$J_{\rho}=\rho v,\qquad J_{v}=\rho+v^{2}.$

These are the currents of the Leroux system [86, 87] (the particular case
$\alpha=\beta=\frac{1}{2}$ is the one considered in [84]), which is known to
be a Temple class system.

####  2.3.2 The general case: Riemann variables

For the problem under investigation one could expect that in addition to the
generic complexity of the analysis of a coupled system of conservation
equations, one has to face the further complication due to the implicit
dependence of the currents on the densities, which goes through the auxiliary
variables ${\bf z}$. Actually, quite unexpectedly, what seems a drawback of
the equations turns out to be the main feature which allows to solve them.
Indeed the variables ${\bf z}$ happen to be Riemann variables for our
conservation laws, i.e. they diagonalize the system of eqs.(2.22) and simplify
substantially their analysis. From now on we want to think of both $\bm{\rho}$
and ${\bf J}$ as functions of ${\bf z}$ .

We first notice that, solving eqs.(2.11,4.27) for the densities $\bm{\rho}$ in
terms of ${\bf z}$ and ${\bf J}$ and replacing them into eqs.(2.9,4.29), the
currents are the solution of the following linear system of equations

$\displaystyle\frac{J_{\circ}}{z_{\alpha}}+\frac{J_{\bullet}}{z_{\alpha}-1}-\frac{J_{\circ}+J_{\bullet}}{z_{\alpha}-\alpha}+1=0$
(2.24)
$\displaystyle\frac{J_{\bullet}}{z_{\beta}}+\frac{J_{\circ}}{z_{\beta}-1}-\frac{J_{\circ}+J_{\bullet}}{z_{\beta}-\beta}+1=0.$
(2.25)

Now differentiate the l.h.s. of eq.(2.9) with respect to $t$, differentiate
the l.h.s. of eq.(2.24) with respect to $x$ and sum the obtained results.
Thanks to the conservation laws eqs.(2.22) the derivatives
$\partial_{t}\bm{\rho}$ and $\partial_{x}{\bf J}$ cancel and one remains with

$\partial_{t}z_{\alpha}+v_{\alpha}({\bf z})\partial_{x}z_{\alpha}=0,\qquad
v_{\alpha}({\bf
z})=\frac{\left(\frac{J_{\circ}}{z_{\alpha}^{2}}+\frac{J_{\bullet}}{(z_{\alpha}-1)^{2}}-\frac{J_{\circ}+J_{\bullet}}{(z_{\alpha}-\alpha)^{2}}\right)}{\left(\frac{\rho_{\circ}}{z_{\alpha}^{2}}+\frac{\rho_{\bullet}}{(z_{\alpha}-1)^{2}}+\frac{1-\rho_{\circ}-\rho_{\bullet}}{(z_{\alpha}-\alpha)^{2}}\right)}.$
(2.26)

In the same way one obtains the equation for $z_{\beta}$

$\partial_{t}z_{\beta}+v_{\beta}({\bf z})\partial_{x}z_{\beta}=0,\qquad
v_{\beta}({\bf
z})=\frac{\left(\frac{J_{\bullet}}{z_{\beta}^{2}}+\frac{J_{\circ}}{(z_{\beta}-1)^{2}}-\frac{J_{\circ}+J_{\bullet}}{(z_{\beta}-\beta)^{2}}\right)}{\left(\frac{\rho_{\bullet}}{z_{\beta}^{2}}+\frac{\rho_{\circ}}{(z_{\beta}-1)^{2}}+\frac{1-\rho_{\circ}-\rho_{\bullet}}{(z_{\beta}-\beta)^{2}}\right)}.$
(2.27)

The speeds $v_{\alpha}$ and $v_{\beta}$ are the eigenvalues of the
linearization matrix $\partial_{\rho_{j}}J_{i}$, and on general grounds they
can also be written as

$v_{\alpha}=\partial_{\rho_{i}}J_{i}|_{z_{\beta}}=\frac{\partial_{z_{\alpha}}J_{i}({\bf
z})}{\partial_{z_{\alpha}}\rho_{i}({\bf z})},\qquad
v_{\beta}=\partial_{\rho_{i}}J_{i}|_{z_{\alpha}}=\frac{\partial_{z_{\beta}}J_{i}({\bf
z})}{\partial_{z_{\beta}}\rho_{i}({\bf z})}.$ (2.28)

A close inspection of their expression allows to conclude that

$v_{\beta}({\bf z})\geq v_{\alpha}({\bf z}),$ (2.29)

with the equality holding for $1-z_{\alpha}-z_{\beta}=0$, which is a non empty
set only for $\alpha+\beta\geq 1$. We conclude that for $\alpha+\beta<1$, the
system in (2.22) is strictly hyperbolic on the whole physical domain
$\mathcal{D}$, whereas for for $\alpha+\beta\geq 1$ it is degenerate
hyperbolic on the locus $1-z_{\alpha}-z_{\beta}=0$, i.e.
$\rho_{\ast}=0,\rho_{\circ}\leq\alpha,\rho_{\bullet}\leq\beta$ (green segments
in Figs. 2.2(a),2.2(b)) and strictly hyperbolic on the rest of the physical
domain.

Using eqs.(2.26,2.27) we can easily work out the rarefaction fans. These are
continuous solutions of eqs.(2.22), which depend only on the self-similarity
variable $\xi=x/t$, ${\bf z}(x,t)={\bf z}(\xi=x/t)$. They are given by the
solutions of the equations

$\begin{split}(v_{\alpha}-\xi)\partial_{\xi}z_{\alpha}&=0\\\
(v_{\beta}-\xi)\partial_{\xi}z_{\beta}&=0.\end{split}$ (2.30)

Locally we have four possibilities.

  1. 1.

The trivial solution, namely both $z_{\alpha}$ and $z_{\beta}$ are constant.

  2. 2.

Both $\partial_{\xi}z_{\alpha}\neq 0,\partial_{\xi}z_{\beta}\neq 0$. In this
case we must have $v_{\alpha}-\xi=v_{\beta}-\xi=0$, and in particular
$v_{\alpha}=v_{\beta}$. As mentioned above, this is possible only if
$1-z_{\alpha}-z_{\beta}=0$. In this case we get a _rarefaction fan_ of
equation

$\rho_{\circ}(\xi)=z_{\alpha}(\xi)=\frac{1+\xi}{2},\qquad\rho_{\bullet}(\xi)=z_{\bullet}(\xi)=\frac{1-\xi}{2}.$
(2.31)

Notice that the condition $1-z_{\alpha}-z_{\beta}=0$ implies absence of
$\ast$-particles, and the solution (2.31) corresponds to the fan solution of
the single species TASEP (upon identification of $\circ$-particles with empty
sites).

  3. 3.

$z_{\alpha}$ constant, $\partial_{\xi}z_{\beta}\neq 0$. In this case the
rarefaction fan ($\beta$–fan) is given by

$v_{\beta}(z_{\alpha},z_{\beta}(\xi,z_{\alpha}))=\xi.$ (2.32)

This equation, combined with the expression $\bm{\rho}(z_{\alpha},z_{\beta})$,
allows to write a $\beta$-fan in parametrized form ($z_{\alpha}$ is kept
constant while $z_{\beta}$ varies), see Fig. 2.4(b). One can show that (at
fixed $z_{\alpha}$) $v_{\beta}$ is a decreasing function of $z_{\beta}$, while
$\rho_{\bullet}$ is an increasing function of $z_{\beta}$ . This means that
$z_{\beta}(\xi,z_{\alpha})$ and $\rho_{\bullet}(\xi,z_{\alpha})$ are
decreasing functions of $\xi$.

  4. 4.

$z_{\beta}$ constant, $\partial_{\xi}z_{\alpha}\neq 0$. This case is similar
to the previous one, but the role of $\alpha$ and $\beta$ variables is
exchanged. So we speak of an $\alpha$–fan, given by

$v_{\alpha}(z_{\alpha}(\xi,z_{\beta}),z_{\beta})=\xi.$ (2.33)

One can show that (at fixed $z_{\beta}$) $v_{\alpha}$ and $\rho_{\circ}$ are
increasing function of $z_{\alpha}$. This means that
$z_{\alpha}(\xi,z_{\beta})$ and $\rho_{\circ}(\xi,z_{\beta})$ are increasing
functions of $\xi$.

Projection in the $\bm{\rho}$–plane of the three types of fans as well as an
example of a $\beta$–fan are represented in Fig. 2.4.

$\rho_{\circ}$$\rho_{\bullet}$$1$$1$ (a)

$v_{\beta}^{-}$$v_{\beta}^{+}$$\rho_{\bullet}$$\rho_{\ast}$$\rho_{\circ}$$\bm{\rho}$$\xi$
(b)

Figure 2.4:  (a) Projection in the $\bm{\rho}$–plane of the three types of
rarefactions fans in the case $\alpha,\beta<1$: in green a TASEP-like
rarefaction fan, in blue an $\alpha$–fan, in red a $\beta$–fan. (b) A plot of
a $\beta$–fan. The local densities correspond to the width of the
corresponding colored regions. In particular the line separating the yellow
region from the violet region represents the plot of the density
$\rho_{\bullet}(\xi)$. The extremes of the fan are given by :
$v_{\beta}^{-}=v_{\beta}(z_{\alpha},1-z_{\alpha}),v_{\beta}^{+}=v_{\beta}(z_{\alpha},0)$.

####  2.3.3 Shocks

It is well known that a smooth solution of a nonlinear conservation laws like
eqs.(2.22) may develop a shock discontinuity in finite time. One needs
therefore to admit the notion of _weak solution_ , i.e. solution in the sense
of distributions, which need not even be continuous. The discontinuity
associated to a shock with trajectory $x_{s}(t)$, has to satisfy the _Rankine-
Hugoniot jump relations_. If we denote by $[{\bm{\rho}}]$ and $[{\bf J}]$
respectively the discontinuities of the densities and of the currents across
the shock, i.e.

