Large deviations conditioned on large deviations II: Fluctuating hydrodynamics

TL;DR

This paper investigates the large deviation behavior of density fluctuations conditioned on current in diffusive particle systems, deriving a microscopic large deviation function and connecting it to fluctuating hydrodynamics via Hamilton-Jacobi equations.

Contribution

It provides a microscopic calculation of the conditioned large deviation function and links it to a fluctuating hydrodynamics framework through Hamilton-Jacobi equations.

Findings

Derived the conditioned large deviation function for density fluctuations.

Connected microscopic results to fluctuating hydrodynamics via Hamilton-Jacobi equations.

Applicable to general diffusive systems with boundary reservoirs.

Abstract

For diffusive many-particle systems such as the SSEP (symmetric simple exclusion process) or independent particles coupled with reservoirs at the boundaries, we analyze the density fluctuations conditioned on current integrated over a large time. We determine the conditioned large deviation function of density by a microscopic calculation. We then show that it can be expressed in terms of the solutions of Hamilton-Jacobi equations, which can be written for general diffusive systems using a fluctuating hydrodynamics description.

Equations

$${\partial }_t\rho (x,t)=-{\partial }_{x}j(x,t)\text{with}j(x,t)=-D(\rho (x,t)){\partial }_x\rho (x,t)+\eta (x,t)$$

$$⟨\eta (x,t)\eta (x^{\prime },t^{\prime })⟩=\frac{1}{L}\sigma (\rho (x,t))\delta (x-x^{\prime })\delta (t-t^{\prime })$$

$$P(Q_T=qT)\sim e^{-TL\Phi (q)}\text{and}P[\rho (x)]\sim e^{-L\mathcal{F}[\rho (x)]}$$

$$Q_T^{(\alpha )}={\int }_0^{1}dx\alpha (x){\int }_0^{T}dtj(x,t)\text{with}{\int }_0^{1}dx\alpha (x)=1.$$

$${\mathcal{P}}_t^{(\alpha )}[\rho (x)|Q=qT]\sim e^{-L{\psi }_t^{(\alpha )}[\rho (x),q]}$$

$${\psi }_t^{(\alpha )}[\rho (x),q]\equiv {\psi }_{\text{qs}}[\rho (x),q]={\psi }_{\text{left}}^{(\alpha )}[\rho (x),q]+{\psi }_{\text{right}}^{(\alpha )}[\rho (x),q]$$

$${\psi }_{\text{left}}^{(\alpha )}[\rho (x),q]=$$

$${\psi }_{\text{right}}^{(\alpha )}[\rho (x),q]=$$

$${\partial }_t\rho (x,t)={\partial }_x\left\{D(\rho ){\partial }_x\rho -\sigma (\rho )\left({\Phi }^{\prime }(q)\alpha (x)+{\partial }_x\frac{\delta {\psi }_{\text{right}}^{(\alpha )}}{\delta \rho (x,t)}\right)+\eta (x,t)\right\}$$

$${\partial }_t\rho (x,t)={\partial }_x\left\{D(\rho ){\partial }_x\rho -\sigma (\rho )\left({\Phi }^{\prime }(q)\alpha (x)-{\partial }_x\frac{\delta {\psi }_{\text{left}}^{(\alpha )}}{\delta \rho (x,t)}\right)+\eta (x,t)\right\}$$

$$\displaystyle\int_{0}^{1}dx\,\bigg{\{}\frac{\sigma(\rho)}{2}\left(\partial_{x}\frac{\delta\psi^{(\alpha)}_{\textrm{left}}}{\delta\rho(x,t)}-\Phi^{\prime}(q)\alpha(x)+\frac{D(\rho)\partial_{x}\rho}{\sigma(\rho)}\right)^{2}$$

$$={\Phi }^{\prime }(q)q-\Phi (q)$$

$$\displaystyle\int_{0}^{1}dx\,\bigg{\{}\frac{\sigma(\rho)}{2}\left(\partial_{x}\frac{\delta\psi^{(\alpha)}_{\textrm{right}}}{\delta\rho(x,t)}+\Phi^{\prime}(q)\alpha(x)-\frac{D(\rho)\partial_{x}\rho}{\sigma(\rho)}\right)^{2}$$

$$={\Phi }^{\prime }(q)q-\Phi (q)$$

$${\mathcal{Q}}_N^{(\mathbf{\lambda })}=\sum _{i=0}^L{\lambda }_i\times \text{ensurestackMath}\text{stackanchor}[1pt]\text{number of jumps from site i to i+1}\text{during the time interval [0,N] }$$

$$\sum _{i=0}^L{\lambda }_i=1.$$

$${\mathcal{M}}_{\kappa }^{(\mathbf{\lambda })}({\mathbf{n}}^{\prime },\mathbf{n})=\{\begin{array}{ll}{e}^{\kappa {\sum }_{i=0}^L{\lambda }_i{\mathcal{J}}_i\left({\mathbf{n}}^{\prime },\mathbf{n}\right)}{\mathcal{M}}_0({\mathbf{n}}^{\prime },\mathbf{n})& \text{for n′=n,}\\ -{\sum }_{{\mathbf{n}}^{\prime \prime }\ne \mathbf{n}}{\mathcal{M}}_0({\mathbf{n}}^{\prime \prime },\mathbf{n})& \text{for n′=n}\end{array}$$

$${\mathcal{J}}_i({\mathbf{n}}^{\prime },\mathbf{n})=\{\begin{array}{ll}1,& \text{if a particle jumps from i to i+1,}\\ -1,& \text{if a particle jumps from i+1 to i,}\\ 0,& \text{if no particle jumps between i and i+1.}\end{array}$$

$$\mu (\kappa )=\underset{N\to \infty }{\lim }\frac{\log ⟨e^{\kappa {\mathcal{Q}}_N^{(\mathbf{\lambda })}}⟩}{N},$$

$$\sum _{\mathbf{n}}{\mathcal{M}}_{\kappa }^{(\mathbf{\lambda })}({\mathbf{n}}^{\prime },\mathbf{n}){\mathcal{R}}^{(\kappa ,\mathbf{\lambda })}(\mathbf{n})=$$

$$\sum _{{\mathbf{n}}^{\prime }}{\mathcal{L}}^{(\kappa ,\mathbf{\lambda })}({\mathbf{n}}^{\prime }){\mathcal{M}}_{\kappa }^{(\mathbf{\lambda })}({\mathbf{n}}^{\prime },\mathbf{n})=$$

$$P_0^{(\kappa ,\mathbf{\lambda })}(\mathbf{n})={\mathcal{L}}^{(\kappa ,\mathbf{\lambda })}(\mathbf{n}){\mathcal{R}}^{(0,\mathbf{\lambda })}(\mathbf{n})$$

$$P_N^{(\kappa ,\mathbf{\lambda })}(\mathbf{n})={\mathcal{R}}^{(\kappa ,\mathbf{\lambda })}(\mathbf{n})$$

$$P_{\text{qs}}^{(\kappa ,\mathbf{\lambda })}(\mathbf{n})={\mathcal{L}}^{(\kappa ,\mathbf{\lambda })}(\mathbf{n}){\mathcal{R}}^{(\kappa ,\mathbf{\lambda })}(\mathbf{n}).$$

$${\mathcal{J}}_{i-1}({\mathbf{n}}^{\prime },\mathbf{n})-{\mathcal{J}}_i({\mathbf{n}}^{\prime },\mathbf{n})=n_i^{\prime }-n_i$$

