Distinct Spatiotemporal Dynamics of Thermoelectric Transport Across Superconducting Transition

Rajae Malek; Qing-Dong Jiang; Haiwen Liu

arXiv:2508.20792·cond-mat.supr-con·August 29, 2025

Distinct Spatiotemporal Dynamics of Thermoelectric Transport Across Superconducting Transition

Rajae Malek, Qing-Dong Jiang, Haiwen Liu

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TL;DR

This paper explores how heat transport dynamics change across the superconducting transition, revealing distinct linear and nonlinear relaxation behaviors above and below Tc with implications for experimental probing.

Contribution

It introduces two classes of relaxation diffusion equations capturing the fundamental shift in heat transport mechanisms across the superconducting transition.

Findings

01

Type I relaxation leads to uniform heat current above Tc.

02

Type II relaxation shows inhomogeneous, propagating hot spots below Tc.

03

Distinct dynamics highlight a fundamental change in transport across the transition.

Abstract

We investigate the relaxation dynamics of heat transport in superconductors, shaped by the interplay of diffusion, nonlinearity, and magnetic fields. Focusing on regimes near the critical temperature Tc, we analyze two classes of relaxation diffusion equations that give rise to qualitatively distinct dynamics, which we denote as Type I (linear) and Type II (nonlinear). Type I relaxation, characteristic of the normal state above Tc, results in a steady and spatially uniform heat current governed by linear diffusion. By contrast, Type II relaxation, relevant below Tc, exhibits non steady dynamics marked by pronounced spatial inhomogeneities and an evolving pattern, in which an initially localized hot spot propagates transiently through the system. The striking distinction between these regimes underscores a fundamental shift in transport mechanisms across the superconducting phase…

Tables1

Table 1. Table 1 : Comparison of exact and numerical heat current averages across different k x k_{x} domains at varying time instances. The exact averages (denoted as Average ex ) are computed from the analytical solution, while the numerical averages (denoted as Average ap ) are obtained using the δ − \delta_{-} Ziti method. The table shows the results for k x k_{x} domains [ − 10 , 10 ] , [ − 20 , 20 ] [-10,10],[-20,20] , and [ − 40 , 40 ] [-40,40] at times t = 1 , t = 2 t=1,t=2 , and t = 20 t=20 .

$k_{x}$	$[- 10, 10]$	$[- 20, 20]$	$[- 40, 40]$
$t = 1$
Average_ex	$1.035$	$1.0496$	$1.0521$
Average_ap	$1.0351$	$1.0495$	$1.0521$
$t = 2$
Average_ex	$1.0351$	$1.0498$	$1.0521$
Average_ap	$1.0352$	$1.05$	$1.0522$
$t = 20$
Average_ex	$1.0351$	$1.0498$	$1.0521$
Average_ap	$1.0352$	$1.05$	$1.0522$

Equations100

[\frac{\partial}{\partial t} - D_{φ} (ℏ\nabla - 2 i e A)^{2} + \frac{1}{τ _{s}}] R_{0} (r, t; r_{0}, t_{0}) = δ (r - r_{0}) δ (t - t_{0}) .

[\frac{\partial}{\partial t} - D_{φ} (ℏ\nabla - 2 i e A)^{2} + \frac{1}{τ _{s}}] R_{0} (r, t; r_{0}, t_{0}) = δ (r - r_{0}) δ (t - t_{0}) .

R_{0} (r, t; r_{0}, t_{0}) \equiv \frac{1}{2 π} \int_{- \infty}^{\infty} e^{iω (t - t_{0})} R_{0} (r, r_{0}) d ω,

R_{0} (r, t; r_{0}, t_{0}) \equiv \frac{1}{2 π} \int_{- \infty}^{\infty} e^{iω (t - t_{0})} R_{0} (r, r_{0}) d ω,

[iω - D_{φ} (ℏ\nabla - 2 i e A)^{2} + \frac{1}{τ _{s}}] R_{0} (r, r_{0}) = δ (r - r_{0}) .

[iω - D_{φ} (ℏ\nabla - 2 i e A)^{2} + \frac{1}{τ _{s}}] R_{0} (r, r_{0}) = δ (r - r_{0}) .

[iω - D_{φ} (ℏ\nabla - 2 i e A)^{2} + \frac{1}{τ _{s}}] R_{0} (r, r_{0})

[iω - D_{φ} (ℏ\nabla - 2 i e A)^{2} + \frac{1}{τ _{s}}] R_{0} (r, r_{0})

= n, p_{x} \sum C (n, p_{x}) e^{i \frac{p _{x} . x}{ℏ}} [iω + \frac{1}{τ _{s}} + (n + \frac{1}{2}) ℏ ω_{B}] χ_{n} (y)

= δ (x - x_{0}) δ (y - y_{0}) .

C (n, p_{x}) = \frac{1}{2 ^{n} . n !} e^{- i p_{x} \frac{x _{0}}{ℏ}} . \frac{H _{n} ( \frac{y _{0} - y _{1}}{a _{B}} ) . exp ( - \frac{( y _{0} - y _{1} ) ^{2}}{2 a _{B}^{2}} )}{iω + \frac{1}{τ _{s}} + ( n + \frac{1}{2} ) ℏ ω _{B}}

C (n, p_{x}) = \frac{1}{2 ^{n} . n !} e^{- i p_{x} \frac{x _{0}}{ℏ}} . \frac{H _{n} ( \frac{y _{0} - y _{1}}{a _{B}} ) . exp ( - \frac{( y _{0} - y _{1} ) ^{2}}{2 a _{B}^{2}} )}{iω + \frac{1}{τ _{s}} + ( n + \frac{1}{2} ) ℏ ω _{B}}

R_{0} (r, t; r_{0}, t_{0}) = \frac{1}{2 π} \int_{- \infty}^{\infty} d ω e^{iω (t - t_{0})} . p_{x} \sum \frac{ω _{B}}{2 π ℏ D _{ϕ}} e^{i p_{x} \frac{x - x _{0}}{ℏ}} \times exp (- \frac{( y - y _{1} ) ^{2} + ( y _{0} - y _{1} ) ^{2}}{2 a _{B}^{2}})

R_{0} (r, t; r_{0}, t_{0}) = \frac{1}{2 π} \int_{- \infty}^{\infty} d ω e^{iω (t - t_{0})} . p_{x} \sum \frac{ω _{B}}{2 π ℏ D _{ϕ}} e^{i p_{x} \frac{x - x _{0}}{ℏ}} \times exp (- \frac{( y - y _{1} ) ^{2} + ( y _{0} - y _{1} ) ^{2}}{2 a _{B}^{2}})

\times n \sum \frac{1}{2 ^{n} . n !} \frac{H _{n} ( \frac{y - y _{1}}{a _{B}} ) . H _{n} ( \frac{y _{0} - y _{1}}{a _{B}} )}{iω + \frac{1}{τ _{s}} + ( n + \frac{1}{2} ) ℏ ω _{B}} .