$[{\bm{\rho}}]=\bm{\rho}(x_{s}^{+})-\bm{\rho}(x_{s}^{-}),\qquad[{\bf J}]={\bf
J}(x_{s}^{+})-{\bf J}(x_{s}^{-})$

then the Rankine-Hugoniot jump relations read

$[{\bm{\rho}}]-v_{s}[{\bf J}]=0,\qquad v_{s}=\text{shock's speed}.$ (2.34)

The Rankine-Hugoniot jump relations, allow to express the speed of the shock
in terms of the discontinuity and at the same time put a constraint on the
admissible discontinuities, the Hugoniot condition:

$\det\left(\begin{array}[]{cc}[\rho_{\circ}]&[J_{\circ}]\\\\[5.0pt]
[\rho_{\bullet}]&[J_{\bullet}]\end{array}\right)=0.$ (2.35)

In order to analyze the Hugoniot condition, we consider
$\bm{\rho}(x_{s}^{-})=\bm{\rho}^{-}$ as fixed. For strictly hyperbolic systems
of $N$ conservation laws, it is known that the set of
$\bm{\rho}^{+}=\bm{\rho}(x_{s}^{+})$ satisfying the Hugoniot condition passes
through $\bm{\rho}^{-}$ and locally decompose around $\bm{\rho}^{-}$ in $N$
different branches [92], each one called a _shock curve_. In our case we
expect two shock curves passing through any point $\rho^{-}$, which correspond
to two different kinds of shocks. Using eq.(2.9) and eq.(2.24) we see that if
$z_{\alpha}^{+}=z_{\alpha}^{-}=z_{\alpha}$, then we have

$\begin{split}\frac{[\rho_{\circ}]}{z_{\alpha}}+\frac{[\rho_{\bullet}]}{z_{\alpha}-1}-\frac{[\rho_{\circ}]+[\rho_{\bullet}]}{z_{\alpha}-\alpha}=0\\\
\frac{[J_{\circ}]}{z_{\alpha}}+\frac{[J_{\bullet}]}{z_{\alpha}-1}-\frac{[J_{\circ}]+[J_{\bullet}]}{z_{\alpha}-\alpha}=0.\end{split}$
(2.36)

This means that the matrix in the eq.(2.35) has the left null vector
$(\frac{1}{z_{\alpha}}-\frac{1}{z_{\alpha}-\alpha},\frac{1}{z_{\alpha}-1}-\frac{1}{z_{\alpha}-\alpha})$
and therefore its determinant vanishes. We conclude that a straight line at
$z_{\alpha}$ constant is a shock curves. In the same way we find that the
lines at $z_{\beta}$ constant are also shock curves. In conclusion we have
found that we have two kind of admissible shocks

  * •

$\beta$-shocks:
$z_{\alpha}(\bm{\rho}^{-})=z_{\alpha}(\bm{\rho}^{+})=z_{\alpha}$ with speed
$v_{s,\beta}(z_{\alpha};z_{\beta}^{-},z_{\beta}^{+})$;

  * •

$\alpha$-shocks:
$z_{\beta}(\bm{\rho}^{-})=z_{\beta}(\bm{\rho}^{+})=z_{\beta}$, with speed
$v_{s,\alpha}(z_{\beta};z_{\alpha}^{-},z_{\alpha}^{+})$.

The corresponding shocks speeds
$v_{s,\beta}(z_{\alpha};z_{\beta}^{-},z_{\beta}^{+})$ and
$v_{s,\alpha}(z_{\beta};z_{\alpha}^{-},z_{\alpha}^{+})$ do not have
particularly transparent expressions except for some particular cases. In the
case $\alpha=\beta=1$, a $\beta$–shock is a discontinuity of the density
$\rho_{\bullet}$, with $\rho_{\circ}$ constant across the discontinuity, while
an $\alpha$–shock is the other way round, i.e. a discontinuity of the density
$\rho_{\circ}$, with $\rho_{\bullet}$ constant across the discontinuity. Shock
curves coincide with rarefaction curves so this is an example of a
conservation system of Temple class [68].

A further analysis of the Hugoniot condition allows to conclude that in the
bulk of the physical domain $\mathcal{D}$, the only possible shocks are
$\alpha$ and $\beta$–shocks. There exists however one more class of shocks
when both sides of the discontinuity lie on the boundary line $\rho_{\ast}=0$.
These shocks have speed $v_{s}=\frac{[J_{\bullet}]}{[\rho_{\bullet}]}$, where
the current $J_{\bullet}$ is given by eq.(2.16).

It is possible to show that at fixed $z_{\alpha}$ the current $J_{\bullet}$ is
a concave function of the density $\rho_{\bullet}$ (see fig. 2.5 ). This
implies that, for a fixed value of the densities on one side of the shock, say
$\bm{\rho}^{-}$, the speed of a $\beta$–shock is a decreasing function of
$\rho_{\bullet}^{+}$.

$\rho_{\bullet}$$J_{\bullet}$0.20.40.60.810.10.20.30.40.5(a)$\rho_{\bullet}$$J_{\bullet}$0.20.40.60.810.050.10.150.20.25(b)
Figure 2.5: Current $J_{\bullet}$ as function of $\rho_{\bullet}$ at constant
$z_{\alpha}=.2$. (a) Fixed $\alpha=4$ and decreasing $\beta$ (blue line
$\beta=5$, green line $\beta=1$, red line $\beta=0.3$). (b) Fixed $\beta=0.3$
and decreasing $\alpha$ (blue line $\alpha=3$, green line $\alpha=1$, red line
$\alpha=0.3$).

This property can be conveniently reformulated in terms of the ${\bf z}$
variables. At constant $z_{\alpha}$, $\rho_{\bullet}$ is an increasing
function of $z_{\beta}$, hence at fixed $z_{\alpha}$ and $z_{\beta}^{-}$, the
speed of $\beta$–shock $v_{s,\beta}(z_{\alpha};z_{\beta}^{-},z^{+}_{\beta})$
is a decreasing function of $z^{+}_{\beta}$. In particular it takes its
minimum for the largest $z_{\beta}$ allowed, i.e.
$z_{\beta,\textrm{max}}=1-z_{\alpha}$

$v_{s,\beta}(z_{\alpha};z_{\beta},z^{+}_{\beta})\geq
v_{s,\beta}(z_{\alpha};z_{\beta},1-z_{\alpha})\qquad 0\leq z^{+}_{\beta}\leq
1-z_{\alpha}.$ (2.37)

The current $J_{\circ}$ is a convex function of $\rho_{\circ}$ at fixed
$z_{\beta}$. A similar reasoning as the one presented above allows to conclude

$v_{s,\alpha}(z_{\beta};z_{\alpha},z^{+}_{\alpha})\leq
v_{s,\alpha}(z_{\beta};z_{\alpha},1-z_{\beta})\qquad 0\leq z^{+}_{\alpha}\leq
1-z_{\beta}.$ (2.38)

Now, from an explicit computation, we notice that

$v_{s,\alpha}(z_{\beta};z_{\alpha},1-z_{\beta})=v_{s,\beta}(z_{\alpha};z_{\beta},1-z_{\alpha})=z_{\alpha}-z_{\beta}.$

This allows to conclude that

$v_{s,\alpha}(z_{\beta};z_{\alpha},\widetilde{z_{\alpha}})\leq
v_{s,\beta}(z_{\alpha};z_{\beta},\widetilde{z_{\beta}}).$ (2.39)

In words this means that, for a fixed value of the densities on one side of
the shock, the speed of any $\beta$–shock is larger than the speed of any
$\alpha$–shock.

Since $\lim_{\widetilde{z_{\alpha}}\rightarrow
z_{\alpha}}v_{s,\alpha}(z_{\beta};z_{\alpha},\widetilde{z_{\alpha}})=v_{\alpha}(z_{\alpha},z_{\beta})$
and $\lim_{\widetilde{z_{\beta}}\rightarrow
z_{\beta}}v_{s,\beta}(z_{\alpha};z_{\beta},\widetilde{z_{\beta}})=v_{\beta}(z_{\alpha},z_{\beta})$,
we get also

$\displaystyle v_{\alpha}(z_{\alpha},z_{\beta})\leq
v_{s,\beta}(z_{\alpha};z_{\beta},\widetilde{z_{\beta}})$ (2.40) $\displaystyle
v_{\beta}(z_{\alpha},z_{\beta})\geq
v_{s,\alpha}(z_{\beta};z_{\alpha},\widetilde{z_{\alpha}}).$ (2.41)

The Hugoniot condition is not sufficient to select the physical shocks. Indeed
eqs.(2.22) has to be thought of as the zero viscosity limit of a set of
conservation laws which contains a diffusive term, a term which comes from the
microscopic corrections to the currents and depends on the derivatives of the
densities. Inviscid limits of viscous solutions are typically characterized by
entropy conditions, which for shocks take the form of the Liu entropy
criterion [93, 92]. Let the right densities $\bm{\rho}^{+}$ lie on a Hugoniot
curve emanating from $\bm{\rho}^{-}$, then for all densities
$\widetilde{\bm{\rho}}$ lying on the same Hugoniot curve in between
$\bm{\rho}^{-}$ and $\bm{\rho}^{+}$ the Liu condition states that

$v_{s}(\bm{\rho}^{-},\bm{\rho}^{+})\leq
v_{s}(\bm{\rho}^{-},\widetilde{\bm{\rho}}).$ (2.42)

Physically this condition can be understood as a stability condition: if under
a perturbation the shock were to split by inserting an intermediate state
$\widetilde{\bm{\rho}}$, then a violation of condition (2.42) would imply that
the shock between $\bm{\rho}^{-}$ and $\widetilde{\bm{\rho}}$ would move away
from the original shock between $\bm{\rho}^{-}$ and $\bm{\rho}^{+}$. In the
case of $\beta$–shocks the Liu condition reads

$v_{s,\beta}(z_{\alpha};z_{\beta}^{-},z_{\beta}^{+})\leq
v_{s,\beta}(z_{\alpha};z_{\beta}^{-},\widetilde{z_{\beta}}),\qquad\forall\widetilde{z_{\beta}}\in[\min(z_{\beta}^{-},z_{\beta}^{+}),\max(z_{\beta}^{-},z_{\beta}^{+})]$
(2.43)

Since, as mentioned above, at fixed $z_{\alpha}$ the current $J_{\beta}$ is a
concave function of the density $\rho_{\bullet}$,
$\partial_{\rho_{\bullet}}^{2}J_{\beta}|_{z_{\alpha}}<0$, we conclude that the
Liu constraint means $\rho_{\bullet}^{-}<\rho_{\bullet}^{+}$. This can be also
formulated in terms of the $z$ variables. Indeed, since
$\partial_{z_{\beta}}\rho_{\bullet}|_{z_{\alpha}}>0$, we must have
$z_{\beta}^{-}<z_{\beta}^{+}$. A similar analysis can be performed for
$\alpha$–shocks.

Here is a summary of our results. They are schematized in the figure on the
right, where we have reported the direction of the possible
shock–discontinuities.

  * •

For an $\alpha$–shocks (blue oriented line), we need
$\rho_{\circ}^{-}>\rho_{\circ}^{+}$ or equivalently
$z_{\alpha}^{-}>z_{\alpha}^{+}$.

  * •

For a $\beta$–shocks (red oriented line), we need
$\rho_{\bullet}^{-}<\rho_{\bullet}^{+}$ or equivalently
$z_{\beta}^{-}<z_{\beta}^{+}$.

