下面我会先**对比**“我上一版整合输出”与**你这份文件**（8b58…md），然后**严格按该文件的要点结构**把“完整内容”重写一遍，并且做到：**每个要点都给出公式 + 逐项物理解释**（可直接粘到 LaTeX）。我会以该文件为主干，并把我上一版中补足的电路闭合、TTE 细化等内容融入进去，但不偏离文件核心。

---

## 1) 我上次输出 vs 该文件：差异与补全方向

**共同点（核心一致）：**

* 都把电池写成连续时间状态空间系统 (\mathbf{x}(t)=[z,v_p,T_b,S]^T)，输入 (\mathbf{u}(t)=[L,C,N,\Psi,T_a]^T)。这与文件完全一致。
* 都采用四条耦合 ODE：SOC 守恒、极化 RC、热平衡、SEI/Arrhenius 老化。
* 都采用恒功率负载（CPL）功耗映射 (P_{\rm tot}(t)) 并用二次方程闭合求 (I(t))。

**我上次输出相对“缺失/弱化”的精华点（文件里明确强调）：**

1. **奇异摄动/多尺度**：把“快变量放电”与“慢变量老化”分离，并指出单次 TTE 内 (S(t)) 可视作准静态，但跨周期需动态更新。
2. **判别式 (\Delta) 的奇点物理含义**：(\Delta) 是最大可传输功率极限；(\Delta<0) 对应“电压崩塌/意外关机”。并且 TTE 终止条件不仅是 (V_{\rm term}\le V_{\rm cut})，还包括 (\Delta\to 0)。
3. **不确定性建模**：把未来负载建成均值回复随机过程，并用 Monte Carlo 生成 TTE 分布而非单点。
4. **策略洞察**：低温下信号强度 (\Psi) 可能超过亮度成为主导驱动（Sobol），以及 OS 的 Thermal-Aware Throttling。

下面的“完整内容”会把这些**逐点展开到论文级**。

---

# Full Write-up (expanded from the file): A Multi-scale Coupled Electro–Thermal–Aging Framework

## 1. Modeling Philosophy: A Continuous-Time State-Space System

Smartphone battery drain is modeled as a **nonlinear continuous-time dynamical system** to capture feedback loops among **power demand**, **temperature rise**, and **capacity degradation**. In contrast to discrete regressions, a state-space formulation preserves physical interpretability and is directly aligned with the requirement that SOC be returned as a function of time under realistic usage conditions (screen, processor, network, temperature, aging). 

### 1.1 State and Input Vectors

Define the state vector and usage inputs as 
[
\mathbf{x}(t)=
\begin{bmatrix}
z(t)\
v_p(t)\
T_b(t)\
S(t)
\end{bmatrix},
\qquad
\mathbf{u}(t)=
\begin{bmatrix}
L(t)\
C(t)\
N(t)\
\Psi(t)\
T_a(t)
\end{bmatrix}.
]
**State meanings (physics):**

* (z(t)\in[0,1]): SOC (fraction of usable charge remaining).
* (v_p(t)) (V): polarization voltage (electrochemical transient “memory”).
* (T_b(t)) (K): internal battery temperature.
* (S(t)\in[0,1]): SOH (capacity-fade factor due to aging).

**Input meanings (usage/environment):**

* (L(t)): normalized screen brightness.
* (C(t)): normalized CPU load.
* (N(t)): normalized network throughput/activity intensity.
* (\Psi(t)): normalized signal strength (weak signal (\Rightarrow) higher modem power).
* (T_a(t)): ambient temperature.

---

## 2. Governing Equations: The Multi-Physics Core (with Multi-scale Separation)

The core model is a set of coupled ODEs: 
[
\boxed{
\begin{aligned}
\frac{dz}{dt} &= -\frac{I(t)}{3600 , Q_{\mathrm{eff}}(T_b,S)}
&& \text{(Charge conservation)} [4pt]
\frac{dv_p}{dt} &= \frac{I(t)}{C_1}-\frac{v_p(t)}{R_1C_1}
&& \text{(Polarization transient)} [4pt]
\frac{dT_b}{dt} &= \frac{1}{C_{\mathrm{th}}}\Big[I(t)^2R_0 + I(t)v_p-hA(T_b-T_a)\Big]
&& \text{(Thermal balance)} [4pt]
\frac{dS}{dt} &= -\Gamma |I(t)|\exp!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b}\right)
&& \text{(Aging kinetics)}
\end{aligned}}
]

