Continuity properties of transport coefficients in simple maps
Abstract
We consider families of dynamics that can be described in terms of PerronFrobenius operators with exponential mixing properties. For piecewise expanding interval maps we rigorously prove continuity properties of the drift and of the diffusion coefficient under parameter variation. Our main result is that has a modulus of continuity of order , i.e. is Lipschitz continuous up to quadratic logarithmic corrections. For a special class of piecewise linear maps we provide more precise estimates at specific parameter values. Our analytical findings are verified numerically for the latter class of maps by using exact formulas for the transport coefficients. We numerically observe strong local variations of all continuity properties.
1 Introduction
In simple deterministic dynamical systems physical quantities like transport coefficients can be fractal functions of control parameters. This finding was first reported for a onedimensional piecewise linear map lifted periodically onto the whole real line, for which the diffusion coefficient was computed by using Markov partitions and topological transition matrices [26, 27, 29]. A generalization of this result was obtained for a map with both drift and diffusion by deriving exact analytical solutions for the transport coefficients [15, 9]. Further maps modeling chemical reactiondiffusion [14] and anomalous diffusion [33] yielded also fractal transport coefficients. Recent work aimed at physically more realistic models like (Hamiltonian) particle billiards, for which computer simulations yielded transport coefficients that are nonmonotonic under parameter variation [31]. Ref. [32] contains a summary of this line of research.
These results asked for a more detailed characterization of the “fractality” of transport coefficients. A first attempt in this direction was reported by Klages and Klauß [30], who used standard techniques from the theory of fractal dimensions for characterizing the drift and diffusion coefficients of the map studied in [15]. They numerically computed a noninteger box counting dimension for these curves which varied with the parameter interval, leading to the notion of a “fractal fractal dimension”. These results were questioned by Koza [35], who computed the oscillation of these graphs at specific Markov partition parameter values. His work suggested a dimensionality of one by conjecturing that there exist nontrivial logarithmic corrections to the usual power law behaviour in the oscillation.
This research reveals the need to study the parameter dependence of transport coefficients in a rigorous mathematical setting, which can be formulated as follows: Given a parametrized family of chaotic dynamical systems on an interval with unique invariant physical measures together with a family of sufficiently regular observables one has, under suitable mixing assumptions on the systems , a law of large numbers and a central limit theorem for the partial sum processes , namely
and
where . For suitable choices of the observables , the process is just the deterministic random walk generated by a lift of the map to the real axis, and and are the drift and diffusion coefficient of this random walk respectively.
There are a few rigorous results in the literature describing the dependence of and of quantities like for various classes of systems. Without going into the details they can be summarized as follows: If the maps and the observables depend smoothly on and if the topological conjugacy class of is not changed when is varied, then (and hence ) depends differentiably on [4, 8, 10, 19, 20, 39, 40]. If the topological class changes, quantities like may behave less regular and have a modulus of continuity not better than , even for very simple maps like symmetric tent maps [3]. On the other hand, this modulus of continuity is the rule for systems whose PerronFrobenius operator (acting on a suitable space of “regular” densities) has a spectral gap [22, 24].
The goal of this paper is to explicitly relate these mathematical results to transport coefficients. We do so by rigorously proving continuity properties of and under parameter variation for certain classes of deterministic maps. In Section 2 we give a general estimate for families of dynamics (deterministic or not), which can be described in terms of PerronFrobenius operators with exponential mixing properties. The applicability of these general results to piecewise expanding interval maps and in particular to the class of piecewise linear maps discussed in [26, 27, 29, 15, 32] is checked in Section 3. The main result is that has a modulus of continuity of order , i.e. is Lipschitz continuous up to quadratic logarithmic corrections. In Section 4 we summarize the general results for transport coefficients in the special case of piecewise linear maps and provide more precise estimates for special parameters. Our analytical findings are verified by numerical computations in Section 5, for which we use exact analytical formulas of the transport coefficients [15]. Particularly, we numerically analyze local variations of these properties. Our work corrects and amends previous results reported in [30, 35].
2 The general setting
Let be a compact interval, normalized Lebesgue measure on , the space of Lebesgueintegrable functions from to , and the space of equivalence classes of functions of bounded variation. We use the following simplified notation for the two corresponding norms:
(1) 
where
(2) 
is the variation of as a function from (i.e. extended by on ). If is differentiable as a function from integration by parts shows easily that . is obviously a seminorm, and as , it is actually a norm. This and more details on functions of bounded variation can be found in [25, section 2.3]. The monograph [2] is a comprehensive reference for most of the background material needed in this section.
We consider a family of nonsingular maps . Nonsingular means that the PerronFrobenius operator is well defined, i.e.
(3) 
By definition, for all , and we assume
Hypothesis 1
.
Our main assumption is that the maps in are uniformly exponentially mixing in the following sense:
Hypothesis 2
Each has a unique invariant probability density (so ), and there are constants and such that, for all ,
(4) 
Observe the following consequences of Hypothesis 1 and 2:
(5) 
and
(6) 
Indeed, as by Hypothesis 2 for each probability density , and by Hypothesis 1. Hence (6) follows from the definition (2) of , and then (5) is an immediate consequence.
