Nonequilibrium and Fluctuation Relation

Abstract

A review on the fluctuation relation, fluctuation theorem and related topics.

Appendices

About Certain Comments on CH

In the above sections quotes from [8] have been reproduced without really commenting them. The reason is that the quotes were written when the Chaotic Hypothesis had been just developed and many had not yet had the time to really study the subject.

But the same comments have appeared in a second edition of the just quoted book [87], which I have seen only very recently (after completion of the present text). Since the comments have had some resonance the next few lines try to clarify some of the issues.

The Author of [87] criticizes the use of the Anosov systems as paradigm of chaotic motions. The full section from pages 344 to 347 discusses the merits and demerits of Anosov systems. On p. 344 begins

... has discussed the possibility that the useful properties exhibited by certain oversimplified and quite rare dynamical systems, termed “Anosov systems”, have counterparts in the more usual thermostatted systems studied with nonequilibrium simulation methods. Anosov systems are oversimplifications, like square clouds or spherical chickens...

This seems to refer to the proposal that the “Axiom A” systems should be the right paradigm for generic chaotic systems [19, 88]: a proposal which however is not centered on Anosov systems. The Axiom A systems are systems which have an the attracting set ${\mathcal {A}}$ on which motion has strong chaotic properties (is essentially hyperbolic).

And the CH just proposes, in its final formulation, (1996), that for many purposes the axiom A paradigm can be strengthened and simplified by requiring in addition that ${\mathcal {A}}$ is an attracting surface, possibly of dimension lower than that of phase space, on which the motion is an Anosov system. Even in time reversible cases ${\mathcal {A}}$can be different from its time reversal image. This is explicitly stated with related problems and examples in [20, 21] and in several successive publications.

The underlying idea being that it is not possible to distinguish, in a system of physical interest, a fractal of Hausdorff dimension $=10^6+3.1415...$ from a surface of exactly $10^6$ dimensions.

In summary the Chaotic Hypothesis only assumes that the dynamics under consideration behaves (in some respects) like Anosov dynamics. This is after all not too astonishing if the most relevant degrees of freedom are chaotic like those of Anosov systems.

Most of the subsequent criticism in [87] is anchored on keeping the identification between the CH and the proposal that the whole dynamical system is an Anosov system.

On p. 346 the fluctuation theorem is called a “retrospective result” and identified with the true Fluctuation Theorem, Sect. 6 above, claiming: “These same “results” were actually given earlier by Denis Evans and several of his coworkers, for more general circumstances and through more elementary arguments.”

but no reference is made here to the applicability of the “earlier retrospective result” to stationary nonequilibria to which the Fluctuation Theorem applies, see$^{46}$.

Then on p. 347 the view is found that:

“Theoretical constructs such as “measures”, should be viewed with a healthy suspicion until algorithms for evaluating them are supplied. The chaos inherent in interesting differential equations guarantees that our only access to the “strange sets” which constitute attractors and repellers will be representative time series from dynamical simulations. In no way can we construct, or even conceive of constructing, a Sinai-Ruelle-Bowen measure for an interesting system.”

However for most purposes by the CH Hamiltonian systems should be considered Anosov systems (literally, except of course the integrable ones). Hence the assumption that the attracting set is the full phase space is not always unreasonable.

Furthermore it is useful to stress that there are easy examples of systems satisfying the CH, with equal or disjoint attracting and repelling surfaces, time reversible, with as many degrees of freedom and negative Lyapunov exponents as wished (unrelated to the number of positive ones) and whose SRB measure is explicitly and completely constructed [11, Sect. 10.2].

Local Fluctuations: An Example

The phase space contraction in the evolution of a macroscopic system is typically a macroscopic quantity: whether it is the amount of heat ceded to the thermostats or the amount of work performed by the systems.

Therefore the average phase space contraction $\sigma _+$ which controls the large fluctuations, Sect. 6 and the occurrence of “anomalous” patterns, Sect. 12, cannot be really observed in measurements on macroscopic systems.

Avoiding comments on the many experimental fluctuations observations which claim to check the FR,⁴⁷ the question asked here is whether a kind of fluctuation relation could be defined, and constrain quantities depending on events that can be observed in very small parts of the system.

In other words is it possible to give a meaning to a local fluctuation relation? [2, Chap. 4.9].

The following relies on Sect. 15: it is inserted as it provides a quite interesting example on how to make use of the symbolic dynamics representation of the Anosov systems.