$${\mathcal{J}}_i({\mathbf{n}}^{\prime },\mathbf{n})={\mathcal{J}}_0({\mathbf{n}}^{\prime },\mathbf{n})-\sum _{j=1}^i(n_j^{\prime }-n_j).$$

$$\left[e^{\kappa {\sum }_{i=1}^L{n}_i^{\prime }{\sum }_{j\ge i}^L{\lambda }_j}\right]{\mathcal{M}}_{\kappa }^{(\mathbf{\lambda })}({\mathbf{n}}^{\prime },\mathbf{n})\left[e^{-\kappa {\sum }_{i=1}^L{n}_i{\sum }_{j\ge i}^L{\lambda }_j}\right]=$$

$${\mathcal{R}}^{(\kappa ,\mathbf{\lambda })}(\mathbf{n})\left[e^{\kappa {\sum }_{i=1}^L{n}_i{\sum }_{j\ge i}^L{\lambda }_j}\right]=$$

$${\mathcal{L}}^{(\kappa ,\mathbf{\lambda })}(\mathbf{n})\left[e^{-\kappa {\sum }_{i=1}^L{n}_i{\sum }_{j\ge i}^L{\lambda }_j}\right]=$$

$$P_{\text{qs}}^{(\kappa ,\mathbf{\lambda })}(\mathbf{n})\equiv P_{\text{qs}}^{(\kappa )}(\mathbf{n}).$$

$${\mathcal{P}}_{\tau }^{(\mathbf{\lambda })}(\mathbf{n}|\mathcal{Q}=vN)\simeq P_{\tau }^{({\varphi }^{\prime }(v),\mathbf{\lambda })}(\mathbf{n})\text{for large time N,}$$

$$\varphi (v)=v\kappa -\mu (\kappa )\text{with }{\mu }^{\prime }(\kappa )=v.$$

$$P({\mathcal{Q}}_N^{(\mathbf{\lambda })}=vN)\sim e^{-N\varphi (v)}\text{ for large time N.}$$

$${\mathcal{Q}}_N^{(\mathbf{\lambda })}\simeq L{Q}_T^{(\alpha )}\text{for large L,}$$

Full text

††journal: J. Stat. Phys.

1 Introduction

In recent years, there has been a growing interest Jack2010 ; Orland2015 ; Hirschberg2015 ; Schutz2016 ; Popkov2011 ; Popkov2010 ; TOUCHETTE2009 ; Chetrite2015 ; Maes1999 in characterizing trajectories conditioned on rare events. Such questions appear in a wide range of topics, including protein folding Mey2014 , chemical reactions Delarue2017 ; Dykman1994 , stochastic models of gene expression Horowitz2017 , atmospheric activities Laurie2015 , glassy systems Garrahan2007 ; Garrahan2009 , disordered media Dorlas2001 , etc.. One motivation is to find efficient algorithms where the rare events become typical such that they are computationally easy to generate Orland2015 ; Delarue2017 ; Limer2018 ; Giardina2011 ; cloning . To calculate the probability of rare events one needs to understand how these rare events are created and how they relax. In these activities, a major interest concerns conditioning on an atypical value of an empirical observable, such as the integrated current, the empirical density, the entropy production, and the activity Lebowitz1999 ; TOUCHETTE2017 ; Maes2008 ; Maes20082 ; Mehl2008 ; Speck2012 ; Agranov2019 . How do the fluctuations get affected by such conditioning, and what is the effective dynamics in this conditioned ensemble? These are questions we address in this present work.

For Markov processes, in particular, for systems with few degrees of freedom and evolving according to a Langevin equation, it is known Sadhu20182 ; Jack2010 ; Jack2015 ; Chetrite2013 ; Chetrite2015 ; TOUCHETTE2017 ; Lecomte20072 ; Garrahan2007 ; Garrahan2009 ; Evans2004 how to describe the conditioned dynamics when the time window for the empirical observable is large. In this paper, we see how the same ideas Sadhu20182 can be extended to systems with many degrees of freedom.

We work here in the steady state of a one-dimensional diffusive system of length $L$ coupled with reservoirs at the boundary, as shown in Figure 1. Under a diffusive re-scaling of space $i$ and time $\tau$, defining $\left(\frac{i}{L},\frac{\tau}{L^{2}}\right)\equiv(x,t)$ for large $L$, these systems are described Bertini2014 ; Bertini2009 ; Derrida2007 ; Eyink1990 ; Spohn1991 ; Sadhu2016 by a hydrodynamic density $\rho(x,t)$ and current $j(x,t)$ whose time evolution is governed by a fluctuating hydrodynamics equation

[TABLE]

Our interest is in the statistics of the steady state density $\rho(x)$ when conditioned on the integrated current $Q_{T}$ over a time interval $[0,T]$, for large $T$. Individual statistics of $\rho(x)$ and $Q_{T}$ have already been studied Derrida2007 ; Tailleur2008 ; Bertini2001 ; Bodineau2004 ; Bodineau2005 ; Derrida2001 ; Hurtado2010 ; Hurtado2014 ; Bertini2006Current ; Bertini2007Current ; Bertini2005Current . For the conserved dynamics (1b), when the density is bounded inside the bulk, the probability of $Q_{T}$ for large $T$ is independent of where the current is measured, and $P(Q_{T})$ has a large deviation description Bodineau2004 ; Bodineau2005 ; Bertini2005Current . On the other hand, in the steady state, the probability of $\rho(x)$ has also a large deviation form when the system size $L$ is large Derrida2001 ; Derrida2007 ; Tailleur2008 ; Bertini2001 . These two probabilities are given by

[TABLE]

for $T$ and $L$ both being large, where $\Phi(q)$ and $\mathcal{F}[\rho(x)]$ are the corresponding large deviation functions. Here, the precise meaning of the symbol $\sim$ is that $\lim_{L\rightarrow\infty}\lim_{T\rightarrow\infty}\frac{\log P(Q_{T}=qT)}{L\;T}=-\Phi(q)$ and $\lim_{L\rightarrow\infty}\frac{\log P[\rho(x)]}{L}=-\mathcal{F}[\rho(x)]$.

Their joint statistics, and equivalently, the conditioned probability of $\rho(x)$ for a given value of $Q_{T}$ depends on where the latter is measured. Taking this into account, we define an empirical observable

[TABLE]

Here, $\alpha(x)$ is arbitrary except for the normalization111The normalization for $\alpha(x)$ in (3) ensures that the probability $P(Q_{T}^{(\alpha)}=qT)$ has the large deviation form (2) with the same large deviation function $\Phi(q)$, independent of $\alpha(x)$. in (3).

We denote by $\mathcal{P}_{t}^{(\alpha)}[\rho(x)|Q]$ the conditioned probability of a density profile $\rho(x)$ at time $t$ given that $Q_{T}^{(\alpha)}$ takes value $Q$ while the system is in its steady state. For large $T$ and $L$, it has the large deviation form

[TABLE]

where $\psi_{t}^{(\alpha)}[\rho(x),q]$ is the conditioned large deviation function.

Our goal is to determine this large deviation function $\psi_{t}^{(\alpha)}[\rho(x),q]$. In this paper, we will give this function at $t=0$, in the intermediate quasi-stationary state ($t\gg 1$ and $T-t\gg 1$), and at $t=T$ (indicated in Figure 2) for diffusing independent particles and for the symmetric simple exclusion process Derrida2007 . Our results for the latter are in a perturbation expansion in small density. A hydrodynamic description of these two examples is given by (1b) with $D(\rho)=1$ and $\sigma(\rho)=2\rho$ for the diffusing independent particles, and $D(\rho)=1$ and $\sigma(\rho)=2\rho(1-\rho)$ for the symmetric simple exclusion process.

