# Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-
Presutti Model

Eldad Bettelheim<EMAIL_ADDRESS>Racah Institute of Physics,
Hebrew University of Jerusalem, Jerusalem 91904, Israel  Naftali R. Smith
<EMAIL_ADDRESS>Department of Solar Energy and Environmental Physics,
Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev,
Sede Boqer Campus, 8499000, Israel  Baruch Meerson<EMAIL_ADDRESS>Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904,
Israel

###### Abstract

We investigate non-stationary heat transfer in the Kipnis-Marchioro-Presutti
(KMP) lattice gas model at long times in one dimension when starting from a
localized heat distribution. At large scales this initial condition can be
described as a delta-function, $u(x,t=0)=W\delta(x)$. We characterize the
process by the heat, transferred to the right of a specified point $x=X$ by
time $T$,

$J=\int_{X}^{\infty}u(x,t=T)\,dx\,,$

and study the full probability distribution $\mathcal{P}(J,X,T)$. The
particular case of $X=0$ has been recently solved [Bettelheim et al. Phys.
Rev. Lett. 128, 130602 (2022)]. At fixed $J$, the distribution $\mathcal{P}$
as a function of $X$ and $T$ has the same long-time dynamical scaling
properties as the position of a tracer in a single-file diffusion. Here we
evaluate $\mathcal{P}(J,X,T)$ by exploiting the recently uncovered complete
integrability of the equations of the macroscopic fluctuation theory (MFT) for
the KMP model and using the Zakharov-Shabat inverse scattering method. We also
discuss asymptotics of $\mathcal{P}(J,X,T)$ which we extract from the exact
solution, and also obtain perturbatively directly from the MFT equations.

###### Contents

  1. I Introduction
  2. II Formulation of the MFT problem
  3. III Solution of the MFT problem
    1. III.1 Scattering amplitudes and their dynamics
    2. III.2 Solving the scattering problem at $t=0$ and $t=1$
    3. III.3 Recovering the rate function $s(J,X)$ from the scattering amplitudes
      1. III.3.1 Full solution for $X=0$, and for $X>0$ with sufficiently small $\lambda$
      2. III.3.2 Solution for $X<\sqrt{8}$ at all $\lambda$
      3. III.3.3 Solution for $X>\sqrt{8}$ at all $\lambda$
  4. IV Asymptotic limits
    1. IV.1 $\lambda\ll 1$: linear theory
    2. IV.2 $\lambda\gg 1$, $X\gg 1$
      1. IV.2.1 From adiabatic perturbation theory
      2. IV.2.2 From exact rate function
  5. V Discussion

##  I Introduction

The Kipnis-Marchioro-Presutti (KMP) model KMP  is an important member of a
family of simple continuous-time Markov processes in space, which describe
transport of matter or energy and are called diffusive lattice gases Spohn ;
KipnisLandim . The KMP model is considered a prototypical model of energy
transfer by diffusion: not in the least because, for this model, the Fourier’s
law of heat diffusion at a coarse-grained level was proven rigorously KMP .
The model’s definition is quite simple: There are immobile agents occupying
the whole lattice and carrying continuous amounts of a scalar quantity which
can be viewed as energy. At each (exponentially distributed) random move the
total energy of a randomly chosen pair of nearest neighbors is redistributed
among them randomly according to uniform distribution, and the process
continues. Because of its simplicity the KMP model has become a paradigmatic
model in the studies of fluctuations, including large deviations, of heat
transfer in classical macroscopic nonequilibrium systems Bertini2005 ;
BodineauDerrida ; DG2009b ; Lecomte ; HurtadoGarrido ; KrMe ; Pradosetal ;
MS2013 ; Peletier ; JonaLasinioreview ; ZarfatyM ; Spielberg ; Guttierez ;
Frassek ; Grabsch ; BSM1 .

To set up our ideas, let us consider the KMP model on a one-dimensional
lattice. Suppose that only one agent has energy $W$ (or a few neighboring
agents have total energy $W$) at $t=0$. Due to the interactions with the
neighbors, the energy will spread in a random fashion throughout the system.
Looking for universality, we will focus on times much longer than the inverse
rate of the elemental energy exchange between the two neighbors (equal to
$1/2$), and on distances much larger than the lattice constant (equal to $1$).
As it was shown already in the original paper of KMP KMP , the mean, or
expected, coarse-grained temperature $\bar{u}(x,t)$ obeys the heat diffusion
equation

$\partial_{t}\bar{u}(x,t)=\partial_{x}^{2}\bar{u}(x,t)\,.$ (1)

The initial coarse-grained temperature profile is a delta-function,

$\bar{u}(x,t=0)=W\delta(x),$ (2)

so the mean-field solution for the coarse-grained temperature is

$\bar{u}(x,t)=\frac{W}{\sqrt{4\pi t}}\,\exp\left(-\frac{x^{2}}{4t}\right)\,.$
(3)

However, in individual realizations of the stochastic process, the coarse-
grained temperature $u(x,t)$ will of course fluctuate around the mean-field
solution. These fluctuations are exemplified by Fig. 1 which shows a snapshot,
and a coarse-grained version of it, of a Monte Carlo simulation of the
microscopic KMP model in this setting. To characterize fluctuations of the
heat transfer, we have recently evaluated BSM1  the full probability
distribution $\mathcal{P}_{0}(J_{0},T)$ of the total amount of heat $J_{0}$,
observed at time $t=T\gg 1$ on the right half-line:

$J_{0}=\int_{0}^{\infty}u(x,t=T)\,dx\,.$ (4)

Figure 1: A Monte-Carlo simulation of the microscopic KMP model. Initially
only one agent, at $x=0$, had a nonzero energy $W=1$. Plotted is the simulated
energy profile $u$ as a function of $x$ at time $t=3\times{10}^{4}$ (bars),
its spatial average over each 50 consecutive lattice sites (solid line), and
the theoretical mean-field profile (3) (dashed line).

Our work BSM1  continued a line of studies of large deviations of non-
stationary mass or energy transfer in different diffusive lattice gas models
DG2009a ; DG2009b ; KrMe ; MS2013 ; MS2014 ; VMS2014 . In parallel, there has
been much interest in the large deviations of the position of a tracer (or a
tagged particle) in single-file systems such as the simple symmetric exclusion
process (SSEP) KMS1 ; KMS2 ; IMS1 ; IMS2 ; Grabsch ; Poncet1 ; Poncet2 ;
Illien . (The SSEP is another well-known diffusive lattice gas model Spohn ;
KipnisLandim ; Schuetz ; Chou , where each particle can hop with equal
probabilities to one of the adjacent lattice sites, but a hop to a site
occupied by another particle is forbidden.) However, prior to our work BSM1 ,
the full statistics of the corresponding quantities were determined only for
_annealed_ initial conditions, which assume a “pre-thermalization” of the
lattice gas (or of the regions of $x<0$ and $x>0$ thereof at different
densities or temperatures) DG2009a ; DG2009b ; IMS1 ; IMS2 ; Grabsch , see
also the more recent Ref. MMS22 .

The evaluation of the full distribution $\mathcal{P}_{0}(J_{0},T)$ for the
quenched (that is deterministic) initial condition (2) was achieved in Ref.
BSM1  by employing the macroscopic fluctuation theory (MFT) JonaLasinioreview
for the KMP model, and by uncovering and exploiting complete integrability of
the MFT equations for the KMP model by the Zakharov-Shabat inverse scattering
method. In this work we extend the approach of Ref. BSM1  to a more general
characterization of the stochastic heat transfer in the KMP model. We specify
an arbitrary point $x=X$ and evaluate the full time-dependent probability
distribution $\mathcal{P}(J,X,T)$ of the total amount of heat,

$J=\int_{X}^{\infty}u(x,t=T)\,dx\,$ (5)

observed at time $t=T\gg 1$ to the right of $x=X$. Because of the energy
conservation in the KMP model, the distribution $\mathcal{P}(J,X,T)$ must have
compact support: $0<J<W$. The mean, or expected value of $J$ at fixed $X$
readily follows from the deterministic solution (3):

$\bar{J}=\bar{J}(X,T)=\int_{X}^{\infty}\bar{u}(x,T)\,dx=\frac{W}{2}\text{erfc}\left(\frac{X}{\sqrt{4T}}\right)\,,$
(6)

where $\text{erfc}\,z=1-\text{erf}\,z$, and $\text{erf}\,z$ is the error
function. For $X=0$ one obtains $\bar{J}=W/2$. The main objective of this work
is to determine the full statistics of fluctuations of $J$ around the mean
value $\bar{J}$.

The presence of a new parameter $0<X<\infty$ allows for a more detailed
characterization of the system, as we show here. Importantly, when the
position $x=X$ is varied at fixed $J$, it becomes an analog of the position of
a tracer in single-file processes such as the SSEP Grabsch . Indeed, in a
single-file process, the total number of particles to the right of the tracer
position $x=X$ does not change with time, and this can be mimicked by keeping
$J=\text{const}$ in the KMP model. As a result, as we discuss below, the
probability distribution $\mathcal{P}(J,X,T)$ as a function of $X$ at
$J=\text{const}$ has the same dynamical scaling properties as the probability
distribution of the position $x=X$ of a tracer in single-file processes, such
as the SSEP.

