Capturing non-Markovian dynamics in non-equilibrium stochastic systems using flow matching

# Capturing non-Markovian dynamics in non-equilibrium stochastic systems using flow matching
Source: [https://arxiv.org/html/2606.06658](https://arxiv.org/html/2606.06658)
Bhargav Sriram Siddani Lawrence Berkeley National Laboratory bsiddani@lbl\.gov &John B\. Bell Lawrence Berkeley National Laboratory jbbell@lbl\.gov &Alejandro L\. Garcia San Jose State University alejandro\.garcia@sjsu\.edu &Ishan Srivastava Lawrence Berkeley National Laboratory isriva@lbl\.gov

###### Abstract

Hydrodynamic models of stochastic particle systems represented by coarse\-grained stochastic partial differential equations \(SPDE\), such as the regularized Dean\-Kawasaki \(DK\) equation, do not accurately capture the short\-time system dynamics that is dominated by non\-Markovian effects, and low particle density regimes where the distributions are highly non\-Gaussian\. We develop a generative flow matching method that directly models the probability distribution of fluxes from particle simulations that explicitly incorporates non\-Markovian and non\-Gaussian effects\. As a demonstration, we use this method to simulate the Kramers first passage time problem for a system of non\-interacting Brownian particles\. We show the model accurately captures the short\-time behavior and provides better predictions of the statistical moments of the number density when compared against the solution of the Markovian baseline, regularized DK equation\.

## 1Introduction

A wide range of phenomena in areas as diverse as fluid dynamics, chemistry, biology and the social sciences can be described by stochastic particle systems\. For small to modest sized systems, the dynamics can be modeled directly using particle\-based simulations, such as molecular dynamics\. However, for very large systems, direct simulation of particle dynamics becomes prohibitively expensive\. One approach to reducing the computational cost of many\-particle system simulations, originally introduced by DeanDean \([1996](https://arxiv.org/html/2606.06658#bib.bib33)\)and KawasakiKawasaki \([1994](https://arxiv.org/html/2606.06658#bib.bib34)\), is to coarse\-grain the particle dynamics to obtain a stochastic partial differential equation \(SPDE\) for the number density, similar to the formulation of the dynamic density functional theory\. Development and analysis of these types of coarse\-grained SPDE models has been an active area of research for the past several decades\. SeeIllien \([2025](https://arxiv.org/html/2606.06658#bib.bib24)\); Te Vrugt and Wittkowski \([2023](https://arxiv.org/html/2606.06658#bib.bib25)\)for recent reviews of work in this area\.

Here, we consider a representative system ofNNnon\-interacting Brownian particles under the influence of an external potential,V​\(x\)V\(x\)\. In the overdamped limit, the dynamics of the position,𝑿i​\(t\)\\bm\{X\}\_\{i\}\(t\), of theit​hi^\{th\}particle is given by the stochastic ODE

d​𝑿i=−μ​∇V​\(𝑿i\)​d​t\+d​𝑩ti,i=1,2,…,N,d\\bm\{X\}\_\{i\}=\-\\mu\\nabla V\(\\bm\{X\}\_\{i\}\)dt\+d\\bm\{B\}\_\{t\}^\{i\},\\,i=1,2,\.\.\.,N,\(1\)whereμ\\muis a Stokes mobility and the𝑩t\\bm\{B\}\_\{t\}are independent Brownian motions\. An SPDE that describes the evolution of the particles can be formally derived using Itô’s formulaDean \([1996](https://arxiv.org/html/2606.06658#bib.bib33)\)but the resulting system, which includes the divergence of space\-time white noise, is highly irregular\. Typically, a coarse\-grained regularized version of the SPDE is introduced by incorporating a high\-frequency cutoff to the white noise\. This results in a hydrodynamic equation for number density,qq, of the form

d​q=∇⋅12​∇q​d​t\+∇⋅q​d​𝑾M\+∇⋅\(μ​q​∇V\)​d​t,dq=\\nabla\\cdot\\frac\{1\}\{2\}\\nabla q\\,dt\+\\nabla\\cdot\\sqrt\{q\}\\,d\\bm\{W\}^\{M\}\\,\+\\nabla\\cdot\\left\(\\mu\\,q\\nabla V\\right\)dt\\,,\(2\)whered​𝑾Md\\bm\{W\}^\{M\}represents the regularized noise\. Here we regularize the noise by discretizing Eq\. \([2](https://arxiv.org/html/2606.06658#S1.E2)\) on a one\-dimensional finite volume mesh using an Euler\-Maruyama discretization that gives

