Fluctuation-dissipation relation for chaotic non-Hamiltonian systems

0.66 MB·English

by  Matteo Colangeli

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Fluctuation-dissipation relation for chaotic non-Hamiltonian
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a systems
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] Matteo Colangeli1, Lamberto Rondoni1,2, Angelo Vulpiani3,4
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1DipartimentodiMatematica,PolitecnicodiTorino,CorsoDucadegliAbruzzi24,10129
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t Torino,Italy
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s 2INFN,SezionediTorino,ViaP.Giuria1,10125Torino,Italy
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m 3DipartimentodiFisica,Universita`diRomaSapienza,p.leAldoMoro2,00185Roma,Italy
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d 4IstitutodeiSistemiComplessi(ISC-CNR),ViadeiTaurini19,00185Roma,Italy
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c E-mail:[email protected]
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3 Abstract. In dissipative dynamical systems phase space volumes contract, on average.
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6 Therefore, the invariant measure on the attractor is singular with respect to the Lebesgue
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1 measure. As noted by Ruelle, a generic perturbation pushes the state out of the attractor,
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2
hence the statistical features of the perturbation and, in particular, of the relaxation, cannot
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be understood solely in terms of the unperturbed dynamics on the attractor. This remark
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r seemsto seriouslylimitthe applicabilityofthe standardfluctuationdissipationprocedurein
a
thestatisticalmechanicsofnonequilibrium(dissipative)systems. Inthispaperweshowthat
thesingularcharacterofthesteadystatedoesnotconstituteaseriouslimitationinthecaseof
systemswithmanydegreesoffreedom. Thereasonisthatonetypicallydealswithprojected
dynamics,andtheseareassociatedwithregularprobabilitydistributionsinthecorresponding
lowerdimensionalspaces.
Fluctuation-dissipationrelationforchaoticnon-Hamiltoniansystems 2
1. Introduction
Sinceitsearlydevelopments,duemainlytotheworksofL.OnsagerandR. Kubo[1,2,3,4],
the fluctuation-dissipation theorem (FDT) represents a cornerstone in the construction of
a theory of nonequilibrium phenomena [5]. This celebrated result was developed in the
context of Hamiltonian dynamical systems, slightly perturbed out of their thermodynamic
equilibrium, and it was later extended to stochastic systems obeying a Langevin Equation
[6, 7]. The importance of the FDT rests on the fact that it sheds light on the crucial relation
between the response R (t) of a system to an external perturbation and a time correlation
V
function computed at equilibrium. In other words, having perturbed a given Hamiltonian
H with an external field h , to obtain the perturbed Hamiltonian H − h V, where V is an
0 e 0 e
observableconjugatedwithh ,theFDTallowsustocomputenonequilibriumquantities,such
e
asthetransportcoefficients[8,9],solelyintermsoftheunperturbedequilibriumstate. Onthe
other hand, a generic dynamical system is not Hamiltonian: for phenomenological practical
pruposes, one typically deals with dissipative dynamics, as in the important case of viscous
hydrodynamics[6].
The invariant measure of a chaotic dissipative system, µ say, is singular with respect
to the Lebesgue measure and is usually supported on a fractal attractor. This is not just a
mathematical curiosity, it is a potential source of difficulties for the applicability of the FDT
in dissipative systems. Indeed, the standard FDT ensures that the statistical features of a
perturbationarerelated tothestatisticalpropertiesoftheunperturbedsystem,butthat cannot