$\rho_{\circ}$$\rho_{\bullet}$$1$$1$

####  2.3.4 Riemann’s problem

With the result for the rarefaction curves and the shock curves at our
disposal, it is rather simple to describe the general solution of the Riemann
problem, i.e. the solution of eqs.(2.22) with domain wall initial conditions

$\bm{\rho}(x,t=0)=\left\\{\begin{array}[]{l}\bm{\rho}^{L}=(\rho_{\circ}^{L},\rho_{\bullet}^{L})\qquad
x<0\\\ \bm{\rho}^{R}=(\rho_{\circ}^{R},\rho_{\bullet}^{R})\qquad
x>0.\end{array}\right.$ (2.44)

By uniqueness, the solution of the Riemann problem has to take the form
$\bm{\rho}(x,t)=\bm{\rho}(\xi)$ with $\xi=x/t$ and is given by a sequence of
rarefaction waves and/or shocks. It is best described in terms of the
variables ${\bf z}^{L}=(z_{\alpha}^{L},z_{\beta}^{L}),{\bf
z}^{R}=(z_{\alpha}^{R},z_{\beta}^{R})$. We have four possible situations
(these are schematically summarized in figure 2.6).

$\bullet$ $z^{L}_{\alpha}>z^{R}_{\alpha},z^{L}_{\beta}<z^{R}_{\beta}$. The
solution is composed of two shocks: an $\alpha$-shock with ${\bf
z}^{-}=(z^{L}_{\alpha},z^{L}_{\beta})$ and ${\bf
z}^{+}=(z^{R}_{\alpha},z^{L}_{\beta})$ at position
$\xi_{\alpha}=v_{s,\alpha}(z^{L}_{\beta};z^{L}_{\alpha},z^{R}_{\alpha})$,
followed by a $\beta$–shock with ${\bf z}^{-}=(z^{R}_{\alpha},z^{L}_{\beta})$
and ${\bf z}^{+}=(z^{R}_{\alpha},z^{R}_{\beta})$ at position
$\xi_{\beta}=v_{s,\beta}(z^{R}_{\alpha};z^{L}_{\beta},z^{R}_{\beta})$. This
result follows from the inequality (2.39)

$\xi_{\alpha}=v_{s,\alpha}(z^{L}_{\beta};z^{L}_{\alpha},z^{R}_{\alpha})\leq
v_{s,\beta}(z^{R}_{\alpha};z^{L}_{\beta},z^{R}_{\beta})=\xi_{\beta}.$

$\bullet$ $z^{L}_{\alpha}>z^{R}_{\alpha},z^{L}_{\beta}>z^{R}_{\beta}$. The
solution is composed of an $\alpha$-shock and a $\beta$–fan. The
$\alpha$–shock has ${\bf z}^{-}=(z^{L}_{\alpha},z^{L}_{\beta})$ and ${\bf
z}^{+}=(z^{R}_{\alpha},z^{L}_{\beta})$ and it is located at position
$\xi_{\alpha}=v_{s,\alpha}(z^{L}_{\beta};z^{L}_{\alpha},z^{R}_{\alpha})$. The
$\beta$–fan starts with value $(z^{R}_{\alpha},z^{L}_{\beta})$ at
$\xi_{\beta,1}=v_{\beta}(z^{R}_{\alpha},z^{L}_{\beta})$ and ends with value
$(z^{R}_{\alpha},z^{R}_{\beta})$ at
$\xi_{\beta,2}=v_{\beta}(z^{R}_{\alpha},z^{R}_{\beta})$. This result follows
from the inequality

$\xi_{\alpha}=v_{s,\alpha}(z^{L}_{\beta};z^{L}_{\alpha},z^{R}_{\alpha})\leq
v_{\beta}(z^{R}_{\alpha},z^{L}_{\beta})=\xi_{\beta,1}\leq
v_{\beta}(z^{R}_{\alpha},z^{R}_{\beta})=\xi_{\beta,2}.$

The first inequality is just inequality (2.41), while the second one follows
from the fact that $v_{\beta}$ is a decreasing function of $z_{\beta}$.

$\bullet$ $z^{L}_{\alpha}<z^{R}_{\alpha},z^{L}_{\beta}<z^{R}_{\beta}$. The
solution is composed of an $\alpha$–fan and a $\beta$–shock. The $\alpha$–fan
starts with value $(z^{L}_{\alpha},z^{L}_{\beta})$ at
$\xi_{\alpha,1}=v_{\alpha}(z^{L}_{\alpha},z^{L}_{\beta})$ and ends with value
$(z^{R}_{\alpha},z^{L}_{\beta})$ at
$\xi_{\alpha,2}=v_{\alpha}(z^{R}_{\alpha},z^{L}_{\beta})$. The $\beta$ shock
has ${\bf z}^{-}=(z^{R}_{\alpha},z^{L}_{\beta})$ and ${\bf
z}^{+}=(z^{R}_{\alpha},z^{R}_{\beta})$ and is located at position
$\xi_{\beta}=v_{s,\beta}(z^{R}_{\alpha};z^{L}_{\beta},z^{R}_{\beta})$.

$\bullet$ $z^{L}_{\alpha}<z^{R}_{\alpha},z^{L}_{\beta}>z^{R}_{\beta}$. The
solution is composed of fans. One has to distinguish two cases depending
whether $z_{\alpha}^{R}+z_{\beta}^{L}$ is larger or smaller than $1$. If
$z_{\alpha}^{R}+z_{\beta}^{L}<1$, the solution consists of an $\alpha$–fan
starting with value $(z_{\alpha}^{L},z_{\beta}^{L})$ at
$\xi_{\alpha,1}=v_{\alpha}(z^{L}_{\alpha},z^{L}_{\beta})$ and ending with
value $(z_{\alpha}^{R},z_{\beta}^{L})$ at
$\xi_{\alpha,2}=v_{\alpha}(z^{R}_{\alpha},z^{L}_{\beta})$ followed by a
$\beta$–fan starting at
$\xi_{\beta,1}=v_{\beta}(z^{R}_{\alpha},z^{L}_{\beta})$ and ending at
$\xi_{\beta,2}=v_{\beta}(z^{R}_{\alpha},z^{R}_{\beta})$. If $\alpha+\beta>1$
then it is possible to have $z_{\alpha}^{R}+z_{\beta}^{L}>1$. In this case the
$\alpha$-fan cannot reach the value $(z_{\alpha}^{R},z_{\beta}^{L})$, which
lies outside the physical domain. It ends at the value
$(1-z_{\beta}^{L},z_{\beta}^{L})$, followed by a degenerate fan till
$(z_{\alpha}^{R},1-z_{\alpha}^{R})$ and then by a $\beta$–fan till
$(z_{\alpha}^{R},z_{\beta}^{R})$.

$1$$1$$z_{\alpha}$$z_{\beta}$LRRRRR Figure 2.6: Projection in the ${\bf
z}$–plane of the different type of solutions of the Riemann problem. The left
values of the ${\bf z}$ variables are represented by the point $L$. The points
$R$ represent the different distinct possibilities for the right values of the
${\bf z}$ variables. Continuous lines represent fans, while dashed lines
represent shocks. In green is indicated the possible TASEP-like fan, in red
either an $\alpha$–shock or an $\alpha$–fan and in blue either a $\beta$–shock
or a $\beta$–fan.

###  2.4 Monte Carlo simulations

We come back to the original microscopic stochastic model and compare the
predictions of the hydrodynamic equations with numerical simulations. We have
simulated our model on a finite lattice of integer coordinates, running on the
interval $[-L,L]$, with $L=2100$. The system is initialized in a random
configuration sampled from a product measure of local densities
$\bm{\rho}^{L}$ on sites of coordinate $i<0$ and $\bm{\rho}^{R}$ on sites of
coordinates $i\geq 0$. At the left and right boundaries the particles are
chosen neither to leave nor to enter the system. This means that we expect to
find three distinct regions: two kinetic waves coming from the boundaries and
a kinetic wave originating from the discontinuity at the origin. Whereas we
make no prediction on the boundary waves, we expect that as long as they don’t
meet the bulk one, they do not influence the latter.

Let us introduce the height function, which is defined up to an arbitrary
additive constant by

$h_{i}\left(\frac{n}{t},t\right)-h_{i}\left(-1,t\right):=\frac{1}{t}\sum_{-t<k\leq
n}\chi_{i}\left(k,t\right)\quad\quad-t<n<t,$ (2.45)

From our assumption eq.(2.21), it follows that at large time and for each
sample, the height function should converge to the deterministic shape

$h_{i}(\xi)-h_{i}(-1)=\int_{-1}^{\xi}\rho_{i}(\xi^{\prime})d\xi^{\prime},$
(2.46)

where $\rho_{i}(\xi)$ is the solution of the Riemann problem found in Section
2.3.4.

(a) $(\rho_{\circ}^{L}=0.45,\rho_{\bullet}^{L}=0.11)$,
$(\rho_{\circ}^{R}=0.23,\rho_{\bullet}^{R}=0.67)$

(b) $(\rho_{\circ}^{L}=0.23,\rho_{\bullet}^{L}=0.67)$,
$(\rho_{\circ}^{R}=0.45,\rho_{\bullet}^{R}=0.11)$

(c) $(\rho_{\circ}^{L}=0.8,\rho_{\bullet}^{L}=0.1)$,
$(\rho_{\circ}^{R}=0.2,\rho_{\bullet}^{R}=0.7)$

(d) $(\rho_{\circ}^{L}=0.2,\rho_{\bullet}^{L}=0.1)$,
$(\rho_{\circ}^{R}=0.5,\rho_{\bullet}^{R}=0.4)$

Figure 2.7: Height function of the $\bullet$-particles (green lines) and of
the $\circ$-particles (red lines). Dashed lines represent the numerical
values, continuous lines represent the theoretical prediction. At the bottom
of each plot we have reported the predicted densities.