Like many previous studies of the statistics of energy transfer DG2009b ; KrMe
; MS2013 ; MS2014 ; VMS2014 ; BSM1 ; MMS22  and of the tracer position KMS1 ;
KMS2 , we will employ the MFT JonaLasinioreview . The MFT is rooted in the
fluctuational hydrodynamics of diffusive lattice gases Spohn ; KipnisLandim :
a coarse-grained description of the lattice gas on length scales much larger
than the lattice constant 1, and time scales much larger than the inverse rate
1/2 of the energy exchange between the neighboring agents. In our problem the
corresponding small parameter is $1/\sqrt{T}\ll 1$ BSM1 . The fluctuational
hydrodynamics has the form of a stochastic partial differential equation. For
a general one-component diffusive lattice gas this equation has the form Spohn
; KipnisLandim

$\partial_{t}u=\partial_{x}\left[D(u)\partial_{x}u+\sqrt{\sigma(u)}\,\eta\right]\,,$
(7)

where $u(x,t)$ is the coarse-grained density or temperature, $D(u)$ is the
diffusion (or heat diffusion) coefficient, $\sigma(u)$ is the gas
compressibility, and $\eta(x,t)$ is a delta-correlated Gaussian noise,
$\left<\eta(x,t)\right>=0$ and
$\quad\left<\eta(x,t)\eta(x^{\prime},t^{\prime})\right>=\delta(x-x^{\prime})\delta(t-t^{\prime})$.
For the KMP model, where $u(x,t)$ is the coarse-grained temperature, one
obtains $D=1$ and $\sigma(u)=2u^{2}$, and Eq. (7) becomes

$\partial_{t}u=\partial_{x}\left(\partial_{x}u+\sqrt{2}u\eta\right)\,.$ (8)

Without the noise term Eq. (8) becomes the heat diffusion equation (1).

The MFT JonaLasinioreview  represents a variant of the optimal fluctuation
method. It is a large-deviation theory based on a saddle-point evaluation of
the path integral for the stochastic process, described by Eq. (7). In our
problem, the small parameter of the saddle-point evaluation is again
$1/\sqrt{T}$. The saddle-point evaluation leads to a constrained optimization
problem, where the (integral) constraint comes from conditioning the process
on the specified transferred heat (5) subject to the initial condition (2).
The integral constraint (5) is conveniently taken into account via a Lagrange
multiplier DG2009b . We will present a complete Hamiltonian formulation of the
MFT problem shortly. The solution of this problem describes the _optimal path_
of the process: the most likely time histories of the coarse-grained
temperature field and of the noise field which dominate the probability
distribution $\mathcal{P}(J,X,T)$ that we are after.

The MFT, being stripped of unnecessary microscopic details, is an appealing
tool for studying large deviations in lattice gases on large scales and at
long times. However, the MFT equations – two coupled nonlinear partial
differential equations – are usually very hard to solve exactly, without
reliance on additional small or large parameters. Ref. BSM1  reported a first
major advance in this area of statistical mechanics by finding an exact
solution to the MFT problem for the distribution $\mathcal{P}(J,X,T)$ in the
particular case of $X=0$. As already mentioned above, the exact solution came
from uncovering and exploiting complete integrability of the MFT equations by
the Zakharov-Shabat inverse scattering method. A second major advance in this
area has been recently reported in Ref. MMS22 . The authors employed the MFT
for determining the full statistics of $J$ defined in Eq. (5) for the SSEP
with a step-like annealed initial condition. Earlier this problem was exactly
solved (by a Bethe ansatz) in its microscopic formulation, and the long-time
asymptotic was extracted, in Ref. DG2009b . The authors of Ref. MMS22
reproduced this asymptotic by combining a generalization of the canonical
Cole-Hopf transformation with the inverse scattering method.

The recent works BSM1  and MMS22  [and Refs. JKM ; KLD1 ; KLD2  where complete
integrability was uncovered JKM  and exploited KLD1 ; KLD2  for the
determination of full short-time statistics of the interface height as
described by the Kardar-Parisi-Zhang (KPZ) equation KPZ ], have shown a great
potential for the combination of the optimal fluctuation methods, such as the
MFT, with the inverse scattering method for obtaining exact solutions for
different large-deviation statistics. The present work is a natural next step
in exploring this potential.

In Sec. II we present the formulation of the MFT problem for
$\mathcal{P}(J,X,T)$ DG2009b ; MS2013  and briefly discuss the scaling
behavior of $\mathcal{P}(J,X,T)$ and its relation with the universal scaling
behavior of the tracer statistics in single-file processes. In Sec. III we
solve the MFT problem by the inverse scattering method. In Sec. IV we discuss
some asymptotic limits of $\mathcal{P}(J,X,T)$. Our results are briefly
summarized in Sec. V.

##  II Formulation of the MFT problem

We rescale the variables $t$, $x$ and $u$ by $T$, $\sqrt{T}$ and $W/\sqrt{T}$,
respectively. As a result, $X$ and $J$ become rescaled by $\sqrt{T}$ and by
$W$, respectively. The optimal path of the process is described by two coupled
Hamilton’s equations: partial differential equations for the rescaled
temperature field $u(x,t)$ and the conjugate “momentum density” field $p(x,t)$
(the latter describes the optimal history of the noise $\eta(x,t)$ conditioned
on the transferred heat $J$ at given $X$).

We also introduce the (minus) momentum density gradient field
$v(x,t)=-\partial_{x}p(x,t)$. As was shown earlier DG2009b ; MS2013 ; BMS1,
the MFT equations can be written as

$\displaystyle\partial_{t}u$ $\displaystyle=$
$\displaystyle\partial_{x}(\partial_{x}u+2u^{2}v)\,,$ (9)
$\displaystyle\partial_{t}v$ $\displaystyle=$
$\displaystyle\partial_{x}(-\partial_{x}v+2uv^{2})\,.$ (10)

The initial condition, after the rescaling, is

$u(x,t=0)=\delta(x)\,.$ (11)

The rescaled transferred heat at $t=T$ is constrained by the equation

$\int_{X}^{\infty}u(x,t=1)\,dx=J\,.$ (12)

The particular case $X=0$ was solved in Ref. BSM1 . Being interested in $X\neq
0$ we can, without loss of generality, set $X>0$. The constrained optimization
problem yields, aside from Eqs. (9) and (10), a second boundary condition in
time DG2009b ,

$v(x,t=1)=-\lambda\,\delta(x-X)\,,$ (13)

where $\lambda$ is a Lagrange multiplier, to be ultimately determined from Eq.
(12). This MFT formulation possesses a nontrivial symmetry

$v(x,t)=-\lambda\,u\left(X-x,1-t\right)\,.$ (14)

which generalizes the symmetry found for the particular case $X=0$ in Ref.
BSM1 .

Once $u(x,t)$ and $v(x,t)$ are determined, the rescaled mechanical action can
be evaluated DG2009b ; KrMe ; MS2013 :

$\displaystyle s=\int_{0}^{1}dt\int_{-\infty}^{\infty}dx\,u^{2}v^{2}.$ (15)

This action determines the probability density ${\mathcal{P}}(J,X,T)$ up to a
pre-exponential factor. In the original variables we have

$\ln{\mathcal{P}}(J,X,T)\simeq-\sqrt{T}\,s\left(\frac{J}{W},\frac{X}{\sqrt{T}}\right)\,,$
(16)

which describes the scaling behavior of ${\mathcal{P}}(J,X,T)$. As we will see
in the following, at fixed $J$ and small $X/\sqrt{T}$, the rate function $s$
is quadratic in $X/\sqrt{T}$, describing Gaussian fluctuations, where the
standard deviation of $X$ scales with time as $T^{1/4}$, that is sub-
diffusively. This scaling coincides with the one universally observed for the
tracer position in single file diffusion processes, see Ref. KMS1 ; KMS2  and
references therein. This coincidence is of course not accidental. Indeed, the
mathematical formulation of the MFT problem that we have just presented, is
very similar to the MFT formulation for the statistics of the position of a
tracer KMS1 ; KMS2 . The differences are in the particular $u$-dependence of
the multiplicative noise term in the Langevin equation (8), and in the delta-
function initial condition (11) (rather than a step-like one) that we adopted
here. Neither of them alters the subdiffusive scaling $T^{1/4}$ of the
standard deviation of $X$. Finite or large $X/\sqrt{T}$ corresponds to non-
Gaussian large deviations which are already non-universal, and which we will
explore. At fixed $X$ and small $J/W$, the rate function $s$ is also quadratic
and therefore describes Gaussian fluctuations which are universal for a whole
family of lattice gases. For a finite $J/W$ (large deviations) the
distribution is non-Gaussian.

It was observed BSM1  that Eqs. (9) and (10) coincide with the derivative
nonlinear Shrödinger equation (DNLSE) in imaginary time and space DNLSE . With
real time and space, the DNLSE describes nonlinear electromagnetic wave
propagation in plasmas and other systems KN . The initial-value problem for
the DNLSE is completely integrable by using the Zakharov-Shabat inverse
scattering method (ISM) adapted by Kaup and Newell for the DNLSE KN . The heat
transfer statistics presents an additional difficulty, as it involves a
boundary-value problem in time. Ref. BSM1  overcame this difficulty in the
particular case $X=0$ by (i) making use of a shortcut (see, e.g. Ref. Vivoetal
) which makes it possible to determine the rate function $s(J,X)$ even in the
absence of the complete solutions $u(x,t)$ and $v(x,t)$, and (ii) exploiting
the symmetry relation (14). Here we extend the approach of Ref. BSM1  to an
arbitrary $X$.