A simple example, in a system with time reversal symmetry, will be discussed in which a local entropy production rate can be defined and checked to satisfy a local version of FR. A general view on the matter can be found in [89, 90].

The analysis deals again with maps rather than flows.⁴⁸

Consider a system with a translation invariant spatial structure, e.g. a periodic chain, or a d-dimensional square lattice $[-L,L]^d$ with periodic boundary, of $(2L)^d$ weakly interacting Anosov maps.

The phase space of the system is $M=\{\mathbf{x}=(\ldots x_1,x_0,x_1,x_2,\ldots )\}={\mathcal {M}}_0^{(2L)^d}$, where $x_i\in {\mathcal {M}}_0$ are points in a manifold ${\mathcal {M}}_0$: to fix ideas we take${\mathcal {M}}_0$to be a torus, on which an Anosov map ${\overline{S}}_0$ acts; then define the “coupled map”:

$$\begin{aligned} {\overline{S}}_\varepsilon (\mathbf{x})_i={\overline{S}}_0 x_i+\varepsilon g(x_{i-1},x_i,x_{i+1}), \ i=...,0,1,\ldots \end{aligned}$$

(B.1)

where g is a smooth perturbation, i.e. a smooth periodic function on ${\mathcal {M}}_0^3$.

If $\varepsilon$ is small and the perturbation has short range it is proved in [36, 93, 94] that, defining the map ${\overline{S}}_\varepsilon$ as in Eq. (B.1) with periodic boundary condition (i.e. identifying the site $-L$ with L), the map ${\overline{S}}_\varepsilon$ remains, if $\varepsilon$ is small enough, still Anosov. It is conjugated to ${\overline{S}}_0$, via a Hölder continuous correspondence ${\varvec{\Theta }}_\varepsilon$, see$^{34}$, by associating points x and $x'$ with the same history under ${\overline{S}}_0$ and ${\overline{S}}_\varepsilon$. Furthermore there is $\varepsilon _0>0$ such that the above holds for $|\varepsilon |<\varepsilon _0$uniformly in the system size L.⁴⁹

Here the purpose is to study whether a local version of the FR can hold at least in an example derived from ${\overline{S}}_\varepsilon$: but the ${\overline{S}}_\varepsilon$ is, in general, not reversible. A related reversible map $S^{rev}_\varepsilon$ can be easily constructed on the “doubled” phase space ${\mathcal {M}}={\mathcal {M}}_0\times {\mathcal {M}}_0$ by setting:

$$\begin{aligned} S_\varepsilon ^{rev}(\mathbf{x},\mathbf{y})= ({\overline{S}}_\varepsilon (\mathbf{x}), ({\overline{S}}_\varepsilon )^{-1}(\mathbf{y})) \end{aligned}$$

(B.2)

which is reversible for the time reversal map $I:(\mathbf{x},\mathbf{y})=(\mathbf{y},\mathbf{x})$. In the rest of this section this system will be considered in more detail.

A Markovian partition ${\mathcal {P}}^L_0$ for ${\overline{S}}^L_0\times ({\overline{S}}^L_0)^{-1}$ will be chosen to be the product of partitions ${\mathcal {P}}_{-\frac{L}{2}},\ldots ,{\mathcal {P}}_{\frac{L}{2}-1}$ for the single site maps ${\overline{S}}_0$ and ${\overline{S}}_0^{-1}$; and ${\mathcal {P}}^L_\varepsilon$ will be the partition ${\varvec{\Theta }}_\varepsilon {\mathcal {P}}^L_0$ existing and defined by the structural stability map ${\varvec{\Theta }}_\varepsilon$, conjugating ${\overline{S}}_0^{rev}$ to ${\overline{S}}_\varepsilon ^{rev}$ [11, Sec.10.2].

Hence the history of a point x will be a sequence of labels $\sigma _{i;j}$ with $i\in M$ and $j\in (-\infty ,\infty )$: naturally i can be called a “space label” while j a “time label”. The superscript rev will be omitted in what follows, to simplify notations.

The analysis in Sect. 15 applied to the Anosov map $S_\varepsilon$, will give a representation of the volume distribution $\mu _0$ and of the SRB distribution $\mu _{srb,\varepsilon }$ for $S_\varepsilon$ via, respectively, suitable potentials ${\varvec{\Phi }}_\varepsilon ,{\varvec{\Phi }}_{\varepsilon }^\pm$.