In Figs. 2.7,2.8 we illustrate some representative results of our simulations.
We have run each simulation up to a time $t$ indicated at the top of each
plot. At that time we find that the left boundary wave has not yet reached the
site $i=-t$ and the right boundary wave has not reached the site $i=t$, so we
can safely use eq.(2.45) as a definition of the height function and plot it
for $-1\leq\xi\leq 1$ (for convenience we chose $h_{i}(-1)=1$). First we
illustrate the simulation of the system for $\alpha=0.3$, $\beta=0.8$ (Figs.
2.7(a)-2.7(c)). In Fig. 2.7(a) the left densities are
$(\rho_{\circ}^{L}=0.45,\rho_{\bullet}^{L}=0.11)$ and the right densities are
$(\rho_{\circ}^{R}=0.23,\rho_{\bullet}^{R}=0.67)$, which in terms of the ${\bf
z}$ variables correspond to a uniform $z_{\alpha}^{L}=z_{\alpha}^{R}=0.15$ and
$z_{1}^{L}=0.1<z_{1}^{R}=0.6$. This choice of boundary densities is expected
to lead to the formation of a single $\beta$-shock. Indeed the plot of the
height functions for the $\bullet$-particles and $\circ$-particles shows two
linear regions, corresponding to the two regions of constant densities,
separated at the predicted shock speed, situated at $\xi=v_{s,\beta}=0.217$.
In Fig. 2.7(b) the boundary densities are inverted w.r.t. Fig. 2.7(a). In this
case we don’t expect any shock to persist, and indeed the result is compatible
with a single fan. In Fig. 2.7(c) we explore the separation of two shocks. The
boundary densities are $(\rho_{\circ}^{L}=0.8,\rho_{\bullet}^{L}=0.1)$,
$(\rho_{\circ}^{R}=0.2,\rho_{\bullet}^{R}=0.7)$, which correspond to
$z_{\alpha}^{L}=0.265>z_{\alpha}^{R}=0.139$ and
$z_{\beta}^{L}=0.097<z_{\beta}^{R}=0.627$. In this case the solution of the
Riemann problem predicts an $\alpha$–shock of speed $v_{s,\alpha}=0.008$
followed by a $\beta$-shock of speed $v_{s,\beta}=0.186$. The $\beta$ shock is
clearly visible at the bend of $h_{\bullet}$ (green line), the $\alpha$ shock
is less visible because of the small bends in both height functions. In Fig.
2.7(d), we have $\alpha=0.7,\beta=1.2$ and we show the result of the
simulation where the theoretical analysis predicts an $\alpha$–fan and a
$\beta$–shock, the shock being visible at the bend of $h_{\bullet}$ (green
line), located at $\xi=v_{s,\beta}=0.49$.

$\rho_{\circ}$$\rho_{\bullet}$$L$$R$$1$$1$

Figure 2.8: Situation diplaying an $\alpha$ and a $\beta$–fan with a depletion
region in the middle,
$(\rho_{\circ}^{L}=0.1,\rho_{\bullet}^{L}=0.5,\rho_{\ast}^{L}=0.4),(\rho_{\circ}^{R}=0.5\rho_{\bullet}^{R}=0.05,\rho_{\ast}^{R}=0.45)$.
_On the left:_ height function of $\bullet$-particles (green lines) and of
$\ast$-particles. Continuous lines represent the numerical values, dashed
lines represent the theoretical prediction. At the bottom we have reported the
predicted densities. Notice the flat central region of $h_{\ast}$
corresponding to a region of vanishing $\rho_{\ast}$. _On the right:_ the
projection of this configuration in the densities plane: in blue the $\beta$
fan, in green the TASEP-like fan with vanishing $\rho_{\ast}$ and in red the
$\alpha$ fan.

Finally in Fig.2.8 we explore the formation of a TASEP–like fan corresponding
to a region where the density of $\ast$–particles vanishes. We chose
$\alpha=1.7,\beta=2$,
$(\rho_{\circ}^{L}=0.1,\rho_{\bullet}^{L}=0.5,\rho_{\ast}^{L}=0.4),(\rho_{\circ}^{R}=0.5,\rho_{\bullet}^{R}=0.05,\rho_{\ast}^{R}=0.45)$
so that initially the $\ast$-particles are present everywhere in the system.
The plot of $h_{\ast}$ (purple line) shows a central flat region, which
corresponds to a region of vanishing $\ast$-particles density.

###  2.5 Conclusion

In this article we have investigated the hydrodynamic behavior of an exactly
solvable two species exclusion process, consisting of two kinds of particles
moving in opposite directions on a one dimensional lattice and swapping their
position when adjacent. By making the assumption of local stationarity and
exploiting the knowledge of the particle currents on periodic geometry, we
have written down the hydrodynamic conservation laws of the model and
investigated their solutions: rarefaction fans, shocks and the solution of the
Riemann problem. Then such predictions have been shown to be in agreement with
numerical simulations of the microscopic model. The macroscopic conservation
laws are shown to belong to a class of conservation laws called Temple
systems, which possess coinciding rarefaction and shock curves.

While this property has important implications for the mathematical analysis
of the conservation equations, it is not clear to us what is its physical
underpinning. We believe it would be interesting to investigate how generic
this property is among exclusion processes, and whether it is somehow related
to the integrability of the underlying microscopic model. On this line of
thoughts, natural candidates to investigate seem to be the multispecies
integrable generalizations (with more than $2$ species) introduced in [77].

Another interesting issue would be to understand the behavior of our model
when restricted to a finite lattice in contact with boundary particle
reservoirs. On general grounds, novel feautures are expected to emerge like
for example the phenomenon of boundary induced phase transitions [21]. For the
case of a single conserved quantity the well-known max–min principle of Krug
[21], later generalized by Popkov and Schütz [22] allows to determine the
dependence of the current on the boundary couplings and hence the phase
diagram of the model. However, it is not known how to generalize the max–min
principle in the case of more than one conserved species. Preliminary
numerical investigations based on the model presented here seems to indicate
that the Riemann problem may play a relevant role in order to tackle this
problem [94].

##  Chapter 3 Integrable tools for the exclusion process

Exactly solvable models are rare gems in theoretical physics. In the domain of
far from equilibrium statistical physics, they are paving the way for its
exploration. In that regard, the exclusion process within its different
variants is playing this role in a way often compared to the role the Ising
model played for the equilibrium counterpart. In this chapter we are
considering two popular boundary conditions for the exclusion process: The
infinite $\mathbb{Z}$ lattice, and the periodic boundary conditions. For the
former, we are interested in the finite time probability distribution of the
position of a finite number of particles conditioned on a given initial
configuration. In the latter, the main objective is to find the expression of
the currents in the steady state in a system with multiple species with
arbitrary rates. The two problems seem quite distinct at the first glance, in
particular, because the first does not possess a stationary state while the
second does. Yet, it happens that a similar technique works for both as a
starting point: writing the stationary measure as Bethe vector. This measure
will not be a probability measure for the first and will not lead to Bethe
equations, while for the second it will. Exact probability calculations on the
line gained importance after the seminal work of [39] revealing a connection
between properly scaled finite probabilities for TASEP and the largest
eigenvalue distribution for the Gaussian Unitary Ensemble.

The novelty of this chapter is the section 3.1.3 where we present a general
framework for calculating the finite time conditional probability for an
arbitrary number of species with arbitrary hopping rates. This leads to
explicit determinantal formulas in particular cases. The rest of section 3.1
serves a pedagogical purpose: we start by reviewing the simplest version of
the problem, which is TASEP with a single species, this roughly follows [34],
the complexity is then increased gradually by visiting briefly the problem
with second class particles of unity rates, that has been discussed in [37]
before reaching the general multi-species situation in section 3.1.3. This
section will the core of a near-future independent publication.

Section 3.2 is devoted to reviewing how the Bethe Ansatz is applied for
finding the currents in the exclusion process on the ring, most of the results
obtained here are needed for the other chapters. The complexity is again
increased gradually by first considering the problem of a single defect and
treating it with Coordinate Bethe Ansatz following [33], and then reviewing
the arbitrary number of defects using the Algebraic Bethe Ansatz, following
[46].

###  3.1 Exclusion process on the line

####  3.1.1 Exact solution for TASEP on the line

Consider a single component TASEP with a finite number of particles $N$,
defined on the integers line $\mathbb{Z}$. Let
$\\{x_{1},...,x_{N}\\}\subset\mathbb{Z}$ the initial positions of particles
arranged in increasing order: $x_{i}<x_{i+1}$. Let
$\\{y_{1},...,y_{N}\\}\subset\mathbb{Z}$ the final position of these particles
at time $t$. The state space will be:
$\mathcal{S}=\\{X\subset\mathbb{Z},\\#X=N\\}$.

We would like to calculate the conditional probability
$P(\\{y_{1},...,y_{N}\\};t|\\{x_{1},...,x_{N}\\}$. By abuse of notation, we
omit the initial condition dependence. This probability evolves according to
the Master equation: If there is no neighboring particles, $y_{i}<y_{i+1}-1$
for all $i$, then:

$\frac{d}{dt}P(\\{y_{1},...,y_{N}\\};t)=\sum_{i=1}^{N}P(\\{y_{1},..,y_{i}-1,..,y_{N}\\};t)-NP(\\{y_{1},...,y_{N}\\};t)$
(3.1)

In case two particles are neighbors, say for instance: $y_{j+1}=y_{j}+1$, and
all the others are apart, then there will be two terms missing on right-hand
side:

$\frac{d}{dt}P(\\{y_{1},..,y_{j},y_{j}+1,..,y_{N}\\};t)=\sum_{i\neq
j+1}P(\\{y_{1},..,y_{i}-1,..,y_{N}\\};t)-(N-1)P(\\{y_{1},...,y_{N}\\};t)$
(3.2)

It’s possible to make equation 3.1 account for equation 3.2 if we allow the
probability function to assign values for non-physical configurations with
consecutive particles, these values have to be:

$P(\\{y_{1},..,y_{j},y_{j+1}=y_{j},..,y_{N}\\};t)=P(\\{y_{1},..,y_{j},y_{j}+1,..,y_{N}\\};t)$
(3.3)

Now with this condition, it’s easy to check that equation 1 becomes valid even
with neighboring particles.

##### Bethe Ansatz Solution

The probability distribution can be seen as a vector in the infinite dimension
space of functions $P(\\{.\\})\in\mathbb{R}^{\mathcal{S}}$. Solving the
ordinary differential equation is equivalent to diagonalizing the Markov
matrix. So we need to consider the spectral problem:

$\lambda P(\\{.\\})=MP(\\{.\\})$ (3.4)

One Particle: Let’s first examine this equation for the almost trivial case of
one particle $N=1$, No Ansatz is required here. We have a functional equation:

$\lambda P(\\{y\\})=P(\\{y-1\\})-P(\\{y\\})$ (3.5)

Which admits plane waves as solutions:

$P_{p}(\\{y\\})=e^{ipy}$ (3.6)

The parameter $p$ is analogous to the momentum and is associated with the
eigenvalue:

$\lambda_{p}=e^{-ip}-1$ (3.7)

And the associated time evolution will be:

$P_{p}(\\{y\\};t)=e^{\lambda_{p}t}e^{ipy}$ (3.8)

Of course, these solutions are not physical, since they are not normalizable,
and not even real. However, we can decompose our initial condition in terms of
these waves: P(x,t=0)=
$\delta_{x,y}=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-ipx}e^{ipy}dp$. The bounded
interval of the integral is due to the integer dependence of the left side,
it’s a Fourier series decomposition. Now we can write the time evolution of
this initial condition:

$\begin{split}P(\\{y\\};t)&=\frac{1}{2\pi
i}\int_{0}^{2\pi}e^{-ipx}P_{p}(\\{y\\};t)dp\\\ &=\frac{1}{2\pi
i}\int_{0}^{2\pi}e^{(e^{-ip}-1)t}e^{-ip(x-y)}dp\\\ &=\frac{1}{2\pi
i}\oint_{0}e^{(z-1)t}z^{(x-y-1)}dz\\\
&=e^{-t}\frac{t^{y-x}}{(y-x)!}\end{split}$ (3.9)

Where the last step integral was performed by residues theorem.