qjn\+1−qjnΔ​t=−\(𝑭¯j\+1/2B−𝑭¯j−1/2B\)Δ​x−\(𝑭~j\+1/2B−𝑭~j−1/2B\)Δ​x−\(𝑭j\+1/2V−𝑭j−1/2V\)Δ​x\\frac\{q\_\{j\}^\{n\+1\}\-q\_\{j\}^\{n\}\}\{\\Delta t\}=\-\\frac\{\(\\bar\{\\bm\{F\}\}^\{B\}\_\{j\+1/2\}\-\\bar\{\\bm\{F\}\}^\{B\}\_\{j\-1/2\}\)\}\{\\Delta x\}\-\\frac\{\(\\tilde\{\\bm\{F\}\}^\{B\}\_\{j\+1/2\}\-\\tilde\{\\bm\{F\}\}^\{B\}\_\{j\-1/2\}\)\}\{\\Delta x\}\-\\frac\{\(\{\\bm\{F\}\}^\{V\}\_\{j\+1/2\}\-\{\\bm\{F\}\}^\{V\}\_\{j\-1/2\}\)\}\{\\Delta x\}\(3\)where the numerical fluxes are given by

𝑭¯j\+1/2B=−qj\+1n−qjn2​Δ​x,𝑭~j\+1/2B=−A​\(qjn,qj\+1n\)​Zj\+1/2nΔ​x​Δ​t,𝑭j\+1/2V=−qjn\+qj\+1n2​μ​V′​\(xj\+1/2\)\.\\bar\{\\bm\{F\}\}^\{B\}\_\{j\+1/2\}=\-\\frac\{q\_\{j\+1\}^\{n\}\-q\_\{j\}^\{n\}\}\{2\\Delta x\},\\;\\tilde\{\\bm\{F\}\}^\{B\}\_\{j\+1/2\}=\-A\(q\_\{j\}^\{n\},q\_\{j\+1\}^\{n\}\)\\frac\{Z\_\{j\+1/2\}^\{n\}\}\{\\sqrt\{\\Delta x\\Delta t\}\},\\,\{\\bm\{F\}\}^\{V\}\_\{j\+1/2\}=\-\\frac\{q\_\{j\}^\{n\}\+q\_\{j\+1\}^\{n\}\}\{2\}\\mu V^\{\\prime\}\(x\_\{j\+1/2\}\)\.HereZj\+1/2nZ\_\{j\+1/2\}^\{n\}are normally distributed random numbers, and

A​\(q1,q2\)=max​\(q1,0\)\+max​\(q2,0\)2A\(q\_\{1\},q\_\{2\}\)=\\sqrt\{\\frac\{\\mathrm\{max\}\(q\_\{1\},0\)\+\\mathrm\{max\}\(q\_\{2\},0\)\}\{2\}\}approximatesq\\sqrt\{q\}at cells faces while avoiding numerical issues ifqqbecomes negative\. Further details regarding the regularization and finite\-volume discretization are discussed inDjurdjevacet al\.\([2025](https://arxiv.org/html/2606.06658#bib.bib4)\)\.

The regularized DK equation has been shown to correctly predict long\-time system statistics, compared to particle simulations, when the number of particles is sufficiently large in every finite\-volume cell\. However, there are significant differences in statistics when there are regions with low particle countsDjurdjevacet al\.\([2025](https://arxiv.org/html/2606.06658#bib.bib4)\)\. Moreover, it is now well\-known that the dynamics of such diffusive, stochastic systems are significantly influenced by memory effects where the initial conditions impact the evolution of the system trajectories at early timesDerrida and Gerschenfeld \([2009](https://arxiv.org/html/2606.06658#bib.bib27)\); Leibovich and Barkai \([2013](https://arxiv.org/html/2606.06658#bib.bib28)\); Banerjeeet al\.\([2022](https://arxiv.org/html/2606.06658#bib.bib26)\)\. Classical Mori\-Zwanzig theory from statistical physics predicts coarse\-graining to introduce a memory term into the dynamics \(cf\.Grabert \([2006](https://arxiv.org/html/2606.06658#bib.bib3)\)\)\. In the present context, the memory reflects the sensitivity of the coarse\-graining of the particles dynamics to initial conditions, which is lost in the regularized coarse\-grained DK setting\.