Let $[0,\tau ]$ be a time interval and $\Lambda =[-\frac{1}{2}L,\frac{1}{2}L]^d=M$, $|\Lambda ]=L^d$. Via the Jacobian matrix $J_\Lambda (x)={\partial }_x (S_\varepsilon x)$ define the phase space contraction and the time averaged contraction per site as, respectively:

$$\begin{aligned} \begin{aligned} \eta _{\Lambda ,\varepsilon }(x)&=-\frac{1}{|\Lambda |}\log |\det ({\partial }_x (S_\varepsilon x))|\\ \eta _{\Lambda ,\varepsilon ,\tau }&=\frac{1}{|\Lambda |}\lim _{\tau \rightarrow \infty } \frac{1}{\tau }\sum _{j=0}^\tau \eta _{\Lambda ,\varepsilon }(S_{\varepsilon }^j x) \end{aligned} \end{aligned}$$

(B.3)

The limit of $\eta _{L,\varepsilon ,+}$ in Eq. (B.3) as $\tau \rightarrow \infty$ exists with probability 1 with respect to the volume $\mu _{vol}$, as well to the SRB distribution $\mu _{srb,\varepsilon }$, and is x-independent aside x’s in a set of 0 volume: because the statistical properties of the volume distribution are those of the Gibbs distribution with potential ${\varvec{\Phi }}^+_\varepsilon$, hence enjoy strong ergodicity properties, as any SRB distribution, with respect to time translations.⁵⁰

The phase space contraction $\sum _{t=0}^\tau \eta _{L,\varepsilon }(S_\varepsilon ^t x)$ can be expressed, see Sect. 15, via the potentials ${\varvec{\Phi }}^+_{\varepsilon },{\varvec{\Phi }}^-_{\varepsilon },{\varvec{\Phi }}_{\varepsilon }^\varphi$, where ${\varvec{\Phi }}^\varphi _{\varepsilon }$ is a potential that describes the interpolation between ${\varvec{\Phi }}_{\varepsilon }^-$ to ${\varvec{\Phi }}_{\varepsilon }^+$ and which is therefore “localized” [see comment to Eq. (15.1)] in the sense that ${\varvec{\Phi }}^\varphi _{\varepsilon }({\varvec{\sigma }}_I)\ne 0$ only if I contains the sites 0 or L and $|{\varvec{\Phi }}_{\varepsilon ,I}^\varphi ({\varvec{\sigma }}_{I})|\le C e^{-\kappa |I|}$ for some $C,\kappa >0$). Given the symbolic history ${\varvec{\sigma }}$ of x, the Eq. (15.6) can be expressed as:

$$\begin{aligned} \begin{aligned} \frac{1}{\tau L^d} \sum _{K\subset M\times [0,\tau ]} ({\varvec{\Phi }}^+_{\varepsilon ,K}({\varvec{\sigma }}_K)-{\varvec{\Phi }}^-_{\varepsilon ,K}({\varvec{\sigma }}_K))+\ldots \end{aligned} \end{aligned}$$

(B.4)

where $K=I\times [a,b]$ is a parallelepiped in $\Lambda \times [0,\tau ]$, and ${\varvec{\Phi }}^z_{\varepsilon ,K}{\mathop {=}\limits ^{def}}\sum _{t\in [a,b]}{\varvec{\Phi }}^z_{I+t}$ for $z=\pm ,\varphi$, and the $\ldots$ indicate a correction $\sum _K {\varvec{\Phi }}^\varphi _{\varepsilon ,K}({\varvec{\sigma }}_K)$,⁵¹. t A natural mathematical definition of the “local average phase space contraction” could be the $-\frac{1}{\tau \Lambda _0}\sum _{t=0}^\tau \log J_{\Lambda _0}(S^x)$ where $J_{\Lambda _0}(x)=|\det ({\partial }_i (S_\varepsilon x)_{i'})|$. But this is a quantity difficult to express in a useful way.