Two particles: The situation is a bit more complicated for $N=2$. The
eigenvalue problem for the Markov matrix is:

$\lambda
P(\\{y_{1},y_{2}\\})=P(\\{y_{1}-1,y_{2}\\})+P(\\{y_{1},y_{2}-1\\})-2P(\\{y_{1},y_{2}\\})$
(3.10)

We can again check that $e^{i(p_{1}y_{1}+p_{2}y_{2})}$ is indeed a family of
solutions for the equation. However, this family is not compatible with the
boundary condition:

$P(\\{y_{1},y_{2}=y_{1}\\})=P(\\{y_{1},y_{2}=y_{1}+1\\})$ (3.11)

At this point, the Bethe Ansatz is needed. The solutions can be expressed as a
superposition of the possible permutations of the momentum of the plane waves:

$P_{\\{p_{1},p_{2}\\}}(\\{y_{1},y_{2}\\})=A_{1,2}e^{i(p_{1}y_{1}+p_{2}y_{2})}+A_{2,1}e^{i(p_{2}y_{1}+p_{1}y_{2})}$
(3.12)

With this combination, the boundary condition can be verified for a right
choice of the coefficients, namely:

$\frac{A_{1,2}}{A_{2,1}}=-\frac{1-e^{ip_{1}}}{1-e^{ip_{2}}}$ (3.13)

And the corresponding eigenvalue would be:

$\lambda_{\\{p_{1},p_{2}\\}}=e^{-ip_{1}}+e^{-ip_{2}}-2=\lambda_{p_{1}}+\lambda_{p_{2}}$
(3.14)

Now we need to write the initial condition as a superposition of this family:

$\delta_{x_{1},y_{1}}\delta_{x_{2},y_{2}}=\frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}f(p_{1},p_{2})(e^{i(p_{1}y_{1}+p_{2}y_{2})}-\frac{1-e^{ip_{2}}}{1-e^{ip_{1}}}e^{i(p_{2}y_{1}+p_{1}y_{2})})dp_{1}dp_{2}$
(3.15)

Actually this integral is not well defined because of the presence of a
singularity at $p_{1}=0$. Let’s write it as an integral in the complex plane:

$\delta_{x_{1},y_{1}}\delta_{x_{2},y_{2}}=\oint_{0}\oint_{0}f(z_{1},z_{2})(z_{1}^{y_{1}-1}z_{2}^{y_{2}-1}-\frac{1-z_{2}}{1-z_{1}}z_{1}^{y_{2}-1}z_{2}^{y_{1}-1})dz_{1}dz_{2}$
(3.16)

To define this integral, we need to tell whether the 1 is included in the unit
circle. If we choose to exclude it, which means taking the contour integral
only around zero, then the naive expression for $f$:
$f(z_{1},z_{2})=z_{1}^{-x_{1}}z_{2}^{-x_{2}}$ would yield a solution for the
previous equation. Finally, we can write the time-evolved solution:

$\begin{split}P(\\{y_{1},y_{2}\\};t)=&\oint\oint
e^{(z_{1}^{-1}+z_{2}^{-1}-2)t}(z_{1}^{y_{1}-x_{1}-1}z_{2}^{y_{2}-x_{2}-1}-\frac{1-z_{2}}{1-z_{1}}z_{1}^{y_{2}-x_{1}-1}z_{2}^{y_{1}-x_{2}-1})dz_{1}dz_{2}\\\
=&(e^{-t}\oint e^{\frac{t}{z_{1}}}z_{1}^{y_{1}-x_{1}-1}dz_{1})(e^{-t}\oint
e^{\frac{t}{z_{2}}}z_{2}^{y_{2}-x_{2}-1}dz_{2})\\\ &-(e^{-t}\oint
e^{\frac{t}{z_{1}}}\frac{z_{1}^{y_{2}-x_{1}-1}}{1-z_{1}}dz_{1})(e^{-t}\oint
e^{\frac{t}{z_{2}}}(1-z_{2})z_{2}^{y_{1}-x_{2}-1}dz_{2})\end{split}$ (3.17)

This suggests to define the function:

$F_{p}(n;t):=e^{-t}\oint e^{\frac{t}{z}}\frac{z^{n-1}}{(1-z)^{p}}dz$ (3.18)

So that we can write the previous expression as:

$\begin{split}P_{\\{p_{1},p_{2}\\}}(\\{y_{1},y_{2}\\};t)&=F_{0}(y_{1}-x_{1};t)F_{0}(y_{2}-x_{2};t)-F_{1}(y_{2}-x_{1};t)F_{-1}(y_{1}-x_{2};t)\\\
&=Det((F_{i-j}(y_{i}-x_{j};t))_{ij})\end{split}$ (3.19)

##### Properties of the functions $F_{p}(n,t)$

First of all these functions can be expressed in series form, by developing
the exponential in the integral, exchanging the integral and the sum, and then
integrating each term by the residues theorem. Consider first that $p>0$

$\begin{split}F_{p}(n;t)&=e^{-t}\oint\sum_{k=0}^{\infty}\frac{t^{k}}{z^{k}k!}z^{n-1}\sum_{m=0}^{\infty}C_{m+p-1}^{p-1}z^{m}dz\\\
&=e^{-t}\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}\oint\frac{t^{k}}{k!}C_{m+p-1}^{p-1}z^{m+n-k-1}dz\\\
&=e^{-t}\sum_{k=n}^{\infty}\frac{t^{k}}{k!}C_{k-n+p-1}^{p-1}=e^{-t}\sum_{k=0}^{\infty}C_{k+p-1}^{p-1}\frac{t^{k+n}}{(k+n)!}\end{split}$
(3.20)

For $p\leq 0$ the power series expansion above is still actually valid, but
one has to extend the definition of $C_{a}^{b}$ to negative numbers. This is
possible using the $\Gamma$ function:

$C_{a}^{b}=\frac{\Gamma(a+1)}{\Gamma(b+1)\Gamma(a-b+1)}$

This can be used to show that for $p\leq 0$:

$C_{p+m-1}^{m}=\begin{cases}(-1)^{m}C_{-p}^{m}\quad&\text{if}\quad m<p+1\\\
0\quad&\text{if}\quad m\geq p+1\\\ \end{cases}$ (3.21)

This can be used to show that the power series expansion of
$\frac{1}{(1-z)^{p}}$ is reduced to the finite expected Binomial expansion for
$p\leq 0$. We can as well rewrite the expression of $F_{p}(n,t)$:

$F_{p}(n,t)=e^{-t}\sum_{k=0}^{-p}(-1)^{k}C_{-p}^{k}\frac{t^{k+n}}{(k+n)!}$
(3.22)

Remark: The function $F_{p}(n,t)$ is related to the confluent hypergeometric
function ${}_{1}F_{1}$

$F_{p}(n;t)=\frac{e^{-1}t^{n}}{n!}{}_{1}F_{1}(p,n+1;t)$ (3.23)

Where
${}_{1}F_{1}(a,b;t)=\sum_{k=0}^{\infty}\frac{(a)_{k}}{(b)_{k}}\frac{t^{k}}{k!}$
And $(a)_{k}=a(a+1)...(a+k-1)$.

Other properties which are easy to show:

  * •

$\frac{d}{dt}F_{p}(n,t)=F_{p-1}(n-1;t)=F_{p}(n-1,t)-F_{p}(n,t)$ (3.24)

  * •

$\int_{0}^{t}F_{p}(n,s)ds=F_{p+1}(n+1;t)-F_{p+1}(n+1;0)=F_{p+1}(n+1;t)-C_{-n+p-1}^{p-1}$
(3.25)

  * •

$\sum_{n=n_{1}}^{n_{2}}F_{p}(n;t)=F_{p+1}(n;t)-F_{p+1}(n+1;t)$ (3.26)

Arbitrary number of particles: An $N$ particle calculation follows the same
logic as the two particles, but with some subtle details. The Bethe wave
function would be composed of $N!$ term:

$P_{\mathbf{p}}(\mathbf{y})=\sum_{\sigma\in\mathcal{S}_{N}}A_{\sigma}\prod_{i=1}^{N}e^{ip_{\sigma(i)}y_{i}}$
(3.27)

Where $\mathbf{p}={\\{p_{1},...,p_{N}\\}}$,
$\mathbf{y}={\\{y_{1},...,y_{N}\\}}$ and $S_{N}$ is the symmetric group. The
restriction on the coefficients generalizes to:

$\frac{A_{\sigma}}{A_{\sigma\tau_{i,i+1}}}=-\frac{1-e^{ip_{\sigma{(i)}}}}{1-e^{ip_{\sigma{(i+1)}}}}$
(3.28)

Where $\tau_{i,i+1}$ is the transposition applied on positions $i$ and $i+1$.

Since each permutation can be written as a product of the transposition of
neighboring sites, then it’s enough to fix one of the coefficients in order to
fix all the others. Say for instance that $A_{Id}=1$. We know that the
decomposition of a permutation in terms of transpositions is not unique, so
for this approach to be self-consistent, the value of the coefficient should
be independent of the transposition path. The most trivial check is the
invariance under a double application of the same transposition, which is
obviously verified here. It happens that we don’t need to check for all the
possible paths and it’s enough to verify that the path is taken by next
neighbor transpositions:

$\tau_{1,3}=\tau_{1,2}\tau_{2,3}\tau_{1,2}=\tau_{2,3}\tau_{1,2}\tau_{2,3}$
(3.29)

This is the Yang-Baxter equation. We can check that it is verified:

$A_{\tau_{1,3}}=A_{\tau_{1,2}\tau_{2,3}\tau_{1,2}}=A_{\tau_{2,3}\tau_{1,2}\tau_{2,3}}=-\frac{1-e^{ip_{3}}}{1-e^{ip_{1}}}$
(3.30)

This can be generalized to any permutation $\tau_{i,j}$. One can notice that
the map $\sigma\rightarrow A_{\sigma}$ is not a group homomorphism(if it were,
there would have been no need to check for Yang-Baxter, it would be trivially
verified), the homomorphism property applies only for permutations with
independent support. Say $\sigma_{1}$ and $\sigma_{2}$ are two such
permutations, then:

$A_{\sigma_{1}\sigma_{2}}=A_{\sigma_{1}}A_{\sigma_{2}}$ (3.31)

By deduction on cyclic permutations, one can reach an explicit formula for
$A_{\sigma}$ that is valid for any permutation:

$A_{\sigma}=\epsilon(\sigma)\prod_{i=1}^{N}(1-z_{\sigma(i)})^{\sigma(i)-i}$
(3.32)

Where $\epsilon(\sigma)$ is the signature of $\sigma$. It’s easy to show that
this formula verifies indeed eq. 3.28. Now we need to check the validity of
the (naive) composition of the initial condition in terms of the Bethe wave
vectors. For this we need to show that:

$\sum_{\sigma\neq
Id}\oint_{0}...\oint_{0}A_{\sigma}\prod_{i=1}^{N}z_{\sigma(i)}^{y_{i}-x_{\sigma(i)}-1}dz_{1}...dz_{N}=0$
(3.33)

It’s actually possible to show that each term in the sum is zero. Take for
instance the term corresponding to some $\sigma\neq Id$. there must exists for
it an index $\tilde{i}$ such that $\tilde{i}>\sigma(\tilde{i})$ so it follows:

$y_{\tilde{i}}>y_{\sigma(\tilde{i})}\geq x_{\sigma(\tilde{i})}$ (3.34)

So the factor $z_{\sigma(\tilde{i})}^{y_{\tilde{i}}-x_{\sigma(\tilde{i})}-1}$
would have no poles, and since $A_{\sigma}$ is holomorphic too in a
neighborhood of zero, the corresponding contour integral around zero would
vanish. Now we need to examine the time evolution. Let’s write down the
eigenvalue equation:

$\begin{split}\lambda_{\mathbf{p}}P_{\mathbf{p}}(\mathbf{y})&=\sum_{i=1}^{N}\sum_{\sigma\in\mathcal{S}_{N}}e^{-ip_{\sigma(i)}}A_{\sigma}\prod_{i=1}^{N}e^{ip_{\sigma(i)}y_{i}}-NP_{\mathbf{p}}\\\
&=\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{\sigma:\sigma(i)=j}e^{-ip_{j}}A_{\sigma}\prod_{i=1}^{N}e^{ip_{\sigma(i)}y_{i}}-NP_{\mathbf{p}}\\\
&=\sum_{j=1}^{N}e^{-ip_{j}}\sum_{i=1}^{N}\sum_{\sigma:\sigma(i)=j}A_{\sigma}\prod_{i=1}^{N}e^{ip_{\sigma(i)}y_{i}}-NP_{\mathbf{p}}\end{split}$
(3.35)

So the second and the third sum can be united again into a sum over all the
permutations and we find again that the eigenvalue for a Bethe wave has the
same form as for two particles:

$\lambda_{\mathbf{p}}=\sum_{i=1}^{N}(e^{-ip_{i}}-1)=\sum_{i=1}^{N}\lambda_{p_{i}}$
(3.36)

All the ingredients are now ready for the time-evolved vector:

$\begin{split}P(\\{\mathbf{y}\\};t)&=\oint_{0}...\oint_{0}(\prod_{j=1}^{N}e^{\frac{t}{z_{j}}-t})\sum_{\sigma\in\mathcal{S}_{N}}\epsilon(\sigma)\prod_{i=1}^{N}(1-z_{\sigma(i)})^{\sigma(i)-i}z_{\sigma(i)}^{y_{i}-x_{\sigma(i)}-1}dz_{1}...dz_{N}\\\
&=\sum_{\sigma\in\mathcal{S}_{N}}\epsilon(\sigma)\prod_{i=1}^{N}\oint_{0}e^{\frac{t}{z_{\sigma(i)}}-t}(1-z_{\sigma(i)})^{\sigma(i)-i}z_{\sigma(i)}^{y_{i}-x_{\sigma(i)}-1}dz_{\sigma(i)}\\\
&=\sum_{\sigma\in\mathcal{S}_{N}}\epsilon(\sigma)\prod_{i=1}^{N}F_{i-\sigma(i)}(y_{i}-x_{\sigma(i)};t)\\\
&=Det((F_{i-j}(y_{i}-x_{j};t))_{ij})\end{split}$ (3.37)

Remarks

  1. 1.

It’s possible to prove the final result of 3.37 is indeed the solution of the
Master equation just by using the properties of the functions $F_{p}(n;t)$ and
elementary operations on the determinant. This is done in [34]

  2. 2.

It’s obviously possible to extend this method to ASEP. The computations are a
bit more tedious, partly because the plane waves are not the only eigenvectors
of the Markov matrix. Families of bound states appear as eigenvectors as well.
To see this, one has to get a bit into the details of the procedure: the extra
term in the master equation will require an extra term in the boundary
condition, so it becomes: $P(y,y)+qP(y+1,y+1)=(1+q)P(y,y+1)$ where $q$ is the
backward hopping parameter. Applying this condition on the Bethe wave vector
yields: $A_{12}(1+qz_{1}z_{2}-z_{2}(1+q))=-A_{21}(1+qz_{1}z_{2}-z_{1}(1+q))$,
beside the usual solutions similar to the TASEP case, one can choose to
satisfy the previous equation for instance by setting $A_{21}=0$ and
$1+qz_{1}z_{2}-z_{2}(1+q)=0$ this leads to a constraint between $z_{1}$ and
$z_{2}$ forcing them to leave the unit circle and producing a one-parameter
family of solutions that would extend the base on which one needs to
decomposing the initial condition. This has been done for two particles
in[34], and generalized to an arbitrary number of particles in [35].

####  3.1.2 Adding second class particles

We consider TASEP on the line with a finite number ofirst-class particles,
denoted by $A$ and second class particles, denoted by $B$, with unity hopping
rates. The space state is now $\mathcal{S}\times\\{A,B\\}^{N}$. Where $N$ is
the total number of particles. For a given set of positions of the particles,
the probabilities of the different permutations can be represented by a vector
in $(\mathbb{C}^{2})^{\otimes N}$ (Notations don’t assume particles type
conservation). The master equation is the same as single species TASEP when
neighboring particles are absent. Let’s write the equation for 2 adjacent
particles:

$\frac{d}{dt}\mathbf{P}(y,y+1;t)=\mathbf{P}(y-1,y+1;t)-M\mathbf{P}(y,y+1;t)$
(3.38)

Where $\mathbf{P}(y_{1},y_{2};t)=\begin{pmatrix}P^{AA}(y_{1},y_{2};t)\\\
P^{AB}(y_{1},y_{2};t)\\\ P^{BA}(y_{1},y_{2};t)\\\
P^{BB}(y_{1},y_{2};t)\end{pmatrix}$ and $M=\begin{pmatrix}1&0&0&0\\\
0&2&0&0\\\ 0&-1&1&0\\\ 0&0&0&1\end{pmatrix}$

This requires new boundary conditions for $P^{AB}$ and $P^{BA}$ so that the
non-neighboring master equation gets reduced to the neighboring one on the
boundaries. These conditions are:

$P^{AB}(y,y)=0$ (3.39) $P^{BA}(y,y)=P^{BA}(y,y+1)+P^{AB}(y,y+1)$ (3.40)

As usual we write the Bethe vector as:

$\mathbf{P}_{\\{p_{1},p_{2}\\}}(y_{1},y_{2})=\mathbf{A}_{1,2}e^{i(p_{1}y_{1}+p_{2}y_{2})}+\mathbf{A}_{2,1}e^{i(p_{2}y_{1}+p_{1}y_{2})}$
(3.41)

Where the coefficients are vector now. The matrix that allows transiting
between them is found thanks to the boundary restrictions:

$\mathbf{A}_{2,1}=R(p_{1},p_{2})\mathbf{A}_{1,2}$ (3.42)

Where:

$R(p_{1},p_{2})=\begin{pmatrix}-\frac{1-e^{ip_{2}}}{1-e^{ip_{1}}}&0&0&0\\\
0&-1&0&0\\\
0&-\frac{e^{ip_{1}}-e^{ip_{2}}}{1-e^{ip_{1}}}&-\frac{1-e^{ip_{2}}}{1-e^{ip_{1}}}&0\\\
0&0&0&-\frac{1-e^{ip_{2}}}{1-e^{ip_{1}}}\end{pmatrix}$ (3.43)

More generally, we can conclude how to apply a transposition on a coefficient:

$\mathbf{A}_{\sigma.\tau_{i,i+1}}=R(p_{\sigma(i)},p_{\sigma(i+1)})\mathbf{A}_{\sigma}$
(3.44)

This is enough to provide all the coefficients in terms of $\mathbf{A}_{Id}$.
To make sure that for each permutation $\sigma$, there is a well-defined
$\mathbf{A}_{\sigma}$, one can show that the $R$ matrix verifies the Yang-
Baxter Equation:

$R_{23}(p_{1},p_{2})R_{12}(p_{1},p_{3})R_{23}(p_{2},p_{3})=R_{12}(p_{2},p_{3})R_{23}(p_{1},p_{3})R_{12}(p_{1},p_{2})$
(3.45)

Where $R_{23}=Id\otimes R$ and $R_{12}=R\otimes Id$

It’s not hard to verify that the initial condition can still be written as an
integral of Bethe vectors in a similar fashion to single species. Let
$\mathbf{P}(x_{1},x_{2};0)$ the initial probability vector, then we have:

$\delta_{x_{1},y_{1}}\delta_{x_{2},y_{2}}\mathbf{A}_{Id}=\oint_{0}\oint_{0}(z_{1}^{x_{1}}z_{2}^{x_{2}})(z_{1}^{y_{1}-1}z_{2}^{y_{2}-1}-R(p_{1},p_{2})z_{1}^{y_{2}-1}z_{2}^{y_{1}-1}))\mathbf{A}_{Id}dz_{1}dz_{2}$
(3.46)

$\delta_{\mathbf{x},\mathbf{y}}\mathbf{P}(\mathbf{x};0)=\oint_{0}\oint_{0}(\prod_{i=1}^{N}z_{i}^{x_{i}})(z_{1}^{y_{1}-1}z_{2}^{y_{2}-1}-R(p_{1},p_{2})z_{1}^{y_{2}-1}z_{2}^{y_{1}-1})\mathbf{A}_{Id}dz_{2}$
(3.47)

$\mathbf{P}_{\mathbf{p}}(\mathbf{y})=\sum_{\sigma\in\mathcal{S}_{N}}\mathbf{A}_{\sigma}\prod_{i=1}^{N}e^{ip_{\sigma(i)}y_{i}}$
(3.48)

So the time evolution:

$\mathbf{P}(\mathbf{y};t)=\oint...\oint(\prod_{j=1}^{N}e^{\frac{t}{z_{j}}-t})\sum_{\sigma\in\mathcal{S}_{N}}\mathbf{A}_{\sigma}\prod_{i=1}^{N}z_{\sigma(i)}^{y_{i}-x_{i}-1}dz_{i}$
(3.49)

It’s possible to write an explicit formula for $\mathbf{P}$ in the case where
the order of species of the final configuration is the same as the initial
one, this has been done in [37](No exchange theorem). In the following
section, we will be examining the case where the hopping rates are arbitrary
per species, and the number of species is arbitrary too.

####  3.1.3 Multispecies exclusion process with arbitrary hopping rates

Let’s treat the most general situation with $N$ particles of multiple species.
The position of the particles shall be denoted by Latin letters, while their
species by Greek ones, so Let $I=\\{i_{1}<...<i_{N}\\}$ be the set of
positions corresponding to the species
$\mathbf{\alpha}=\\{\alpha_{1}<...<\alpha_{N}\\}$. Let $r_{\alpha}$ the
hopping rate of particles of species $\alpha$ and $r_{\alpha,\beta}$ the rate
of the ordered swap $\alpha\leftrightarrow\beta$. of a particle of type
$\alpha$ followed by one of type $\beta$.

Let $G(I,\mathbf{\alpha}|J,\mathbf{\beta};t)$ be the probability of having the
state $(I,\mathbf{\alpha})$ at time $t$ starting from the initial state
$(J,\mathbf{\beta})$ The master equation for non neighboring particles:

$\frac{d}{dt}G(I,\mathbf{\alpha}|J,\mathbf{\beta};t)=\sum_{k=1}^{N}r_{\alpha_{k}}G({i_{1}..,i_{k}-1,..i_{N}},\mathbf{\alpha}|J,\mathbf{\beta};t)-(\sum_{k=1}^{N}\alpha_{k})G(I,\mathbf{\alpha}|J,\mathbf{\beta};t)$
(3.50)

For two neighboring particles:

$\frac{d}{dt}G(i,i+1,12)=r_{\alpha}G(i-1,i+1,12)+r_{\beta\alpha}G(i,i+1,21)-(r_{\alpha,\beta}+r_{\beta})G(i,i+1,12)$
(3.51)

$\frac{d}{dt}G(i,i+1,12)=r_{\alpha}G(i-1,i+1,12)+r_{\beta}G(i,i,12)-(r_{\beta}+r_{\alpha})G(i,i+1,12)$
(3.52)

So the boundary condition needs to be:

$r_{\beta}G(i,i,12)=(r_{\alpha}-r_{\alpha,\beta})G(i,i+1,12)+r_{\beta\alpha}G(i,i+1,21)$
(3.53)

For the Bethe vector, let’s first notice that vector $z_{1}^{i}z_{2}^{j}$ does
not yield the same eigenvalue as $z_{2}^{i}z_{1}^{j}$ (for non-neighboring
particles). To conserve this property, one needs to resale the Bethe vector in
the following way:

$\psi^{I}_{\bm{\alpha}}=r_{\bm{\alpha}}^{I}\sum_{\sigma\in\mathcal{S}_{N}}F_{\bm{\alpha}}^{\sigma}\sigma[\mathbf{z}^{I}]$
(3.54)

Where $r_{\bm{\alpha}}^{I}=\prod_{k=1}^{n}r_{\alpha_{k}}^{i_{k}}$,
$\mathbf{z}^{I}=\prod_{k}z_{k}^{i_{k}}$ and
$\sigma[\mathbf{z}^{I}]=\prod_{k}z_{\sigma(k)}^{i_{k}}$. Applied on the Markov
matrix, the corresponding eigenvalue is:

$\lambda=\sum_{k}(z_{k}^{-1}-r_{k})$ (3.55)

As usual, we need to establish how the coefficients of the Bethe vector are
related so that the boundary conditions are respected. Let’s consider a 2
particle Bethe vector:

$\psi_{\alpha,\beta}^{i,j}=r_{\alpha}^{i}r_{\beta}^{j}(F_{\alpha,\beta}^{()}z_{1}^{i}z_{2}^{j}+F_{\alpha,\beta}^{(12)}z_{2}^{i}z_{1}^{j})$
(3.56)

Inserting it into 3.53, we get:

$-(F_{\alpha,\beta}^{()}+F_{\alpha,\beta}^{(12)})+(r_{\alpha}-r_{\alpha,\beta})(F_{\alpha,\beta}^{()}z_{2}+F_{\alpha,\beta}^{(12)}z_{1})+\frac{r_{\alpha}}{r_{\beta}}r_{\beta,\alpha}(F_{\beta,\alpha}^{()}z_{2}+F_{\beta,\alpha}^{(12)}z_{1})=0$
(3.57)

This can be written in a matrix form:

$\begin{pmatrix}F_{\alpha,\beta}^{(12)}&F_{\beta,\alpha}^{(12)}\end{pmatrix}\begin{pmatrix}(r_{\alpha}-r_{\alpha,\beta})z_{1}-1\\\
\frac{r_{\alpha}}{r_{\beta}}r_{\beta,\alpha}z_{1}\end{pmatrix}+\begin{pmatrix}F_{\alpha,\beta}^{()}&F_{\beta,\alpha}^{()}\end{pmatrix}\begin{pmatrix}(r_{\alpha}-r_{\alpha,\beta})z_{2}-1\\\
\frac{r_{\alpha}}{r_{\beta}}r_{\beta,\alpha}z_{2}\end{pmatrix}=0$ (3.58)

And by the symmetry $\alpha\leftrightarrow\beta$ we get another equation:

$\begin{pmatrix}F_{\alpha,\beta}^{(12)}&F_{\beta,\alpha}^{(12)}\end{pmatrix}\begin{pmatrix}\frac{r_{\beta}}{r_{\alpha}}r_{\alpha,\beta}z_{1}\\\
(r_{\beta}-r_{\beta,\alpha})z_{1}-1\end{pmatrix}+\begin{pmatrix}F_{\alpha,\beta}^{()}&F_{\beta,\alpha}^{()}\end{pmatrix}\begin{pmatrix}\frac{r_{\beta}}{r_{\alpha}}r_{\alpha,\beta}z_{2}\\\
(r_{\beta}-r_{\beta,\alpha})z_{2}-1\end{pmatrix}=0$ (3.59)

The two previous equations can be regrouped:

$\begin{split}\begin{pmatrix}F_{\alpha,\beta}^{(12)}&F_{\beta,\alpha}^{(12)}\end{pmatrix}\begin{pmatrix}(r_{\alpha}-r_{\alpha,\beta})z_{1}-1&\frac{r_{\beta}}{r_{\alpha}}r_{\alpha,\beta}z_{1}\\\
\frac{r_{\alpha}}{r_{\beta}}r_{\beta,\alpha}z_{1}&(r_{\beta}-r_{\beta,\alpha})z_{1}-1\end{pmatrix}=\\\
-\begin{pmatrix}F_{\alpha,\beta}^{()}&F_{\beta,\alpha}^{()}\end{pmatrix}\begin{pmatrix}(r_{\alpha}-r_{\alpha,\beta})z_{2}-1&\frac{r_{\beta}}{r_{\alpha}}r_{\alpha,\beta}z_{2}\\\
\frac{r_{\alpha}}{r_{\beta}}r_{\beta,\alpha}z_{2}&(r_{\beta}-r_{\beta,\alpha})z_{2}-1\end{pmatrix}\end{split}$
(3.60)

In vector notation:

$F^{(12)}M(z_{2})+F^{()}M(z_{1})=0$ (3.61)

For two species It’s convenient to define
$F^{\sigma}:=(F^{\sigma}_{\alpha,\alpha},F^{\sigma}_{\alpha,\beta},F^{\sigma}_{\beta,\alpha},F^{\sigma}_{\beta,\beta})\in\mathbb{C}^{2}\otimes\mathbb{C}^{2}$
and $M$ will become:

$M(z)=\begin{pmatrix}r_{\alpha}z-1&0&0&0\\\
0&(r_{\alpha}-r_{\alpha,\beta})z-1&\frac{r_{\beta}}{r_{\alpha}}r_{\alpha,\beta}z&0\\\
0&\frac{r_{\alpha}}{r_{\beta}}r_{\beta,\alpha}z&(r_{\beta}-r_{\beta,\alpha})z-1&0\\\
0&0&0&r_{\beta}z-1\end{pmatrix}$ (3.62)

We need to generalize this to an arbitrary number of species $m$, but with
only 2 particles, then it’s convenient to define the line vector
$F^{\sigma}\in\mathbb{C}^{m}\otimes\mathbb{C}^{m}$

So we can define the $\check{R}$ matrix:

$F^{(12)}=F^{()}\check{R}(z_{1},z_{2})$ (3.63)

Where:

$\check{R}(z_{1},z_{2})=-M(z_{1})M(z_{2})^{-1}$ (3.64)

In a generalized tensor form:

$M(z)_{\alpha\beta}^{\mu\nu}=((r_{\alpha}-r_{\alpha,\beta})z-1)\delta_{\alpha}^{\mu}\delta_{\beta}^{\nu}+\frac{r_{\alpha}}{r_{\beta}}r_{\beta,\alpha}z\delta_{\alpha}^{\nu}\delta_{\beta}^{\mu}$
(3.65)
$F^{\sigma\tau_{i,i+1}}=F^{\sigma}\check{R}(z_{\sigma(i)},z_{\sigma(i+1)})$
(3.66)

##### Integrability and restrictions on the rates

The matrix $\check{R}$ needs to obey the Braided Yang-Baxter equation, namely:

$\check{R}_{23}(z_{1},z_{2})\check{R}_{12}(z_{1},z_{3})\check{R}_{23}(z_{2},z_{3})=\check{R}_{12}(z_{2},z_{3})\check{R}_{23}(z_{1},z_{3})\check{R}_{12}(z_{1},z_{2})$
(3.67)

Explicit expansion of this equation shows that we need to impose hierarchy
over the species. A given species can hope only over lower ones in the
hierarchy:

$r_{\alpha,\beta}=0\quad\text{if}\quad\alpha>\beta$ (3.68)

In addition to this restriction, an additional one is needed, for
$\alpha>\beta>\gamma$

$r_{\gamma,\alpha}-r_{\beta,\alpha}=r_{\gamma}-r_{\beta}$ (3.69)

Which means:

$r_{\alpha,\beta}=(r_{\alpha}+\nu_{\beta})\mathds{1}_{\beta>\alpha}$ (3.70)

with $\nu_{\beta}$ being a parameter depending only on $\beta$

With these restrictions, the $M$ matrix simplifies to:

$M(z)=\begin{pmatrix}r_{\alpha}z-1&0&0&0\\\
0&-1-\nu_{\beta}z&(r_{\beta}(r_{\alpha}+\nu_{\beta})z)/ra&0\\\
0&0&-1+r_{\beta}z&0\\\ 0&0&0&r_{\beta}z-1\end{pmatrix}$ (3.71)

And the $\check{R}$ matrix:

$\check{R}(z_{1},z_{2})=\left(\begin{array}[]{cccc}-\frac{z_{2}r_{\alpha}-1}{z_{1}r_{\alpha}-1}&0&0&0\\\
0&-\frac{z_{2}\nu_{\beta}+1}{z_{1}\nu_{\beta}+1}&\frac{r_{\beta}(r_{\alpha}+\nu_{\beta})(z_{1}-z_{2})}{r_{\alpha}(-1+r_{\beta}z_{1})(1+\nu_{\beta}z_{1})}&0\\\
0&0&-\frac{z_{2}r_{\beta}-1}{z_{1}r_{\beta}-1}&0\\\
0&0&0&-\frac{z_{2}r_{\beta}-1}{z_{1}r_{\beta}-1}\\\ \end{array}\right)$