We shall start by a microscopic calculation on these models where we determine the conditioned probability in terms of the left and right eigenvectors associated to the largest eigenvalue of a tilted matrix Sadhu20182 . Then, taking a hydrodynamic limit of the conditioned probability we derive $\psi_{t}^{(\alpha)}[\rho(x),q]$. For example, in the quasi-stationary state (regime III in Figure 2), we will see that

[TABLE]

up to an additive constant222We shall show that the $\alpha(x)$ dependence cancels in the expression (5) for $\psi_{\textrm{qs}}[\rho(x),q]$. In other regions of Figure 2, e.g. at $t=0$ and $t=T$, $\psi_{t}^{(\alpha)}[\rho(x),q]$ depends on $\alpha(x)$. (subscript “qs” denotes “quasi-stationary”), where $\psi_{\textrm{left}}^{(\alpha)}$ and $\psi_{\textrm{right}}^{(\alpha)}$ are related Sadhu20182 to the left and right eigenvectors of a tilted matrix (see Section 2.5 for a precise definition). We will see later that these two functions $\psi_{\textrm{left}}^{(\alpha)}[\rho(x),q]$ and $\psi_{\textrm{right}}^{(\alpha)}[\rho(x),q]$ have in fact the following physical interpretation in terms of the large deviation function defined in (4).

[TABLE]

In addition, we shall see that the conditioned dynamics for large $T$, can be effectively described by a fluctuating hydrodynamics equation with an additional driving field, which can be expressed in terms of $\psi_{\textrm{left}}^{(\alpha)}$ and $\psi_{\textrm{right}}^{(\alpha)}$. For example, in the quasi-stationary regime, the path $\rho(x,t)$ leading to a fluctuation at $t_{0}$ with $1\ll t_{0}$ and $1\ll T-t_{0}$ (illustrated in Figure 3), follows, for $t<t_{0}$,

[TABLE]

In the later part of this paper, we shall show that these results are consistent with a macroscopic approach starting from (1b) for a rather general $D(\rho)$ and $\sigma(\rho)$. In this approach, $\psi_{\textrm{left}}^{(\alpha)}$ and $\psi_{\textrm{right}}^{(\alpha)}$ are solution of the Hamilton-Jacobi equations

[TABLE]

One should note that all our results for the conditioned process are in the large $T$ limit.

We present this paper in the following order. In Section 2, we discuss the microscopic framework for analyzing the conditioned probability in a general lattice gas model and introduce conditioned large deviation function in the hydrodynamic limit. Our calculation is in a weighted ensemble, which is known Sadhu20182 ; Jack2010 ; Jack2015 ; Chetrite2013 ; Chetrite2015 ; TOUCHETTE2017 to be equivalent to the conditioned process in the large $T$ limit (through an equivalence of ensembles). We use this procedure to derive $\psi_{t}^{(\alpha)}[\rho(x),q]$ for the diffusing independent particles in Section 3, and for the symmetric simple exclusion process in Section 4. Using this microscopic approach we describe the conditioned dynamics in Section 5. In Section 6 and Section 7, we show how our expressions of the large deviation function $\psi_{t}^{(\alpha)}[\rho(x),q]$ fit with the Hamilton-Jacobi equations derived from a macroscopic approach starting from the fluctuating hydrodynamics description (1b).

2 Microscopic analysis using the tilted matrix

Let us first recall a few earlier results Sadhu20182 ; Jack2010 ; Jack2015 ; Chetrite2013 ; Chetrite2015 ; Chetrite20152 ; TOUCHETTE2017 ; Lecomte20072 ; Garrahan2009 in a Markov process conditioned on an empirical measure by writing them for the two examples: (a) diffusing independent particles and (b) the symmetric simple exclusion process. These are defined on a finite one-dimensional lattice of $L$ sites where particles jump between neighboring sites following a continuous time $\tau$ update rule (see Figure 4 and Figure 5). The jump rates at the boundary correspond to coupling to reservoirs of density $\rho_{a}$ and $\rho_{b}$ Derrida2007 . In both examples, a configuration is specified by the set of occupation variables $\mathbf{n}\equiv\{n_{1},\ldots,n_{L}\}$.

The microscopic analogue of the empirical observable (3) is

[TABLE]

where $\boldsymbol{\lambda}\equiv\{\lambda_{0},\ldots,\lambda_{L}\}$ with $\lambda_{i}$ being real valued parameters. A jump from site $i+1$ to $i$ is counted as a jump from site $i$ to $i+1$ with a negative sign. (Here, $i=0$ denotes the left reservoir and $i=L+1$ denotes the right reservoir.) Similar to the condition in (3) we consider

[TABLE]

2.1 Tilted matrix

Instead of conditioning on the value of $\mathcal{Q}_{N}^{(\boldsymbol{\lambda})}$, it is more convenient Sadhu20182 ; Jack2010 ; Jack2015 ; Chetrite2013 ; Chetrite2015 ; Chetrite20152 to work in the ensemble where events are weighted by a factor $e^{\kappa{\mathcal{Q}_{N}^{(\boldsymbol{\lambda})}}}$. By analogy Evans2004 ; Evans20042 ; Chetrite2013 ; Chetrite2015 ; TOUCHETTE2017 ; Jack2010 ; Sadhu20182 with equilibrium thermodynamics, we shall refer to this ensemble as the canonical ensemble and the ensemble where $\mathcal{Q}_{N}^{(\boldsymbol{\lambda})}$ is fixed as the micro-canonical ensemble. For large $N$, these two ensembles are equivalent (see Section 2.3).

In the canonical ensemble, we need to introduce the following tilted matrix Sadhu20182

[TABLE]

where $\mathcal{M}_{0}(\mathbf{n}^{\prime},\mathbf{n})$ is the transition rate from configuration $\mathbf{n}$ to $\mathbf{n}^{\prime}$ in the original (without condition) dynamics and for this transition,

[TABLE]

The scaled cumulant generating function of $\mathcal{Q}_{N}^{(\boldsymbol{\lambda})}$, defined by

[TABLE]

is the largest eigenvalue of $\mathcal{M}_{\kappa}^{(\boldsymbol{\lambda})}$ Sadhu20182 such that

[TABLE]

At times of our interest, namely $\tau=0$, $\tau=N$, and in the quasi-stationary regime, the probability $P_{\tau}^{(\kappa,\boldsymbol{\lambda})}(\mathbf{n})$ of a configuration $\mathbf{n}$ in the canonical ensemble can be expressed (see Sadhu20182 ; Chetrite2015 ; Roche2004 ) in terms of these eigenvectors up to normalization constants.