##  III Solution of the MFT problem

Let us outline the key ideas behind the approach that we shall use to solve
the MFT problem: the ISM. A key property related to the integrability of Eqs.
(9) and (10) is the existence of a corresponding Lax pair. This means than
Eqs. (9) and (10) are equivalent to the compatibility condition of a system of
two linear differential equations defined by a pair of matrices. For the
latter system, one can define scattering amplitudes which depend on $u$ and
$v$. As explained below, one can then solve for the time evolution of the
scattering amplitudes, which turns out to be very simple. By relating the
fields $u$ and $v$ to the scattering amplitudes, at $t=0$ and $t=1$, the ISM
will enable us to find the heat excess $J=J(\lambda,X)$ which suffices for the
calculation of $s=s(J,X)$.

###  III.1 Scattering amplitudes and their dynamics

Adapting the derivation of Kaup and Newell KN  to imaginary time and space, we
consider the linear system

$\begin{cases}\partial_{x}\mathbold{\psi}(x,t,k)=U(x,t,k)\mathbold{\psi}(x,t,k)\,,\\\\[2.84526pt]
\partial_{t}\mathbold{\psi}(x,t,k)=V(x,t,k)\mathbold{\psi}(x,t,k)\,,\end{cases}$
(17)

where $\mathbold{\psi}(x,t,k)$ is a column vector of dimension 2, the matrices

$\displaystyle U(x,t,k)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}-ik/2&-iv\sqrt{ik}\\\ -iu\sqrt{ik}&ik/2\\\
\end{pmatrix},$ (18) $\displaystyle V(x,t,k)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}k^{2}/2-ikuv&-i(\sqrt{ik})^{3}v+i\sqrt{ik}\,\partial_{x}v-i\sqrt{ik}2v^{2}u\\\
-i(\sqrt{ik})^{3}u+i\sqrt{ik}\,\partial_{x}u-i\sqrt{ik}2u^{2}v&-k^{2}/2+ikuv,\\\
\end{pmatrix}$ (19)

constitute the Lax pair, and $k$ is a spectral parameter. It is now
straightforward to check that the compatibility condition
$\partial_{t}\partial_{x}\mathbold{\psi}=\partial_{x}\partial_{t}\mathbold{\psi}$,
which corresponds to

$\partial_{t}U-\partial_{x}V+\left[U,V\right]=0,$ (20)

is indeed equivalent to Eqs. (9) and (10).

It is useful to define the matrix $\mathcal{T}(x,y,t,k)$ as the $x$-propagator
of the system (17), namely, the solution to the equation

$\partial_{x}\mathcal{T}(x,y,t,k)=U(x,t,k)\mathcal{T}(x,y,t,k)$ (21)

with the initial condition $\mathcal{T}(x,x,t,k)=I$ (the identity matrix). At
$x\to\pm\infty$, the fields $u(x,t)$ and $v(x,t)$ vanish, and us a result, the
matrix $U$ takes the simple form,

$U(x\to\pm\infty,t,k)=\begin{pmatrix}-ik/2&0\\\ 0&ik/2\end{pmatrix}\,.$ (22)

As a result, the entries of $\mathcal{T}(x,y,t,k)$ oscillate as a function of
$x$ and $y$ with angular frequencies whose magnitude is $k/2$ at
$x\to\pm\infty$, $y\to\pm\infty$. Therefore, it is natural to define the full-
space propagator $G(t,k)$ as follows:

$G(t,k)=\lim_{\begin{array}[]{c}x\to\infty\\\
y\to-\infty\end{array}}\begin{pmatrix}e^{ikx/2}&0\\\
0&e^{-ikx/2}\end{pmatrix}\mathcal{T}(x,y,t,k)\begin{pmatrix}e^{-iky/2}&0\\\
0&e^{iky/2}\end{pmatrix}.$ (23)

The matrix $G(t,k)$ plays an important role in the ISM, because its ($u$\- and
$v$-dependent) entries are the scattering amplitudes of the system (17). The
time evolution of $G(t,k)$ is remarkably simple. Indeed, one finds that the
time evolution of the matrix $\mathcal{T}(x,y,t,k)$ satisfies:

$\partial_{t}\mathcal{T}(x,y,t,k)=V(x,t,k)\mathcal{T}(x,y,t,k)-\mathcal{T}(x,y,t,k)V(y,t,k)\,.$
(24)

As one can verify directly, Eq. (24) is compatible with (21) (i.e.
$\partial_{t}\partial_{x}\mathcal{T}=\partial_{x}\partial_{t}\mathcal{T}$) due
to Eq. (20). The matrix $V(x,t,k)$ too becomes very simple in the limit
$x\to\pm\infty$,

$V(x\to\pm\infty,t,k)=\frac{k^{2}}{2}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,.$ (25)

Plugging Eq. (25) into (24), one finds the time evolution of
$\mathcal{T}(x\to\infty,y\to-\infty,t,k)$ which in turn, using (23), yields
that of $G(t,k):$

$\displaystyle G(t,k)=\begin{pmatrix}a(t,k)&\tilde{b}(t,k)\\\
b(t,k)&\tilde{a}(t,k)\\\
\end{pmatrix}=\begin{pmatrix}a(0,k)&\tilde{b}(0,k)e^{k^{2}t}\\\
b(0,k)e^{-k^{2}t}&\tilde{a}(0,k)\\\ \end{pmatrix}$ (26)

In Eq. (26) we have introduced here a notation for the matrix elements of
$G(t,k)$.

###  III.2 Solving the scattering problem at $t=0$ and $t=1$

Let us find the matrix $\mathcal{T}(x,y,0,k)$ at $t=0$. This is achieved by
solving Eq. (21) at $t=0$, which reads (using $u(x,0)=\delta(x)$)

$\partial_{x}\mathcal{T}\left(x,y,t,k\right)=\begin{pmatrix}-\frac{ik}{2}&-iv\left(x,0\right)\sqrt{ik}\\\
-i\delta\left(x\right)\sqrt{ik}&\frac{ik}{2}\end{pmatrix}\mathcal{T}\left(x,y,t,k\right)\,.$
(27)

The general solution to Eq. (27) is found by solving it in the regions $x<0$
and $x>0$ and then matching the solutions at $x=0$. Then, taking into account
the initial condition $\mathcal{T}(x,x,t,k)=I$, we obtain

$\mathcal{T}(x,y,0,k)=\begin{cases}\begin{pmatrix}e^{-ik(x-y)/2}&-i\sqrt{ik/2}\,e^{-ik(x-y)/2}I_{v}(x,y)\\\
0&e^{ik(x-y)/2}\end{pmatrix}\,,&xy>0\,,\\\ \\\
\begin{pmatrix}e^{ik(y-x)/2}\left[1\pm
ikI_{v}(x,0)\right]&-i\sqrt{ik}e^{ik(y-x)/2}\left[I_{u}(x,y)\pm
ikI_{u}(0,y)I_{u}(x,0)\right]\\\ \pm i\sqrt{ik}e^{ik(x+y)/2}&e^{ik(x-y)/2}\pm
ike^{ik(x+y)/2}I_{u}(0,y)\end{pmatrix}\,,&xy<0\,,\end{cases}$ (28)

where

$\displaystyle I_{v}(x,y)=\int_{y}^{x}v(z,0)e^{ik(z-y)}dz,\qquad
I_{u}(x,y)=\int_{y}^{x}u(z,1)e^{-ik(z-y)}dz,$ (29)

and in the second case in (28), the sign $\pm$ is the same as the sign of $y.$
We now compute $G(0,k)$, by plugging Eq. (28) into (23), with the result:

$G(0,k)=\begin{pmatrix}1-ikQ_{+}(k)&-i\sqrt{ik}\left[Q(k)-ikQ_{-}(k)Q_{+}(k)\right]\\\
-i\sqrt{ik}&1-ikQ_{-}(k)\end{pmatrix}$ (30)

in terms of $Q_{\pm}(k)$ which are the Fourier transforms of $v(z,0)$
restricted to $z>0$ and $z<0$, respectively, and of their sum:

$Q_{-}(k)\\!=\\!\int_{-\infty}^{0}\\!v(z,0)e^{ikz}dz,\quad
Q_{+}(k)=\int_{0}^{\infty}\\!v(z,0)e^{ikz}dz\,,\quad
Q(k)=Q_{+}(k)+Q_{-}(k)\,.$ (31)

It is useful to compare this result to the one obtained at $t=1$. Here we have
$v(x,1)=-\lambda\delta(x-X)$, which enables us to solve Eq. (21) at $t=1$. The
solution is very similarly to the $t=0$ case, and one gets