bethecaseingeneral, indissipativesystems. Thereasonisthat,givenaninitialstatex(0)on
theattractorandagenericperturbationδx(0),theperturbedinitialstatex(p)(0) = x(0)+δx(0)
Fluctuation-dissipationrelationforchaoticnon-Hamiltoniansystems 3
and its time evolution may lie outside the support of µ, hence their statistical properties
cannot be expressed by µ, which attributes vanishing probability to such states. In the
cases considered by Ruelle [10], the perturbation δx(0) and its time evolution δx(t) can
be decomposed as the sum of two parts, δx (t) and δx (t), respectively perpendicular and
⊥ k
parallel tothe“fibres”oftheattractor,
δx(t) = δx (t)+δx (t)
⊥ k
which makes it natural to expect that the statistical features of δx (t) be related to the
k
dynamics on the attractor, while it is easy to construct examples in which δx (t) is not
⊥
described bytheunperturbeddynamics.
Fromthemathematicalpointofview,thisfactisrathertransparent. Ontheotherhand,it
shouldnotbeaconcerninstatisticalmechanics,exceptinpathologicalcases. Indeed,aseries
of numerical investigationsof chaotic dissipativesystemsshowsthat the standard FDT holds
underrathergeneralconditions,mainlyiftheinvariantmeasureisabsolutelycontinuouswith
respect to Lebesgue, cf. Ref.[6] for a review. Moreover, although dissipative systems have
singular invariant measures, any small amount of noise produces smooth invariant measures,
which allow generalized FDTs to be expressed solely in terms of the unperturbed states,
analogously to the standard equilibrium case. Apart from technical aspects, the intuitive
reasonforwhichtheFDTinsystemswithnoisecanbeexpressedonlyintermsoftheinvariant
measure, isthatxp(0)remainswithinthesupportofthismeasure.
In this paper, we want to take advantage of the fact that a similar situation is realized
withoutany noise, if one works in the projected space of the physicallyrelevantobservables.
Indeed,marginalsofsingularphasespacemeasures,onspacesofsufficientlylowerdimension
Fluctuation-dissipationrelationforchaoticnon-Hamiltoniansystems 4
thanthephasespace, areusuallyregular[11,12].
Our paper is organized as follows: Section 2 is devoted to a short presentation of some
general results on FDT for chaotic dissipative systems. In Sec. 3 we discuss the numerical
results fortwo dissipativechaoticmaps, showingthat thesingularcharacter oftheirinvariant
measuresdoesnotpreventtheresponseofstandardobservablestobeexpressedonlyinterms
oftheinvariantmeasure, as inthestandardcase. Conclusionsare drawninSec. 4.
2. Someresults onFDT inchaoticdissipativesystems
Let us concisely recall Ruelle’s approach to linear response in deterministic dissipative
dynamical systems [10]. Let (M,St,µ) be a dynamical system, with M its compact phase
space, St : M → M a one parameter group of diffeomorphisms and µ the invariant natural
measure. Following Ruelle [10], who considers axiom A systems, one may show that the
effectofaperturbationδF(t) = δF (t)+δF (t)ontheresponseofageneric(smoothenough)
k ⊥
observableAattainstheform:
t t
δA(t) = R(A)(t−τ)δF (τ)dτ + R(A)(t−τ)δF (τ)dτ (1)
k k ⊥ ⊥
Z0 Z0
where the subscript refers to the dynamics on the unstable tangent bundle (along the
k
attractor), while refers to thetransversal directions, cf. left panel of Fig. 1. Ruelle’s central
⊥
remarkisthatR(A) maybeexpressedintermsofacorrelationfunctionevaluatedwithrespect
k
totheunperturbeddynamics,whileR(A) dependsonthedynamicsalongthestablemanifold,
⊥
hence it may not be determined by µ, and should be quite difficult to compute numerically
[6].
Fluctuation-dissipationrelationforchaoticnon-Hamiltoniansystems 5
Xs Xs
δF
δF⊥ δF
δFk
Xu Xu
Figure 1. Left panel: In Ruelle’s approach, the perturbation is expressed as the sum of
one component parallel to the unstable manifold and one parallel to the stable manifold.
Right panel: In the present work, the reference frame is rotated so that the direction of the
perturbationcoincideswithoneofthebasisvectors.