However it is also possible to propose a different definition of local average phase space contraction based on the representation Eq. (B.4) of the average of the logarithms of the full Jacobian. The latter can be expressed as Eq. (B.4) up to a quantity uniformly bounded in L: and the contribution to Eq. (B.4) from the parallelepipeds K’s entirely contained in $\Lambda _0\times [0,\tau ]$ is:

$$\begin{aligned} \eta _{\Lambda _0,\varepsilon ,\tau }^{loc} {\mathop {=}\limits ^{def}}\frac{1}{\tau L_0^d} \sum _{K\subset \Lambda _0\times [0,\tau ]} \Big ({\varvec{\Phi }}_{+,K}({\varvec{\sigma }})- {\varvec{\Phi }}_{-,K}({\varvec{\sigma }})\Big ) \end{aligned}$$

(B.5)

see Eqs. (15.1), (15.4); the $\eta _{\Lambda _0,\varepsilon ,\tau }^{loc}$ can be, heuristically, called the “local contraction rate”. It can be uniformly bounded (in $\tau ,L$).

Given $\Lambda _0$ let $\eta ^{loc}_{\Lambda _0,\varepsilon ,+}$ be the time average $\eta _{\Lambda _0,\varepsilon ,\tau }^{loc}$ define:

$$\begin{aligned} p'=\frac{1}{\tau }\frac{\eta _{\Lambda ,\varepsilon ,\tau }(x)}{\eta _{\Lambda ,\varepsilon ,+}},\quad p=\frac{1}{\tau }\frac{\eta _{\Lambda _0,\varepsilon ,\tau }^{loc}(x)}{\eta _{\Lambda _0,\varepsilon ,+}^{loc}} \end{aligned}$$

(B.6)

and remark that $\eta _{\Lambda _0,\varepsilon ,+}^{loc}=\eta _{\Lambda ,\varepsilon ,+}+o(L_0^{-1})$ (because of the SRB distribution representation of a Gibbs process).

It can also be shown that to leading order as $L_0,L,\tau \rightarrow \infty$ the large deviation rates for $p',p$ in Eq. (B.6) have the form $\tau L^d\zeta _\infty (p'),\ \tau L_0^d\zeta ^0_\infty (p)$, with $\zeta _\infty =\zeta ^0_\infty$ because $\zeta _\infty$ is obtained as a thermodynamic limit of a kind of partition function: for a proof see [95, (5.14)].

Therefore by the FT applied to $S_\varepsilon$ it is $L^d\zeta _\infty (p')-L^d\zeta _\infty (-p')= L^d p' \eta _{\Lambda ,\varepsilon ,+}$ and, since $\eta _{\Lambda _0,\varepsilon ,+}^{loc}= \eta _{\Lambda ,\varepsilon ,+}+o(L_0^d)$, the large deviations rate for p in Eq. (B.6) satisfies a FR of the form

$$\begin{aligned} \begin{aligned}&L^d_0(\zeta _\infty (p)-\zeta _\infty (-p))=p \,L_0^d\eta _{\Lambda _0,\varepsilon ,+}^{loc}\\&= p\, r\, (L^d\eta _{\Lambda ,\varepsilon ,+}) \end{aligned} \end{aligned}$$

(B.7)

with $r=\frac{L_0^d}{L^d}$ and $|p|\le p^*, p^*\ge 1$, up to corrections of $O(L_0^{d-1})$: which means that the global and local large fluctuations rates are proportional and trivially related by a rescaling which equals $r=(\frac{L_0}{L})^d$ up to a correction bounded $\kappa ^{-1} L_0^{d-1}$ with $\kappa$ bounding the range of the SRB potential, as in Eq. (15.1).

The universal slope 1 in the global FR is modified into $r=(\frac{L_0}{L})^d$ in the local FR. The Eq. (B.7) can be proved for the system in Eq. (B.2).

However p in Eq. (B.6) is not related to a measurable quantity, as it cannot be hoped to be able to measure directly the local phase space contraction .defined as in Eq. (B.5).

Still the phase space contraction is often related to the amount of heat ceded or the work done on the surroundings by a system in a stationary state, as exemplified in the case of Eq. (4.4): hence it is tempting to test, in cases in which the latter quantities are accessible to local measurements, whether Eq. (B.7) holds. This is attempted in some simulations [45].

The interest of the above special example lies in the statements independence on the total size of the systems: they also mean that the fluctuation theorems may lead to observable consequences if one looks at the far more probable microscopic fluctuations of the local entropy production rate,[36, 93, 94]. For more details see [34].