$\displaystyle\mathcal{T}(x,y,1,k)$
$\displaystyle\qquad=\begin{cases}\begin{pmatrix}e^{-ik(x-y)/2}&0\\\
-i\sqrt{ik}\,e^{ik(x-y)/2}I_{u}(x,y)&e^{ik(x-y)/2}\end{pmatrix}\,,&\left(x-X\right)\left(y-X\right)>0\,,\\\
\\\ \begin{pmatrix}e^{-ik(x-y)/2}\pm\lambda ike^{ik(x+y-2X)/2}I_{u}(X,y)&\pm
i\lambda\sqrt{ik}e^{-ik(x+y-2X)/2}\\\
-i\sqrt{ik}e^{ik(x-y)/2}\left[I_{u}(x,y)\pm\lambda
ikI_{u}(X,y)I_{u}(x,X)\right]&e^{ik(x-y)/2}\left[1\pm\lambda
ikI_{u}(x,X)\right]\end{pmatrix}\,,&\left(x-X\right)\left(y-X\right)<0\,,\end{cases}$
(32)

where in the second case, the sign $\pm$ is to be taken as the opposite of the
sign of $y-X.$ Similarly to the $t=0$ case, we now compute

$G(1,k)=\begin{pmatrix}1+\lambda ikR_{+}(k)&+i\lambda\sqrt{ik}\,e^{ikX}\\\
-i\sqrt{ik}\,e^{-ikX}\left[R(k)+\lambda ikR_{-}(k)R_{+}(k)\right]&1+\lambda
ikR_{-}(k)\end{pmatrix}$ (33)

where

$R_{+}(k)=\int_{-\infty}^{X}u(z,1)e^{-ikz}dz,\quad
R_{-}(k)=\int_{X}^{\infty}u(z,1)e^{-ikz}dz,\quad R(k)=R_{+}(k)+R_{-}(k)\,.$
(34)

###  III.3 Recovering the rate function $s(J,X)$ from the scattering
amplitudes

In this section we calculate the rate function $s(J,X)$ by relating the
scattering amplitudes at $t=0$ and $t=1$. We find that several cases must be
considered.

(i) In the simplest case, in which $\lambda$ is sufficiently small, the
derivation closely follows that of BSM1 . This case completely covers the case
$X=0$. (ii) For $X\neq 0$, and larger values of $\lambda$, technical problems
are encountered when evaluating some of the integrals that appear in the
solution in case (i). These are fairly simple to circumvent, by deforming the
integration contours in the complex plane. This completely covers the case
$X<\sqrt{8}$. (iii) For $X>\sqrt{8}$ a third regime, of very large $\lambda$,
exists. This regime is rather interesting as it involves multiple solutions at
a given $\lambda$, i.e., there is a regime of $\lambda$’s for which $s$ and
$J$ are multi-valued functions of $\lambda$. Nevertheless, $s$ remains a
single-valued, analytic function of $J$, unlike in many other systems in which
additional branches of the rate function lead to singularities which have the
character of dynamical phase transitions ZarfatyM ; Baek15 ; JKM ; Baek17 ;
Baek18 ; Shpielberg2016 ; Spielberg ; LeDoussal2017 ; SKM2018 . The reason for
this rather unusual behavior is that $s(J,X)$ is not a convex function of $J$
for $X>\sqrt{8}$.

####  III.3.1 Full solution for $X=0$, and for $X>0$ with sufficiently small
$\lambda$

Comparing the upper-right elements of $G(0,k)$ from Eq. (30) and of $G(1,k)$
from Eq. (33), using Eq. (26), we obtain

$ikQ(k)-ikQ_{-}(k)\times ikQ_{+}(k)=-\lambda ik\,e^{ikX-k^{2}}\,.$ (35)

Eq. (35) provides a key step towards the solution to our problem because,
after solving it, $J=J(\lambda,X)$ is obtained from $Q_{+}(0)$ with the use of
the symmetry (14), and this, as shown below, suffices in order to determine of
the rate function $s(J,X)$. We now turn to the solution to Eq. (35). We begin
by completing the squares in Eq. (35), by writing

$\displaystyle ikQ_{\pm}(k)=1-e^{\Phi_{\pm}(k)-\Phi_{\pm}(0)}\,,$ (36)

where the constant $-\Phi_{\pm}(0)$ in the exponent is found by requiring that
$Q_{\pm}(k)$ is regular at the origin. This equation turns Eq. (35) into

$\displaystyle e^{\Phi_{+}(k)+\Phi_{-}(k)-\Phi_{+}(0)+\Phi_{-}(0)}=1+i\lambda
ke^{ikX-k^{2}},$ (37)

which has the solution

$\Phi_{\pm}(k)=\pm\int_{-\infty}^{\infty}\frac{\ln\left(1+i\lambda
k^{\prime}e^{ik^{\prime}X-k^{\prime 2}}\right)}{k^{\prime}-k\mp
i0^{+}}\frac{dk^{\prime}}{2\pi i}$ (38)

Eq. (38) is derived by noting that $Q_{\pm}(k)$ are analytic in the upper and
lower half complex $k$ plane respectively and are well-behaved when $k$ is
allowed to reach infinity through the respective half-planes111These
properties of $Q_{\pm}(k)$ follow from their definitions (31) together with
the expected Gaussian decay of $v(z\to\pm\infty,0)$ and boundedness of
$v(z\to\pm 0,0)$. We then use the well-known decomposition
$f(k)=f_{+}(k)+f_{-}(k)$ of a general function $f(k)$ into functions analytic
in the upper and lower half-planes, $f_{\pm}(k)$, respectively, given by

$f_{\pm}(k)=\int\frac{f(k^{\prime})}{k^{\prime}-k\mp
i0^{+}}\frac{dk^{\prime}}{2\pi i}\,.$ (39)

The decomposition (39) is applied to the logarithm of Eq. (37), leading to Eq.
(38).

Taking the derivative of Eq. (36) with respect to $k$, setting $k$ to $0$ and
dividing by $i$ we obtain:

$\displaystyle Q_{\pm}(0)=\pm\int_{-\infty}^{\infty}\frac{\ln\left(1+i\lambda
ke^{ikX-k^{2}}\right)}{(k\pm i0^{+})^{2}}\frac{dk}{2\pi}$ (40)

which, using the Sokhotski–Plemelj formula, can be also written as

$\displaystyle
Q_{\pm}(0)=\pm\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{-\infty}^{\infty}\frac{\ln\left(1+i\lambda
ke^{ikX-k^{2}}\right)}{k^{2}}\frac{dk}{2\pi}\mp\frac{\lambda}{2}\,.$ (41)

Using Eqs. (12), (14) and (31) alongside with the conservation law
$\int_{-\infty}^{\infty}u(x,t)\,dx=1$, we determine $J=J(\lambda,X)$:

$J\left(\lambda,X\right)=\int_{X}^{\infty}u\left(x,t=1\right)\,dx=\frac{1}{\lambda}\int_{-\infty}^{0}v\left(x,0\right)\,dx=\frac{Q_{+}\left(0\right)}{\lambda}=\int_{-\infty}^{\infty}\frac{\ln\left(1+i\lambda
ke^{ikX-k^{2}}\right)}{(k+i0^{+})^{2}}\frac{dk}{2\pi\lambda}$ (42)

which can also be written as

$J\left(\lambda,X\right)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{-\infty}^{\infty}\frac{\ln\left(1+i\lambda
ke^{ikX-k^{2}}\right)}{k^{2}}\frac{dk}{2\pi\lambda}-\frac{1}{2}\,.$ (43)

In order to proceed, we utilize a useful shortcut, which enables us to obtain
the rate function $s=s(J,X)$ given that we have calculated $J=J(\lambda,X)$.
The shortcut makes use of the relation $\partial_{J}s=\lambda$, a property
that follows from the fact that $J$ and $\lambda$ are conjugate variables, see
e.g. Ref. Vivoetal . This enables us to calculate $s(J,X)$ bypassing Eq. (15)
[which would require us to know the entire optimal path $u(x,t)$]. Using
$J\left(\lambda,X\right)=Q_{+}\left(0\right)/\lambda$ from Eq. (42), we have

$\frac{\partial s}{\partial\lambda}=\frac{\partial s}{\partial
J}\frac{\partial J}{\partial\lambda}=\lambda\frac{\partial
J}{\partial\lambda}=\frac{\partial
Q_{+}\left(0\right)}{\partial\lambda}-\frac{Q_{+}\left(0\right)}{\lambda}\,.$
(44)

Using Eq. (41), we integrate Eq. (44) with respect to $\lambda$ to get

$s(\lambda,X)=Q_{+}(0)+\int_{-\infty}^{\infty}\frac{\text{Li}_{2}\left(-ik\lambda
e^{ikX-k^{2}}\right)}{2\pi k^{2}}\,dk\,+\frac{\lambda}{2}.$ (45)

where $\text{Li}_{2}\left(z\right)=\sum_{k=1}^{\infty}z^{k}/k^{2}$ is the
dilogarithm function, $Q_{+}(0)$ is given by Eq. (41), and the integration
constant was determined from $s(\lambda=0,X)=0.$ Equations (42) and (45) give
the rate function $s(J,X)$ in a parametric form.

