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Conditional Langevin Green–Kubo and Einstein relation

Physical scope

Conditional one-dimensional prediction for the specified stationary inertial velocity Langevin model with constant mass, damping, temperature and additive noise. It does not assert the Einstein relation for nonlinear dynamics, arbitrary baths, general mechanical systems, or measured material coefficients.

Statement

Assume the adopted velocity bath Adopted inertial velocity Langevin bath on $t \geq 0$ and the Axiom of Choice The Axiom of Choice. Let $X_{t} = X_{0} + \int_{0}^{t} V_{s} d s$ be the position coordinate in that model, and prepare $V_{0}$ in its specified stationary Gaussian law. For zero external force, define the stationary velocity covariance by $C_{V V} (τ) : = E [(V_{s} - E V_{s}) (V_{s + τ} - E V_{s + τ})], s \geq \max (0, - τ) .$ Stationarity makes it independent of $s$ , and the model gives $C_{V V} (τ) = \frac{k_{B} T_{b}}{m} e^{- γ ∣ τ ∣} .$ For a deterministic force applied at time $s \leq t$ , the linear response kernel is $R_{V F} (t - s) = \frac{1}{m} e^{- γ (t - s)} = \frac{1}{k_{B} T_{b}} C_{V V} (t - s) .$ Here $R_{V F}$ is defined by the exact mean-response formula $E [V_{t}] = \int_{0}^{t} R_{V F} (t - s) F (s) d s$ for deterministic forcing from the stated zero-mean stationary preparation; its units are $k g^{- 1}$ . The constant-force mobility below is a distinct steady-response coefficient. The diffusion coefficient defined by centered displacement variance satisfies the Green–Kubo identity $D : = \lim_{t \to \infty} \frac{Var (X_{t} - X_{0})}{2 t} = \frac{k_{B} T_{b}}{m γ} in m^{2} s^{- 1} .$ More precisely, $D = \int_{0}^{\infty} C_{V V} (τ) d τ$ . The constant-force mobility $μ : = \lim_{t \to \infty} \frac{E [V_{t}]}{F_{0}} = \frac{1}{m γ} in s k g^{- 1}$ for constant $F_{0} \neq 0$ . Hence $D = μ k_{B} T_{b}$ . The centered displacement divided by $\sqrt{t}$ converges in distribution to $N (0, 2 D)$ . These are conditional model deductions, not observed transport data or a universal law for arbitrary systems.

Facts & Assumptions

Given: The adopted one-dimensional Langevin equation and stationary preparation, Choice, and the exact mathematical OU velocity-response theorem.

[A1]

Adopted inertial velocity Langevin bath: $m > 0$ is in kilograms, $T_{b} > 0$ in kelvins, $k_{B}$ in joules per kelvin, $γ > 0$ in seconds $^{- 1}$ , $σ^{2} = 2 γ k_{B} T_{b} / m$ , and the stationary initial velocity has variance $k_{B} T_{b} / m$ .

[A2]

The Axiom of Choice: every family of nonempty sets has a choice function, as required by the Brownian and Itô suppliers.

[F1]

Exact OU velocity response, diffusion and Gaussian scaling: under $σ^{2} = 2 γ k_{B} Θ / m$ and $V_{0} \sim N (0, k_{B} Θ / m)$ , the OU covariance is $(k_{B} Θ / m) e^{- γ ∣ τ ∣}$ , the integrated displacement variance has limit $D = k_{B} Θ / (m γ)$ , and the constant-input response coefficient is $1 / (m γ)$ .

Proof

Given: [A1]–[A2] and [F1].

1.1A1algebra

Substituting $σ^{2} = 2 γ k_{B} T_{b} / m$ from [A1] into the stationary variance $V = σ^{2} / (2 γ)$ gives $V = k_{B} T_{b} / m$ . Its units are $J / k g = m^{2} s^{- 2}$ , the square of a velocity.

2.1A2F1step 1.1algebra

Apply [F1] with $Θ = T_{b}$ and input $u = F$ . Its covariance and response formulas give $C_{V V} (τ) = (k_{B} T_{b} / m) e^{- γ ∣ τ ∣}$ and $R_{V F} (t - s) = m^{- 1} e^{- γ (t - s)}$ . Since $(k_{B} T_{b})^{- 1} C_{V V} (t - s) = m^{- 1} e^{- γ (t - s)}$ , the claimed equilibrium response relation follows.

2.2A2F1A1step 1.1algebra

The same substitution in the displacement-variance limit of [F1] gives $D = k_{B} T_{b} / (m γ)$ . The covariance in the Statement gives $\int_{0}^{\infty} C_{V V} (τ) d τ = (k_{B} T_{b} / m) \int_{0}^{\infty} e^{- γ τ} d τ = k_{B} T_{b} / (m γ) = D$ , the Green–Kubo identity for this model. Its constant-input response gives $μ = 1 / (m γ)$ , so $D = μ k_{B} T_{b}$ . The unit of $μ$ is $(m s^{- 1}) / N = s k g^{- 1}$ , and multiplication by $k_{B} T_{b}$ in joules gives $m^{2} s^{- 1}$ .

3.1A2F1A1step 2.2∎

The centered-displacement distributional limit in [F1] gives $[X_{t} - X_{0} - E (X_{t} - X_{0})] / \sqrt{t} \Rightarrow N (0, 2 D)$ . Every conclusion is conditional on [A1]; no observed value or empirical certainty is inferred.

Conditional Langevin Green–Kubo and Einstein relation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources