We derive rigorous results on the link between the principle of maximum
entropy production and the principle of maximum Kolmogorov–Sinai entropy for
a Markov model of the passive scalar diffusion called the Zero Range Process.
We show analytically that both the entropy production and the
Kolmogorov–Sinai entropy, seen as functions of a parameter

A major difficulty in the modelling of nonlinear geophysical or astrophysical
processes is the taking into account of all the relevant degrees of freedom.
For example, fluid motions obeying Navier–Stokes equations usually require of
the order of

When the number of free parameters is small, their prescription can be successfully done
empirically through calibrating experiments or by a posteriori tuning

The goal of the present paper is to derive rigorous results on the link
between MEP and MKS using a Markov model of the passive scalar diffusion
called the Zero Range Process

The equation describing the transport of a passive scalar like temperature in
a given velocity field

Taking the continuous limit of this process, it may be checked that the
fugacity

We will first show that numerically

This Markovian process is a stochastic process with an infinite number of states in
bijection with

The probability to find

For a system subject to internal forces

In the case of the ZRP, irreversibility is created by the fact
that

Thus, when the stationary state is reached, i.e. when

There are several ways to introduce the Kolmogorov–Sinai entropy which is a
mathematical quantity introduced by Kolmogorov and developed by the renowned
mathematicians Sinai and Billingsley

Thus the Kolmogorov–Sinai entropy takes the following form:

For the ZRP, we show in the Appendix that it can be written as

We start by pointing to some interesting properties of

Note that for

First we remark that

Moreover,

Entropy production calculated using Eq. (

These observations are confirmed by the results presented in Figs.

Entropy production (left) and KS entropy (right) as functions of

2-D plot representing

In Fig.

Such numerical investigations help to understand why

From Eq. (

Using Eqs. (

Let us start by computing

After some tedious but straightforward calculations, we get at the first
order in

We remark that we can strictly find the same result by solving the
hydrodynamics continuous approximation given by Eq. (

One can verify this numerically: we first calculate the exact values of the
entropy production function of

In order to find the optimal resolution

This equation has a unique positive solution because the leading coefficient
is positive for

We have shown how a simple 1-D Markov process, the Zero Range Process, can be
used to obtain rigorous results on the problem of parametrization of the
passive scalar transport problem, relevant to many geophysical applications
including temperature distribution in climate modelling. Using this model, we
have derived rigorous results on the link between a principle of maximum
entropy production and the principle of maximum Kolmogorov–Sinai entropy
using a Markov model of the passive scalar diffusion called the Zero Range
Process.
The Kolmogorov–Sinai entropy seen as a function of the convective velocity
admits a unique maximum. We show analytically that both have the same Taylor
expansion at the first order in the deviation from equilibrium. The behaviour
of these two maxima is explored as a function of the resolution

The application of this principle to passive scalar
transport parametrization is therefore expected to provide both the value of
the optimal flux, and of the optimal number of degrees of freedom
(resolution) to describe the system. It would be interesting to apply it to a
more realistic passive scalar transport problem, to see if it would yield a model
that could be numerically handled (i.e. corresponding to a number of bows that
is small enough to be handled by present computers). In view of applications
to atmospheric convection, it would be interesting to apply this procedure to
the case of an active scalar, coupled with a Navier–Stokes equation for the
velocity. In such a case, the role of

In this appendix, we compute the Kolmogorov–Sinai entropy for the Zero Range
Process, starting from its definition in Eq. (

if

if

if

the same applies if

finally, if

Using Eq. (

We thus obtain that

This expression, though complicated at first sight, can be simplified. Indeed,
interested in the function

M. Mihelich thanks IDEEX Paris Saclay for financial support.Edited by: J. M. Redondo Reviewed by: two anonymous referees