[TABLE]

• •

at time $\tau=0$,

[TABLE]

• •

at time $\tau=N$,

[TABLE]

• •

and in the quasi-stationary regime, i.e. $1\ll\tau$ with $1\ll N-\tau$,

[TABLE]

2.2 Dependence on $\lambda$

In our examples, the number of particles is conserved inside the bulk of the system, which means

[TABLE]

for all $1\leq i\leq L$ and equivalently

[TABLE]

This conservation of particles leads to a symmetry of the tilted matrix and its eigenvectors. To see this, we use the above relation in (11). Here, using $\sum_{i=1}^{L}\lambda_{i}\sum_{j=1}^{i}n_{j}=\sum_{i=1}^{L}n_{i}\sum_{j\geq i}^{L}\lambda_{j}$ and the normalization (10) we get

[TABLE]

where $\mathcal{M}_{\kappa}$ is the tilted matrix for the case where $\lambda_{0}=1$ and $\lambda_{i}=0$ for rest of the sites. Denoting the eigenvectors of $\mathcal{M}_{\kappa}$ as $(\mathcal{R}^{(\kappa)},\,\mathcal{L}^{(\kappa)})$, we get

[TABLE]

This gives the $\mathbf{\lambda}$-dependence of the probabilities (2.1). For example, using (15) in (14c) we see that in the quasi-stationary regime, the probability $P_{\textrm{qs}}^{(\kappa,\boldsymbol{\lambda})}$ is independent $\boldsymbol{\lambda}$ (given the normalization in (10)),

[TABLE]

2.3 An equivalence of ensembles: canonical vs. micro-canonical

For large time $N$, there is Sadhu20182 ; Jack2010 ; Jack2015 ; Chetrite2013 ; Chetrite2015 ; TOUCHETTE2017 an equivalence between the canonical ensemble and the micro-canonical ensemble, where $\mathcal{Q}_{N}^{(\boldsymbol{\lambda})}$ is fixed. In this equivalence, the conditioned probability $\mathcal{P}_{\tau}^{(\boldsymbol{\lambda})}(\mathbf{n}|\mathcal{Q})$ of a configuration $\mathbf{n}$ at time $\tau$ given $\mathcal{Q}_{N}^{(\boldsymbol{\lambda})}=\mathcal{Q}$ is related to the probability $P_{\tau}^{(\kappa,\boldsymbol{\lambda})}(\mathbf{n})$ in the canonical ensemble by

[TABLE]

where $\phi(v)$ is the Legendre transform of the eigenvalue $\mu(\kappa)$, defined by

[TABLE]

This also means Derrida2007 ; Bodineau2004 ; Bodineau2005 ; Bertini2005Current that the probability of $\mathcal{Q}_{N}^{(\boldsymbol{\lambda})}$ has a large deviation form given by

[TABLE]

In this work, our results for the micro-canonical ensemble are obtained from the canonical ensemble using the equivalence (17).

2.4 Hydrodynamic limit

Our main interest is the probability in the hydrodynamic limit, which is defined in the scaled coordinates $(x,t)\equiv(\tfrac{i}{L},\tfrac{\tau}{L^{2}})$ for large $L$, with $N=L^{2}T$ and $\lambda_{i}=\frac{1}{L}\alpha(x)$ such that

[TABLE]

with $Q_{T}^{(\alpha)}$ given in (3). In this hydrodynamic limit,

[TABLE]

and they are related (due to (18)) by

[TABLE]

This scaling (20) is well known Bodineau2004 ; Bodineau2005 ; Bertini2005Current ; Bertini2007Current and this will be confirmed in our examples.

For the probability of occupation variables $\mathbf{n}$, taking a hydrodynamic limit means Spohn1991 ; KIPNISLANDIM ; Derrida2007 coarse-graining the system over boxes of width $w$ with $1\ll w\ll L$, such that each box has a uniform density $\rho(x)$ which varies smoothly on the hydrodynamic scale $x$. If $P(\mathbf{n})$ is the probability of a microscopic configuration $\mathbf{n}$, then the probability $P[\rho(x)]$ of a hydrodynamic density profile $\rho(x)$ is

[TABLE]

where the summation is over all $\mathbf{n}$ that correspond to the profile $\rho(x)$ (see for example Section 3).

The right eigenvector $\mathcal{R}^{(\kappa,\boldsymbol{\lambda})}$ has an interpretation of a probability (see (14b)) and therefore, its hydrodynamic limit is similarly defined by

[TABLE]

Using this construction of the hydrodynamic limit in (23), we see that the probability in (2.1) leads to the probability of density $\rho(x)$, which

• •

at $t=0$ is given by

[TABLE]

• •

at $t=T$ is given by

[TABLE]

• •

and in the quasi-stationary regime ($1\ll t$ and $1\ll T-t$) is given by

[TABLE]

up to normalization constants.

• Remark:

Note that to obtain both (24a) and (24c) we have replaced the sum $\sum_{\mathbf{n}\in\rho(x)}\mathcal{L}(\mathbf{n})\mathcal{R}(\mathbf{n})$ by $\ell[\rho(x)]\sum_{\mathbf{n}\in\rho(x)}\mathcal{R}(\mathbf{n})$ for large $L$. We will see that this relation is satisfied for the two models that we consider in this paper. It is expected to remain valid for more general diffusive systems Bodineau_private .

2.5 Large deviation function

In our examples we shall see that the hydrodynamic limit of the eigenvectors have a large deviation form given by

[TABLE]

[TABLE]

with the large deviation function

[TABLE]

• •

at $t=0$,

[TABLE]

• •

at $t=T$,

[TABLE]

• •

and in the quasi-stationary regime, i.e. $t\gg 1$ and $T-t\gg 1$,

[TABLE]

where all these equalities are up to an additive constant (independent of $\rho(x)$).

• Remark:

From (15) we see that the large deviation functions in (25) have a simple dependence on $\alpha(x)$, given by

[TABLE]

3 Independent particles

In this section, we analyze the simple case of a system of independent particles with transition rates defined in Figure 4. Considering the symmetry (15) it is sufficient to analyze the case $\lambda_{0}=1$, and $\lambda_{i}=0$ for $i\geq 1$. In the rest of our analysis, we shall consider this case, unless explicitly stated otherwise.

In this case the tilted matrix (11) is

[TABLE]

where $\Omega(\mathbf{n})$ is the component of an arbitrary state vector $\boldsymbol{\Omega}$ in the configuration space.

The eigenvalue equations (13) become

[TABLE]

One can check that the right and left eigenvectors are of the form

[TABLE]

where $(a_{i},b_{i})$ are positive numbers to be determined. With this ansatz the eigenvalue equations (29) lead to

[TABLE]

[TABLE]

with $a_{0}=\rho_{a}$, $a_{L+1}=\rho_{b}$, and $b_{0}=1=b_{L+1}$. As the occupation variables $n_{i}$ are arbitrary, for the equations to be satisfied, their coefficients must vanish. This leads to a set of coupled linear equations for $a_{i}$ and $b_{i}$, which can be easily solved, and we get, for $1\leq i\leq L$,

[TABLE]

with the largest eigenvalue

[TABLE]

3.1 Hydrodynamic limit

In the hydrodynamic limit, from (32) it is easy to confirm (20), which gives

[TABLE]

For the hydrodynamic limit of the right eigenvector, defined in (23a), we decompose the system into $M=L/w$ boxes each containing $w$ sites and define

[TABLE]

where the sum is over all configurations with $\rho_{m}w$ particles in the $m$-th box. Then, using (30a) we get

[TABLE]

where $m_{i}$ denotes the site index of the $i$-th site of the $m$th box, and $\delta_{i,j}$ is the Kronecker delta. Using an identity

[TABLE]

and that $a_{i}$ is slowly varying such that $a_{i}\simeq a(\frac{mw}{L})$ when the site $i$ is in the $m$-th box, and defining $\rho_{m}=\rho(\frac{mw}{L})$ we get

[TABLE]

Then, for $1\ll w\ll L$, using the Stirling’s formula we get (following the definition (23a, 25a, 28a))

