Non-equilibrium is dominant in geophysical and climate phenomena. However the study of non-equilibrium is much more difficult than equilibrium, and the relevance of probabilistic simplified models has been emphasized. Large deviation rates have been used recently in climate science. In this paper, after recalling progress during the last decades in understanding the role of large deviations in a class of non-equilibrium systems, we point out differences between equilibrium and non-equilibrium. For example, in non-equilibrium (a) large deviation rates may be extensive but not simply additive. (b) In non-equilibrium there are generically long-range space correlations, so large deviation rates are non-local. (c) Singularities in large deviation rates denote the existence of phase transitions often not possible in equilibrium. To exemplify, we shall refer to lattice gas models like the symmetric simple exclusion process and other models which are playing an important role in the understanding of non-equilibrium physics. The reasons why all this may be of interest in climate physics will be briefly indicated.

This paper is an enlarged version of a seminar in the series

The theory is based on variational principles determining for each model the large deviation rates (LDRs) of thermodynamic variables like a density or a time average like the time-averaged current flowing through the system in stationary states. Models in climate science are considerably more complex than those for which MFT has been proved to hold. However, MFT may provide a guide for more complex problems, and it shows that there are considerable differences in fluctuations from an equilibrium or a non-equilibrium state.

In the last decades, the theory of large deviations has become a main
tool in statistical mechanics especially in the study of
non-equilibrium. MFT has been formulated and verified in
probabilistic models of lattice gases which macroscopically lead to
hydrodynamic diffusion equations, which in turn represent laws of large
numbers. Therefore, large deviations from hydrodynamic behavior have
been studied. Here, we shall follow the formulation given in

Non-equilibrium is dominant in geophysical and climate phenomena, and
we refer to the following papers for the relevance of large deviation
estimates in this domain

Non-equilibrium includes an enormous variety of phenomena, so we cannot hope to formulate a unique theory having a generality comparable to classical thermodynamics. We have to restrict to subclasses of problems. One difficulty is to define suitable thermodynamic functionals in far from equilibrium situations. Large fluctuations have offered a way out, as large deviation rates provide genuine thermodynamic functionals in non-equilibrium stationary states.

Studying irreversible processes is much more difficult than understanding equilibrium phenomena. In equilibrium statistical mechanics, we do not have to solve any equation of motion, and the Gibbs distribution provides the basis for the calculation of macroscopic quantities and their fluctuations. In non-equilibrium, we cannot bypass the dynamics even in the study of stationary states, which we may consider as the simplest beyond equilibrium. Examples are the heat flow in an iron rod whose endpoints are thermostated at different temperatures or the stationary flow of electrical current in a given potential difference.

For such states, the fluctuations exhibit novel and rich features with
respect to the equilibrium situation. As experimentally observed

In this paper, we consider a class of systems that behave macroscopically as diffusions. The systems considered locally
deviate only slightly from equilibrium so that small gradients and
linear response to external fields are reasonable
approximations. Microscopically, this implies that the system reaches
a local equilibrium in a time which is short compared to the times
typical of macroscopic evolution. So what characterizes situations in
which this description applies is a separation of scales both in space
and time. Far from equilibrium states are those which exhibit
differences over the size of the whole system. In other words, far from
equilibrium is obtained from building up small local
differences. Local equilibrium is the first assumption we make. For
the relevance of local equilibrium in climate models, see

This restriction allows us to understand some typical phenomena induced by non-equilibrium in somewhat ideal cases which however give a hint of what may happen in more realistic cases.

The content of the paper is as follows: in the next section, we recall the use by
Einstein of the Boltzmann relation between entropy and probability to
estimate the probability of a thermodynamic fluctuation in
equilibrium. In Sect. 3, we describe the essentials of the
macroscopic fluctuation theory in the version reviewed in

The first explicit large deviation theory in equilibrium states is
presumably Einstein's theory of opalescence

Starting from what he calls the Boltzmann principle,

However, Boltzmann's principle does acquire some content independent of any elementary theory if one assumes and generalizes from molecular kinetics the proposition that the irreversibility of physical processes is only apparent.

It follows from Eq. (

In 1931,

To study the fluctuations in states far from equilibrium, let us
analyze the meaning of the difference

The concept of minimal work is meaningful also in non-equilibrium and
can be taken as a generalization of the free energy. This is
essentially the starting point of the macroscopic fluctuation
theory. In the following section, we shall present the basic ideas of
the MFT, stating explicitly the main assumptions, following mainly

The MFT, as we mentioned, was inspired by non-equilibrium microscopic
probabilistic models, or the so-called lattice gases, in particular the
symmetric simple exclusion process (SSEP) for which the macroscopic
dynamics are diffusive and can be proved rigorously. Also the
probabilities of large deviations can be obtained and the rates
computed. For a general introduction to lattice gases, we refer to