##### Representation in terms of operators

Let’s define the action of a permutation over a variable:

$\sigma x_{i}=x_{\sigma(i)}\sigma$ (3.72)

This allows us to write

$F^{\sigma\tau_{i}}\sigma\tau_{i}=F^{\sigma}\check{R}_{i}(z_{\sigma(i)},z_{\sigma(i+1)})\sigma\tau_{i}=F^{\sigma}\sigma\check{R}_{i}(z_{i},z_{i+1})\tau_{i}$
(3.73)

We define the operator:

$\pi_{i}:=\check{R}_{i}(z_{i},z_{i+1})\tau_{i}$ (3.74)

Which acts from the left on an $F^{\sigma}\sigma$:

$F^{\sigma\tau_{i}}\sigma\tau_{i}=F^{\sigma}\sigma\pi_{i}$ (3.75)

So this operator obviously allows to construct $F^{\sigma}\sigma$ starting
from $F^{Id}Id$. Let $\sigma=\prod_{i}\tau_{k_{i}}$ a decomposition of
$\sigma$ in terms of transpositions, then, by defining:

$\pi_{\sigma}:=\prod_{i}\pi_{k_{i}}$ (3.76)

We can write:

$F^{\sigma}\sigma=F^{e}e\pi_{\sigma}$ (3.77)

Of course, the operators $\pi$ need to abide by the Braided equation:

$\pi_{i}\pi_{i+1}\pi_{i}=\pi_{i+1}\pi_{i}\pi_{i+1}$ (3.78)

We can notice that $\sigma\rightarrow\pi_{\sigma}$ is a group morphism:
$\pi_{\sigma\acute{\sigma}}=\pi_{\sigma}\pi_{\acute{\sigma}}$

Now we can write the Bethe wave vector:

$\begin{split}\psi^{I}_{\bm{\alpha}}&=r_{\bm{\alpha}}^{I}\sum_{\sigma\in\mathcal{S}_{N}}F_{\bm{\alpha}}^{\sigma}\sigma\mathbf{z}^{I}\\\
&=r_{\bm{\alpha}}^{I}F_{\bm{\alpha}}^{e}(\sum_{\sigma\in\mathcal{S}_{N}}\pi_{\sigma})\mathbf{z}^{I}\\\
&=r_{\bm{\alpha}}^{I}F_{\bm{\alpha}}^{e}\Pi_{0}\mathbf{z}^{I}\end{split}$
(3.79)

Where $\Pi_{0}=\sum_{\sigma\in\mathcal{S}_{N}}\pi_{\sigma}$ One can show
easily that $\Pi_{0}$ satisfies the following property:

$\Pi_{0}\pi_{\sigma}=\pi_{\sigma}\Pi_{0}=\Pi_{0}$ (3.80)

Now let’s write the propagator:

$\begin{split}G(I,\mathbf{\alpha}|J,\mathbf{\beta};t)&=r_{\bm{\alpha}}^{I}r_{\bm{\beta}}^{-J}\oint...\oint
e^{\lambda(\mathbf{z})t}\mathbf{z}^{-J}(\Pi_{0})_{\bm{\alpha}}^{\bm{\beta}}\mathbf{z}^{I}\prod_{i=1}^{N}\frac{dz_{i}}{2\pi
iz_{i}}\\\ &=r_{\bm{\alpha}}^{I}r_{\bm{\beta}}^{-J}\oint...\oint
e^{\lambda(\mathbf{z})t}\Phi(I,\mathbf{\bm{\alpha}}|J,\mathbf{\bm{\beta}})\prod_{i=1}^{N}\frac{dz_{i}}{2\pi
iz_{i}}\end{split}$ (3.81)

Where

$\Phi(I,\mathbf{\bm{\alpha}}|J,\mathbf{\bm{\beta}}):=\mathbf{z}^{-J}(\sum_{\sigma\in\mathcal{S}_{N}}\pi_{\sigma})_{\bm{\alpha}}^{\bm{\beta}}\mathbf{z}^{I}$
(3.82)

##### Graphical representation of the operator $\Pi_{0}$

It’s possible to see $(\Pi_{0})_{\bm{\alpha}}^{\bm{\beta}}$ as a partition
function of a vertex model constrained by an initial configuration
$\bm{\beta}$ and a final configuration $\bm{\alpha}$. To illustrate this idea,
let’s first consider two arbitrary species
$\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet$ and
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$ ,
with
${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}>{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
we can identify 5 non zero elements of the $\check{R}$ matrix in the
base$\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet},{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet},{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}\\}$

$=h(x,y)$$x$$y$$y$$x$$=g_{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}(x,y)$$x$$y$$y$$x$$=g_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}(x,y)$$x$$y$$y$$x$$=g_{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}(x,y)$$x$$y$$y$$x$$=0$$x$$y$$y$$x$$=f(x,y)$$x$$y$$y$$x$

Where the weights are:

$f(x,y):=\check{R}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}}_{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}}=\frac{r_{{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}}(r_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}}+\nu_{{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}})(x-y)}{r_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}}(-1+r_{{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}}x)(1+\nu_{{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}}x)}$
(3.83)

$g_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bullet}(x,y):=\check{R}^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bullet}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bullet}}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bullet{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bullet}}=-\frac{yr_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bullet}-1}{xr_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\bullet}-1}$
(3.84)

$h(x,y)=:\check{R}^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}}_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}}=-\frac{y\nu_{{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}}+1}{x\nu_{{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}}+1}$
(3.85)

So if we have $m$ species, we would have $3\tbinom{m}{2}+2m$ non zero
vertices.

The quantity that we can want to compute is
$\Phi(I,\mathbf{\bm{\alpha}}|J,\mathbf{\bm{\beta}})$ which is a sum over
permutations. Each non-zero term of this sum corresponds to one or more than
one possible cconnection between the initial and final order of species using
the above building blocks such that there is a path of one color connecting
particles of the same color. This can be best understood through examples. So
we will consider two examples on three particles.

##### Examples on 3 particles

Consider two red particles and one green with
${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}>{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$.
We would like to calculate:
$\Phi(I,{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}|J,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet})$
So we need to build diagrams that connect the initial configuration
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}$
drawn on the top of the diagram to the final configuration
${\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
drawn at the bottom of the diagram. The diagrams should be so that there is a
path of green color connecting green particles and red paths connecting red
particles. Besides the building blocks illustrated above, we can use vertical
lines. In this example, the only permutations that can achieve this are the
ones for which $\sigma(3)=1$, so there are two permutations
$\tau_{12}\tau_{23}$ and $\tau_{13}$

$\Phi(I,{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}|J,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\bullet})=\mathbf{z}^{-J}(\leavevmode\hbox
to71.46pt{\vbox to82.86pt{\pgfpicture\makeatletter\hbox{\hskip
35.73091pt\lower-41.42876pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ \pgfsys@beginscope\pgfsys@invoke{
}{{}} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}{}\pgfsys@moveto{-28.45276pt}{0.0pt}\pgfsys@lineto{-28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@lineto{28.45276pt}{28.45276pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}{}\pgfsys@moveto{14.22638pt}{14.22638pt}\pgfsys@lineto{28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
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}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}{}\pgfsys@moveto{14.22638pt}{-42.67914pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{28.45276pt}{-42.67914pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.98576pt}{-42.67914pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par\par
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \par
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \par\par}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})\mathbf{z}^{I}$

Now we can compute each term:

$\mathbf{z}^{-J}\leavevmode\hbox to71.46pt{\vbox
to82.86pt{\pgfpicture\makeatletter\hbox{\hskip
35.73091pt\lower-41.42876pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ \pgfsys@beginscope\pgfsys@invoke{
}{{}} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}{}\pgfsys@moveto{-28.45276pt}{0.0pt}\pgfsys@lineto{-28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{} {}{} {}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@lineto{28.45276pt}{28.45276pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}{}\pgfsys@moveto{14.22638pt}{14.22638pt}\pgfsys@lineto{28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}{}\pgfsys@moveto{14.22638pt}{14.22638pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{28.45276pt}{14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.98576pt}{14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\par{}{{}}{}{{}}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.94514pt}{33.7902pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$z_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.50761pt}{33.7902pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$z_{3}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{-28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-32.3979pt}{33.7902pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$z_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\par{}{{}}{}{{}}{}\pgfsys@moveto{-28.45276pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-32.3979pt}{-36.2913pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$z_{3}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{28.45276pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.50761pt}{-36.2913pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$z_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{0.0pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.94514pt}{-36.2913pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$z_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\par{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}\definecolor[named]{pgffillcolor}{rgb}{0,1,0}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@moveto{29.52914pt}{28.45276pt}\pgfsys@curveto{29.52914pt}{29.04723pt}{29.04723pt}{29.52914pt}{28.45276pt}{29.52914pt}\pgfsys@curveto{27.85829pt}{29.52914pt}{27.37637pt}{29.04723pt}{27.37637pt}{28.45276pt}\pgfsys@curveto{27.37637pt}{27.85829pt}{27.85829pt}{27.37637pt}{28.45276pt}{27.37637pt}\pgfsys@curveto{29.04723pt}{27.37637pt}{29.52914pt}{27.85829pt}{29.52914pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{1.07639pt}{28.45276pt}\pgfsys@curveto{1.07639pt}{29.04723pt}{0.59447pt}{29.52914pt}{0.0pt}{29.52914pt}\pgfsys@curveto{-0.59447pt}{29.52914pt}{-1.07639pt}{29.04723pt}{-1.07639pt}{28.45276pt}\pgfsys@curveto{-1.07639pt}{27.85829pt}{-0.59447pt}{27.37637pt}{0.0pt}{27.37637pt}\pgfsys@curveto{0.59447pt}{27.37637pt}{1.07639pt}{27.85829pt}{1.07639pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\par{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-28.45276pt}\pgfsys@moveto{1.07639pt}{-28.45276pt}\pgfsys@curveto{1.07639pt}{-27.85829pt}{0.59447pt}{-27.37637pt}{0.0pt}{-27.37637pt}\pgfsys@curveto{-0.59447pt}{-27.37637pt}{-1.07639pt}{-27.85829pt}{-1.07639pt}{-28.45276pt}\pgfsys@curveto{-1.07639pt}{-29.04723pt}{-0.59447pt}{-29.52914pt}{0.0pt}{-29.52914pt}\pgfsys@curveto{0.59447pt}{-29.52914pt}{1.07639pt}{-29.04723pt}{1.07639pt}{-28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}{}\pgfsys@moveto{28.45276pt}{-28.45276pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par{}{{}}{} {}{} {}{}
{}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.45276pt}{-28.45276pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{-28.45276pt}{0.0pt}\pgfsys@lineto{-28.45276pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@rgb@stroke{0}{1}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{1}{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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(3.86)

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