[TABLE]

with

[TABLE]

For the left eigenvector, the hydrodynamic limit (23b) is simple to construct from (30b), which (following the definition (23b, 25b, 28b)) leads to $\ell^{(\kappa)}[\rho(x)]\sim e^{-L\,V_{\textrm{left}}^{(\kappa)}[\rho(x)]}$ with

[TABLE]

• Remark:

The $\kappa=0$ corresponds to the case without a condition on the empirical observable. So, the steady state large deviation function of density $\mathcal{F}_{\textrm{ni}}[\rho(x)]=V_{\textrm{right}}^{(0)}[\rho(x)]$ is

[TABLE]

3.2 Large deviation function

Expressions for $\psi^{(\kappa,\alpha)}_{\textrm{left}}$ and $\psi^{(\kappa,\alpha)}_{\textrm{right}}$ for an arbitrary $\alpha(x)$ follow from (28). Using this result in (27) we get the large deviation function at three different times, all of which are of the form

[TABLE]

with the average density profile

at $t=0$,

[TABLE] 2. 2.

at $t=T$,

[TABLE] 3. 3.

and in the quasi-stationary regime,

[TABLE]

4 Symmetric simple exclusion process

In this section, we analyze the one-dimensional symmetric simple exclusion process with the transition rates defined in Figure 5. We indicate how to perform a low-density expansion. Our results will be limited to the first two terms in this expansion, although it is straightforward to extend our approach to higher orders. Considering the symmetry (15) we will analyze only the case $\lambda_{0}=1$, and $\lambda_{i}=0$ for $i\geq 1$.

4.1 A representation of the eigenvectors

A configuration can be specified by the position of the particles. In a configuration with $m$ occupied sites $\{i_{1},\ldots,i_{m}\}$ we denote the component of the eigenvectors as

[TABLE]

We normalize such that the component of both eigenvectors in the empty configuration is $1$.

4.2 A perturbation solution for small density

For finite $L$, the Perron-Frobenius theorem VANKAMPEN2007193 assures that the largest eigenvalue of the tilted matrix is non-degenerate. Expressions of the eigenvalue and eigenvectors exist Lazarescu_2015 ; Vanicat , but it is hard to extract from them the large scale behaviors. Here, we use a perturbation expansion in small $\rho_{a}$ and $\rho_{b}$, where it is possible to systematically solve the eigenvalue equation order by order. We write

[TABLE]

For $\rho_{a}=\rho_{b}=0$, all $\mathcal{R}_{n}^{(\kappa)}=0$ except for the empty configuration and one has $\mu_{0}(\kappa)=0$. It is also clear that at order $n$ in $\rho_{a}$ and $\rho_{b}$, the $\mathcal{R}_{n}^{(\kappa)}$ of configurations with more than $n$ occupied sites vanish. Therefore,

[TABLE]

and so on. Here we present the solution up to only the second order.

The equations one needs to solve up to the second order in $\rho_{a}$ and $\rho_{b}$ are given in the Appendix A. Since at this order $\mathcal{R}_{2}^{(\kappa)}$ of configurations with $3$ or more occupied sites vanish, the hierarchy closes. For arbitrary $L$, we get the solution

[TABLE]

[TABLE]

As the probabilities (2.1) are always a product of a left and a right eigenvector, we will only need the following orders for the left eigenvector.

[TABLE]

• Remarks:

• –

For the symmetric simple exclusion process, the eigenvalue equation can be systematically solved to arbitrary order. The “miracle” which makes an explicit solution possible in practice is that at every order, the eigenvectors are low degree polynomials of the site indices $i$. This is special to the exclusion process, and may not apply to other diffusive systems.

• –

At the first order in $\rho_{a}$ and $\rho_{b}$, the expressions of eigenvalue and eigenvectors coincide with that of the independent particles case in Section 3.

4.3 Cumulants of the occupation variable

One can then derive the cumulants of the occupation variables from the probability (2.1). For example, at time $\tau=N$, using (14b) and the representation (39) we get

[TABLE]

where the normalization $\mathcal{N}=1+\sum_{i}\mathcal{R}^{(\kappa)}(i)+\sum_{i}\sum_{j\neq i}\mathcal{R}^{(\kappa)}(i,j)+\cdots$.

Using this with the perturbation solution of the eigenvectors we can construct a perturbation expansion of the cumulants for the low density limit. For example, at the second order in $\rho_{a}$ and $\rho_{b}$, the average occupation of a site $i$ at time $\tau=N$ is

[TABLE]

4.4 Hydrodynamic limit

It is straightforward to take the hydrodynamic limit of the expressions (42) and (43), which can be used to derive the hydrodynamic limit of the cumulants of the occupation variables. For large $L$, we define

[TABLE]

and

[TABLE]

Similarly, from (42) we define

[TABLE]

and

[TABLE]

• Remark:

In taking the hydrodynamic limit of the perturbation expansion we have assumed that the limits of large $L$ and small density can be exchanged. We will see that the resulting large deviation functions are consistent with the macroscopic analysis in Section 7.2.

4.4.1 Cumulants of density

At time $\tau=0$, in the quasi-stationary regime, and at $\tau=L^{2}T$ (the hydrodynamic time $t=T$) the cumulants of the occupation variable are of the form (see (44b))

[TABLE]

for large $L$, up to the second order in $\rho_{a}$ and $\rho_{b}$, where

• •

at $t=0$,

[TABLE]

• •

in the quasi-stationary state,

[TABLE]

• •

at $t=T$,

[TABLE]

Explicit expressions of the cumulants are given in Appendix B.

4.4.2 Large deviation function

Here, we show how to derive a small density expansion of the large deviation function in (26) from the above expansion of the cumulants of the density.

Take an arbitrary probability distribution $P(\mathbf{n})$ where the occupation variables are either [math] or $1$. Then, one can write

[TABLE]

where $h_{i}$ is a real valued parameter. If one expands this product, averages over $P(\mathbf{n})$, and uses the fact that cumulants of order $k$ scale like $\rho_{a}$ and $\rho_{b}$ to the power $k$, then one gets, for finite $L$,

[TABLE]

In the hydrodynamic limit, when $h_{i}\simeq h(\frac{i}{L})$, $\left\langle n_{i}\right\rangle\simeq\bar{\rho}(\frac{i}{L})$, and $\left\langle n_{i}n_{j}\right\rangle_{c}\simeq{1\over L}c(\frac{i}{L},\frac{j}{L})$, we get for the generating functional

[TABLE]

with

[TABLE]

Then, $P[\rho(x)]$ has Derrida2007 ; TOUCHETTE2009 the large deviation form (26) with the large deviation function $\psi[\rho(x)]$ given by the Legendre transformation

[TABLE]

where $h(x)$ is the solution of $\frac{\delta\mathcal{G}[h]}{\delta h(x)}=\rho(x)$.

The small density expansion of (59, 60) is then straightforward to get from the perturbation expansion of the cumulants. It gives

[TABLE]

4.4.3 Small density expansion of $\chi(\kappa)$, $V_{\textrm{left}}^{(\kappa)}$ and $V_{\textrm{right}}^{(\kappa)}$

It is straightforward to take the hydrodynamic limit of (41), which confirms (20) at the level of a small density expansion with

[TABLE]

This is in agreement with earlier findings in Roche2004 ; Bodineau2004 .