The macroscopic dynamics of diffusive systems are described by
hydrodynamic equations often provided by conservation laws and
constitutive equations; that is equations expressing the current in
terms of the thermodynamic variables. More precisely on the basis of
a local equilibrium assumption, at the macroscopic level, the system is
completely described by a local density

For diffusive systems, the constitutive equations take the following form:

These equations have to be supplemented by appropriate boundary
conditions on

We assume that the microscopic evolution is given by a Markov process

The macroscopic equations are supposed to derive from an underlying
microscopic dynamics through an appropriate scaling limit where the
microscopic time is divided by a factor

The hydrodynamic equations represent laws of large numbers with
respect to the probability measure

Classically we should start from molecules interacting with realistic forces and evolving with Newtonian dynamics. This is beyond the reach of present-day mathematical theory, and much simpler models have to be adopted in the reasonable hope that some essential features are adequately captured.

We further assume that the stationary measure

Let us denote by

At the level of large deviations, Eq. (

The physical situation we are considering is the following. The
system is in the stationary state

In a SNS the spontaneous emergence of a macroscopic fluctuation takes place most likely following a trajectory which is the time reversal of the relaxation path according to the time-reversed hydrodynamics.

The above statement follows from assuming the existence of a time-reversed dynamics and from our general hypotheses. In equilibrium, a fluctuation emerges following the time reversal of the relaxation trajectory. As illustrated inFrom Eqs. (

The functional

The associated Hamilton equations are

The quasi-potential or non-equilibrium free energy satisfies the
associated Hamilton–Jacobi equation:

For lattice gases, the following expression has been derived

By minimizing first with respect to the current

In a stationary state, it is natural to consider the fluctuations of
the local time-averaged current,

This is a more general form of a large fluctuation principle proposed
by

In general, the appearance of singularities in the large deviation
rates denotes the presence of a non-equilibrium phase
transition. Actually the variational principle of Bodineau and Derrida
may provide several time-independent solutions which in fact have been
found in models discussed in

We are concerned with

We introduce the

We define

In

The simple exclusion process SSEP is the most studied system in far from equilibrium situations and has a role similar to the Ising model in the study of phase transitions. The SSEP in one-dimensional lattice is a process in which particles perform symmetric random walks subject to hard core exclusion. In non-equilibrium, the boundary conditions at the two ends of the lattice are different; an external field may act on the system, or both, so that a current is flowing through the system.

Large deviations functions have also been
calculated via the MFT for other models, like the zero-range process

We consider the variational problem defining

If an external field is present, the large deviation rate has been
calculated by

The time evolution depends on the initial condition. Basic work on non-stationary problems for SSEP is due to

Step initial condition with a density

Define the cumulant generating function

The variational equations are

In

The MFT shows that once macroscopic evolution equations like hydrodynamics are available and a separation of scales holds, a self-consistent macroscopic description of non-equilibrium phenomena can be obtained through a study of rare fluctuations. The origin of the probabilistic behavior may be due to the influence of a smaller scale on a larger one or to chaotic properties of the underlying dynamical system.

The discovery that a purely macroscopic theory could reproduce in the
case of SSEP the large deviation function for non-equilibrium
stationary states obtained from a microscopic calculation in

We have illustrated the theory in an idealized case: we have
considered (i) purely diffusive systems and (ii) simplified stochastic
models. However as the Ising model allowed us to understand a lot
about phase transitions and the critical point, we believe that the
SSEP and other solvable models are providing a guide to what may
happen out of equilibrium. Furthermore the MFT applies to all variants
of SSEP or of other diffusive models that macroscopically have the
same transport coefficients

The general approach of MFT has been extended to systems with more
than one conservation law

The experience so far indicates that the phenomenon of long-range space correlations is not limited to purely diffusive systems and is rather general in non-equilibrium.

In climate science, the models are comparatively more
complicated. However the application of large deviation theory to
rather complex models as in

Formally, the equations of the MFT can be derived also from assuming an
extension to non-equilibrium of the so-called fluctuating
hydrodynamics (FHs). The idea of fluctuating hydrodynamics goes back
to Landau

From the previous equations, we obtain

It would be interesting to provide a rigorous foundation to fluctuating hydrodynamics; however this requires as a first step to give a clear mathematical meaning to the stochastic partial differential equations on which it is founded.

No data sets were used in this article.

The author has declared that there are no competing interests.

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This article is part of the special issue “Interdisciplinary perspectives on climate sciences – highlighting past and current scientific achievements”. It is not associated with a conference.

I wish to acknowledge my long-standing collaboration on the topics of this paper with Lorenzo Bertini, Alberto De Sole, Davide Gabrielli and Claudio Landim. I thank Vera Melinda Galfi and Valerio Lucarini for a critical reading of the paper and for very useful comments.

This paper was edited by Vera Melinda Galfi and reviewed by Valerio Lucarini and one anonymous referee.