\frac{\partial}{\partial t} φ (r, t) - D_{φ} (ℏ\nabla - 2 i e A)^{2} φ (r, t) + \frac{1}{τ _{s}} φ (r, t) = ξ (r, t)

\frac{\partial}{\partial t} φ (r, t) - D_{φ} (ℏ\nabla - 2 i e A)^{2} φ (r, t) + \frac{1}{τ _{s}} φ (r, t) = ξ (r, t)

⟨ ξ (r, t) ξ (r^{'}, t^{'})⟩ = k_{B} T δ (r - r^{'}) δ (t - t^{'}) .

⟨ ξ (r, t) ξ (r^{'}, t^{'})⟩ = k_{B} T δ (r - r^{'}) δ (t - t^{'}) .

{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\frac{\partial}{\partial t}+i\frac{2e}{\hbar c}\Phi(\vec{r},t)\right)}\varphi(\vec{r},t)-D_{\varphi}\left(\hbar\nabla-2ie\vec{A}\right)^{2}\varphi(\vec{r},t)+\frac{1}{\tau_{s}}\varphi(\vec{r},t)=\xi(\vec{r},t)

{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\frac{\partial}{\partial t}+i\frac{2e}{\hbar c}\Phi(\vec{r},t)\right)}\varphi(\vec{r},t)-D_{\varphi}\left(\hbar\nabla-2ie\vec{A}\right)^{2}\varphi(\vec{r},t)+\frac{1}{\tau_{s}}\varphi(\vec{r},t)=\xi(\vec{r},t)

φ (r, t) = \int R (r, t; r^{'}, t^{'}) ξ (r^{'}, t^{'}) d r^{'} d t^{'}

φ (r, t) = \int R (r, t; r^{'}, t^{'}) ξ (r^{'}, t^{'}) d r^{'} d t^{'}

C (r, t; r^{'}, t^{'})

C (r, t; r^{'}, t^{'})

= k_{B} T \int R (r, t; r_{0}, t_{0}) R^{*} (r^{'}, t^{'}; r_{0}, t_{0}) d r_{0} d t_{0} .

R \approx R_{0} + R_{1}

R \approx R_{0} + R_{1}

\left[{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\frac{\partial}{\partial t}+i\frac{2e}{\hbar c}\Phi(\vec{r},t)\right)}-D_{\varphi}\left(\hbar\nabla-2ie\vec{A}\right)^{2}+\frac{1}{\tau_{s}}\right]\left(R_{0}+R_{1}\right)=\delta(\vec{r}-\vec{r_{0}})\delta(t-t_{0}).

\left[{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\frac{\partial}{\partial t}+i\frac{2e}{\hbar c}\Phi(\vec{r},t)\right)}-D_{\varphi}\left(\hbar\nabla-2ie\vec{A}\right)^{2}+\frac{1}{\tau_{s}}\right]\left(R_{0}+R_{1}\right)=\delta(\vec{r}-\vec{r_{0}})\delta(t-t_{0}).

⎩ ⎨ ⎧ [\frac{\partial}{\partial t} - D_{φ} (ℏ\nabla - 2 i e A)^{2} + \frac{1}{τ _{s}}] R_{0} = δ (r - r_{0}) δ (t - t_{0}) . [\frac{\partial}{\partial t} + (- D_{φ} (ℏ\nabla - 2 i e A)^{2} + \frac{1}{τ _{s}})] R_{1} = - (i \frac{2 e}{ℏ c} Φ (r, t)) R_{0}

⎩ ⎨ ⎧ [\frac{\partial}{\partial t} - D_{φ} (ℏ\nabla - 2 i e A)^{2} + \frac{1}{τ _{s}}] R_{0} = δ (r - r_{0}) δ (t - t_{0}) . [\frac{\partial}{\partial t} + (- D_{φ} (ℏ\nabla - 2 i e A)^{2} + \frac{1}{τ _{s}})] R_{1} = - (i \frac{2 e}{ℏ c} Φ (r, t)) R_{0}

R_{1} (r, t; r^{'}, t^{'}) = - (i \frac{2 e}{ℏ c}) \times \int R_{0} (r, t; r_{1}, t_{1}) R_{0} (r_{1}, t_{1}; r^{'}, t^{'}) Φ (r_{1}, t_{1}) d r_{1} d t_{1}

R_{1} (r, t; r^{'}, t^{'}) = - (i \frac{2 e}{ℏ c}) \times \int R_{0} (r, t; r_{1}, t_{1}) R_{0} (r_{1}, t_{1}; r^{'}, t^{'}) Φ (r_{1}, t_{1}) d r_{1} d t_{1}

C (r, t; r^{'}, t^{'}) \approx k_{B} T \int (R_{0} + R_{1}) \cdot (R_{0}^{*} + R_{1}^{*}) d r_{0} d t_{0} .

C (r, t; r^{'}, t^{'}) \approx k_{B} T \int (R_{0} + R_{1}) \cdot (R_{0}^{*} + R_{1}^{*}) d r_{0} d t_{0} .