Reversible Heating

Imagine a rarefied gas enclosed in a cubic container of side L described by a canonical distribution at inverse temperature $\beta ^{-1}$. The potential energy is $\sum _{i=1}^N mg z_i +\sum _{i,j} v(x_i-x_j){\mathop {=}\limits ^{def}}M\,g\,H +V$, with M=total mass and H the height of the center of mass. The initial free energy if $F(\beta ,g)=-\beta ^{-1} \log \int e^{-\beta (V+gP)} d^{3N}pd^{3N}q$. The entropy can be computed via Gibbs’ formula $S_0=-\int \rho (p,q)\log \rho (p,q)d^{3N}pd^{3N}q$.

The gas is set out of equilibrium by changing the gravity g to a new value $g'$ for instance suddenly at time $t=0$ or following a given prescription $t\rightarrow g(t), t\in [0,\tau ]$ with $g(\tau )=g', \tau <\infty$. Then it is let to evolve.

Since the evolution is Hamiltonian (although not autonomous) $\rho (p,q)$ evolves in $\rho (p,q;t)$ and the latter tends, as $t\rightarrow \infty$, to a new equilibrium state in the gravity potential $m g'z$; but $-\int \rho (p,q;t)\log \rho (p,q;t)d^{3N}pd^{3N}q$ remains equal to $S_0$. Therefore at the end of the evolution the new distribution $\rho (p,q,\infty )$ will be an equilibrium state of the system in the modified gravity field.

It will not be, however, any more a canonical Gibbs state at temperature $\beta ^{-1}$ in a gravity field with acceleration $g'$; if the system is ergodic on the energy surface then the final distribution reached at infinite time after suddenly increasing the gravity g to a new value $g'$ will be (integration over $p',q'$ only)”

$$\begin{aligned} \mu ^\infty (dpdq)=\int \mu _\beta (dp'dq') \mu ^{mc}_{E(p',q')}(dpdq) \end{aligned}$$

(C.1)

where $\mu ^{mc}_{E}(dpdq)$ is the microcanonical distribution with energy E and $E'(p',q')= K(p')+V(q')+Mg'H(q')$ is the sum of the kinetic energy, internal potential energy and energy of the center of mass in a gravity acceleration $g'$.

The distribution Eq. (C.1) will be equivalent to a canonical Gibbs distribution (with temperature different from $\beta ^{-1}$) only in the thermodynamic limit: in the finite system that we are considering it will be different by corrections vanishing in the thermodynamic limit. Yet the new state will be a stationary state close to a canonical (or any other equivalent) equilibrium state,

To estimate, actually to define, the temperature $\beta ^{'-1}$ of the new state imagine to identify the above $\mu ^\infty$ with a canonical distribution $\mu _{\beta '}$, i.e. neglect the finite volume corrections. Computing the Gibbs entropy $S_\infty$ of the new equilibrium (reached after infinite time) and make use of the identity between the Gibbs entropies of the initial and final states:

$$\begin{aligned} \begin{aligned}&F=-\frac{1}{\beta }\log \int e^{-\beta (V+gP)} d^{3N}pd^{3N}q=-T \log Z_0\\&S=-{\partial }_T F=-\log Z_0-\beta \langle V+ gP \rangle \\&{\partial }_g S|_\beta =\beta ^{2}(\langle P V \rangle -\langle P \rangle \langle V \rangle ) +g(\langle P^2 \rangle -\langle P \rangle ^2) \end{aligned} \end{aligned}$$

(C.2)

with $T=\beta ^{-1}$, $P=MgH$ and ${\partial }_g S|_\beta \ne 0$ at $g=0$: thus the new equilibrium cannot have the same entropy as the initial state if the temperature remained the same unless $\beta '\ne \beta$ (because in general ${\partial }_g S|_\beta \ne 0$, e.g. if $V\simeq 0$ it is $\langle P^2 \rangle > \langle P \rangle ^2$). If the final state has to become a canonical distribution at some temperature (e.g. the above estimated $\beta ^{'-1}$) then the system will have to be attached to a thermostat and some heat exchange will take place and the entire transformation will be irreversible: in any event, if the system container was really adiabatic and at any finite (or infinite) time the gravity acceleration was dropped back to the initial value, then the system should in the same time return to the initial canonical state. See also [96] for the analysis of equally interesting cases.