### 2.1 Detailed Physical Interpretation (term-by-term)

#### (a) SOC equation: (\dot z)

[
\frac{dz}{dt}=-\frac{I(t)}{3600,Q_{\mathrm{eff}}(T_b,S)}.
]

* The numerator (I(t)) (A) is discharge current.
* (Q_{\mathrm{eff}}) (Ah) is **effective deliverable capacity**, reduced by cold temperature and aging.
* The factor 3600 converts Ah to Coulombs (since (1,\mathrm{Ah}=3600,\mathrm{C})).
  **Meaning:** SOC decays faster when current increases or when the usable capacity shrinks (cold/aged battery).

#### (b) Polarization equation: (\dot v_p)

[
\frac{dv_p}{dt}=\frac{I(t)}{C_1}-\frac{v_p}{R_1C_1}.
]
This is a 1st-order RC branch (Thevenin model):

* (R_1C_1) is a polarization time constant ((\tau)), representing charge-transfer/diffusion relaxation.
* A sudden increase in (I(t)) produces a transient rise in (v_p), which reduces terminal voltage and creates “after-effects” even if load later decreases.

#### (c) Thermal balance: (\dot T_b)

[
\frac{dT_b}{dt}=
\frac{1}{C_{\mathrm{th}}}\Big[I^2R_0 + Iv_p - hA(T_b-T_a)\Big].
]

* (I^2R_0): **Joule heating** from ohmic resistance.
* (I v_p): **polarization heat** (irreversible losses associated with overpotential).
* (hA(T_b-T_a)): convective heat removal to ambient.
* (C_{\mathrm{th}}): effective thermal capacitance (J/K).
  **Meaning:** heavy usage raises temperature, which in turn modifies resistance and capacity (see Section 4), creating a closed feedback loop.

#### (d) Aging kinetics: (\dot S)

[
\frac{dS}{dt}=-\Gamma |I|\exp!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b}\right).
]
This is an SEI-growth-inspired Arrhenius law:

* Higher current magnitude (|I|) accelerates degradation.
* Higher temperature increases reaction rate via (\exp(-E_{\mathrm{sei}}/(R_gT_b))).
  **Meaning:** the model explains why sustained heavy use (high (I), high (T_b)) causes faster long-term capacity fade.

### 2.2 Singular Perturbation (Multi-scale “O-Award Edge”)

The file explicitly introduces a **fast–slow decomposition**: discharge/thermal/polarization evolve on minutes–hours, while aging (S(t)) evolves over many cycles. 

Formally, define a small parameter (\varepsilon \ll 1) such that
[
\frac{dS}{dt}=\varepsilon,g(\cdot),\qquad
\frac{dz}{dt},\frac{dv_p}{dt},\frac{dT_b}{dt}=O(1).
]
**Implementation rule:**

* **Within a single TTE prediction**, treat (S(t)\approx S_0) as quasi-static to improve numerical robustness.
* **Across repeated discharge cycles**, update (S(t)) dynamically by integrating (\dot S) to capture long-term aging.
  This is exactly the “multi-scale approach” described in the file. 

---

## 3. Component-Level Power Mapping and Current Closure (CPL + Signal Strength)

Smartphones are approximately **constant-power loads (CPL)**: the OS and power-management circuitry maintain nearly constant *power* demands for a given workload, so current must be solved implicitly rather than assumed constant. 

### 3.1 Total Power Demand with Signal Sensitivity

The file’s core mapping is 
[
P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}
+k_LL(t)^{\gamma}
+k_CC(t)
+k_N\frac{N(t)}{\Psi(t)^{\kappa}}.
]
**Interpretation of each component:**

* (P_{\mathrm{bg}}): baseline background drain (OS tasks, sensors, idle radio).
* (k_LL^\gamma): display power; (\gamma>1) reflects nonlinear brightness-power response.
* (k_CC): compute power; linear is a first-order approximation of dynamic power scaling under normalized load.
* (k_N N/\Psi^\kappa): network power with **power amplification under weak signal**—when (\Psi) drops, transmit gain/baseband effort rises nonlinearly to maintain throughput. 