C (r, t; r^{'}, t^{'})

C (r, t; r^{'}, t^{'})

+ k_{B} T \int [R_{0} (r, t; r_{0}, t_{0}) R_{1}^{*} (r^{'}, t^{'}; r_{0}, t_{0}) + R_{1} (r, t; r_{0}, t_{0}) R_{0}^{*} (r^{'}, t^{'}; r_{0}, t_{0})] d r_{0} d t_{0}

+ O (R_{1}^{2}) .

C_{0} (r, t; r^{'}, t^{'}) = k_{B} T \int R_{0} (r, t; r_{0}, t_{0}) R_{0}^{*} (r^{'}, t^{'}; r_{0}, t_{0}) d r_{0} d t_{0},

C_{0} (r, t; r^{'}, t^{'}) = k_{B} T \int R_{0} (r, t; r_{0}, t_{0}) R_{0}^{*} (r^{'}, t^{'}; r_{0}, t_{0}) d r_{0} d t_{0},

C_{1} (r, t; r^{'}, t^{'}) = k_{B} T \int [R_{0} (r, t; r_{0}, t_{0}) R_{1}^{*} (r^{'}, t^{'}; r_{0}, t_{0}) + R_{1} (r, t; r_{0}, t_{0}) R_{0}^{*} (r^{'}, t^{'}; r_{0}, t_{0})] d r_{0} d t_{0} .

C_{1} (r, t; r^{'}, t^{'}) = k_{B} T \int [R_{0} (r, t; r_{0}, t_{0}) R_{1}^{*} (r^{'}, t^{'}; r_{0}, t_{0}) + R_{1} (r, t; r_{0}, t_{0}) R_{0}^{*} (r^{'}, t^{'}; r_{0}, t_{0})] d r_{0} d t_{0} .

C (r, t; r^{'}, t^{'}) ≃ C_{0} (r, t; r^{'}, t^{'}) + C_{1} (r, t; r^{'}, t^{'}) + O (R_{1}^{2}),

C (r, t; r^{'}, t^{'}) ≃ C_{0} (r, t; r^{'}, t^{'}) + C_{1} (r, t; r^{'}, t^{'}) + O (R_{1}^{2}),

J^{h}=\left(\mathcal{A}+\mathcal{\tilde{A}}\right)C_{1}(\vec{r},t;\vec{r^{\prime}},t^{\prime})\Big{|}_{\underset{\vec{r}=\vec{r^{\prime}}}{t=t^{\prime}}}

J^{h}=\left(\mathcal{A}+\mathcal{\tilde{A}}\right)C_{1}(\vec{r},t;\vec{r^{\prime}},t^{\prime})\Big{|}_{\underset{\vec{r}=\vec{r^{\prime}}}{t=t^{\prime}}}

\displaystyle\mathcal{A}=\left(\nabla-\frac{2ie}{\hbar{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c}}\vec{A}(\vec{r})\right)\left(\frac{\partial}{\partial t^{\prime}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-}i\frac{2e}{\hbar c}\Phi(\vec{r^{\prime}})\right),

\displaystyle\mathcal{A}=\left(\nabla-\frac{2ie}{\hbar{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c}}\vec{A}(\vec{r})\right)\left(\frac{\partial}{\partial t^{\prime}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-}i\frac{2e}{\hbar c}\Phi(\vec{r^{\prime}})\right),

\displaystyle\mathcal{\tilde{A}}=\left(\nabla^{\prime}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+}\frac{2ie}{\hbar{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c}}\vec{A}(\vec{r^{\prime}})\right)\left(\frac{\partial}{\partial t}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+}i\frac{2e}{\hbar c}\Phi(\vec{r})\right).

\displaystyle\frac{-1}{\hbar^{2}D_{\phi}}\mathcal{A}C_{1}(\vec{r},t;\vec{r^{\prime}},t^{\prime})\Big{|}_{\underset{\vec{r}=\vec{r^{\prime}}}{t=t^{\prime}}}

\displaystyle\frac{-1}{\hbar^{2}D_{\phi}}\mathcal{A}C_{1}(\vec{r},t;\vec{r^{\prime}},t^{\prime})\Big{|}_{\underset{\vec{r}=\vec{r^{\prime}}}{t=t^{\prime}}}

\displaystyle\hskip 5.69046pt=\left(\nabla-\frac{2ie}{\hbar{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c}}\vec{A}(\vec{r})\right)\left(\frac{\partial}{\partial t^{\prime}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-}i\frac{2e}{\hbar c}\Phi(\vec{r^{\prime}})\right)C_{1}(\vec{r},t;\vec{r^{\prime}},t^{\prime})\Big{|}_{\underset{\vec{r}=\vec{r^{\prime}}}{t=t^{\prime}}},

\displaystyle\hskip 5.69046pt=\left(\partial_{x}+\frac{2ie}{\hbar{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c}}By\right)\left(\frac{\partial}{\partial t^{\prime}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-}i\frac{2e}{\hbar c}\Phi(\vec{r^{\prime}})\right)\left(\frac{-i2e}{\hbar c}\right)\times\int\left(-E_{y}y_{1}\right)R_{0}(\vec{r},t;\vec{r}_{1},t_{1})C_{0}^{*}(\vec{r^{\prime}},t^{\prime};\vec{r_{1}},t_{1})dx_{1}dy_{1}dt_{1}.

C_{0}^{*} (r^{'}, t^{'}; r_{1}, t_{1})

C_{0}^{*} (r^{'}, t^{'}; r_{1}, t_{1})

= \frac{k _{B} T}{2 π} \int_{- \infty}^{\infty} e^{i ω_{2} (t_{1} - t^{'})} d ω_{2} . \int d k_{x_{2}} \frac{ω _{B}}{2 π ℏ D _{ϕ}} e^{i k_{x_{2}} (x_{1} - x^{'})} \times exp (- \frac{( y ^{'} - y _{k_{2}} ) ^{2}}{2 a _{B}^{2}}) exp (- \frac{( y _{1} - y _{k_{2}} ) ^{2}}{2 a _{B}^{2}})

\times n \sum \frac{1}{2 ^{n} . n !} \frac{H _{n} ( \frac{( y ^{'} - y _{k_{2}} )}{a _{B}} ) . H _{n} ( \frac{y _{1} - y _{k_{2}}}{a _{B}} )}{ω _{2}^{2} + ( \frac{1}{τ _{s}} + ( n + \frac{1}{2} ) ℏ ω _{B} ) ^{2}} .