To illustrate these facts, the Authors of Ref.[13] study a 2-dimensional model, which
consists of a chaotic rotator on a plane and, for such a system, succeed to numerically
estimate the R(A) term in eq.(1). Nevertheless, in the next Section, we argue that R(A) may
⊥ ⊥
spoil the generalized FDT only if the perturbation is carefully oriented with respect to the
stable and unstable manifolds. This is only possible in peculiar situations, such as those of
Ref.[13],inwhichtheinvariantmeasureistheproductofaradialandandangularcomponent
and, furthermore, the perturbation lies on the radial direction, leaving the angular dynamics
unaffected.
AdifferentapproachtotheFDThasbeenproposedin[14],whichconcernsdeterministic
dynamicsperturbedbystochasticcontributions. Here,theinvariantmeasureµcanbeassumed
to have density ρ: dµ(x) = ρ(x)dx. Then, if the initial conditions are modified by an
impulsiveperturbation x → x +δx , the invariant density ρ(x ) is replaced by a perturbed
0 0 0 0
initial density ρ (x ;δx ) = ρ(x − δx ), where the subscript 0 denotes the initial state,
0 0 0 0 0
Fluctuation-dissipationrelationforchaoticnon-Hamiltoniansystems 6
right after the perturbation. This state is not stationary and evolves in time, producing time
dependent densities ρ (x ;δx ), which are assumed to eventually relax back to ρ(x ). Given
t 0 0 0
the transition probability W(x ,0 → x,t) determined by the dynamics, the response of
0
coordinatex isexpressed by:
i
δx (t) = x [ρ(x −δx )−ρ(x )]W(x ,0 → x,t)dx dx (2)
i i 0 0 0 0 0
Z Z
and onemayintroducetheresponsefunctionR as [14]:
ij
δx (t) ∂logρ
i
R (t) = = − x (t) (3)
ij i
δx (0) ∂x
j (cid:28) j (cid:12)t=0(cid:29)
(cid:12)
(cid:12)
which is a correlation function computed with resp(cid:12)ect to the unperturbed state. It is worth
to note that it makes no difference in the derivation of eq.(3) whether the steady state is an
equilibriumstateornot;itsuffices that ρbedifferentiable.
Let us consider again Eq.(2) and, for sake of simplicity, assume that all components of
δx(0) vanish,except thei-th component. Then,theresponseofx mayalso bewrittenas:
i
δx (t) = x [ρ(x −δx )−ρ(x )]W(x ,0 → x,t)dx dx dx
i i 0 0 0 0 0 j i
( )
Z Z j6=i
Y
≡ x B (x ,δx ,t)dx (4)
i i i 0 i
Z
whereB (x ,δx ,t),defined bythetermwithincurlybrackets, may alsobewrittenas:
i i 0
B (x ,δx ,t) = ρ (x ;δx )−ρ(x ) (5)
i i 0 t i 0 i
e e
whereρ(x ) and ρ arethemarginalprobabilitydistributionsdefined by:
i t
e e
ρ(x ) = ρ(x) dx , ρ (x ;δx ) = ρ (x;δx ) dx .
i j t i 0 t 0 j
Z j6=i Z j6=i
Y Y
e e
As projected singular measures are expected to be smooth, especially if the dimension
of the projected space is sensibly smaller than that of the original space, one may adopt the
Fluctuation-dissipationrelationforchaoticnon-Hamiltoniansystems 7
same procedure also for dissipative deterministic dynamical systems. Indeed, the response
functionB (x ,δx ,t)inEq.(5)isalsoexpectedtobesmooth,andtomaketheresponseofx
i i 0 i
computablefromtheinvariantmeasureonly. Inthenextsectionweinvestigatethispossibility.
3. Coarsegraining analysis
In termsofphasespaceprobabilitymeasures, theresponseformulaEq.(2)reads:
δx (t) = x dµ (x;δx )− x dµ(x) (6)
i i t 0 i
Z Z
wheredµ (x;δx ) isthetimeevolvingperturbedmeasurewhoseinitialstateisgivenby
t 0
dµ (x ;δx ) = ρ (x ;δx ) dx = ρ(x −δx ) dx .
0 0 0 0 0 0 0 0 0 0
Because dissipative dynamical systems do not have an invariant probability density, it is
convenienttointroduceacoarsegraininginphasespace,toapproximatethesingularinvariant
measureµby meansofpiecewiseconstantdistributions.
Let us consider a d-dimensional phase space M, with an ǫ-partition made of a finite set
of d-dimensional hypercubes Λ (ǫ) of side ǫ and centers x . Introduce the ǫ-coarse graining
k k
ofµand ofµ defined bytheprobabilitiesP (ǫ) andP (ǫ;δx ) ofthehypercubesΛ (ǫ):
t k t,k 0 k
P (ǫ) = dµ(x), P (ǫ;δx ) = dµ (x;δx ) . (7)
k t,k 0 t 0
ZΛk(ǫ) ZΛk(ǫ)
Thisleads tothecoarsegrained invariantdensityρ(x;ǫ):
P (ǫ)/ǫd ifx ∈ Λ (ǫ)
k k
ρ(x;ǫ) = ρ (x;ǫ), with ρ (x;ǫ) =  (8)
k k