Figure 2 shows $s(J,X)$ alongside with the asymptotic $J\to\bar{J}$, which
corresponds to $\lambda\to 0$. This asymptotic can be obtained either from the
exact rate function (42) and (45), or from a perturbative expansion applied
directly to the MFT equations KrMe ; BSM1 . Figure 3 shows
$\text{Re}\,Q_{+}(k)$ and $\text{Im}\,Q_{+}(k)$ versus $k$ at $X=1$. This
figure also shows the same quantities computed by solving Eqs. (9) and (10)
numerically with a back-and-forth iteration algorithm CS . The analytical and
numerical curves are in good agreement.

Figure 2: Action $s(J,X=3)$ as a function of $J$ for $X=3$. Dashed line is the
asymptote at $J\to\bar{J}=\text{erfc}(3/2)/2=0.016947\dots$, given by Eqs.
(68) below. The inset shows an enlargement of the asymptotic region.

Figure 3: Analytical results for the real and imaginary parts of $Q_{+}(k)$,
described by Eqs. (36) and (38) (solid lines), versus numerical results
(dashed lines), for $\lambda=X=1$, corresonding to $J=0.31\dots$.

We now deal with the question of the uniqueness of the solution. Let us denote
$a_{+}\left(k\right)=a\left(k,0\right)$ and
$a_{-}\left(k\right)=\tilde{a}\left(k,0\right)$ [using the notation introduced
in (26)]. From Eqs. (30) and (33), we have

$a_{\pm}\left(k\right)=1-ikQ_{\pm}\left(k\right)=1+\lambda
ikR_{\pm}\left(k\right)\,.$ (46)

Above, we had reached the equation (35) and found a solution to it, Eq. (36).
In terms of $a_{\pm}$ these two equations read

$\displaystyle 1-a_{+}\left(k\right)a_{-}\left(k\right)=-\lambda
ik\,e^{ikX-k^{2}},$ (47) $\displaystyle
a_{\pm}\left(k\right)=e^{\Phi_{\pm}(k)-\Phi_{\pm}(0)},$ (48)

respectively. However, there are more solutions to Eq. (47). Given any
solution $a_{\pm}\left(k\right)$ to Eq. (47), it is easily seen that, for any
$k_{1},k_{2}\in\mathds{C}$, where $k_{1}$ is in the upper half plane and
$k_{2}$ in the lower half plane,

$\hat{a}_{\pm}\left(k\right)=\left(\frac{k_{2}}{k_{1}}\frac{k-k_{1}}{k-k_{2}}\right)^{\pm
1}a_{\pm}\left(k\right)$ (49)

is also a solution, since only the product
$a_{+}\left(k\right)a_{-}\left(k\right)$ enters in (47). This transformation
can be applied any number of times. We must now find principles to select
which of these solutions corresponds to the proper solution to the MFT
equations, i.e., to histories $u$ and $v$ that are real and nonnegative, which
minimize the action $s$ constrained on a given $J$. One useful check is to
note that the factors that appear in the transformation (49) lead, in general,
to exponential decays rather than a Gaussian decay of $u(x,1)$ or $v(x,0)$ at
$x\to\pm\infty$ due to the poles introduced by the Fourier transforms of these
functions. On the other hand, one may show that the solution must decay as a
Gaussian at infinity. This is because at $x\to\pm\infty$, $u$ and $v$ are
small and therefore the nonlinear terms in the MFT equations (9) and (10) are
negligible, so they become the diffusion and anti-diffusion equations
respectively, whose solutions are Gaussian at all times (for a localized
initial condition). The exponential decay is however avoided, if $k_{1}$ and
$k_{2}$ solve the equation

$\displaystyle 1+i\lambda k_{i}e^{ik_{i}X-k_{i}^{2}}=0\ ,$ (50)

then the pole cancels out and the inclusion of the extra factor in
$a_{\pm}(k)$ does not alter the Gaussian drop-off at infinity. Another
scenario is that $k_{i}$ is real (or infinitesimally below or above the real
axis). These observations will be useful below.

####  III.3.2 Solution for $X<\sqrt{8}$ at all $\lambda$

In the analysis up to now, we tacitly assumed that in the term
$\ln\left(1+i\lambda ke^{ikX-k^{2}}\right)$, branch cuts of the logarithm are
never encountered in the integrals given above. This is indeed the case for
$X=0$ and also at sufficiently small $\lambda$ for $X\neq 0$. However, at
larger $\lambda$, branch cuts are encountered at $X\neq 0$, i.e., for some
$k\in\mathds{R}$,

$\displaystyle 1+i\lambda ke^{ikX-k^{2}}\in\mathds{R}_{-}\,.$ (51)

Let us find the values of $k$ and $\lambda$ at which these branch cuts cross
the real axis, by solving $1+i\lambda ke^{ikX-k^{2}}=0$ for real $k$ and
$\lambda.$ The solutions are $k=\pm k_{R,n}$ where

$\displaystyle k_{R,n}=\frac{\left(n+\frac{1}{2}\right)\pi}{X},\qquad
n=0,1,2,\dots,$ (52)
$\displaystyle\lambda_{c}\left(X,n\right)=\frac{\left(-1\right)^{n}X}{\left(n+\frac{1}{2}\right)\pi}\exp\left[\frac{\left(n+\frac{1}{2}\right)^{2}\pi^{2}}{X^{2}}\right]\,.$
(53)

It follows that for each $n$ for which $\lambda>\lambda_{c}(X,n)$, there is an
additional branch cut crossing the real axis at $k=\pm k_{R,n}$.

The branch cut crossing the real axis would cause $\Phi_{\pm}$ to be non-
analytic as a function of $\lambda$ if the real axis would be retained as the
contour of integration in Eq. (38). This would lead to non-analyticity in all
the physical quantities ($J$, $S$, etc.). To retain analyticity, one must
change the contour of integration in Eq. (38), and the equations stemming from
it, Eqs. (40), (42), (45), such as to avoid crossing any branch cut. Suggested
contours are shown in Fig. 4, but any contour that does not cross the branch
cuts would do.

Figure 4: Contour plot of $\arg ike^{ikX-k^{2}}$ for $\lambda=10$ and $X=2$
(a) and $X=3$ (b), in the complex $k$ plane. Dotted and solid lines correspond
to $ike^{ikX-k^{2}}\in\mathds{R}_{\pm}$ respectively. The solid lines are
branch cuts of $\ln\left(1+i\lambda ke^{ikX-k^{2}}\right).$ A convenient
choice for integration contours are the red lines while the points at which
the branch cuts cross the real axis are denoted by $\pm k_{R0}$ in accordance
with Eq. (52). In panel (b) the branch points $\kappa_{i}$ are denoted, while
in panel (a) three branch points are arbitrarily denoted as $k_{i}$,
$i=0,1,2$.

####  III.3.3 Solution for $X>\sqrt{8}$ at all $\lambda$

At $X>\sqrt{8}$, it is impossible to avoid the branch cuts altogether at
sufficiently large $\lambda$, see below. In fact, the analytical continuation
of the formula of Eq. (38) leads to a complex $J.$ We show these statements
and obtain the proper solutions for any value $\lambda>0$ below (at
$\lambda<0$, the solution for the case $X<\sqrt{8}$ is valid here as well).

It is convenient at this point to use a different method to analytically
continue the solutions from small $\lambda$, as opposed to the one used for
$X<\sqrt{8}$, which consisted of changing the contour of integration. Let us
first define for the case $X<\sqrt{8},$ three critical values of $\lambda$,
which will be important in the following

$\lambda_{c0}=\lambda_{c}\left(X,0\right)<\lambda_{c1}=\frac{4e^{\frac{1}{2}+\frac{X^{2}-X\sqrt{X^{2}-8}}{8}}}{X-\sqrt{X^{2}-8}}<\lambda_{c2}=\frac{4e^{\frac{1}{2}+\frac{X^{2}+X\sqrt{X^{2}-8}}{8}}}{X+\sqrt{X^{2}-8}}$
(54)

At $\lambda_{c1}$ and $\lambda_{c2}$, the number of purely imaginary solutions
to the equation (50) changes.