J_{x}^{h} = - k_{B} T E_{y} \frac{e}{c} \frac{ω _{B}}{π} \int_{- \infty}^{+ \infty} y_{1} d y_{1} \int_{- \infty}^{+ \infty} d k_{x_{1}} (k_{x_{1}} + \frac{2 e B}{ℏ c} y)

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Full text

Distinct Spatiotemporal Dynamics of Thermoelectric Transport Across Superconducting Transition

Rajae Malek

Shanghai Research Center for Quantum Sciences, Shanghai 201315, China

Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 212, China

Qing-Dong Jiang

[email protected]

Shanghai Research Center for Quantum Sciences, Shanghai 201315, China

Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 212, China

Haiwen Liu

[email protected]

Center for Advanced Quantum Studies, School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China

Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University, Beijing 100875, China

(August 28, 2025)

Abstract

We investigate the relaxation dynamics of heat transport in superconductors, shaped by the interplay of diffusion, nonlinearity, and magnetic fields. Focusing on regimes near the critical temperature $T_{c}$ , we analyze two classes of relaxation–diffusion equations that give rise to qualitatively distinct dynamics, which we denote as Type-I (linear) and Type-II (nonlinear). Type-I relaxation, characteristic of the normal state above $T_{c}$ , results in a steady and spatially uniform heat current governed by linear diffusion. By contrast, Type-II relaxation, relevant below $T_{c}$ , exhibits non-steady dynamics marked by pronounced spatial inhomogeneities and an evolving pattern, in which an initially localized hot spot propagates transiently through the system. The striking distinction between these regimes underscores a fundamental shift in transport mechanisms across the superconducting phase transition and provides experimentally relevant predictions in light of emerging techniques for probing local dissipation.

I Introduction

To understand the nature of superconductivity, researchers have extensively investigated various related phenomena, with superconducting fluctuations being of particular importance and providing crucial insights into the behavior of superconducting systems through decades of study [1]. These fluctuations, especially near the superconducting transition temperature $T_{c}$ , can significantly influence the properties of materials, particularly in how they respond to external fields and thermal perturbations [2]. Landmark contributions in this regard was provided by Aslamazov and Larkin [3] and Maki [4], who demonstrated that fluctuations in the normal state of superconductors lead to a dramatic divergence in electrical conductivity as the system approaches $T_{c}$ . This observation paved the way for further exploration of how fluctuations at the critical temperature can impact transport properties like electric and heat conductivity, providing new perspectives on the behavior of superconducting systems near phase transitions [5]. Building on these early insights, the study of superconducting fluctuations has evolved to incorporate advanced methods from both the phenomenological Ginzburg-Landau (GL) theory and microscopic approaches [6, 1]. These fluctuations are key factors in determining experimentally observable quantities such as electrical conductivity, heat current, and the Nernst coefficient [7]. The Nernst effect has been identified as an excellent experimental probe for studying the response of superconducting fluctuations near $T_{c}$ under thermal gradients and magnetic fields [8, 9, 10, 11, 12, 13, 14, 15]. While steady-state transport properties near $T_{c}$ have been extensively studied, the underlying spatiotemporal dynamics governing how heat propagates and energy dissipates across the transition remains less understood.

Our work leverages the time-dependent Ginzburg-Landau (TDGL) framework to delve deeper into the dynamics of superconducting fluctuations, particularly the relaxation processes that govern the evolution of the superconducting order parameter. The TDGL framework allows us to explore how the superconducting state responds to thermal fluctuations and external perturbations, providing a versatile tool for understanding both linear and nonlinear effects in these systems [2, 16, 17]. A central aspect of this work is the classification of relaxation processes near $T_{c}$ into two distinct regimes: Type-I and Type-II relaxation. Type-I relaxation, typically occurring above $T_{c}$ [16], describes dynamics governed by a linear relaxation-diffusion equation, characterized by the slow relaxation of superconducting fluctuations within the normal state. In contrast, Type-II relaxation, relevant below $T_{c}$ [18], involves inherently nonlinear dynamics (often referred to as “quenched” dynamics). In this regime, the interplay between thermal fluctuations, magnetic fields, and, crucially, the nonlinear self-interaction of the order parameter dictates the system’s evolution. Thus, the relaxation dynamics of superconducting systems are inherently complex, involving the interplay of dissipation, coherence, and nonlinear effects.[19, 18]. The TDGL framework provides a powerful and flexible approach to modeling the evolution of propagators, correlations, and physical quantities such as heat current for both Type-I and Type-II relaxations, capturing the interplay between dissipation and coherence, and handling nonequilibrium dynamics under magnetic field[20, 10]. By solving these equations, either numerically or analytically, we can gain valuable insights into the dynamics of superconducting systems and their response to external perturbations. The thermoelectric Nernst coefficient plays a critical role in distinguishing between these two relaxation processes [9]. The sharp difference of heat propagation between Type-I and Type-II relaxations provides experimentalists with a measurable quantity that exhibits distinctive signatures, offering a direct connection between the theoretical understanding of these processes and experimental observations.

In this work, we examine thermoelectric transport across the superconducting transition. Our results reveal a pronounced contrast between the two dynamical regimes: above $T_{c}$ , Type-I relaxation produces steady, spatially uniform heat transport, while below $T_{c}$ , Type-II relaxation gives rise to non-steady dynamics marked by spatially inhomogeneous and evolving patterns. Using the $\delta_{-}$ Ziti method [21, 22, 23], we numerically solve the propagator equations Eq. (1) and Eq. (16), from which physical observables such as the heat current are obtained. This framework reveals the distinct transport characteristics of the two regimes, capturing both linear and nonlinear dynamics in superconducting systems.

