Polaron测度的强耦合极限与Pekar过程
概率论
2018-06-20 v1
摘要
{\it{Polaron测度}}定义为关于有限区间[ − T , T ] [-T,T] [ − T , T ] P \mathbb P P P ^ ϵ , T = Z ϵ , T − 1 exp { 1 2 ∫ − T T ∫ − T T ϵ \e − ϵ ∣ t − s ∣ ∣ ω ( t ) − ω ( s ) ∣ \d s \d t } \d P \widehat{\mathbb P}_{\epsilon,T}= Z_{\epsilon,T}^{-1}\,\, \exp\bigg\{\frac{1}{2}\int_{-T}^T\int_{-T}^T\frac{\epsilon\e^{-\epsilon|t-s|}}{|\omega(t)-\omega(s)|} \,\d s \,\d t\bigg\}\d\mathbb P P ϵ , T = Z ϵ , T − 1 exp { 2 1 ∫ − T T ∫ − T T ∣ ω ( t ) − ω ( s ) ∣ ϵ \e − ϵ ∣ t − s ∣ \d s \d t } \d P Z ϵ , T Z_{\epsilon,T} Z ϵ , T ϵ > 0 \epsilon>0 ϵ > 0 Z ϵ , T Z_{\epsilon,T} Z ϵ , T g 0 = lim ϵ → 0 [ lim T → ∞ log Z \eps , T 2 T ] = sup \heap ψ ∈ H 1 ( R 3 ) ∥ ψ ∥ 2 = 1 { ∫ R 3 ∫ R 3 \d x \d y ψ 2 ( x ) ψ 2 ( y ) ∣ x − y ∣ − 1 2 ∥ ∇ ψ ∥ 2 2 } . g_0=\lim_{\epsilon \to 0}\bigg[\lim_{T\to\infty}\frac{\log Z_{\eps,T}}{2T}\bigg]=\sup_{\heap{\psi\in H^1(\R^3)}{\|\psi\|_2=1}} \bigg\{\int_{\mathbb R^3}\int_{\mathbb R^3}\d x\d y\,\frac {\psi^2(x) \psi^2(y)}{|x-y|} -\frac 12\big\|\nabla \psi\big\|_2^2\bigg\}. g 0 = ϵ → 0 lim [ T → ∞ lim 2 T log Z \eps , T ] = \heap ψ ∈ H 1 ( R 3 ) ∥ ψ ∥ 2 = 1 sup { ∫ R 3 ∫ R 3 \d x \d y ∣ x − y ∣ ψ 2 ( x ) ψ 2 ( y ) − 2 1 ∇ ψ 2 2 } . P ^ \eps = lim T → ∞ P ^ \eps , T {\widehat {\mathbb P}}_{\eps}=\lim_{T\to\infty}\widehat{\mathbb P}_{\eps,T} P \eps = lim T → ∞ P \eps , T P ^ \eps , T \widehat{\mathbb P}_{\eps,T} P \eps , T ( 2 T ) − 1 / 2 ( ω ( T ) − ω ( − T ) ) (2T)^{-1/2}(\omega(T)-\omega(-T)) ( 2 T ) − 1/2 ( ω ( T ) − ω ( − T )) σ 2 ( \eps ) \sigma^2(\eps) σ 2 ( \eps ) lim \eps → 0 lim T → ∞ P ^ \eps , T = lim \eps → 0 P ^ \eps \lim_{\eps\to 0} \lim_{T\to\infty} \widehat{\mathbb P}_{\eps,T}=\lim_{\eps\to 0} \widehat {\mathbb P}_\eps lim \eps → 0 lim T → ∞ P \eps , T = lim \eps → 0 P \eps 1 2 Δ + ( ∇ ψ ψ ) ⋅ ∇ \frac 12 \Delta+ \bigg(\frac{\nabla\psi}\psi\bigg)\cdot \nabla 2 1 Δ + ( ψ ∇ ψ ) ⋅ ∇ g 0 g_0 g 0 ψ \psi ψ
引用
@article{arxiv.1806.06865,
title = {Strong coupling limit of the Polaron measure and the Pekar process},
author = {Chiranjib Mukherjee and S. R. S. Varadhan},
journal= {arXiv preprint arXiv:1806.06865},
year = {2018}
}