Let us assume for now

$\lambda_{c2}<\min_{n=2,4,6,\dots}\lambda_{c}\left(X,n\right)\,,$ (55)

as this makes the analysis a little simpler. At $\lambda_{c0}$ two branch cuts
cross the real axis. The branch points are denoted by $\kappa_{1}$,
$\kappa_{2}$ and the points which the branch cuts cross the real axis are
denoted by $\pm k_{R0}$. The branch points $\kappa_{1}$ and $\kappa_{2}$,
which satisfy $1+i\lambda\kappa_{i}e^{i\kappa_{i}X-\kappa_{i}^{2}}=0$, are
reflections across the imaginary axis of each other,
$\kappa_{1}=-\kappa_{2}^{*}$. As $\lambda$ increases beyond $\lambda_{c0},$
rather than changing the contour of integration, one could instead represent
the solution as

$\displaystyle\hat{a}^{(1)}_{\pm}\left(k\right)=\left(\frac{k_{R}^{2}}{\kappa_{1}\kappa_{2}}\frac{(k-\kappa_{1})(k-\kappa_{2})}{k_{R}^{2}-k^{2}}\right)^{\pm
1}a_{\pm}\left(k\right),$ (56)

while keeping the contour of integration in the definition of $a_{\pm}$, Eq.
(48), through $\Phi,$ Eq. (38), to be the real axis.
$\hat{a}^{(1)}_{\pm}\left(k\right)$ is a legitimate solution to Eq. (47) as it
is reached through a repeated use of the transformation described in Eq.
(49)222 One can take $k_{R}$ to have an infinitesimally small negative
imaginary part, in order to keep $\hat{a}^{(1)}_{\pm}\left(k\right)$ analytic
in the upper half complex $k$ plane.. It may be easily shown that that for
$\lambda<\lambda_{c2}$ this is equivalent to just changing the contour of
integration defining $a_{\pm}$ to avoid the branch cuts, while for
$\lambda>\lambda_{c2}$ it is impossible to find such a contour. This allows us
to define for $\lambda_{c1}<\lambda<\lambda_{c2}$ two more solutions:

$\displaystyle\hat{a}^{(2)}_{\pm}\left(k\right)=\left(\frac{k_{R}^{2}}{\kappa_{3}\kappa_{2}}\frac{(k-\kappa_{3})(k-\kappa_{2})}{k_{R}^{2}-k^{2}}\right)^{\pm
1}a_{\pm}\left(k\right),$ (57)
$\displaystyle\hat{a}^{(3)}_{\pm}\left(k\right)=\left(\frac{k_{R}^{2}}{\kappa_{3}\kappa_{1}}\frac{(k-\kappa_{3})(k-\kappa_{1})}{k_{R}^{2}-k^{2}}\right)^{\pm
1}a_{\pm}\left(k\right).$ (58)

Symmetry to reflection across the imaginary axis, which is present since all
$\kappa_{i}$ are purely imaginary at $\lambda_{c1}<\lambda<\lambda_{c2}$,
ensures that all three solutions produce real physical fields $u(x,1)$ and
$v(x,0)$. Only $\hat{a}^{(1)}$ respects this symmetry for
$\lambda<\lambda_{c1}$, where $\kappa_{1}$ and $\kappa_{2}$ are mirror images
of each other. When $\lambda$ increases beyond $\lambda_{c2}$ the branch point
$\kappa_{2}$ remains imaginary, while $\kappa_{1}$ and $\kappa_{3}$ are mirror
images of each other. This means that beyond $\lambda_{c2}$ only
$\hat{a}^{(3)}$ yields real physical fields. Figure 5 shows
$\text{Re}\,Q_{+}(k)$ and $\text{Im}\,Q_{+}(k)$ versus $k$ at $X=3.5$,
$\lambda=20>\lambda_{c2}$, obtained by substituting Eqs. (48) and (38) in Eq.
(58). This figure also shows the same quantities computed by solving Eqs. (9)
and (10) numerically with a back-and-forth iteration algorithm CS . The
analytical and numerical curves are in good agreement.

Figure 5: Analytical results for the real and imaginary parts of $Q_{+}(k)$
for $X>\sqrt{8}$ and $\lambda>\lambda_{c2}$, described by Eqs. (48), (38) and
(58) (solid lines), versus numerical results (markers). Parameters are
$\lambda=20$ and $X=3.5$, corresponding to $J=0.70\dots$.

To avoid having to write additional terms in the equation for $a_{\pm}$ each
time additional branch cuts cross the real axis, it is now convenient, after
establishing which solutions are real, to change the contour of integration
such that it does not cross any of these other branch cuts. One may choose the
curve that satisfies ${\rm Re}\lambda ke^{ikX-k^{2}}=0$ and passes through the
point $i\frac{X+\sqrt{X^{2}-8}}{4}$, see Figure 4 (b). For
$\lambda>\lambda_{c2},$ this curve contains the branch points $\kappa_{1}$ and
$\kappa_{3}$, and in computing the integral, the branch cut of the integrand
must be drawn as to not cross this curve. This change of contour of
integration from the real axis to this contour affects the change:

$\displaystyle
a_{\pm}\to\left(\frac{k_{R}^{2}}{\kappa_{1}\kappa_{2}}\frac{(k-\kappa_{1})(k-\kappa_{2})}{k_{R}^{2}-k^{2}}\right)^{\mp
1}a_{\pm},$ (59)

for all $\lambda>\lambda_{c0}$, and leaves $a_{\pm}$ unchanged for
$0<\lambda<\lambda_{c0}$. In this new representation (namely, for this choice
of contour of integration), we have the following formulas:

$a_{\pm}\left(k\right)=\begin{cases}e^{\Phi_{\pm}(k)-\Phi_{\pm}(0)},&\lambda<\lambda_{c2},\\\\[2.84526pt]
\left(\frac{\kappa_{1}}{\kappa_{3}}\frac{k-\kappa_{3}}{k-\kappa_{1}}\right)^{\pm
1}e^{\Phi_{\pm}(k)-\Phi_{\pm}(0)},&\lambda_{c1}<\lambda<\lambda_{c2},\\\\[5.69054pt]
\left(\frac{\kappa_{2}}{\kappa_{3}}\frac{k-\kappa_{3}}{k-\kappa_{2}}\right)^{\pm
1}e^{\Phi_{\pm}(k)-\Phi_{\pm}(0)},&\lambda>\lambda_{c1}.\end{cases}$ (60)

As $J$ is increased, $\lambda$ grows up to the value $\lambda_{c2}$ along the
branch corresponding to the first line in (60), up to a value $J=J_{1}$. As
$J$ grows above $J_{1}$, $\lambda$ decreases down to a value $\lambda_{c1}$
which corresponds to some $J=J_{2}$. As $J$ is increased even further,
$\lambda$ increases again from $\lambda_{c1}$, eventually reaching
$\lambda\to\infty$ which corresponds to $J\to 1$. This non-monotonic
dependence of $\lambda$ on $J$ implies that $s$ is a non-convex function of
$J$ (this is easy to see from $\partial_{J}s=\lambda$), which is a rather
unusual situation in large-deviation theory. $J_{1}$ and $J_{2}$ are the
inflection points, at which $\partial_{J}^{2}s=0$. These behaviors are
schematically plotted in Fig. 6. Eq. (60) in fact holds independently of the
assumption (55), so it describes the solution at all $X>\sqrt{8}$ and at all
$\lambda>0$.

Figure 6: Schematic plots of (a) $J(\lambda,X)$, (b) $s(\lambda,X)$ and (c)
$s(J,X)$ for $X>\sqrt{8}$ and $\lambda>0$. Interestingly, $J$ and $s$ are not
single-valued functions of $\lambda$, and $s$ is not a convex function of $J$.

We now calculate $Q_{\pm}(0)$, and from it we calculate $J=J(\lambda,X)$ and
$s=s(\lambda,X)$. By following a similar route to the one above for
$X<\sqrt{8}$ , we find for $J(\lambda,X)=\frac{Q_{+}(0)}{\lambda}$ the
following

$J\left(\lambda,X\right)=\int\frac{\ln\left(1+i\lambda
ke^{ikX-k^{2}}\right)}{k^{2}}\frac{dk}{2\pi\lambda}+\begin{cases}0,&J<J_{1},\\\\[2.84526pt]
\frac{i}{\lambda}\left(\frac{1}{\kappa_{1}}-\frac{1}{\kappa_{3}}\right),&J_{1}<J<J_{2},\\\\[5.69054pt]
\frac{i}{\lambda}\left(\frac{1}{\kappa_{2}}-\frac{1}{\kappa_{3}}\right),&J>J_{2},\end{cases}$
(61)

and by integrating again Eq. (44) with $s(\lambda=0,X)=0$, we find for the
action:

$\\!s\left(\lambda,X\right)=\\!\int\\!\ln\left(1+i\lambda
ke^{ikX-k^{2}}\right)\frac{dk}{2\pi
k^{2}}+\\!\int\\!\text{Li}_{2}\left(-ik\lambda
e^{ikX-k^{2}}\right)\frac{dk}{2\pi
k^{2}}+\begin{cases}0,&J<J_{1},\\\\[2.84526pt]
2i\left(\kappa_{3}-\kappa_{1}\right)+X\ln\left(\frac{\kappa_{3}}{\kappa_{1}}\right),&J_{1}<J<J_{2},\\\\[5.69054pt]
2i\left(\kappa_{3}-\kappa_{2}\right)+X\ln\left(\frac{\kappa_{3}}{\kappa_{2}}\right),&J>J_{2},\end{cases}$
(62)

a function which we plot in Fig. 2 for fixed $X=3$ and as a function of $J$.
Note that the integration contour here is similar to the one draw in Fig. 4
panel (b) and as such it is away from the real axis, where the integrands have
a pole and no principal value integral is necessary. Consequently, also the
Sokhotski–Plemelj formula is not employed here in contrast, e. g., to Eq.
(45). To achieve this result one encounters the integral:

$\displaystyle\int^{\lambda}\frac{d\ln\lambda^{\prime}}{\kappa_{i}(\lambda^{\prime})}=\frac{\ln(\lambda)}{\kappa_{i}(\lambda)}+\int^{\kappa_{i}(\lambda)}\frac{\ln\lambda(\kappa_{i}^{\prime})}{\kappa_{i}^{\prime
2}}d\kappa_{i}^{\prime}.$ (63)

Now, by definition, the root $\kappa^{\prime}_{i}$ satisfies the following
equation:

$\displaystyle\ln(\lambda(\kappa^{\prime}_{i}))=\frac{\imath\pi}{2}-\ln\kappa^{\prime}_{i}-\imath\kappa^{\prime}_{i}X+\kappa^{\prime
2}_{i},$ (64)

so that the integal on the right hand side of Eq. (63) may be easily
performed:

$\displaystyle\int^{\lambda}\frac{d\ln\lambda^{\prime}}{\kappa_{i}(\lambda^{\prime})}=-\imath
X\ln\kappa_{i}(\lambda)+\frac{1}{\kappa_{i}(\lambda)}+2\kappa_{i}(\lambda)$
(65)

allowing us to write Eq. (62).