The outline of this paper is as follows: In Section II, we discuss our theoretical results, focusing on the analytical solution of the Type-I relaxation equation, along with the related correlations and heat current in the presence of an electromagnetic field. In section III, we present our numerical results for solving the linear Type-I equation, while section IV focuses on our numerical contribution to solving the nonlinear Type-II relaxation equation. We conclude with brief remarks and perspectives in section V. Appendix A outlines the key steps in constructing the numerical method $\delta_{-}$ Ziti, which is designed to deal with both Type-I and Type-II relaxation equations, which can successfully tackle the analytically intractable nonlinear dynamics inherent in the Type-II regime.

II Theoretical results of Type-I relaxation equation

We consider two distinct regimes: the Type-I relaxation equation, relevant for temperatures above $T_{c}$ , and the Type-II quenched dynamical relaxation, characterizing the regime below $T_{c}$ . In this section, we present the analytical treatment of the Type-I case. The corresponding numerical analyses are reported in Sections III and IV: for Type-I, the numerical simulations are benchmarked against the analytical solutions derived here, while for Type-II—where no closed-form analytical solution exists—numerical simulations provide essential support for our physical interpretations.

II.1 The Type-I relaxation equation with magnetic field.

We begin by considering the Type-I relaxation–diffusion equation, which governs the dynamical evolution of a conventional superconducting system subjected to an external magnetic field in the regime just above the superconducting transition temperature $T_{c}$ . The corresponding propagator, from which the system’s physical properties can be derived, is defined by the following equation: [2, 20]:

[TABLE]

Let $R_{0}(\vec{r},\vec{r_{0}})$ be the Fourier transform of the real time propagator $R_{0}(\vec{r},t;\vec{r_{0}},t_{0})$ :

[TABLE]

where the propagator in frequency space satisfies,

[TABLE]

We aim to solve Eq. (1) by Hermite polynomials by expanding $R_{0}(\vec{r},\vec{r_{0}})=\underset{n,p_{x}}{\sum}C(n,p_{x})e^{i\frac{p_{x}.x}{\hbar}}\chi_{n}(y)$ . For the sake of simplicity, let us introduce the Landau gauge magnetic vector potential: $\vec{A}=(-By,0).$ In this case, the solution of Eq. (1) is similar to the Landau Level solution of 2D electron gas, with $\omega_{B}=4eBD_{\varphi}$ , $a_{B}=\sqrt{\frac{\hbar}{2eB}}$ and $y_{1}=\frac{p_{x}}{2eB}$ . Recall that, the Hermite polynomials $\chi_{n}(y)$ are given by $\chi_{n}(y):=\frac{1}{\pi^{\frac{1}{4}}a_{B}^{\frac{1}{2}}\sqrt{2^{n}n!}}\exp\left(-\frac{(y-y_{1})^{2}}{2a_{b}^{2}}\right)H_{n}\left(\frac{y-y_{1}}{a_{b}}\right)$ . It remains to identify the coefficients $C(p_{x},n)$ based on the Hermite polynomial properties. Indeed:

[TABLE]

Thus, we have the expression of the coefficient $C(n,p_{x})$ :

[TABLE]

and then the Fourier transform of the propagator $R_{0}(\vec{r},\vec{r_{0}})=\underset{n,p_{x}}{\sum}C(n,p_{x})e^{i\frac{p_{x}.x}{\hbar}}\chi_{n}(y)$ reads as follows:

[TABLE]

II.2 The relaxation and correlation with thermal fluctuations

In the following, we consider the relaxation–diffusion equation in the presence of thermal noise, denoted by $\xi(\vec{r},t)$

[TABLE]

where the noise term $\xi(\vec{r},t)$ satisfies local correlation:

[TABLE]

When an electric field $\vec{E}$ is applied, it can be described by a scalar potential $\Phi(\vec{r},t)=-\vec{E}\cdot\vec{r}e^{-i\omega t}$ . In the following calculation, we take the limit $\omega\rightarrow 0$ to subtracte the direct current transport features. Equation Eq. (3) then reads:

[TABLE]

The solution of Eq. (3) can be written as the convolution between the propagator and the noise function:

[TABLE]

and thus the correlation function can be expressed in terms of propagators:

[TABLE]

We decompose the propagator $R$ into $R_{0}$ and $R_{1}$ , where $R_{0}$ is the equilibrium part, and, assuming that the perturbation is small, $R_{1}$ is the linear order perturbation:

[TABLE]

Notice that we are considering linear response regime and therefore have neglected higher-order terms in the expansion. This decomposition allows us to analyze the system’s response to the perturbation in a systematic way, starting from the unperturbed equilibrium state $R_{0}$ and then considering the additional effects due to the perturbation captured by $R_{1}$ . Using Eq. (5), we have:

[TABLE]

which further gives:

[TABLE]

Based on this equation, we derive the expression of $R_{1}$

[TABLE]

Therefore, the total $R\approx R_{0}+R_{1}$ is constructed, where $R_{0}(\vec{r},t;\vec{r^{\prime}},t^{\prime})$ is given by Eq. (2) and $R_{1}(\vec{r},t;\vec{r^{\prime}},t^{\prime})$ is given by Eq. (6). From the definition of the correlation function given by Eq. (4), we have,

[TABLE]

Expanding this expression yields:

[TABLE]

The expansion of the correlation function can now be separated into its equilibrium and perturbative contributions. The leading-order term corresponds to the equilibrium state:

[TABLE]

while the first-order correction captures the system’s response to the external perturbation:

[TABLE]

Putting both contributions together, the total correlation function Eq. (8) reads:

[TABLE]

where higher-order terms are neglected. The term $C_{1}$ governs the linear response of the system and serves as the starting point for constructing physical observables such as the heat current.

According to the Eq.(22) and Eq.(23) in [24], the heat current operator $J^{h}$ is defined as:

[TABLE]

Here,

[TABLE]

We consider an electric field along the $y$ -direction, $\Phi(\vec{r})=-yE_{y}$ , and a magnetic field along the $x$ -axis represented by the vector potential $A_{x}=-By$ . The spatial differential operators $\nabla$ (and $\nabla^{\prime}$ ), combined with $\vec{A}(\vec{r})$ , ensure gauge invariance and capture the influence of the magnetic field on heat transport via the phase of the superconducting order parameter. The time derivatives $\partial_{t}$ (and $\partial_{t^{\prime}}$ ), coupled with the scalar potential $\Phi(\vec{r})$ , describe the system’s response to time-dependent electric fields, linking thermal and electrical effects.