## 2Background

In the present work, we alleviate the shortcomings of the regularized DK equation through data\-driven learning for the Brownian diffusion flux,𝑭B≡𝑭¯B\+𝑭~B\\bm\{F\}^\{B\}\\equiv\\bar\{\\bm\{F\}\}^\{B\}\+\\tilde\{\\bm\{F\}\}^\{B\}\. Specifically we investigate the impact of including memory on the predictive capability of the machine learning \(ML\) model\. Recent studies have used machine learning to model stochastic processes in dynamical systems\. For example, diffusion models were used to learn Lagrangian trajectories of particles in turbulenceLiet al\.\([2024a](https://arxiv.org/html/2606.06658#bib.bib20),[b](https://arxiv.org/html/2606.06658#bib.bib21)\), and a statistics\-informed neural network \(SINN\) frameworkZhuet al\.\([2023](https://arxiv.org/html/2606.06658#bib.bib22)\)was developed to learn non\-Markovian stochastic dynamics using recurrent neural networks trained with statistics\-based loss functions\. Furthermore, machine learning methods have been used to learn and parameterize the memory kernels of non\-Markovian dynamics in the context of generalized Langevin equationsXieet al\.\([2024](https://arxiv.org/html/2606.06658#bib.bib29)\); Sheet al\.\([2023](https://arxiv.org/html/2606.06658#bib.bib30)\); Bassiet al\.\([2024](https://arxiv.org/html/2606.06658#bib.bib31)\)\.

## 3Method

We define the current,J​\(𝒙,t\)J\(\\bm\{x\},t\), as the net number of particles crossing a face between two adjacent computational cells from time stepttto stept\+Δ​tt\+\\Delta t, and which is related to the flux𝑭\\bm\{F\}asJ=\(𝑭⋅𝑨\)​Δ​tJ=\\left\(\\bm\{F\}\\cdot\\bm\{A\}\\right\)\\Delta t, where𝑨\\bm\{A\}is the face area\. The probability density function \(PDF\) ofJJdepends on the number of particles in the cells to the left and right of the face,NLN\_\{L\}andNRN\_\{R\}respectively, and exhibits non\-Gaussian properties when either value is lowDjurdjevacet al\.\([2025](https://arxiv.org/html/2606.06658#bib.bib4)\)\. Furthermore, the currentJ​\(𝒙,t\)J\(\\bm\{x\},t\)is also non\-Markovian, with probability distribution that depends on the history of the systemDerrida and Gerschenfeld \([2009](https://arxiv.org/html/2606.06658#bib.bib27)\); Leibovich and Barkai \([2013](https://arxiv.org/html/2606.06658#bib.bib28)\); Banerjeeet al\.\([2022](https://arxiv.org/html/2606.06658#bib.bib26)\)\. However, the stochastic flux,𝑭~B\\tilde\{\\bm\{F\}\}^\{B\}, used in the regularized DK equation \(Eq\.[2](https://arxiv.org/html/2606.06658#S1.E2)\) is a Gaussian distribution that is uncorrelated in space and time\. Therefore, we use generative modeling based on the flow matching \(FM\) frameworkLipmanet al\.\([2023](https://arxiv.org/html/2606.06658#bib.bib7),[2024](https://arxiv.org/html/2606.06658#bib.bib8)\); Maet al\.\([2024](https://arxiv.org/html/2606.06658#bib.bib9)\)to approximate the non\-Markovian and non\-GaussianJ​\(𝒙,t\)J\(\\bm\{x\},t\)distribution from particle simulations to accurately predict short\-time dynamics\.

### Flow matching overview

Flow matching builds a probability pathpτ,0≤τ≤1,p\_\{\\tau\},\\,0\\leq\\tau\\leq 1,from a known source distributionp0=sp\_\{0\}=sto the target distributionp1=rp\_\{1\}=r\. The goal of flow matching is to learn the parametersθ\\thetaof a velocity field neural network,uτθu\_\{\\tau\}^\{\\theta\}\. Once trained, the velocity fielduτθu\_\{\\tau\}^\{\\theta\}can be used to convert a sample from the source distributionssto a sample from the target distributionrrby solving the ordinary differential equation \(ODE\) fromτ=0\\tau=0toτ=1\\tau=1\. The neural networkuτθu\_\{\\tau\}^\{\\theta\}is trained using the conditional flow matching loss,

