Detailed Usage Log Phase 1: Mathematical Framework & Physics (Gemini 3 Pro) Query: “I need to construct a continuous-time model for smartphone battery dynam- ics that accounts for the ’Voltage Collapse’ phenomenon in cold weather.Specifics:

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shutdown logic

How do I couple a First-Order Thevenin ECM with a lumped thermal model? 2. What is the standard equation for Entropy-based heat generation (Qrev) in Li-ion batteries? 3. How should I model the user’s stochastic behavior (switching apps) mathematically?” AI Output (Summary):Gemini provided the coupled system of ODEs used inSec- tion 3, specifically introdu...

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Resistance-Aware Throttling

Verification and Integrity Statement We, the members of Team #2614784, acknowledge the use of the aforementioned AI tools. We certify that: 1.Scientific Accuracy:All equations, physical laws (e.g., Arrhenius, Ohm’s Law), and parameter values were verified against standard textbooks (Newman et al.) and the NASA dataset. AI was not treated as a source of tr...

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