This operator describes the interplay between diffusion, electromagnetic interactions, and dissipation in the superconducting state. Experimentally, this is directly relevant to phenomena such as the Nernst effect, where a transverse voltage arises in response to a thermal gradient under a magnetic field. The components of $J^{h}$ , including the magnetic-field-modified gradient and the time-dependent electric response, correspond to measurable transport coefficients. Comparing the theoretical form of $J^{h}$ with experimental Nernst signals in superconducting thin films or mesoscopic devices provides a means to test the model’s validity.

Substituting the expression of $C_{1}$ from Eq. (10) into the heat current formula Eq. (12), we obtain:

[TABLE]

The correlation function $C_{0}$ can be conveniently connected to the $R_{0}$ in Eq. (2) by the Fluctuation-Dissipation Theorem, and thus $C_{0}$ can be obtained from the imaginary part of $R_{0}$ , and reads:

[TABLE]

To expand the expression for the heat current, we first substitute Eq. (2) and Eq. (13) into Eq. (10) to obtain the full correlation function $C_{1}(\vec{r},t;\vec{r}^{\prime},t^{\prime})$ . This result is then inserted into Eq. (12), yielding the following analytical expression for the heat current:

[TABLE]

III Numerical results of the linear type-I relaxation equation

In this section, we numerically solve the Type-I equation given by Eq. (1). On the one hand, our numerical approach validates the analytical solution; on the other hand, it provides a practical means to address the nonlinear Type-II relaxation equation in the next section, where an analytical solution is not attainable.

For the convenience of numerical simulation, we introduce the scaled variable $\tilde{R}=\frac{1}{\tau_{s}}R$ . Choosing the Landau gauge, $\vec{A}=(-By,0)$ , Eq. (1) becomes:

[TABLE]

where the dimensionless variables are defined as:

[TABLE]

Our objective is to solve Eq. (15) numerically using the $\delta_{-}$ Ziti method, where the corresponding analytical solution is given by Eq. (2). Observe that, The propagator exhibits exponential decay in time. Indeed, For $n=0$ , the denominator in the summation reduces to $i\omega_{1}+\frac{1}{\tau_{s}}+\frac{1}{2}\hbar\omega_{B}$ , and the integral over $\omega_{1}$ contains the factor $e^{i\omega_{1}(t-t_{1})}$ . This leads to a decay governed by the imaginary part of the pole at $\omega_{1}=-i\left(\frac{1}{\tau_{s}}+\frac{1}{2}\hbar\omega_{B}\right).$ Since $\hbar\omega_{B}$ introduces only a constant frequency shift, the dominant time dependence is given by $\exp\left(-\frac{t}{\tau_{s}}\right).$

To accurately reproduce this behavior, we numerically solve Eq. (15) using the $\delta-$ Ziti method. This approach is particularly well-suited for handling relaxation-type dynamics, allowing for precise treatment of both spatial diffusion and temporal decay. The essential aspects of this numerical method are summarized in the Appendix Eq. (A) , and we refer the interested reader to [21, 22, 23] for a more comprehensive discussion.

The numerical scheme is constructed to balance all terms in the discretized equations. Stability is ensured through careful selection of the time step and spatial discretization parameters, which helps prevent artificial numerical dissipation and preserves the system’s intrinsic decay dynamics.

Fig. 1 shows the time evolution of the numerical solution for various values of $\tau_{s}$ , clearly demonstrating how the relaxation time controls the rate of decay. The strong agreement between numerical and analytical results confirms the robustness of the method for modeling relaxation-driven diffusive processes.

After following the time evolution of $R_{0}$ , we now examine the spatial profile of the numerical propagator at different times. Using the analytical solution at $t=0$ as an initial condition, we aim to observe how the solution evolves in both amplitude and spatial distribution, illustrating the combined effect of relaxation and diffusion.

The Fig. 2 shows the gradual decay and spatial broadening of the propagator over time. The decrease in amplitude reflects the effect of relaxation, while the widening of the distribution is governed by diffusion, consistent with the theoretical expectations.

It is worth to point out that the analytical expression of the heat current in Eq. (14) depends on the spatial coordinate $y$ , although for a sufficiently large system, the current is expected to be uniform along $y$ . To investigate this, we compute the heat current for a range of $k_{x}\in[-40,40]$ and compare the spatial averages of the analytical and numerical results. As shown in Fig. 3, the numerical heat current, despite being derived from a time-space dependent model, remains stable over time and aligns closely with the analytical results.

The average values, reported in Table. 1, exhibit good agreement with relative errors on the order of $10^{-4}$ , validating the consistency of the numerical approach.

The close agreement between the numerical and analytical results strongly supports the accuracy and stability of the used method in capturing heat transport phenomena. This consistency highlights the method’s reliability for systems with known solutions and demonstrates its potential for addressing more complex scenarios, such as the Type-II relaxation equation, where analytical solutions are unavailable. In the next section, we present our numerical findings for Type-II systems, exploring their dynamics and heat flow behavior. Based on the validated framework from the Type-I case, we aim to gain deeper insight into the interplay between diffusion, relaxation, and nonlinearity in Type-II superconductors, shedding light on the fundamental mechanisms governing their behavior.

IV Numerical results of the nonlinear Type-II relaxation equation

For a Type-II superconductor under an external magnetic field, the physical properties are determined by the following nonlinear propagator equation:

[TABLE]

which describes the time evolution of the propagator $R_{0}(\vec{r},t;\vec{r_{0}},t_{0}),$ in the presence of damping and a nonlinear interaction term (see [20]). This equation represents a system with both dissipative and nonlinear features. Due to the complexity introduced by the non-linear term $R_{0}^{3}$ and the damping contribution, an analytical solution to this equation is not feasible. Instead, we employ the $\delta_{-}$ Ziti method to compute the time evolution of the propagator, as shown in Fig. 4. The results reveal a rich interplay between diffusion, relaxation, and nonlinear effects. Initially, at early times $(t\ll\tau_{s})$ , the solution exhibits an initial decay dominated by the diffusion term, consistent with the diffusive nature of the system. In this regime, the nonlinear contribution $\lambda R_{0}^{3}$ remains negligible, and the dynamics follow a classical diffusion process. However, as time progresses, nonlinear effects accumulate: the cubic term $\lambda R_{0}^{3}$ becomes significant enough to inhibit further decay, effectively stabilizing the solution at a saturation value, inversely proportional to $\sqrt{\lambda\tau_{s}}$ . The non-monotonic evolution of the peak heat flux stands in sharp contrast to the exponential decay of Type-I.