ℒCFM\(θ\)=𝔼τ,Zτ,Z1\|\|uτθ\(Zτ\)−uτ\(Zτ\|Z1\)\|\|2,\\mathcal\{L\}\_\{\\text\{CFM\}\}\(\\theta\)=\\mathbb\{E\}\_\{\\tau,Z\_\{\\tau\},Z\_\{1\}\}\|\|u\_\{\\tau\}^\{\\theta\}\(Z\_\{\\tau\}\)\-u\_\{\\tau\}\(Z\_\{\\tau\}\|Z\_\{1\}\)\|\|^\{2\},\(4\)whereτ∼U​\[0,1\]\\tau\\sim U\[0,1\]\(uniform distribution\),Z0∼sZ\_\{0\}\\sim s,Z1∼rZ\_\{1\}\\sim r,Zτ=aτ​Z0\+bτ​Z1Z\_\{\\tau\}=a\_\{\\tau\}Z\_\{0\}\+b\_\{\\tau\}Z\_\{1\}, anduτ​\(Zτ\|Z1\)=a˙τ​Z0\+b˙τ​Z1u\_\{\\tau\}\(Z\_\{\\tau\}\|Z\_\{1\}\)=\\dot\{a\}\_\{\\tau\}Z\_\{0\}\+\\dot\{b\}\_\{\\tau\}Z\_\{1\}is called conditional velocity field\.

### Problem specific details

We use flow matching to learn the PDF ofJ​\(𝒙,t\)J\(\\bm\{x\},t\)conditioned upon the history of the local state of the system fromkkprevious time steps, i\.e\.,\(𝑵Lk,𝑵Rk,𝑱k\)\(\\bm\{N\}\_\{L\}^\{k\},\\bm\{N\}\_\{R\}^\{k\},\\bm\{J\}^\{k\}\), where

𝑵Lk=\{NL​\(𝒙,t\),NL​\(𝒙,t−Δ​t\),…,NL​\(𝒙,t−k​Δ​t\)\},\\displaystyle\\bm\{N\}\_\{L\}^\{k\}=\\\{N\_\{L\}\(\\bm\{x\},t\),N\_\{L\}\(\\bm\{x\},t\-\\Delta t\),\.\.\.,N\_\{L\}\(\\bm\{x\},t\-k\\Delta t\)\\\},𝑵Rk=\{NR​\(𝒙,t\),NR​\(𝒙,t−Δ​t\),…,NR​\(𝒙,t−k​Δ​t\)\},\\displaystyle\\bm\{N\}\_\{R\}^\{k\}=\\\{N\_\{R\}\(\\bm\{x\},t\),N\_\{R\}\(\\bm\{x\},t\-\\Delta t\),\.\.\.,N\_\{R\}\(\\bm\{x\},t\-k\\Delta t\)\\\},𝑱k=\{J​\(𝒙,t−Δ​t\),J​\(𝒙,t−2​Δ​t\),…,J​\(𝒙,t−k​Δ​t\)\}\.\\displaystyle\\bm\{J\}^\{k\}=\\\{J\(\\bm\{x\},t\-\\Delta t\),J\(\\bm\{x\},t\-2\\Delta t\),\.\.\.,J\(\\bm\{x\},t\-k\\Delta t\)\\\}\.Therefore,uτθu\_\{\\tau\}^\{\\theta\}takes\(𝑵Lk,𝑵Rk,𝑱k\)\(\\bm\{N\}\_\{L\}^\{k\},\\bm\{N\}\_\{R\}^\{k\},\\bm\{J\}^\{k\}\)as additional inputs to capture the non\-Markovian effects ofJ​\(𝒙,t\)J\(\\bm\{x\},t\)\. To capture the large tails of the target distribution, we chose the source distributionssto be the Student’s t\-distribution with a zero mean, unity scale, and a value of 4 for degrees of freedom,aτ=cos⁡\(12​π​τ\)a\_\{\\tau\}=\\cos\{\(\\frac\{1\}\{2\}\\pi\\tau\)\}andbτ=sin⁡\(12​π​τ\)b\_\{\\tau\}=\\sin\{\(\\frac\{1\}\{2\}\\pi\\tau\)\}\. Each mini\-batch of training samplesZ1Z\_\{1\}was generated from a one\-dimensional, equilibrium distribution of Brownian walkers across 186 computational cells, where the average number of particles per cell varied between11and5050, and the history lengthkkwas sampled uniformly between0and1010\. The neural network architecture uses two separate branches depending onkk\. A DeepONet\-based architectureLuet al\.\([2021](https://arxiv.org/html/2606.06658#bib.bib6)\)was used fork=0k=0, whereas a Transformer\-based architectureVaswaniet al\.\([2017](https://arxiv.org/html/2606.06658#bib.bib5)\)was used fork\>0k\>0\.