Xk  0 else

LetZ bethenumberofbinsofofform x(q) −ǫ/2,x(q)+ǫ/2 ,q ∈ {1,2,...,Z },inthei-th
i i i i
h (cid:17)
direction. Then,themarginalizationofthecoarsegraineddistributionyieldsthefollowingset
Fluctuation-dissipationrelationforchaoticnon-Hamiltoniansystems 8
ofZ probabilities:
i
p(q)(ǫ) = x(iq)+2ǫ ρ(x;ǫ) dx dx = Prob x ∈ x(q) − ǫ,x(q) + ǫ (9)
i Zx(iq)−2ǫ (Z Yj6=i j) i (cid:16) i h i 2 i 2(cid:17)(cid:17)
each of which is the invariant probability that the coordinate x lie in one of the Z bins.
i i
In an analogous way, one may define the marginal of the evolving coarse grained perturbed
probability p(q)(ǫ;δx ). In both cases, dividingby ǫ, one obtains the coarse grained marginal
t,i 0
probability densities ρ(q)(ǫ) and ρ(q)(ǫ;δx ), as well as the ǫ-coarse grained version of the
i t,i 0
responsefunctionB (x ,δx ,t):
i i 0
1
B(q)(x ,δx ,t,ǫ) = p(q)(ǫ,δx )−p(q)(ǫ) = ρ(q)(ǫ,δx )−ρ(q)(ǫ) (10)
i i 0 ǫ t,i 0 i t,i 0 i
h i
Inthefollowing,wewillshowthatther.h.s.ofEq.(10)tendstoaregularfunctionofx inthe
i
Z → ∞,ǫ → 0,limit. Then,inthelimitofsmallperturbationsδx ,B(q)(x ,δx ,t,ǫ)maybe
i 0 i i 0
expanded as a Taylorseries, to yield an expressionsimilarto standard responsetheory,in the
sensethatitdependssolelyontheunperturbedstate. Thedifference,here,isthattheinvariant
measureissingularand represents anonequilibriumsteady state.
To illustrate this fact, we run a set of N trajectories with uniformly distributed initial
conditionsinthephasespacesoftwosimple,butsubstantiallydifferent,2-dimensionalmaps:
adissipativebakermap, and theHenon map.
Fluctuation-dissipationrelationforchaoticnon-Hamiltoniansystems 9
3.1. The dissipativebaker map
Let M = [0,1]×[0,1]bethephasespace, andconsidertheevolutionequation
x /l
n
  , for0 ≤ x < l;
n


xn+1 xn   ryn 
  = M   =    .(11)
  


y y  (x −l)/r
 n+1   n  n
     , forl ≤ x ≤ 1.
    n


  r +lyn 
  
whoseJacobian determinantisgivenby   


J = r/l , for 0 ≤ x ≤ l;
A
J (x) =  . (12)
M

 J = l/r = J−1 , for l ≤ x ≤ 1.
B A
and shows that the M isdissiaptive for l < 1/2. The map M is hyperbolic, since stable

and unstable manifolds which intersect each other orthogonally are defined at all points
x ∈ M, except in the irrelevant vertical segment at x = l. The directions of these manifolds
coincide, respectively, with the vertical and horizontal directions. It can also be shown that
this dynamical system is endowed with an invariant measure µ which is smooth along the
unstable manifold and singular along the stable one, cf. Figs.2. In particular, µ factorizes as
dµ(x) = dx×dλ(y),similarlytothecaseof[13].
In order to verify whether the functions corresponding to the above introduced
B(q)(x ,δx ,t,ǫ) become regular functions in the fine graining limit, let us consider first
i i 0
an impulsive perturbation, directed purely along the stable manifold, i.e. δx = (0,δy ).
0 0
Ruelle’sworkonsingularmeasuresisclearlyrelevant,inthiscase,becausethesupportofthe
marginal perturbed probability measure, obtained projecting out the y-direction has simply
driftedpreservingitssingularcharacter, whilethestatemayhavefallenoutsidethesupportof
theunperturbedinvariantmeasure,cf. leftpanel ofFig.3.
Fluctuation-dissipationrelationforchaoticnon-Hamiltoniansystems 10
14
12
14 10
12
8
10
ρ(x,y) 8 6
6 4
4 2
2 0
0
1
0.9
0.8
0.7
0 0.1 0.2 0.3 x 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 y
Figure2. InvariantprobabilitydistributionofthemapdefinedbyEq.(11).
Consider now an initial impulsive perturbation with one component, no matter how
small, along the unstable manifold, δx = (δx ,δy ) and rotate the vectors of the basis of
0 0 0
the 2-dimensional plane, so that the coordinate x lies along the direction of the perturbation,
as shown in the right panel of Fig.1. We find that B(q)(x,δx ,t,ǫ) is regular as a function
x 0
of x. Indeed, the projections of µ and of its perturbations onto the direction of δx have a
0
density along all directions except the vertical one, cf. right panel of Fig.3. Hence, a small
perturbationdoesnot takethestateoutsidethecorrespondingprojected support.
As already noted in [13], this Baker map shows that the response to very carefully
selected perturbations, cannot be computed in general from solely the invariant measure.
However, similarly to the case of [13], the factorization of µ makes the present case rather
peculiar. Indeed, for the overwhelming majority of dynamical systems, it looks impossible
to select directions such that the projected measures preserve the same degree of singularity
as the full measures. This is a consequence of the fact that stable and unstable manifolds
have different orientations in different parts of the phase space, provided they exist. Clearly,
the higher the dimensionality of the phase space and the larger the number of projected out