The spatial distribution of the propagator for the Type-II equation is plotted in Fig. 5, providing further insight into the interplay between diffusion and nonlinear effects. This illustration, captured at different time steps $(t=0.1,1$ , and 4 $)$ , highlights how nonlinearity influences the evolution of the spatial profile. At early times $(t=0,0.1,1)$ , the solution shows a high peak at the center $(x=0)$ that gradually decays, reflecting the dominance of linear relaxation. Diffusion also causes a moderate outward spread. As time progresses $(t=2,3,4)$ , the peak decays further, and the distribution broadens significantly, highlighting the growing role of diffusion.

The heat current results presented in Fig. 6 illustrate the time evolution of the numerical heat current, revealing a transition from localized, concentrated regions to a more diffuse distribution. At early times $(t=0.1$ to $t=0.5)$ , as shown in Fig. 6 (a)-(c), the heat current is initially concentrated in four symmetric regions around the origin. This localized pattern arises due to the structured nature of the initial excitation and the effects of the nonlinear self-interaction. The heat current at this stage maintains moderate magnitudes, reflecting a balance between diffusion and nonlinear effects. As time advances $(t>1)$ , the heat current begins to spread across the entire domain. This expansion signifies the increasing role of diffusion as the dominant mechanism. Interestingly, the peak magnitude of the heat current does not decrease monotonically but instead increases over time, reaching its maximum around $t=6$ . This peak enhancement is attributed to the nonlinear self-interaction amplifying energy transfer before dissipation takes over. Beyond $t=6$ , the heat current decreases significantly, indicating that relaxation mechanisms eventually suppress the sustained energy flow.

We consider a prototypical, widely studied superconducting NbN thin film [25] to estimate the experimentally accessible manifestations of Type-I and Type-II relaxation. The material parameters of NbN are taken as a critical temperature $T_{c}\approx 10~\mathrm{K}$ , a zero-temperature coherence length $\xi(0)\approx 5~\mathrm{nm}$ and a film thickness $d\approx 10~\mathrm{nm}$ . The system is under a perpendicular magnetic field $B=1~\mathrm{T}$ and either an in-plane electric field $E_{y}$ or a controlled thermal gradient $\nabla T$ . The Ginzburg–Landau relaxation time [20], $\tau_{s}=\displaystyle\frac{\pi\hbar}{8k_{B}(T-T_{c})}$ , evaluated at $T=1.01\,T_{c}$ (hence $\mid T-T_{c}\mid=0.1~\mathrm{K}$ ) gives $\tau_{s}\approx 30~\mathrm{ps}$ , which sets the characteristic recovery time of order-parameter fluctuations. The corresponding magnetic length is $l_{B}\simeq 25~\mathrm{nm}$ , while the temperature-dependent coherence length becomes $\xi(T)=\displaystyle\frac{\xi(0)}{\sqrt{\mid 1-T/T_{c}\mid}}\approx 50~\mathrm{nm}$ . For Type-I relaxation ( $T>T_{c}$ ) the theoretical estimation of Nernst coefficient reads $N\approx\displaystyle\frac{\alpha_{xy}}{\sigma_{xx}}\sim\frac{(k_{B}/e)\,\xi(T)^{2}}{(c\cdot\,l_{B}^{\,2})}$ , here we assume the Nernst response $\alpha_{xy}\sim\displaystyle\frac{ek_{B}\xi(T)^{2}}{(h\,l_{B}^{\,2})}$ [11] and longitudinal conductivity $\sigma_{xx}\sim\displaystyle\frac{c\cdot e^{2}}{h}$ with $c$ of order $10^{2}$ . Thus near $T_{c}$ and at $B=1~\mathrm{T}$ the experimental Nernst coefficient $N=\displaystyle\frac{E_{y}}{\nabla T}$ is expected to lie in the range $1$ – $10~\mathrm{\mu V\,K^{-1}}$ for NbN-like films. In contrast, Type-II relaxation ( $T<T_{c}$ ) produces a non-steady, spatially inhomogeneous heat flux that dynamically evolves into transient structures such as the quadrupolar patterns predicted in Fig. 6 of the main text. The crossover time from initial decay to nonlinear saturation coincides with $\tau_{s}$ , implying that the relevant dynamics unfold within a window of a few tens to a few hundred picoseconds—well suited to ultrafast pump–probe techniques. Spatially, for type-II relaxation the hot spots inherit the system’s characteristic lengths, giving a size of about $20$ – $50~\mathrm{nm}$ , comfortably within the resolution of SQUID-on-tip or scanning NV-center magnetometry. Numerical results (Fig. 6) further indicate that the instantaneous heat-current peak in the non-steady regime can match, or even overshoot, its initial steady-state amplitude, implying a dynamic sequence of amplification and reconstruction rather than monotonic decay. This provides a clear experimental characteristic for distinguishing Type-I from Type-II behaviour.

V Conclusion

In summary, we find that the thermoelectric responses associated with Type-I and Type-II relaxation exhibit distinct dynamical characteristics. In the case of Type-I relaxation, which occurs when the temperature is above the critical temperature $T_{c}$ , the thermoelectric current reaches a steady state, it is spatially uniform and time-independent. This behavior arises from the predominance of linear relaxation mechanisms, which drive the system toward a homogeneous equilibrium configuration. In contrast, Type-II relaxation emerges at temperatures below $T_{c}$ , where nonlinear effects become dominant. As confirmed by numerical simulations, the resulting thermoelectric current is non-steady state: it exhibits pronounced spatial inhomogeneity and evolves dynamically over time. Initially, localized structures such as quadrupolar patterns may form; these structures subsequently diffuse and reorganize, giving rise to a temporally and spatially varying thermoelectric current.