### Statistical reflection symmetry

Symmetry of particle current dictates that the PDF ofJ\(𝒙,t,NL=a,NR=b\)J\(\\bm\{x\},t,N\_\{L\}=a,N\_\{R\}=b\)is the same as the PDF of−J\(𝒙,t,NL=b,NR=a\)\-J\(\\bm\{x\},t,N\_\{L\}=b,N\_\{R\}=a\)\. In the FM model this implies that

uτθ​\(Zτ,𝑵Lk,𝑵Rk,𝑱k\)=−uτθ​\(−Zτ,ℛ​\(𝑵Lk\),ℛ​\(𝑵Rk\),−𝑱k\),u\_\{\\tau\}^\{\\theta\}\(Z\_\{\\tau\},\\bm\{N\}\_\{L\}^\{k\},\\bm\{N\}\_\{R\}^\{k\},\\bm\{J\}^\{k\}\)=\-u\_\{\\tau\}^\{\\theta\}\(\-Z\_\{\\tau\},\\mathcal\{R\}\(\\bm\{N\}\_\{L\}^\{k\}\),\\mathcal\{R\}\(\\bm\{N\}\_\{R\}^\{k\}\),\-\\bm\{J\}^\{k\}\),\(5\)whereℛ​\(𝑵Lk\)=𝑵Rk\\mathcal\{R\}\(\\bm\{N\}\_\{L\}^\{k\}\)=\\bm\{N\}\_\{R\}^\{k\}, andℛ​\(𝑵Rk\)=𝑵Lk\\mathcal\{R\}\(\\bm\{N\}\_\{R\}^\{k\}\)=\\bm\{N\}\_\{L\}^\{k\}\. This property was enforced by reconstructinguτθu\_\{\\tau\}^\{\\theta\}as:

uτθ​\(Zτ,𝑵Lk,𝑵Rk,𝑱k\)=0\.5​\[vτθ​\(Zτ,𝑵Lk,𝑵Rk,𝑱k\)−vτθ​\(−Zτ,ℛ​\(𝑵Lk\),ℛ​\(𝑵Rk\),−𝑱k\)\],u\_\{\\tau\}^\{\\theta\}\(Z\_\{\\tau\},\\bm\{N\}\_\{L\}^\{k\},\\bm\{N\}\_\{R\}^\{k\},\\bm\{J\}^\{k\}\)=0\.5\\left\[v\_\{\\tau\}^\{\\theta\}\(Z\_\{\\tau\},\\bm\{N\}\_\{L\}^\{k\},\\bm\{N\}\_\{R\}^\{k\},\\bm\{J\}^\{k\}\)\-v\_\{\\tau\}^\{\\theta\}\\left\(\-Z\_\{\\tau\},\\mathcal\{R\}\(\\bm\{N\}\_\{L\}^\{k\}\),\\mathcal\{R\}\(\\bm\{N\}\_\{R\}^\{k\}\),\-\\bm\{J\}^\{k\}\\right\)\\right\],\(6\)wherevτθv\_\{\\tau\}^\{\\theta\}has the optimizable parameters\.