##  IV Asymptotic limits

Our exact results greatly simplify in two asymptotic limits that we present
here. The asymptotic formulas can be obtained either from the exact results,
or directly from the MFT equations by using two different perturbation
theories.

###  IV.1 $\lambda\ll 1$: linear theory

In the limit of small $\lambda$ the optimal path can be found by a
straightforward perturbative expansion around the zeroth-order solution (3)
KrMe . Here, due to the symmetry relation (14), the leading order calculation
in $\lambda\ll 1$ is especially simple BSM1 . Using Eq. (15), we obtain

$s(\lambda,X)\simeq\lambda^{2}\int_{0}^{1}dt\int_{-\infty}^{\infty}dx\,\bar{u}^{2}(x,t)\,\bar{u}^{2}(X-x,1-t)=\frac{\lambda^{2}e^{-\frac{X^{2}}{2}}}{8\sqrt{2\pi}}\,,$
(66)

where we have plugged the zeroth-order solution (3), rescaled by $W$. The same
result (66) follows from the exact rate function, described by Eq. (42) and
(45), via a straightforward expansion in $\lambda$.

Rewriting the shortcut relation $\partial s/\partial J=\lambda$ in the form of
$(\partial s/\partial\lambda)(\partial\lambda/\partial J)=\lambda$, and
combining it with Eq. (66), we can express $\lambda$ in terms of $J$ and $X$:

$\lambda\simeq 4\sqrt{2\pi}e^{\frac{X^{2}}{2}}(J-\bar{J})$ (67)

Equations (66) and (67) yield (still in the rescaled variables) a quadratic in
$J-\bar{J}$ asymptotic which strongly depends on $X$:

$s(J\to\bar{J},X)\simeq 2\sqrt{2\pi}\,e^{\frac{X^{2}}{2}}(J-\bar{J})^{2}\,.$
(68)

The corresponding probability distribution $\mathcal{P}(J,X,T)$ describes
typical, small Gaussian fluctuations of the transferred heat around its mean
value $\bar{J}$. In the original variables

$\ln\mathcal{P}_{\text{Gauss}}\simeq-2\sqrt{2\pi
T}\,e^{\frac{X^{2}}{2T}}\left[\frac{J}{W}-\frac{1}{2}\text{erfc}\left(\frac{X}{\sqrt{4T}}\right)\right]^{2}\,.$
(69)

In the particular case $X=0$ this Gaussian asymptotic coincides with the one
obtained in Ref. BSM1 . Note that Eq. (68) is applicable only for
$|\lambda|\ll 1$, that is for $|J-\bar{J}|\ll e^{-X^{2}/2}$. This condition
becomes restrictive at $X\gg 1$, where the expected value of the transferred
heat is extremely small.

Evaluating Eqs. (67) and (68) at $J=1/2$, one can probe small fluctuations of
the energy median $x=X_{m}$, which is an analog of the tracer position in the
SSEP. At fixed $J$ the applicability condition of the perturbation expansion
$\lambda\ll 1$ demands that $|X_{m}|\ll 1$. Using the small-argument
asymptotic of $\text{erfc}(...)$, we obtain in the leading order
$\lambda\simeq 2\sqrt{2}X_{m}$, and (in the rescaled variables)

$s\simeq\frac{X_{m}^{2}}{\sqrt{2\pi}}\,.$ (70)

This corresponds to a Gaussian asymptotic of the probability distribution
$\rho(X_{m},T)$ of the median. In the original variables

$\ln\rho(X_{m},T)\simeq-\frac{X_{m}^{2}}{\sqrt{2\pi T}}\,.$ (71)

The standard deviation of the energy median position, $\langle
X_{m}^{2}\rangle^{1/2}=(\pi/2)^{1/4}T^{1/4}$, exhibits the familiar one-
dimensional tracer scaling $T^{1/4}$.

###  IV.2 $\lambda\gg 1$, $X\gg 1$

####  IV.2.1 From adiabatic perturbation theory

The large-deviation limit $\lambda\gg 1$ and $X\gg 1$ is especially
interesting. The reason for that is a unique way of the most probable transfer
of a significant amount of energy to large distances, specific for lattice-gas
models (such as the KMP) which have with a positive second derivative of the
compressibility $\sigma(u)$ with respect to $u$ MS2013 . In this case the
optimal path has the form of an adiabatically evolving _traveling doublon_ :
coupled large-amplitude solitary pulses of $u$ and $v$ MS2013 ; ZarfatyM . The
amplitudes of the $u$ and $v$ pulses slowly (adiabatically) vary with time
MS2013 . An adiabatic perturbation theory, which describes this nontrivial
process in our present setting, is a slightly modified version of the
perturbation theory of Ref. MS2013 , where a step-like initial $u$-profile
were considered, and $X$ was zero.

The adiabatic perturbation theory of Ref. MS2013  exploits the strong
inequality $\lambda\gg 1$ which guarantees a large length-scale separation
between the relatively small internal region of the doublon and the large-
scale _exterior_ regions in front of the doublon and behind it. In the
external region one can neglect the diffusion terms in the MFT equations (9)
and (10). The diffusion, however, is important for the description of the
doublon itself (the internal region). Importantly, when viewed from the
exterior regions, the doublon can be effectively described as a single _point-
like singularity_ located at a point $x=X_{d}(t)$ which travels in space. The
singularity carries a finite energy $M(t)$, a finite jump of the momentum
$V(t)=p_{1}-p_{2}$ (where the subscripts 1 and 2 refer to “in front of the
singularity” and “behind the singularity”, respectively), and a finite value
of the Hamiltonian $\mathcal{H}(t)$, corresponding to the canonical MFT
equations (9) and (10). These quantities slowly depend on time. They are
formally defined as

$\\{M(t),\,V(t),\,{\mathcal{H}}(t)\\}=\int_{X_{d}(t)-\varepsilon}^{X_{d}(t)+\varepsilon}\\{u,\,\tilde{v},\,h\\}\,dx,$
(72)

where $\tilde{v}(x,t)=-v(x,t)$,
$h=-\partial_{x}u\,\partial_{x}\tilde{v}+u^{2}v^{2}$ is the density of the
Hamiltonian of Eqs. (9) and (10), and $\varepsilon>0$ can be fixed at an
arbitrary small value as $X\to\infty$ and $\lambda\to\infty$. It was shown by
direct calculations in Ref. MS2013  that

$M(t)\simeq\frac{1}{\tilde{v}_{1}(t)},\quad
V(t)\simeq\frac{1}{u_{2}(t)},\quad\text{and}\quad\mathcal{H}(t)\simeq c(t),$
(73)

where $\tilde{v}_{1}(t)$ is the value of $v$ in front of the doublon, and
$u_{2}(t)$ is the value of $u$ behind the doublon. As we will see shortly, the
doublon velocity $c(t)$ is actually constant. Using Eqs. (57)-(59) and
(62)-(64) of Ref. MS2013  and the relation $\dot{X}_{d}(t)=c(t)$, we arrive at
the equations

$\displaystyle\dot{M}$ $\displaystyle=$ $\displaystyle-cu_{2},$ (74)
$\displaystyle\dot{V}$ $\displaystyle=$ $\displaystyle c\tilde{v}_{1},$ (75)
$\displaystyle\dot{{\cal H}}$ $\displaystyle=$ $\displaystyle 0.$ (76)

Then, using Eqs. (73), we obtain

$\displaystyle\dot{M}$ $\displaystyle=$ $\displaystyle-\frac{c}{V}\,,$ (77)
$\displaystyle\dot{V}$ $\displaystyle=$ $\displaystyle\frac{c}{M}\,,$ (78)
$\displaystyle c$ $\displaystyle=$ $\displaystyle\text{const}\,.$ (79)

The doublon must arrive at $t=1$ at the point $x=X$, which gives $c=X$.
Equations (77) and (78) have the first integral

$M(t)V(t)=k_{1}(X)\,,$ (80)

where the function $k_{1}(X)$ is a priori unknown. Using this first integral
in Eq. (77), we obtain

$M(t)=\exp\left[k_{2}(X)-\frac{Xt}{k_{1}(X)}\right]\,.$ (81)

The a priori unknown function $k_{2}(X)$ must be set to zero in view of the
condition $M(t=0)=1$, and we obtain

$M(t)=\exp\left[-\frac{Xt}{k_{1}(X)}\right]\,.$ (82)

Then Eq. (80) yields

$V(t)=k_{1}(X)\exp\left[\frac{Xt}{k_{1}(X)}\right]\,.$ (83)

Using the two remaining boundary conditions smallboundarylayers ,

$M(t=1)=j\quad\text{and}\quad V(t=1)=\lambda\,,$ (84)

we determine $k_{1}(X)$ and $\lambda$:

$k_{1}(X)=\frac{X}{\ln\frac{1}{j}}\quad\text{and}\quad\lambda=\frac{X}{j\ln\frac{1}{j}}\,.$
(85)

Using the first of these relations in Eq. (82), we see that the energy $M(t)$
of the doublon decays exponentially in time with rate $\ln(1/j)$:

$M(t)=\exp\left(-t\ln\frac{1}{j}\right)\,.$ (86)

In its turn, the doublon momentum $V$ grows exponentially in time with the
same rate:

$V(t)=\frac{X}{\ln\frac{1}{j}}\exp\left(t\ln\frac{1}{j}\right)\,,$ (87)

which completes the task of determining the optimal path of the doublon in the
adiabatic approximation.