For $\lambda_{0}=1$ and $\lambda_{i}=0$ for $i\geq 1$, the large deviation functions $\psi_{\textrm{left}}^{(\kappa,\alpha)}\equiv V_{\textrm{left}}^{(\kappa)}$ and $\psi_{\textrm{right}}^{(\kappa,\alpha)}\equiv V_{\textrm{right}}^{(\kappa)}$ (see (28)). An expression for $V_{\textrm{left}}^{(\kappa)}$ can be derived by taking the hydrodynamic limit (23b, 25b) of $\mathcal{L}^{(\kappa)}$. For a configuration $\mathbf{n}$, where the occupation variables are either [math] or $1$, we write

[TABLE]

using the representation in (39). Then, from the perturbation expansion (40c, 43) and taking the hydrodynamic limit (23b, 25b) we get

[TABLE]

where $\ell^{(\kappa)}(x)$ and $g_{\textrm{left}}^{(\kappa)}(x,y)$ are defined in (46) and (48).

In comparison, it is harder to derive an expression for $V_{\textrm{right}}^{(\kappa)}$ by taking the hydrodynamic limit of the expression (42) for $\mathcal{R}^{(\kappa)}$. It is much easier to derive using the relation (27b) and the result (54, 61) at time $T$. This gives

[TABLE]

• Remark:

We have checked (details in Appendix C) that the result (63), and the expression of $\psi_{\textrm{qs}}^{(\kappa)}$ and $\psi_{T}^{(\kappa)}$ obtained from (61) satisfy the relation (27c).

5 Effective dynamics

In the canonical ensemble, the biased dynamics is Markovian and one can write Sadhu20182 ; Chetrite2015 ; TOUCHETTE2017 the transition rates in terms of the tilted Matrix. For example, in the quasi-stationary regime, the transition rate $W_{\textrm{qs}}^{(\kappa,\boldsymbol{\lambda})}(\mathbf{n}^{\prime},\mathbf{n})$ from a microscopic configuration $\mathbf{n}$ to another configuration $\mathbf{n}^{\prime}$ is given by Sadhu20182

[TABLE]

Similarly, the path leading to a spontaneous fluctuation can be described by a time reversal of (65a). The transition rate $\mathbb{W}_{\textrm{qs}}^{(\kappa,\boldsymbol{\lambda})}(\mathbf{n}^{\prime},\mathbf{n})$ of this time-reversed process can be constructed Stroock14 using the quasi-stationary distribution (14c), which gives

[TABLE]

For the two examples considered in this paper (the independent particles and the symmetric simple exclusion process), it is straightforward to see that the effective dynamics (65a) with (11) correspond to re-weighting the jump rates of particles (see Figure 4 and Figure 5): the jump rate for a particle from site $i$ to $i+1$ is weighted by a factor $e^{E_{i}}$, whereas the jump rate from $i+1$ to $i$ is weighted by $e^{-E_{i}}$, where

[TABLE]

Hydrodynamic limit

In the large $L$ limit, when $x=\frac{i}{L}$ and $\lambda_{i}=\frac{1}{L}\alpha(x)$, (66b) becomes

[TABLE]

In a hydrodynamic description (1b), the effect of such a weak bias can be incorporated (see Bodineau2010 for another example) within the linear response theory, where $e(x)$ acts as an external driving field. This leads to the following dynamics.

[TABLE]

• Relaxation:

A spontaneous fluctuation of hydrodynamic density in the quasi-stationary state relaxes following $\partial_{t}\rho(x,t)=-\partial_{x}j(x,t)$ with

[TABLE]

where $\eta(x,t)$ is a Gaussian white noise of zero mean and covariance (1b).

A somewhat similar analysis based on (65b) leads to the following dynamics:

• Fluctuation:

The path leading to a fluctuation in the quasi-stationary state is described by $\partial_{t}\rho(x,t)=-\partial_{x}j(x,t)$ with

[TABLE]

• Remark:

The time evolution of the most probable density profile leading to a fluctuation and its subsequent relaxation are the zero noise case of (68b) and (68a). We have verified this explicitly for the independent particles starting from their microscopic dynamics. For the symmetric simple exclusion process, we checked this up to the second order in a low density expansion.

6 Macroscopic analysis

The two examples discussed in Section 3 to Section 5 are governed Bertini2002 ; Bodineau2004 ; Derrida2007 ; Sadhu2016 , for large $L$, by the fluctuating hydrodynamics equation (1b). Our goal here is to show that the large deviation functions (34, 35) and (63, 64) are consistent with a macroscopic approach. Besides this, the macroscopic analysis applies for a general class of models where the microscopic details enter in the terms $D(\rho)$ and $\sigma(\rho)$.

Much of the results can be infered by drawing an analogy of (1b) to a Langevin equation in the weak noise limit, as in Sadhu20182 . In this analogy, a simple quantity is the generating function, which for (1b) is defined by

[TABLE]

where $P^{(\alpha)}_{T}[r(x),Q|s(x)]$ is the joint probability of a density profile $\rho(x,T)=r(x)$ at the hydrodynamic time $t=T$, and $Q_{T}^{(\alpha)}$ in (3) to take value $Q$ given the density $\rho(x,0)=s(x)$ at $t=0$. Similarly to the Langevin equation Sadhu20182 , one expects, for large $L$ and $T$, the generating function to have the form

[TABLE]

where $\chi(\kappa)$, $\psi^{(\kappa,\alpha)}_{\textrm{right}}$, and $\psi^{(\kappa,\alpha)}_{\textrm{left}}$ are the same quantities as in (20) and (25). Starting from (70) we now obtain a variational formulation as in Bertini2014 ; Bertini2009 ; Bertini2001 ; Derrida2007 .

6.1 A variational formulation

For large $L$, the probability of a certain time evolution of $\rho(x,t)$ and $j(x,t)$ inside the time window $[0,T]$, which follows (1b) is given by Derrida2007 ; Bertini2014

[TABLE]

where $\sim$ means that sub-leading terms in large $L$ are neglected. Using this, the generating function (69) can be written as a path-integral

[TABLE]

with $t_{i}=0$ and $t_{f}=T$. The path integral in (72a) is over all paths $\{\rho(x,t),j(x,t)\}$ satisfying $\partial_{t}\rho=-\partial_{x}j$ with the initial density profile $\rho(x,0)=s(x)$ and the final density profile $\rho(x,T)=r(x)$.

For large $L$ and $T$, assuming a single optimal path, we get the large deviation form (70) with

[TABLE]

where the optimization is over all paths $(\rho(x,t),j(x,t))$ satisfying the conditions mentioned earlier.

In a rather general class of systems Bertini2005Current , the optimal path for (73) starts at the given density profile $\rho(x,0)=s(x)$ but soon it becomes time independent $\rho(x,t)=\bar{\rho}_{\text{qs}}(x)$ and remains at this density until only close to the final time $T$ where it changes to $\rho(x,T)=r(x)$. (This assumption for the time independence of the optimal profile for $t\gg 1$ and $T-t\gg 1$ is equivalent to assuming the additivity principle Bodineau2004 .) This is illustrated in the schematic in Figure 6. In this paper, we shall only consider situations where this scenario holds. For examples where this breaksdown see Bertini2007Current ; Bodineau2007 .

• Remarks:

The probability (71) does not include the contribution of reservoirs. One way to do it is to consider density profiles $\rho(x)$ which are fixed at the boundary, i.e. $\rho(0,t)=\rho_{a}$ and $\rho(1,t)=\rho_{b}$. This is justified due to the strong coupling with the reservoirs, so that fluctuations of density at the boundary relax to the reservoir density in a time scale much faster than the hydrodynamic time scale. 2. 2.