Since it is our goal to investigate the dependence of various dynamical quantities as functions of , we need to introduce a distance on . At this stage the following one, which was already considered in [22], is most apropriate. It measures the distance between two maps and from in terms of a suitable norm of :
(7) 
This distance can be controlled in terms of a more “handson” distance between the graphs of the maps:
(8) 
Namely (see [22, Lemma 13]),
(9) 
Now, as a warmup exercise, we can prove the following estimate: for let
(10) 
Lemma 1
There exist constants such that
(11) 
Proof.
Remark 1
Suppose now that to each there is associated an “observable” . We make the following assumptions:
Hypothesis 3
Hypothesis 4
There is such that for all .
Corollary 1
There is some such that, for all ,
(14) 
is the “drift” of the partial sum process
under the invariant measure , where . Observe that
(15) 
In view of Hypothesis 2 we can also define the
“diffusion coefficient”
(16) 
Even more, we have the central limit theorem
(17) 
see e.g. [21, 18, 38]. Among physicists (16) is known as the TaylorGreenKubo formula for diffusion [32]. For the dependence of on we prove:
Proposition 1
There is some such that, for all ,
(18) 
Proof.
Observe first that, for all ,
(19) 
Indeed, for differentiable we have , so for differentiable and (19) follows from the product rule of diferentiation. General and are then approximated using mollifiers. It follows that, in view of (4),
(20) 
Let as before, denote (so that ), and fix . For all , eq. (20) implies
(21) 
For we use a different estimate. We decompose
(22) 
where
(23) 
and
(24) 
where the last inequality follows from eq. (12). Next,
(25) 
and
(26) 
Remark 2
Quite often slightly stronger forms of Hypotheses 2 and 4 are satisfied, where the mixing assumption (4) is replaced by
(29) 
and the assumption on the dependence of is strengthened to
(30) 
An inspection of the above estimates shows that and if (29) is assumed. If additionally (30) is assumed, then can be estimated as follows: Let . Then
(31) 
Hence, uniformly in and . But we see no way, in general, to bound the terms in a similar way. However, for particular families of maps (which are all topologically conjugate), we will see in subsection 4.2 that for all and that the estimate for can be made more precise.
3 Checking Hypothesis 1 and 2
3.1 General piecewise expanding maps
In this subsection we show how the general Hypothesis 1 and 2 can be verified in the more particular setting when is a parametrized family of piecewise twice continuously differentiable and expanding interval maps. So, from now on, we look at the following setting:
is a compact parameter space, , and  (T1)  
there is some such that for all .  (T2) 
We start with an abstract result which reduces Hypothesis 2 essentially to a uniform LasotaYorke type inequality.
Lemma 2
Proof.
As and , it is straightforward to check that Hypothesis 1 holds with .
Our next task is to give sufficient conditions for ((LY)). To this end we specialize further and assume from now on that our maps are piecewise expanding (PE) maps in the following sense:
(PE) 
Already in [36] it was proved that each individual (PE)map (even if ) satisfies ((LY)) with constants depending on the map. For parametrized families of maps one can generally find uniform constants, but there are counterexamples where this is not possible [22, 6, 7]. Under the above assumption one can, however, give simple sufficient conditions ensuring the uniform LYinequality. The proof in [36] (see also [25, Proposition 2.1]) shows
(32) 
where
(33) 
From this ((LY)) follows with , , and provided this supremum is finite.
In some cases of interest the , , are not bounded because there are arbitrarily short monotonicity intervals. In such situations, ad hoc arguments are needed. We give an example in the next section.
3.2 Piecewise linear modulo 1 maps
We now look at a particular model dealt with in [12, 13, 16, 17] from a mathematical perspective and in [26, 27, 29, 15, 32] from a physics point of view. Let , for some constants , and for consider
(34) 
Hofbauer [16] showed that these maps have always a
unique invariant probability density
Lemma 3
Proof.
Denote by the family of all functions with
. In [23, eq. (11)] a number
is defined for each pair of and .
In view of [23, eq. (13) and (14)] it suffices to show that for
each there is some such that
.
Let and . Then , , and has monotonicity intervals , , where
In order to estimate in [23, eq. (11)] one has to evaluate certain terms and . In our case, , as is the inverse of the derivative of the th monotone branch. The quantity is an abbreviation for , where denotes a limit from the right. Therefore, using the formula on the bottom of [23, p.1779], we obtain
(36) 
and similarly,
(37) 
It follows that for all . Hence, by [23, eqs. (11) and (12)],
(38) 
∎
Next we check assumption ((T2)) on the Lipschitz dependence of the maps on the parameters, so we estimate . For the proof we extend the maps to the whole real line (keeping the same names) by applying definition (34) to all .
Suppose and denote by and , , the discontinuity points of the two maps as introduced in the proof of Lemma 3. Consider the linear map , and observe that and for all . Let and for some arbitrarily small . Define by if and extend to a diffeomorphism of . Then


for all