Moreover, the analytical and numerical findings of this study offer concrete theoretical model for designing and interpreting experiments [7, 8, 9] on superconducting thin films and mesoscopic devices under varied magnetic fields. These qualitative differences in the thermal transport response provide clear, measurable signatures across $T_{c}$ , With recent advances in local dissipation measurements—particularly the SQUID-on-tip technique [26, 27], which enables direct probing of spatiotemporal current distributions—such signatures should be experimentally accessible. This, in turn, offers a powerful diagnostic for distinguishing between the two relaxation regimes and for testing theoretical predictions concerning nonlinear saturation and characteristic crossover times. Looking ahead, the theoretical framework developed here may also prove valuable for exploring thermal transport phenomena in the quantum transport of topological superconducting systems [28, 29, 30].

VI Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grants No. 12374037), the Innovation Program for Quantum Science and Technology Grant No. 2021ZD0301900, the National Key Research and Development Program of China (Grants No. 2024YFA1409001), and the Fundamental Research Funds for the Central Universities, Cultivation Project of Shanghai Research Center for Quantum Sciences Grant No. LZPY2024, and Shanghai Science and Technology Innovation Action Plan Grant No. 24LZ1400800.

Appendix A Overview of the $\delta_{-}$ Ziti method

For completeness and ease of reading, we present a self-contained overview of the numerical method used in this paper. The $\delta_{-}$ Ziti method has proven effective in handling strongly nonlinear PDEs across various contexts [21, 22, 23]. A key advantage of this approach is its ability to capture transitions between regular and singular behaviors without requiring finer mesh refinement, making it particularly suitable for relaxation equations with nonlinear terms. By employing the $\delta_{-}$ Ziti method, we extend its applicability to new domains while leveraging its accuracy in modeling nonlinear dynamics. Mathematically, the method is based on the classical Galerkin variational formulation, with its core step being the construction of an orthonormal basis in one dimension. This is achieved using the well-known function ${\mathcal{F}}$ defined for any $x\in\mathbb{R}$ as:

[TABLE]

Below, we outline the key steps involved in constructing the basis functions in the one-dimensional case.

A.1 Construction of the basis functions of $\delta_{-}$ Ziti method

Let $I=[a,b]$ be a given interval. For an integer $m>0$ , we define the spatial step size as $h=\frac{b-a}{m}$ , with nodes given by

[TABLE]

We them construct the family $(\varphi_{i})$ as follows:

[TABLE]

where $C$ is the normalization constant:

[TABLE]

The support of each function is given by:

[TABLE]

Moreover, the set $\left(\varphi_{i}\right)_{1\leq i\leq m+1}$ is linearly independent in $L^{2}(\rm I)$ . Using the Gram-Schmidt process, we construct the orthogonal family, denoted $\left(\tilde{\psi}_{i}\right)$ , satisfying the following relation

[TABLE]

where

[TABLE]

Then, we normalize $\tilde{\psi}_{i}(x)$ to obtain the final orthonormal basis:

[TABLE]

A.2 The $\delta_{-}$ Ziti approximation relations

The aim of $\delta_{-}$ ziti approximation involves projecting a given function $f$ onto the finite-dimensional space spanned by the $\left(\psi_{i}\right)$ , and approximating its integral accordingly. This gives

[TABLE]

where

[TABLE]

A key property of this method is its ability to express integrals in terms of the roots $r_{i}$ of the basis functions $\psi_{i}$ :

[TABLE]

These relations are crucial for constructing the numerical scheme associated with our system. The above one-dimensional framework generalizes to higher dimensions via tensorization.

A.3 Construction of the $\delta_{-}$ Ziti scheme

Now, we are ready to outline the important steps in the construction of $\delta_{-}$ ziti scheme. From the formula Eq. (19), we can approximate the propagator as follows:

[TABLE]

where

[TABLE]

The numerical scheme is constructed following these key steps:

(i)

Galerkin Projection: Multiply the given equation by a basis function and integrate over the domain $\Omega$ , following the Galerkin method. 2. (ii)

Application of the Approximation Formula: Apply the approximation formula Eq. (22) as it stands. 3. (iii)

Substituting the Approximation Formula: Injecting the approximation formula Eq. (23) into the resulting equation from the previous step, we obtain:

[TABLE] 4. (iv)

Temporal Discretization: The time interval $[0,T]$ is divided into $N$ uniform sub-intervals $[t^{n},t^{n+1}]$ with a time step $\Delta t=t^{n+1}-t^{n}$ for $n=0,\dots,N-1$ . Denoting $a^{n}_{ij}$ as the approximation of $a_{ij}(t)$ at $t=t^{n}$ , we use a semi-discrete time approximation:

[TABLE]

For the remaining terms, we extend the use of formula Eq. (22) to the bi-dimensional case, allowing us to approximate each term and construct the final numerical scheme.

Bibliography30

The reference list from the paper itself. Each links out to its DOI / PubMed record.

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TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Taxonomy

Distinct Spatiotemporal Dynamics of Thermoelectric Transport Across Superconducting Transition

Abstract

I Introduction

II Theoretical results of Type-I relaxation equation

II.1 The Type-I relaxation equation with magnetic field.

II.2 The relaxation and correlation with thermal fluctuations

III Numerical results of the linear type-I relaxation equation

IV Numerical results of the nonlinear Type-II relaxation equation

V Conclusion

VI Acknowledgments

Appendix A Overview of the δ−\delta_{-}δ−​Ziti method

A.1 Construction of the basis functions of δ−\delta_{-}δ−​Ziti method

A.2 The δ−\delta_{-}δ−​Ziti approximation relations

A.3 Construction of the δ−\delta_{-}δ−​ Ziti scheme

Appendix A Overview of the $\delta_{-}$ Ziti method

A.1 Construction of the basis functions of $\delta_{-}$ Ziti method

A.2 The $\delta_{-}$ Ziti approximation relations

A.3 Construction of the $\delta_{-}$ Ziti scheme