### 3.2 Constant-Power Closure and Quadratic Current Solution

Define terminal voltage through a Thevenin form:
[
V_{\mathrm{term}}(t)=V_{\mathrm{oc}}(z)-v_p-I(t)R_0.
]
Impose the CPL constraint:
[
P_{\mathrm{tot}}(t)=V_{\mathrm{term}}(t),I(t)=\big(V_{\mathrm{oc}}(z)-v_p-I R_0\big)I.
]
Rearranging yields a quadratic in (I):
[
R_0 I^2-\big(V_{\mathrm{oc}}(z)-v_p\big)I + P_{\mathrm{tot}}=0.
]
Thus, the physically admissible root (positive and consistent with discharge) is 
[
I(t)=\frac{V_{\mathrm{oc}}(z)-v_p-\sqrt{\Delta}}{2R_0},
\qquad
\Delta=\big(V_{\mathrm{oc}}(z)-v_p\big)^2-4R_0P_{\mathrm{tot}}.
]

### 3.3 Singularity (Voltage Collapse) and the Discriminant (\Delta)

The file’s critical insight is: (\Delta) represents the **maximum power transfer limit**. 

* If (\Delta>0): the required power can be delivered and (I(t)) is real.
* If (\Delta=0): the system hits the boundary of feasibility (“power limit”).
* If (\Delta<0): no real current can satisfy the constant-power demand, implying **voltage collapse / unexpected shutdown**, especially when:

  * (R_0\uparrow) (cold temperature increases resistance), or
  * (V_{\mathrm{oc}}(z)\downarrow) (low SOC reduces OCV). 

This is a mechanistic explanation for “rapid drain before lunch” days under cold weather or weak signal, matching the problem’s narrative about complex drivers beyond “heavy use.” 

---

## 4. Constitutive Relations (Physics-Based Corrections)

The file lists three key constitutive relations. 
To make the model operational, these relations supply (R_0(T_b)), (Q_{\rm eff}(T_b,S)), and (V_{\rm oc}(z)).

### 4.1 Internal Resistance (Arrhenius)

[
R_0(T_b)=R_{\mathrm{ref}}
\exp!\left[
\frac{E_a}{R_g}\left(\frac{1}{T_b}-\frac{1}{T_{\mathrm{ref}}}\right)
\right].
]

* (E_a) is an activation energy describing temperature sensitivity of impedance.
* When (T_b<T_{\mathrm{ref}}), the exponent is positive, so (R_0) increases sharply—capturing cold-weather performance loss. 

### 4.2 Effective Capacity (Aging + Temperature)

[
Q_{\mathrm{eff}}(T_b,S)=Q_{\mathrm{nom}},S,\Big[1-\alpha_Q,(T_{\mathrm{ref}}-T_b)\Big].
]

* (S) scales nominal capacity to reflect irreversible degradation.
* The bracket term reduces deliverable capacity at low (T_b) (transport limitations and polarization). 

### 4.3 OCV Curve (Modified Shepherd)

[
V_{\mathrm{oc}}(z)=E_0-K\left(\frac{1}{z}-1\right)+A e^{-B(1-z)}.
]

* The rational term (K(1/z-1)) increases curvature near low SOC.
* The exponential term shapes the end-of-discharge “knee.” 

---

## 5. Numerical Implementation and Uncertainty

### 5.1 RK4 with Nested Algebraic Current Solver

The file specifies RK4 and emphasizes that the algebraic current computation is nested inside each RK sub-step. 

Let (\dot{\mathbf{x}}=F(t,\mathbf{x};\mathbf{u}(t))) be the ODE RHS, where (I(t)) is computed from the quadratic root using the current sub-step values of ((z,v_p,T_b,S)). For a step (\Delta t), RK4 is:
[
\begin{aligned}
\mathbf{k}_1 &= F(t_n,\mathbf{x}_n),\
\mathbf{k}_2 &= F!\left(t_n+\frac{\Delta t}{2},\mathbf{x}_n+\frac{\Delta t}{2}\mathbf{k}_1\right),\
\mathbf{k}_3 &= F!\left(t_n+\frac{\Delta t}{2},\mathbf{x}_n+\frac{\Delta t}{2}\mathbf{k}_2\right),\
\mathbf{k}_4 &= F!\left(t_n+\Delta t,\mathbf{x}_n+\Delta t,\mathbf{k}*3\right),\
\mathbf{x}*{n+1} &= \mathbf{x}_n + \frac{\Delta t}{6}\left(\mathbf{k}_1+2\mathbf{k}*2+2\mathbf{k}*3+\mathbf{k}*4\right).
\end{aligned}
]
**Crucial implementation note:** at each evaluation of (F(\cdot)), compute in order
[
P*{\rm tot}(t)\rightarrow R_0(T_b)\rightarrow Q*{\rm eff}(T_b,S)\rightarrow V*{\rm oc}(z)\rightarrow \Delta \rightarrow I(t),
]
then substitute (I(t)) into the ODEs.