Arnold–Euler Geodesics

The analysis of [47] applies to the geodesic flows: which are Hamiltonian flows with Hamiltonian $H(\mathbf{p},\mathbf{q})= \frac{1}{2} g(\mathbf{q})^{-1}{\mathbf{p}}{\mathbf{p}}$. Hence the Lyapunov exponents of $\dot{\mathbf{q}}=g(\mathbf{q}) \mathbf{p}, \dot{\mathbf{p}}=-\frac{1}{2}({\varvec{\partial }}_{\mathbf{q}}g(\mathbf{q})^{-1}){\mathbf{p}}\,{\mathbf{p}}-\nu \mathbf{p}+{\mathbf{f}}$ with $\mathbf{f}$ independent of $\mathbf{p},\mathbf{q}$ are paired to $-\frac{1}{2}\nu$, see footnote$^{18}$, with a general $H(\mathbf{p},\mathbf{q})$.

The Euler flow is a geodesic flow [97, 98], and the canonical coordinates are $\mathbf{u},{\varvec{\delta }}$ where ${\varvec{\delta }}$ is the diffeomorphism bringing the reference state O of the fluid into the actual state A: if a force $\mathbf{f}$ is added which does not change the Hamiltonian nature of the motion (e.g. if, as in [47], $\mathbf{f}(\mathbf{q})$ is locally conservative) and if, furthermore, a viscosity force of the form $-\nu \mathbf{u}$ is also added, then the above shows that pairing takes place (formally) to $-\frac{\nu }{2}$.

A formal proof that the Euler equations for a fluid can be written as Arnold’s geodesic flow is summarized as follows (correcting also typos in the somewhat obscure argument in the appendix theof [52]). Let ${\varvec{\xi }}\rightarrow {\mathbf{x}}={\varvec{\delta }}({\varvec{\xi }})$ be the diffeomorphism in $Sdiff(T)\subset diff(T)$⁵² mapping the reference fluid configuration O into the actual one A. Here and below the differential operators are denoted ${\partial },\ldots$ but are intended to be ${\partial }_{{\varvec{\delta }}({\varvec{\xi }})},\ldots$ when operating on functions of ${\varvec{\delta }}({\varvec{\xi }})$.

A map from O to a configuration $A'\in diff(T)$infinitesimally close to one in Sdiff(T) can be parameterized by ${\varvec{\delta }},{\varvec{\zeta }}$ with ${\varvec{\delta }}\in Sdiff(T)$ and ${\varvec{\zeta }}$ an infinitesimal variation of the form ${\varvec{\partial }}z({\varvec{\delta }}({\varvec{\xi }}))$ for some scalar z.

The coordinates ${\varvec{\delta }},{\varvec{\zeta }}$ can be checked to form a system of coordinates “orthogonal and well adapted”, in the sense of definition 12 in [50], to the surface $J({\varvec{\delta }}({\varvec{\xi }}))=1$ in the space diff(T) of torus diffeomorphisms in the metric $g({\varvec{\delta }},{\varvec{\zeta }})$ attributing to an infinitesimal variation $(\mathbf{w}+\mathbf{z})$ of $({\varvec{\delta }},{\varvec{\zeta }})$ the square length:

$$\begin{aligned} \int d{\varvec{\xi }}(\mathbf{w}({\varvec{\delta }}({\varvec{\xi }}))^2+{\varvec{\partial }}z({\varvec{\delta }}({\varvec{\xi }}))^2)d{\varvec{\xi }} \end{aligned}$$

(D.1)

for the infinitesimal variations $(\mathbf{w},{\varvec{\partial }}z)$ of $({\varvec{\delta }},{\varvec{\zeta }})$ with $\mathbf{w}$ a divergence free field and ${\varvec{\partial }}z$ a gradient field. Furthermore if ${\varvec{\xi }}\rightarrow {\varvec{\delta }}({\varvec{\xi }})$ is an incompressible configuration in Sdiff(T) and ${\varvec{\delta }}'({\varvec{\xi }})={\varvec{\delta }}({\varvec{\xi }})+\mathbf{u}({\varvec{\delta }}({\varvec{\xi }}))+{\varvec{\partial }}z({\varvec{\delta }}({\varvec{\xi }}))$ an infinitesimally close one in diff(T) then

$$\begin{aligned} \begin{aligned}&J({\varvec{\delta }}'({\varvec{\xi }}))-1=\Delta z({\varvec{\delta }}({\varvec{\xi }})),\ \hbox {(``Liouville's theorem'')}\\&\int d{\varvec{\xi }}(J({\varvec{\delta }}'({\varvec{\xi }}))-1)^2=\int d{\varvec{\xi }}(\Delta z({\varvec{\delta }}({\varvec{\xi }})))^2\\&=\int d{\mathbf{x}}(\Delta z({\mathbf{x}}))^2 \end{aligned} \end{aligned}$$

(D.2)

depending only on the “violation of the constraint” $J-1$ in the above orthogonal and well adapted coordinates.