The formula (73) means

[TABLE]

which leads to the well-known result Bodineau2004 ; Derrida2007

[TABLE]

Their solution Bodineau2004 ; Derrida2007 for the independent particles and for the symmetric simple exclusion process are in agreement with (33) and (62).

6.2 Hamilton-Jacobi equation

In (73), the deviation of the optimal path from $\bar{\rho}_{\text{qs}}(x)$ near $t=0$ and $t=T$ (see Figure 6) are important and they contribute to $\psi^{(\kappa,\alpha)}_{\textrm{right}}$ and $\psi^{(\kappa,\alpha)}_{\textrm{left}}$. Here, we show how this variational formula (73) leads to a pair of Hamilton-Jacobi equations for $\psi^{(\kappa,\alpha)}_{\textrm{right}}$ and $\psi^{(\kappa,\alpha)}_{\textrm{left}}$.

We start by deriving the equation for $\psi^{(\kappa,\alpha)}_{\textrm{left}}$. For this, we use

[TABLE]

for $0<t<T$, which can be seen from the definition (69). We consider infinitesimal $t>0$ but large $T$, such that $T-t$ is large. This means, we can use (70) for $G_{T-t}^{(\kappa,\alpha)}[r(x),\rho(x)]$. On the other hand, using the Action formulation (72b) we write, for an infinitesimal $t$,

[TABLE]

where $\rho(x)\simeq s(x)-t\,\partial_{x}j(x)$. Using this in (76) and a saddle point analysis for large $L$ we get

[TABLE]

Expanding $\psi^{(\kappa,\alpha)}_{\textrm{left}}[s(x)-t\,\partial_{x}j(x)]$ in a Taylor series up to linear order in $t$, and then using an integration by parts, we get

[TABLE]

For density profiles which are fixed at the boundary (see the remark 1 in Section 6.1), one can see from (28, 35, 63) that

[TABLE]

for the two systems we study in this paper. Similar conditions occured already in earlier works Bertini2002 ; Bertini2014 . Using this in (78) and optimizing over $j(x)$ leads to

[TABLE]

A similar analysis (by considering small decrement around the time $T$) leads to an analogous equation for $\psi^{(\kappa,\alpha)}_{\textrm{right}}$ .

[TABLE]

These two are the Hamilton-Jacobi equations associated to the variational problem (73).

6.3 Optimal path

In (78) the optimal current

[TABLE]

This means that the optimal path $\rho_{\textrm{opt}}(x,t)$ near $t=0$ follows

[TABLE]

It is straightforward to extend the argument for $t\geq 0$ but $T-t\gg 1$ (region II of Figure 2) and show that the dynamics is the same.

A similar analysis in the derivation of (80b) shows that the optimal path $\rho_{\textrm{opt}}(x,t)$ in region IV of Figure 2 is described by

[TABLE]

In the quasi-stationary state (83) also describes the optimal path leading to a fluctuation and (82) describes the optimal path of relaxation (see illustration in Figure 3).

6.4 Fixed point of the dynamics

Using (80b) we show in Appendix D that along the optimal path (82),

[TABLE]

with $\psi_{\textrm{qs}}^{(\kappa)}$ given in (27c). Since $\sigma(\rho)$ is positive, this means $\frac{d}{dt}\psi_{\textrm{qs}}^{(\kappa)}[\rho_{\textrm{opt}}(x,t)]=0$ if and only if $\partial_{x}\frac{\delta\psi_{\textrm{qs}}^{(\kappa)}[\rho_{\textrm{opt}}(x,t)]}{\delta\rho_{\textrm{opt}}(x,t)}=0$. (The case $\kappa=0$ has been discussed earlier in Bertini2002 .)

The examples we consider here have a unique quasi-stationary density $\bar{\rho}_{\textrm{qs}}$, where $\frac{\delta\psi_{\textrm{qs}}^{(\kappa)}}{\delta\bar{\rho}_{\text{qs}}(x)}=0$. Then (84) implies that $\bar{\rho}_{\textrm{qs}}$ is an attractive fixed point of (82) (see Figure 3).

For $\bar{\rho}_{\textrm{qs}}$ the optimal current (81) is $j_{\text{opt}}=\chi^{\prime}(\kappa)$, which can be seen from (75). Then, we get

[TABLE]

which leads to

[TABLE]

These give conditions for the solution of (80b). It is well-known Goldstein2000 ; Bertini2002 that, there are multiple solutions of a Hamilton-Jacobi equation. For the two examples studied in this work, the relevant solution of (80b) follows the boundary condition (85b) and (see (79))

[TABLE]

6.5 Conditioned stochastic dynamics

In Section 5 we have shown using a microscopic analysis that, in the quasi-stationary state, conditioned dynamics is given by a fluctuating hydrodynamics equation (5). Here, we give a derivation using the macroscopic approach.

In the quasi-stationary state, if $\mathcal{P}[\rho,j|\rho_{i}]$ is the probability of a path $\left\{\rho(x,t),j(x,t)\right\}$ in a time window $[t_{i},t_{f}]$ (for $1\ll t_{i}<t_{f}$ and $1\ll T-t_{f}$) given an initial density $\rho(x,t_{i})=\rho_{i}(x)$, then using (69) and (72b) one can write

[TABLE]

where we denote $\rho(x,t_{f})=\rho_{f}(x)$ and $\rho(x,T)=\rho_{T}(x)$. Then, for large $L$, using (26) and (70) we get

[TABLE]

This expression can be simplified by using

[TABLE]

where the last equality is obtained by using $\partial_{t}\rho=-\partial_{x}j$, integration by parts, and the boundary condition (86). In addition, we use (80a) to write

[TABLE]

Comparing with (71) one can clearly see that the conditioned dynamics in the quasi-stationary state is given by a fluctuating hydrodynamics equation $\partial_{t}\rho(x,t)=-\partial_{x}j(x,t)$ with $j(x,t)$ in (68a). This describes, how a spontaneous fluctuation relaxes in the quasi-stationary state.

On the other hand, (68b) shows the path leading to a fluctuation. This is given by a time reversal of (68a), which can be constructed (for example see eq. 2.15 of Bertini2002 ) by using that $P_{\textrm{qs}}^{(\kappa,\alpha)}[\rho]$ is the steady state of (68a). This gives a fluctuating hydrodynamics equation $\partial_{t}\rho(x,t)=-\partial_{x}\left\{j(x,t)+\sigma(\rho)\partial_{x}\frac{\delta\psi_{\textrm{qs}}^{(\kappa)}}{\delta\rho(x,t)}\right\}$ with the $j(x,t)$ in (68a). Then, using (27c) one gets (68b).

7 Solution in specific examples

Here, we show how to check that the results for $\psi_{\textrm{left}}^{(\kappa,\alpha)}$ and $\psi_{\textrm{right}}^{(\kappa,\alpha)}$ derived in Section 3 and Section 4 using a microscopic analysis, are indeed solution of the Hamilton-Jacobi equations (80b).

7.1 Independent particles

In this case, using (28b) and (33) in (80a) we get

[TABLE]

Expression (35a) for $V^{(\kappa)}_{\textrm{left}}$ gives

[TABLE]

With this, the left hand side of (89) becomes

[TABLE]

From the boundary condition $\rho(0)=\rho_{a}$, $\rho(1)=\rho_{b}$, and using (35b) we see that the above expression agrees with the right hand side of (89).