## 4Results

We tested our numerical method on a variant of the Kramers first passage time problemKramers \([1940](https://arxiv.org/html/2606.06658#bib.bib1)\)\. We simulated51205120independent realizations of a one\-dimensional system consisting of100100finite\-volume cells of sizeΔ​x=1×10−2\\Delta x=1\\times 10^\{\-2\}\. The time step for evolving both the SPDE and the Brownian particle simulation was chosen asΔ​t=3×10−6\\Delta t=3\\times 10^\{\-6\}with homogeneous Dirichlet boundary conditions at the two ends of the domain\. The dynamics includes an external potential of the formV​\(x\)=\(x−α\)2​\(x−β\)2V\(x\)=\(x\-\\alpha\)^\{2\}\(x\-\\beta\)^\{2\}, whereα=0\.3\\alpha=0\.3andβ=0\.7\\beta=0\.7\. The external potential has minima atx=0\.3x=0\.3andx=0\.7x=0\.7, and a local maximum atx=0\.5x=0\.5\. The system was initialized with10310^\{3\}total particles that were distributed in the region fromx=0\.2x=0\.2tox=0\.4x=0\.4based on an equilibrium distribution under the external potential,1𝒵​exp⁡\(−2​μ​V​\(x\)\)\\frac\{1\}\{\\mathcal\{Z\}\}\\exp\(\{\-2\\mu V\(x\)\}\), where𝒵\\mathcal\{Z\}is the normalization constant andμ=2×103\\mu=2\\times 10^\{3\}\. The primary objective in this numerical experiment was to track the rate at which particles transition from the region aroundx=αx=\\alphainto the region aroundx=βx=\\beta, across the energy barrier of heightVbV\_\{b\}atx=0\.5x=0\.5\. We computed the particle statistics in the target region aroundx=βx=\\betawhereV​\(x\)≤12​VbV\(x\)\\leq\\frac\{1\}\{2\}V\_\{b\}\.

Each system was simulated for∼3×103\\sim 3\\times 10^\{3\}steps using the three models: Brownian random\-walkers, regularized DK model, and the FM model where the SPDE flux𝑭B\\bm\{F\}^\{B\}is generated by the FM method\. To emphasize non\-Markovian effects, the simulations were run with the FM models both with a history length ofk=10k=10, and the Markovian limit without history, i\.e\.,k=0k=0\. In the FM method, a uniform 20 step Euler time discretization was used in the ODE solver for sampling the target distribution\. All the simulations were performed on CPUs, and the most expensive non\-Markovian FM model scaled well up to100100cells per CPU, and averaging 9 seconds per time step\.

Figure[1](https://arxiv.org/html/2606.06658#S4.F1)depicts independent realizations of an early\-time state of the system\. The regularized DK model results in high occurrence of nonphysical negative densities as compared to the ML models\. While the DK model well\-predicts mean particle densities in the target region, the non\-Markovian ML model more accurately captures the spatial particle distribution and also the higher\-order statistical moments \(see bottom row\), thus demonstrating significant non\-Markovian effects at early times\.

Code and data availability:The code and data are open\-source and available at the following links:[code](https://github.com/siddanib/PAI26_Submission_76)and[data](https://huggingface.co/datasets/bsiddani/PAI26_Submission_76)\.

![Refer to caption](https://arxiv.org/html/2606.06658v1/figures/Composite_image_cool.jpg)Figure 1:The top row shows a subset of 100 independent one\-dimensional realizations \(along vertical direction\) at timet=5\.1×10−3t=5\.1\\times 10^\{\-3\}simulated using Brownian random walkers, the regularized Dean–Kawasaki \(DK\) model, and the two flow matching \(FM\) models: Markovian and non\-Markovian\. The white patches denote negative values in the simulation\. The bottom row compares the time evolution of the statistics of particle count in the target region aroundx=0\.7x=0\.7\.
## 5Conclusion

In this work, we used a generative flow matching framework to model non\-Gaussian and non\-Markovian effects \(i\.e\., memory\) in coarse\-grained stochastic partial differential equations that represent stochastic particle systems\. Using non\-interacting Brownian particles as an example, we showed that the proposed approach more accurately captures short\-time dynamics than Markovian models such as the DK equation\. Furthermore, our flow matching based method reproduced higher\-order statistics reasonably well at early times of the system evolution\. However, at later times, the model predictions began to deviate significantly from the particle simulations, with the error increasing with time\. While the current methodology is more computationally expensive than non\-interacting Brownian dynamics simulations, it is expected to be significantly more efficient for stochastic particle systems with complex, long\-ranged interactions, where costly neighbor\-list construction and time step constraints incur significant computational costs\.

## Acknowledgments and Disclosure of Funding

This work was supported by the LDRD program of Lawrence Berkeley National Laboratory under U\.S\. DOE Contract No\. DE\-AC02\-05CH11231, and by the U\.S\. DOE, Office of Science \(SC\), Office of ASCR, Applied Mathematics Program under Contract No\. DE\-AC02\-05CH11231\. This research used the resources of the NERSC, a DOE\-SC User Facility, under Contract No\. DE\-AC02\-05CH11231\.

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