Therefore a general theorem on constrained motions, (for the constraint $J({\varvec{\delta }}({\varvec{\xi }}))=1$) can be applied: because the perfection criteria for the constraint are met as a consequence of Eqs. (D.1), (D.2), see Definition 13, Sects. 3.7, and Proposition 13, Sect. 3.8 in [50]. Hence the motions obey Euler-Lagrange equations for the Lagrangian ${\mathcal {L}}({\dot{{\varvec{\delta }}}},{\varvec{\delta }})=\int \frac{1}{2}\dot{{\varvec{\delta }}}({\varvec{\xi }})^2 d{\varvec{\xi }}$ with the ideal holonomic constraint$J({\varvec{\delta }}({\varvec{\xi }}))\equiv 1$.

In other words the motions driven by the Lagrangian ${\mathcal {L}}_\Lambda ({\dot{{\varvec{\delta }}}},{\varvec{\delta }})=\int d{\varvec{\xi }}\Big (\frac{1}{2}\dot{{\varvec{\delta }}}({\varvec{\xi }})^2 + \Lambda (J({\varvec{\delta }}(\xi ))-1)^2\Big )$, in the space $diff(T)\supset Sdiff(T)$, are motions which depend on the auxiliary parameter $\Lambda$ and, if the initial data are an incompressible configuration ${\varvec{\delta }}\in Sdiff(T)$, in the limit as $\Lambda \rightarrow +\infty$ converge, at any prefixed time, to motions driven by the constrained Lagrangian ${\mathcal {L}}$.

Explicitly, let ${\mathbf{u}}({\mathbf{x}})={\dot{{\varvec{\delta }}}}({\varvec{\xi }})$: since at time $\varepsilon >0$${\varvec{\delta }},{\mathbf{u}}$ become ${\varvec{\delta }}_\varepsilon ({\varvec{\xi }})$ and ${\mathbf{u}}_\varepsilon ({\mathbf{x}})={\varvec{\delta }}_\varepsilon ({\mathbf{x}}+\varepsilon {\mathbf{u}}({\mathbf{x}}))$, it follows that and the Euler-Lagrange equations for the constrained ${\mathcal {L}}$ become the Euler equations for a perfect incompressible fluid, obtained as a geodesic flow for the Hamiltonian $H({\mathbf{u}},{\varvec{\delta }})$ defined as

(D.3)

with interpreted as “pressure”.

If a force $\mathbf{f}({\mathbf{x}})=(-{\partial }_2\Phi ({\mathbf{x}}),{\partial }_1\Phi ({\mathbf{x}}))$, with $\Phi$ a given scalar independent of the fluid configuration, acts on the fluid then to Eq. (D.3) a term ${\varvec{\delta }}({\varvec{\xi }})\cdot (-{\partial }_2\Phi ({\varvec{\delta }}({\varvec{\xi }})),{\partial }_1\Phi ({\varvec{\delta }}({\varvec{\xi }})))$ has to be added.

Recalling that ${\mathbf{x}}={\varvec{\delta }}({\varvec{\xi }})$, the equations of motion are the pair:

(D.4)

and by the just mentioned result in [47], display a set of Lyapunov exponents, local and global, paired to $-\nu /2$.

In the Euler–Arnold case the exponents are naturally divided in two classes: the exponents relative to the $\mathbf{u}$ coordinates, ’fluid exponents’, which do not depend (explicitly) from the evolution of the ${\varvec{\delta }}$ coordinates and the others.

It would be natural to think that the other exponents are simply a copy of the first: if so the fluid exponents, alone, would be paired. However if true this cannot be a general property of geodesic flows to which friction and forcing are added (as it fails in the flow with Hamiltonian $H=\frac{1}{2} p^2$, i.e. for the equations $\dot{q}=p, \dot{p}=-\nu p+f$). I neither succeded in proving the just mentioned double degeneracy of the Lyapunov spectrum for the Arnold–Euler equations nor in convincing myself that it is a reasonable hypothesis.

The results just discussed do not apply to the NS flow because the viscosity is $\nu \Delta \mathbf{u}$: however a pairing to a line which is not constant but which depends on the scale in which the motion is studied, compatible with the above simulations, was proposed in [52].