Moreover, one can check that the solution (28b, 35a) is consistent with the boundary condition (85b, 86).

A similar calculation shows that $\psi_{\textrm{right}}^{(\kappa)}$ in (28a, 34a) is a solution of (80b) with the boundary condition (85b, 86).

7.2 Symmetric simple exclusion process

In this case, our solution for $\psi_{\textrm{left}}^{(\kappa,\alpha)}$ and $\psi_{\textrm{right}}^{(\kappa,\alpha)}$ in (28, 63, 64) are for small density. We have explicitly verified, up to the second order in density, that these are solutions of the Hamilton-Jacobi equations (80b) and they satisfy the boundary condition (85b, 86). The analysis is similar to that of the independent particles in Section 7.1. In fact, at the leading order in density, they are identical.

8 Summary

In the present work we have tried to determine the probability of the density (4) in a diffusive many-particle system conditioned on the time-integrated current (3) for large $T$. This is a generalization to extended systems of earlier works on conditioned stochastic processes Sadhu20182 ; Jack2015 ; Chetrite2013 ; Chetrite2015 ; TOUCHETTE2017 ; Lecomte20072 ; Garrahan2009 . We mostly worked with the canonical ensemble where dynamics is weighted by the current (3). However, the equivalence of ensembles allows to make predictions for the conditioned process (see Section 2). We give explicit results for a system of independent particles (Section 3) and the symmetric simple exclusion process (Section 4).

In the hydrodynamic limit, the conditioned probability of the density $\rho(x)$ is characterized by the large deviation function $\psi_{t}^{(\kappa,\alpha)}[\rho]$ in (26). For the two systems considered in this paper, we have calculated $\psi_{t}^{(\kappa,\alpha)}[\rho]$ at three different times of the evolution, namely, at $t=0$, at $t=T$, and in the quasi-stationary regime. These are, in general, related by (see (27))

[TABLE]

with $\mathcal{F}$ defined in (2).

In the second half of the paper, we used a macroscopic approach, where $\psi_{t}^{(\kappa,\alpha)}$ is expressed (see (27)) in terms of $\psi_{\textrm{left}}^{(\kappa,\alpha)}$ and $\psi_{\textrm{right}}^{(\kappa,\alpha)}$, which are solutions of a pair of Hamilton-Jacobi equations (80b). These solutions also act as the potential for an additional field that drives the conditioned process (see (5) and Section 6.5). Using this macroscopic approach we verified the microscopic results for the two specific examples (see Section 7).

The macroscopic formulation is expected to work for a wide class of diffusive systems where the microscopic details enter in the two functions $D(\rho)$ and $\sigma(\rho)$ in (1b). This is in the spirit of the macroscopic fluctuation theory Bertini2014 . It would be interesting find more examples for explicit solutions of $\psi_{\textrm{left}}^{(\kappa,\alpha)}$ and $\psi_{\textrm{right}}^{(\kappa,\alpha)}$ using both the microscopic and the macroscopic approach. For systems on a ring, where the unconditioned state is in equilibrium, the calculation may be simpler. However, due to the periodic boundary condition, the optimal profile in the quasi-stationary state could become time dependent (see Bertini2007Current ; Bodineau2007 ; Nicolas2018 ).

In the unconditioned case ($\kappa=0$), the spatial correlations of density follow simple differential equations Sadhu2016 ; Bertini2007Correlation , whose solutions can be formally expressed in terms of a Green’s function. It would be interesting to see if there are similar equations for the conditioned case, especially in the quasi-stationary state.

Acknowledgements.

We thank T. Bodineau for his useful comments related to (27c) and (90).

Appendix A Tilted matrix for the symmetric simple exclusion process

Here, we explicitly write the eigenvalue equation for the tilted Matrix in a symmetric simple exclusion process of arbitrary length $L$. Our results are for the case $\lambda_{0}=1$ and $\lambda_{i}=0$ for $i\geq 1$. We use the representation (39) for the eigenvectors with a normalization such that the component of both right and left eigenvectors for the empty configuration is $1$. We write the eigenvalue equation up to the two particle sector.

For the right eigenvector, defining $\mathcal{R}^{(\kappa)}(i,j)=0$ for $i=j$, we get the following set of coupled equations.

• •

Empty-particle sector.

[TABLE]

• •

Single-particle sector.

• $\circ$

For $1<i<L$,

[TABLE]

• $\circ$

For $i=1$,

[TABLE]

• $\circ$

For $i=L$,

[TABLE]

• •

Two-particle sector.

• $\circ$

For $1<i<j<L$,

[TABLE]

• $\circ$

For $1=i<j<L$,

[TABLE]

• $\circ$

For $1<i<j=L$,

[TABLE]

Similarly, for the left eigenvector, we define $\mathcal{L}^{(\kappa)}(i,j)=0$ for $i=j$, and get the following set of equations.

• •

Empty particle sector.

[TABLE]

• •

Single particle sector.

• $\circ$

For $1<i<L$,

[TABLE]

• $\circ$

For $i=1$,

[TABLE]

• $\circ$

For $i=L$,

[TABLE]

• •

Two particle sector.

• $\circ$

For $1<i<j<L$,

[TABLE]

• $\circ$

For $1=i<j<L$,

[TABLE]

• $\circ$

For $1<i<j=L$,

[TABLE]

It is easy to see a pattern in the equations and using this we can write the equations for an arbitrary particle sector. It is then possible to systematically solve the equations order by order using the perturbation expansion in small density as given in Section 4.2.

Appendix B Cumulants for the symmetric simple exclusion process

We write the cumulants of density using a set of parameters

[TABLE]

which was used earlier Roche2004 . In terms of these, the cumulants in (54) have the following expression.

• •

At $t=0$,

[TABLE]

• •

In the quasi-stationary state,

[TABLE]

• •

at $t=T$,

[TABLE]

Appendix C Small density expansion of $V_{\textrm{left}}^{(\kappa)}$

Here, we show that the expression of $V_{\textrm{left}}^{(\kappa)}$ in (63) agrees with $\psi_{\textrm{qs}}^{(\kappa)}-\psi_{T}^{(\kappa)}$, up to an additive constant, as expected from (27c). To see this, we use (61a) and we get

[TABLE]

where the subscripts $T$ and qs refer to the cumulants (54) at time $t=T$ and in the quasi-stationary state. (In the third line of the above equation we used that $c_{T}(x,y)$ and $c_{\textrm{qs}}(x,y)$ are symmetric under exchange of $x$ and $y$.) The terms in the first line are constant and therefore ignored. For the rest of the terms we use

[TABLE]

up to the second order in $\rho_{a}$ and $\rho_{b}$, which can be seen from (56) and (57).

Then, it is straightforward to see that, up to the second order in $\rho_{a}$ and $\rho_{b}$,

[TABLE]

and

[TABLE]

This shows that the expression in (95) agrees with $V_{\textrm{left}}^{(\kappa)}$ in (63), up to an additive constant.

Appendix D A H-theorem

To derive (84) we write, along the optimal path (82),

[TABLE]

Then, using (82) and an integration by parts we get

[TABLE]

where we use $\frac{\delta\psi_{\text{qs}}^{(\kappa,\alpha)}}{\delta\rho_{\textrm{opt}}(x,t)}=0$ at the boundary $x=0$ and $1$ (due to (27c) and (86)). To simplify the expression, we use

[TABLE]

which is obtained by subtracting (80b) from (80a) and then using (27c). From the above two equations we get (84).