### 5.2 TTE Definition Consistent with Singularity

The file states that TTE is reached when either terminal voltage hits the cutoff or the discriminant approaches zero. 

Define
[
\mathrm{TTE}=\inf\left{\Delta t>0:
\left[V_{\mathrm{term}}(t_0+\Delta t)\le V_{\mathrm{cut}}\right]
\ \lor
\left[\Delta(t_0+\Delta t)\le 0\right]
\right}.
]
This dual criterion is important: it captures “unexpected shutdown” when the required power becomes infeasible even before SOC formally reaches zero.

### 5.3 Uncertainty Quantification (Monte Carlo + Mean-Reverting Loads)

The file specifies modeling future workloads as a mean-reverting random process and running 1000 simulations to obtain a TTE distribution. 

A minimal continuous-time mean-reverting model is the Ornstein–Uhlenbeck (OU) process for each normalized load component (clipped to ([0,1])):
[
dU(t)=\theta\big(\mu-U(t)\big)dt+\sigma dW_t,\qquad U\in{L,C,N},
]
with (\Psi(t)) optionally modeled similarly (or via a Markov regime for good/poor signal). For each Monte Carlo path (m=1,\dots,M) (e.g., (M=1000)), compute (\mathrm{TTE}^{(m)}). The output is an empirical PDF and confidence interval:
[
\hat f_{\mathrm{TTE}}(\tau),\qquad
\mathrm{CI}*{95%}=\big[\mathrm{quantile}*{2.5%},,\mathrm{quantile}_{97.5%}\big].
]
This aligns with the problem requirement to “quantify uncertainty” rather than report a single deterministic time-to-empty. 

---

## 6. Strategic Insights and Recommendations (Mechanism-Explained)

### 6.1 Global Sensitivity (Sobol Indices)

The file’s key result-style claim is: in sub-zero temperatures, (\Psi) may dominate over screen brightness. 

To formalize, let the model output be (Y=\mathrm{TTE}) and parameters/inputs be ({X_i}\subset{k_L,k_C,k_N,\gamma,\kappa,T_a,\dots}). The first-order Sobol index is
[
S_i=\frac{\mathrm{Var}\big(\mathbb{E}[Y\mid X_i]\big)}{\mathrm{Var}(Y)}.
]
**Physical reason for (\Psi) dominance at low (T):**

* Cold (\Rightarrow R_0\uparrow) (Arrhenius), making the discriminant (\Delta=(V_{\rm oc}-v_p)^2-4R_0P_{\rm tot}) shrink.
* Weak signal (\Rightarrow P_{\rm tot}) increases via (N/\Psi^\kappa).
  Together they push the system toward (\Delta\to 0) more quickly than brightness alone, so (\Psi) can become the “hidden killer” in winter.

### 6.2 OS-Level Recommendation: Thermal-Aware Throttling

The file proposes a “Thermal-Aware Throttling” strategy: when (T_b) exceeds a threshold, the OS should reduce (\Psi)-sensitive background tasks to prevent an “avalanche effect.” 

A mechanistic control rule can be written as:
[
\text{If }T_b>T_{\mathrm{thr}},\quad
N(t)\leftarrow \rho_N N(t),\ \ \rho_N\in(0,1),
]
particularly for tasks with high radio scanning or upload bursts. Why this is physically justified: reducing (N) lowers (P_{\rm tot}), which increases (\Delta), which prevents infeasibility and reduces (I), thereby reducing both (I^2R_0) heating and SOC decay rate.

---

## Bottom line

* 你这份文件最“值钱”的精华是：**(i) 多尺度奇异摄动 + (ii) 判别式奇点解释电压崩塌 + (iii) 均值回复随机负载的 TTE 分布化 + (iv) 低温下信号强度主导的反直觉结论 + OS策略**。这些我已经全部按论文级细节展开，并且**每一点都配了公式与逐项解释**，且完全沿用文件的结构与符号体系。

如果你愿意，我下一步可以把这整段进一步“SIAM/IEEE 化”：把口吻改成更强的被动语态与段落衔接，并补上一个**可直接放附录的 RK4 伪代码**（每一步如何嵌套求 (I)、如何检测 (\Delta\le 0) 终止）。