The rescaled action, according to Ref. MS2013 , is

$s(j,X)\simeq c\ln(cMV)=X\ln\frac{X^{2}}{\ln\frac{1}{j}}\,,\quad X\gg 1.$ (88)

The corresponding large-$X$ asymptotic of the probability distribution is

$\ln\mathcal{P}(J,X,T)\simeq-X\ln\left(\frac{X^{2}}{T\ln\frac{W}{J}}\right)\,,\quad
X\gg\sqrt{T}\,.$ (89)

This expression describes a faster-than-exponential tail as a function of $X$.
This tail depends only logarithmically on $T$, and extremely weakly on $J$ and
$W$.

####  IV.2.2 From exact rate function

For large $X$ we are automatically in the regime of $X>\sqrt{8}$. As the
contour of integration we can use the curve $\arg ike^{ikX-k^{2}}=0$, which at
${\rm Re}k\to\pm\infty$ merges with the line ${\rm Im}k=\frac{X}{2}$. In fact,
one can ascertain that for large $X$ the entire curve $\arg ike^{ikX-k^{2}}=0$
has ${\rm Im}k\simeq\frac{X}{2}$. Then, in view of Eq. (62) for the action, it
can be shown that the integrand is vanishingly small unless $\lambda\sim
e^{\frac{X^{2}}{4}}$. In addition, we shall see a posteriori that we obtain
all possible values of $J=O(1)$ by keeping $\lambda$ much smaller than
$e^{\frac{X^{2}}{4}}$. In fact, we obtain $J_{1}=0$ and $J_{2}=e^{-1}$. Given
all this, the action in this region can be written as:

$\displaystyle
s\left(\lambda,X\right)\simeq\begin{cases}2i\left(\kappa_{3}(\lambda)-\kappa_{1}(\lambda)\right)+X\ln\left(\frac{\kappa_{3}(\lambda)}{\kappa_{1}(\lambda)}\right),&J_{1}<J<J_{2},\\\\[5.69054pt]
2i\left(\kappa_{3}(\lambda)-\kappa_{2}(\lambda)\right)+X\ln\left(\frac{\kappa_{3}(\lambda)}{\kappa_{2}(\lambda)}\right),&J>J_{2}\,.\end{cases}$
(90)

The expression for the current, Eq. (61), also benefits from the vanishing of
the integral, and we can write approximately:

$J\left(\lambda,X\right)\simeq\begin{cases}\frac{i}{\lambda}\left(\frac{1}{\kappa_{1}}-\frac{1}{\kappa_{3}}\right),&J_{1}<J<J_{2},\\\\[5.69054pt]
\frac{i}{\lambda}\left(\frac{1}{\kappa_{2}}-\frac{1}{\kappa_{3}}\right),&J>J_{2}\,.\end{cases}$
(91)

As we shall also see, for $J=O(1)$, that $\lambda\simeq X$ and
$\kappa_{1,2}\sim\frac{1}{\lambda}$. Therefore we can approximate Eq. (50) for
$\kappa_{j},$ $j\in\\{1,2\\}$ as follows:

$\displaystyle-\frac{1}{\lambda}=i\kappa_{j}e^{i\kappa_{j}X}.$ (92)

This algebraic equation for $\kappa_{j}$ can be solved in terms of the Lambert
function $W_{j-2}$:

$\displaystyle\kappa_{j}=-\frac{i}{X}W_{j-2}\left(-\frac{X}{\lambda}\right)\,,\quad
j\in\\{1,2\\}\,.$ (93)

As for $\kappa_{3},$ in the large $X$ limit, where $\lambda\sim X,$ one finds
that $\kappa_{3}$ is of order $X$ and has the following form:

$\displaystyle\kappa_{3}=iX-i\frac{\ln(\lambda X)}{X}.$ (94)

Then $J$ approximates to

$\displaystyle
J\sim\frac{i}{\lambda\kappa_{j}}=-\frac{X}{\lambda}W_{j-2}^{-1}\left(-\frac{X}{\lambda}\right),\quad
j\in\\{1,2\\}$ (95)

Since this equation can give any $J$ from $0$ to $1$, we thus confirm that
$J_{1}=0$, and only the second and third branches are relevant in Eqs. (61)
and (62). We also see that $J_{2}$ is obtained at the branch point of the
Lambert function, $-e^{-1}$. Namely $J_{2}=\frac{-e^{-1}}{W_{0}(-e^{-1})}$.

To find $s(J)$ from Eq. (90) for $s(\lambda)$, we first write
$J=\frac{i}{\lambda\kappa_{j}}=e^{i\kappa_{i}X},$ the last equality being a
consequence of the definition of $\kappa_{i}.$ This gives
$\kappa_{i}=-i\frac{\ln J}{X}$. Them, using again the equality
$J=\frac{i}{\lambda\kappa_{j}}$, we obtain the desired relation for $\lambda$
in terms of $X$ and $J$,

$\displaystyle\quad\lambda=\frac{X}{J\ln\frac{1}{J}}\,,$ (96)

which coincides with our second perturbative relation in Eq. (85). Using this
expression, we can rewrite $\kappa_{j}$ in terms of $J$ only:

$\displaystyle\kappa_{j}=\begin{cases}\frac{i\ln(1/J)}{X}&j\in\\{1,2\\}\\\
iX-i\frac{\ln\left(\frac{X^{2}}{J\ln(1/J)}\right)}{X}&j=3\\\ \end{cases}$ (97)

Then one can substitute this into the action Eq. (90) and arrive at

$s(J,X)=X\ln\left(\frac{X^{2}}{e^{2}\ln\frac{1}{J}}\right)+O(1)\,.$ (98)

A comparison of Eq. (98) with the perturbative result (88) shows that the
latter misses an important subleading factor $e^{-2}$ under the external
logarithm.

##  V Discussion

Here we studied the full distribution $\mathcal{P}(J,X,T)$ of the total heat
$J$ transferred to the right of a point $x=X$ in the KMP model at long times
for an initially localized heat pulse, thereby extending our previous work
BSM1  which was limited to $X=0$. We calculated exactly the rescaled rate
function $s(J,X)$ that describes this distribution, by combining the MFT and
the ISM. In particular we found that for $X>\sqrt{8}$ (in properly rescaled
variables) $s$ is not convex as a function of $J$, a rather unusual situation.
Furthermore, we studied some asymptotics of $s(J,X,T)$ by extracting them from
the exact solution and obtaining them perturbatively from the MFT equations.
In the asymptotic regimes we found the optimal histories of the system
conditioned on $J$. This gives an important additional physical insight, at
present unaccessible for the ISM. The distribution $\mathcal{P}(J,X,T)$ as a
function of $X$ at fixed $J$ has similar long-time dynamical scaling as the
distribution of a tracer particle in a single-file transport of interacting
particles, as in the SSEP for instance. It would be interesting to try to
apply the ISM to other large-deviation problems for the KMP model, in which
non-stationary heat transfer statistics problems were previously solved
analytically only in some limiting cases DG2009b ; KrMe ; MS2013 .

The MFT of lattice gases can be viewed as a particular case of the optimal
fluctuation method (OFM), a versatile tool which can be used to study large
deviations in a broad class of macroscopic systems that also includes
paradigmatic surface growth models (such as the KPZ equation), reaction-
diffusion systems, and many more. Mathematically, the OFM formulation involves
a set of coupled nonlinear partial differential equations whose solution gives
the optimal history of the system, conditioned on the rare event in question.
It is generally very hard to solve these equations exactly. Therefore,
uncovering and exploiting exact integrability of these equations appears to be
a way to make a significant progress in some of these systems. Indeed, the
present work is one of several very recent studies BSM1 ; KLD1 ; KLD2 ; MMS22
; KLD3  which use the ISM in order to obtain exact solutions. We hope that
these works are the beginning of a breakthrough in the study of large
deviations.

Note added. Ref. KLD3  was posted shortly before the present paper was
submitted. The authors of Ref. KLD3  focused on a crossover between different
large-deviation regimes for a model which is mathematically identical to the
KMP model. It appears that their work is closely related to ours, and it will
be interesting to make a detailed comparison of the results.

## Acknowledgment

The research of E.B. and B.M. is supported by the Israel Science Foundation
(grants No. 1466/15 and 1499/20, respectively).

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